Recent zbMATH articles in MSC 82https://www.zbmath.org/atom/cc/822022-05-16T20:40:13.078697ZUnknown authorWerkzeugThe twenty-third international conference on transport theory, Santa Fe, New Mexico, USA, September 15--20, 2013https://www.zbmath.org/1483.000492022-05-16T20:40:13.078697ZFrom the text: This special issue of the \textit{Journal of Computational and Theoretical Transport} (JCTT) contains papers from presentations at the Twenty-Third International Conference on Transport Theory (ICTT-23), held at La Fonda on the Plaza in Santa Fe, New Mexico, September 15--20, 2013.Fluctuations for the partition function of Ising models on Erdös-Rényi random graphshttps://www.zbmath.org/1483.051582022-05-16T20:40:13.078697Z"Kabluchko, Zakhar"https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Löwe, Matthias"https://www.zbmath.org/authors/?q=ai:lowe.matthias"Schubert, Kristina"https://www.zbmath.org/authors/?q=ai:schubert.kristinaSummary: We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with \(N\) vertices and edge probability \(p=p(N)\) that were introduced by \textit{A. Bovier} and \textit{V. Gayrard} [J. Stat. Phys. 72, No. 3--4, 643--664 (1993; Zbl 1100.82515)] and investigated in [\textit{Z. Kabluchko} et al., J. Stat. Phys. 177, No. 1, 78--94 (2019; Zbl 1426.82031)] and [\textit{Z. Kabluchko} et al., ``Fluctuations of the magnetization for Ising models on Erdős-Rényi random graphs -- the regimes of small \(p\) and the critical temperature'', Preprint, \url{arXiv:1911.10624}]. We prove Central Limit Theorems for the partition function of the model and -- at other decay regimes of \(p(N)\) -- for the logarithmic partition function. We find critical regimes for \(p(N)\) at which the behavior of the fluctuations of the partition function changes.Infinite stable Boltzmann planar maps are subdiffusivehttps://www.zbmath.org/1483.051642022-05-16T20:40:13.078697Z"Curien, Nicolas"https://www.zbmath.org/authors/?q=ai:curien.nicolas"Marzouk, Cyril"https://www.zbmath.org/authors/?q=ai:marzouk.cyrilSummary: The infinite discrete stable Boltzmann maps are generalisations of the well-known uniform infinite planar quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than \(\frac{1}{3} \). Our method is based on stationarity and geometric estimates obtained via the peeling process which are of individual interest.Grassmannians and cluster structureshttps://www.zbmath.org/1483.130372022-05-16T20:40:13.078697Z"Baur, Karin"https://www.zbmath.org/authors/?q=ai:baur.karinGrassmannians are classical mathematical structures that appear in the study of Lie theory, algebraic geometry, combinatorics and many other areas. In these notes Baur introduces Grassmannians and their interplay with cluster theory.
The notes have few prerequisites for the first sections, so a student with a good course in linear algebra and some notions of commutative algebra can follow them with no problem. The introduction contains several references and exercises. This proposes a hands-on approach to the reader.
By the end of Section 1, the reader can understand the definition of cluster algebra of type A and the relation between cluster algebras and Postnikov diagrams, that is the essence of J. Scott's key theorem in the area.
In Section 2 the author introduces quiver with potentials arising from dimer models and the boundary algebra, which is the link to study categorification for cluster algebras in the sense of Jensen-King-Su.
By Section 3 the author gives a primer on her latest research on the topic [\textit{K. Baur} et al., Proc. Lond. Math. Soc. (3) 113, No. 2, 213--260 (2016; Zbl 1386.13060); Nagoya Math. J. 240, 322--354 (2020; Zbl 1452.05187); Algebra Number Theory 15, No. 1, 29--68 (2021; Zbl 1459.05346)] while still providing several examples and references. This part is more suitable for graduate students and researchers interested in cluster algebras arising from Grassmannians.
Reviewer: Ana Garcia Elsener (Buenos Aires)Inferring the connectivity of coupled oscillators and anticipating their transition to synchrony through lag-time analysishttps://www.zbmath.org/1483.340752022-05-16T20:40:13.078697Z"Leyva, Inmaculada"https://www.zbmath.org/authors/?q=ai:leyva.inmaculada"Masoller, Cristina"https://www.zbmath.org/authors/?q=ai:masoller.cristinaSummary: The synchronization phenomenon is ubiquitous in nature. In ensembles of coupled oscillators, explosive synchronization is a particular type of transition to phase synchrony that is first-order as the coupling strength increases. Explosive sychronization has been observed in several natural systems, and recent evidence suggests that it might also occur in the brain. A natural system to study this phenomenon is the Kuramoto model that describes an ensemble of coupled phase oscillators. Here we calculate bi-variate similarity measures (the cross-correlation, \(\rho_{ij}\), and the phase locking value, \(\text{PLV}_{ij})\) between the phases, \(\varphi_i(t)\) and \(\varphi_j(t)\), of pairs of oscillators and determine the lag time between them as the time-shift, \(\tau_{ij}\), which gives maximum similarity (i.e., the maximum of \(\rho_{ij}(\tau)\) or \(\text{PLV}_{ij}(\tau))\). We find that, as the transition to synchrony is approached, changes in the distribution of lag times provide an earlier warning of the synchronization transition (either gradual or explosive). The analysis of experimental data, recorded from Rossler-like electronic chaotic oscillators, suggests that these findings are not limited to phase oscillators, as the lag times display qualitatively similar behavior with increasing coupling strength, as in the Kuramoto oscillators. We also analyze the statistical relationship between the lag times between pairs of oscillators and the existence of a direct connection between them. We find that depending on the strength of the coupling, the lags can be informative of the network connectivity.A neural network closure for the Euler-Poisson system based on kinetic simulationshttps://www.zbmath.org/1483.351582022-05-16T20:40:13.078697Z"Bois, Léo"https://www.zbmath.org/authors/?q=ai:bois.leo"Franck, Emmanuel"https://www.zbmath.org/authors/?q=ai:franck.emmanuel"Navoret, Laurent"https://www.zbmath.org/authors/?q=ai:navoret.laurent"Vigon, Vincent"https://www.zbmath.org/authors/?q=ai:vigon.vincentSummary: This work deals with the modeling of plasmas, which are ionized gases. Thanks to machine learning, we construct a closure for the one-dimensional Euler-Poisson system valid for a wide range of collisional regimes. This closure, based on a fully convolutional neural network called V-net, takes as input the whole spatial density, mean velocity and temperature and predicts as output the whole heat flux. It is learned from data coming from kinetic simulations of the Vlasov-Poisson equations. Data generation and preprocessings are designed to ensure an almost uniform accuracy over the chosen range of Knudsen numbers (which parametrize collisional regimes). Finally, several numerical tests are carried out to assess validity and flexibility of the whole pipeline.Lump, lumpoff, rogue wave, breather wave and periodic lump solutions for a \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics and plasma physicshttps://www.zbmath.org/1483.351662022-05-16T20:40:13.078697Z"Wang, Meng"https://www.zbmath.org/authors/?q=ai:wang.meng.1|wang.meng"Tian, Bo"https://www.zbmath.org/authors/?q=ai:tian.bo"Qu, Qi-Xing"https://www.zbmath.org/authors/?q=ai:qu.qixing"Zhao, Xue-Hui"https://www.zbmath.org/authors/?q=ai:zhao.xue-hui"Zhang, Ze"https://www.zbmath.org/authors/?q=ai:zhang.ze"Tian, He-Yuan"https://www.zbmath.org/authors/?q=ai:tian.he-yuanSummary: Under investigation in this paper is a \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics. With the help of symbolic computation, we obtain and discuss the influence of the perturbed effect and disturbed wave velocity along the transverse spatial coordinate on the lump, lumpoff, rogue wave, breather wave and periodic lump solutions: When the value of \(\delta_2\) decreases to \(-1\), the amplitude of the lump wave becomes smaller; When the value of \(\delta_1\) increases to 5, the location of the lump wave moves along the positive direction of the \(y\) (a transverse spatial coordinate) axis; When the value of \(\delta_2\) decreases to 0.5, the location of the stripe soliton moves along the negative direction of the \(y\) axis and the amplitude of the lump wave becomes smaller; When the value of \(\delta_2\) decreases to \(-0.4\), the amplitude of the rogue wave becomes smaller; When the value of \(\delta_1\) increases to 5, breather waves propagate along the positive \(t\) (the temporal coordinate) direction and distance between the adjacent crests becomes shorter; When the value of \(\delta_2\) decreases to \(-1\), breather waves propagate along the negative \(t\) direction and distance between the adjacent crests becomes shorter; When the value of \(\delta_2\) decreases to 0.5, periodic lump waves move along the positive direction of the \(y\) axis. Lump solutions have more parameters than those in the existing literature. Lumpoff wave is generated from the process of the interaction between the lump wave and one stripe soliton. Moving path of the lumpoff wave is investigated via the moving path of the lump wave. Besides, we derive the rogue wave, breather wave and periodic lump solutions.A priori bounds for the kinetic DNLShttps://www.zbmath.org/1483.351992022-05-16T20:40:13.078697Z"Cacciafesta, Federico"https://www.zbmath.org/authors/?q=ai:cacciafesta.federico"Tsutsumi, Yoshio"https://www.zbmath.org/authors/?q=ai:tsutsumi.yoshioSummary: In this note, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which arises as a model of propagation of a plasma taking the effect of the resonant interaction between the wave modulation and the ions into account. In contrast to the standard derivative NLS equation, KDNLS does not conserve the mass and the energy. Nevertheless, the dissipative structure of KDNLS enables us to show an a priori bound in the energy space and a lower bound of the \(L^2\) norm for its solution, as we see in this note. Combined with the local wellposedness result, which we plan to show in a forthcoming paper, these bounds will give a global existence result in the energy space for small initial data.
For the entire collection see [Zbl 1459.37002].Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithmhttps://www.zbmath.org/1483.352172022-05-16T20:40:13.078697Z"Mo, Yifan"https://www.zbmath.org/authors/?q=ai:mo.yifan"Ling, Liming"https://www.zbmath.org/authors/?q=ai:ling.liming"Zeng, Delu"https://www.zbmath.org/authors/?q=ai:zeng.deluThe aim of the paper is to investigate data-driven vector solitons for the coupled nonlinear Schrödinger equation (CNLSE) by means of an improved physics-informed neural networks (PINN) method. One considers an initial and boundary Dirichlet problem for the CNLSE:
\[
iq_t + \frac{1}{2}q_{xx}+{\|q\|} _2^2q =0,\text{ for }x\in [- x_L,x_L], t\in[t_0,t_T],
\]
\[
q(x,t_0)=q^{t_0}(x),\text{ for }x\in [- x_L,x_L],
\]
\[
q(x_L,t)=q(-x_L,t)=q^b(t),\text{ for }t\in[t_0,t_T]
\]
Here, $q(x,t)=(q_1(x,t),q_2(x,t))$ is a complex vector field. The original PINN method for solving CNLSE is briefly reviewed and in order to improve the convergence rate and the approximation ability of the original PINN method a new pre-fixed multistage training algorithm is proposed. In the second section of the article the construction of an improved PINN by pre-fixed multi-stage training is largely described, the flow diagram together with the pre-fixed multi-stage training algorithm are shown. Few words on the efficiency and accuracy of the method end the section. In the third section of the article the performance of the improved method is investigated. One considers the case of a nondegenerate vector solution soliton and the collision of vector soliton solution. References contain 65 titles.
Reviewer: Claudia Simionescu-Badea (Wien)Domain wall motion in axially symmetric spintronic nanowireshttps://www.zbmath.org/1483.352382022-05-16T20:40:13.078697Z"Rademacher, Jens D. M."https://www.zbmath.org/authors/?q=ai:rademacher.jens-d-m"Siemer, Lars"https://www.zbmath.org/authors/?q=ai:siemer.larsSize-dependent thermoelasticity of a finite bi-layered nanoscale plate based on nonlocal dual-phase-lag heat conduction and Eringen's nonlocal elasticityhttps://www.zbmath.org/1483.352602022-05-16T20:40:13.078697Z"Xue, Zhangna"https://www.zbmath.org/authors/?q=ai:xue.zhangna"Cao, Gongqi"https://www.zbmath.org/authors/?q=ai:cao.gongqi"Liu, Jianlin"https://www.zbmath.org/authors/?q=ai:liu.jianlinSummary: The size effects on heat conduction and elastic deformation are becoming significant along with the miniaturization of the device and wide application of ultrafast lasers. In this work, to better describe the transient responses of nanostructures, a size-dependent thermoelastic model is established based on nonlocal dual-phase-lag (N-DPL) heat conduction and Eringen's nonlocal elasticity, which is applied to the one-dimensional analysis of a finite bi-layered nanoscale plate under a sudden thermal shock. In the numerical part, a semi-analytical solution is obtained by using the Laplace transform method, upon which the effects of size-dependent characteristic lengths and material properties of each layer on the transient responses are discussed systematically. The results show that the introduction of the elastic nonlocal parameter of Medium 1 reduces the displacement and compressive stress, while the thermal nonlocal parameter of Medium 1 increases the deformation and compressive stress. These findings may be beneficial to the design of nano-sized and multi-layered devices.Global existence analysis of energy-reaction-diffusion systemshttps://www.zbmath.org/1483.352622022-05-16T20:40:13.078697Z"Fischer, Julian"https://www.zbmath.org/authors/?q=ai:fischer.julian"Hopf, Katharina"https://www.zbmath.org/authors/?q=ai:hopf.katharina"Kniely, Michael"https://www.zbmath.org/authors/?q=ai:kniely.michael"Mielke, Alexander"https://www.zbmath.org/authors/?q=ai:mielke.alexanderMacroscopic approximation of a Fermi-Dirac statistics: Unbounded velocity space settinghttps://www.zbmath.org/1483.352642022-05-16T20:40:13.078697Z"Masmoudi, Nader"https://www.zbmath.org/authors/?q=ai:masmoudi.nader"Tayeb, Mohamed Lazhar"https://www.zbmath.org/authors/?q=ai:tayeb.mohamed-lazharSummary: An approximation by diffusion of a nonlinear Boltzmann equation modeling a Fermi-Dirac statistics is analyzed for an unbounded velocity space and Poisson coupling. A careful analysis of the entropy and entropy-dissipation allows to control the distribution function and to pass to the limit using duality method.Velocity decay estimates for Boltzmann equation with hard potentialshttps://www.zbmath.org/1483.352652022-05-16T20:40:13.078697Z"Cameron, Stephen"https://www.zbmath.org/authors/?q=ai:cameron.stephen-p"Snelson, Stanley"https://www.zbmath.org/authors/?q=ai:snelson.stanleyThe estimates of the ill-posedness index of the (deformed-) continuous Heisenberg spin equationhttps://www.zbmath.org/1483.352662022-05-16T20:40:13.078697Z"Zhong, Penghong"https://www.zbmath.org/authors/?q=ai:zhong.penghong"Chen, Ye"https://www.zbmath.org/authors/?q=ai:chen.ye"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshanSummary: Although the exact treatment of the continuous Heisenberg spin is already known, the exact solution of the deformed system is not found in the literature. In this paper, some traveling wave solutions of the deformed (indicated by the coefficient \(\alpha)\) continuous Heisenberg spin equation are obtained. Based on the exact solution being constructed here, the ill-posedness results are proved by the estimation of the Fourier integral in \(\dot{H}^s\). If \(\alpha \neq 0\), the range of the mild ill-posedness index \(s\) is \((1, \frac{3}{2})\), which is consistent with the result of the formal analysis of the solution. Moreover, the upper bound of the strong ill-posedness index \(s\) jumps at \(\alpha = 0\): if \(\alpha \neq 0\), the upper bound is 2; if \(\alpha = 0\), then the upper bound jumps to \(\frac{3}{2} \).
{\copyright 2021 American Institute of Physics}Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forceshttps://www.zbmath.org/1483.352672022-05-16T20:40:13.078697Z"Carrillo, José A."https://www.zbmath.org/authors/?q=ai:carrillo.jose-antonio"Choi, Young-Pil"https://www.zbmath.org/authors/?q=ai:choi.young-pil"Jung, Jinwook"https://www.zbmath.org/authors/?q=ai:jung.jinwookConvergence analysis of asymptotic preserving schemes for strongly magnetized plasmashttps://www.zbmath.org/1483.352692022-05-16T20:40:13.078697Z"Filbet, Francis"https://www.zbmath.org/authors/?q=ai:filbet.francis"Rodrigues, L. Miguel"https://www.zbmath.org/authors/?q=ai:rodrigues.luis-miguel"Zakerzadeh, Hamed"https://www.zbmath.org/authors/?q=ai:zakerzadeh.hamedSummary: The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes [the first and second authors, SIAM J. Numer. Anal. 54, No. 2, 1120--1146 (2016; Zbl 1342.35392)] for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-in-Cell methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.Graphop mean-field limits for Kuramoto-type modelshttps://www.zbmath.org/1483.352702022-05-16T20:40:13.078697Z"Gkogkas, Marios Antonios"https://www.zbmath.org/authors/?q=ai:gkogkas.marios-antonios"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christianDecay estimates of solutions to the \(N\)-species Vlasov-Poisson system with small initial datahttps://www.zbmath.org/1483.352722022-05-16T20:40:13.078697Z"Wang, Yichun"https://www.zbmath.org/authors/?q=ai:wang.yichunSummary: This paper is concerned with the time decay rates of the \(N -\) species Vlasov-Poisson system with small data in the whole space. The global existence and large time behaviors are obtained in \(\mathbb{R}^3\) and more higher dimensional space. For the proof, the classical (for \(\mathbb{R}^n, n \geq 4)\) and the modified (for \(\mathbb{R}^3)\) vector field method and the bootstrap argument are mainly employed. Compared to the unipolar case, there are some crucial new ideas introduced to handle the multi-species case, such as a new bootstrap assumption with some necessary parameters and the multipolar version of vector field method with new coefficients corresponding to different species charged particles, respectively.Sharp decay estimates for the Vlasov-Poisson system with an external magnetic fieldhttps://www.zbmath.org/1483.352732022-05-16T20:40:13.078697Z"Wu, Man"https://www.zbmath.org/authors/?q=ai:wu.manSummary: In this paper, we establish sharp decay estimates for the Vlasov-Poisson system with an external magnetic field on \(\mathbb{R}_x^3 \times \mathbb{R}_v^3\). Our arguments are based on the modified vector field method developed in Smulevici (2016) for the classical Vlasov-Poisson system in the 3-D case, and hence we extend some results in [\textit{J. Smulevici}, Ann. PDE 2, No. 2, Paper No. 11, 55 p. (2016; Zbl 1397.35033)] to the Vlasov-Poisson system with an external magnetic field.Stability of global Maxwellian for fully nonlinear Fokker-Planck equationshttps://www.zbmath.org/1483.352772022-05-16T20:40:13.078697Z"Liao, Jie"https://www.zbmath.org/authors/?q=ai:liao.jie"Yang, Xiongfeng"https://www.zbmath.org/authors/?q=ai:yang.xiongfengSummary: This paper considers the stability of solutions around a global Maxwellian to the fully non-linear Fokker-Planck equation in the whole space. This model preserves mass, momentum and energy at the same time, and its dissipation is much weaker than that in the simplified model considered in [the authors and \textit{Q. Wang}, ibid. 173, No. 1, 222--241 (2018; Zbl 1398.35017)]. To overcome the new difficulties, the macro-micro decomposition of the solution around the \textit{local Maxwellian} and energy estimates introduced in [\textit{T.-P. Liu} et al., Physica D 188, No. 3--4, 178--192 (2004; Zbl 1098.82618)] and [\textit{T. Yang} and \textit{H.-J. Zhao}, J. Math. Phys. 47, No. 5, 053301, 19 p. (2006; Zbl 1111.82048)] for Boltzmann equation is used. That is, we reformulate the model into a fluid-type system coupled with an equation of the microscopic part. The a priori estimates of the solution could be obtained by the standard energy method. Especially, by careful computation, the viscosity and heat diffusion terms in the fluid-type system are derived from the microscopic part, which give the dissipative mechanism to the system.Asymptotic decoupling and weak Gibbs measures for finite alphabet shift spaceshttps://www.zbmath.org/1483.370202022-05-16T20:40:13.078697Z"Pfister, C.-E."https://www.zbmath.org/authors/?q=ai:pfister.charles-edouard"Sullivan, W. G."https://www.zbmath.org/authors/?q=ai:sullivan.wayne-gThe paper deals with equilibrium states for continuous functions on a large class of finite-alphabet shift spaces. The authors study the decoupling condition on shift spaces and the space of functions of bounded total oscillations on shift spaces. Their properties and examples are presented. Let \(A\) be a finite set and \(L=\mathbb{Z}^d\). As the main result, the authors prove that if a shift space \(X\subset A^{L}\) satisfies the decoupling condition and \(\phi\) is a function with bounded total oscillations on \(X\), then an equilibrium measure \(\nu\) for \(\phi\) is a weak Gibbs measure for \(\phi-P(\phi)\) where \(P(\phi)\) is the topological pressure of \(\phi\). Then they obtain a full large-deviation principle for the empirical measures on \((X, \nu)\). They prove that if \(X\) is a shift space satisfying the decoupling condition then the ergodic measures on \(X\) are entropy dense. An example of a function of bounded total oscillations not satisfying the Bowen property is provided.
Reviewer: Yuki Yayama (Chiilán)Phase space classification of an Ising cellular automaton: the Q2R modelhttps://www.zbmath.org/1483.370232022-05-16T20:40:13.078697Z"Montalva-Medel, Marco"https://www.zbmath.org/authors/?q=ai:montalva-medel.marco"Rica, Sergio"https://www.zbmath.org/authors/?q=ai:rica.sergio"Urbina, Felipe"https://www.zbmath.org/authors/?q=ai:urbina.felipeSummary: An exact classification of the different dynamical behaviors that exhibits the phase space of a reversible and conservative cellular automaton, the so-called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a cycle. In this work, we classify them in four types accordingly to well differentiated topological characteristics. Three of them -- which we call of type S-I, S-II, and S-III -- share a symmetry property, while the fourth, which we call of type AS does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous types of cycles. Moreover, at a combinatorial level, we can determine the number of cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space and, in addition, we realize an exhaustive study of a small Ising system \((4\times 4)\) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation.Fifth-order generalized Heisenberg supermagnetic modelshttps://www.zbmath.org/1483.370822022-05-16T20:40:13.078697Z"Jiang, Nana"https://www.zbmath.org/authors/?q=ai:jiang.nana"Zhang, Meina"https://www.zbmath.org/authors/?q=ai:zhang.meina"Guo, Jiafeng"https://www.zbmath.org/authors/?q=ai:guo.jiafeng"Yan, Zhaowen"https://www.zbmath.org/authors/?q=ai:yan.zhaowenSummary: This paper is concerned with the construction of the fifth-order generalized Heisenberg supermagnetic models. We investigate the integrable structure and properties of the supersymmetric systems. We also establish their gauge equivalent equations with the gauge transformation for two quadratic constraints, i.e., the super fifth-order nonlinear Schrödinger equation and the fermionic fifth-order nonlinear Schrödinger equation, respectively.On uniform second-order nonlocal approximations to diffusion and subdiffusion equations with nonlocal effect parameterhttps://www.zbmath.org/1483.450082022-05-16T20:40:13.078697Z"Yang, Jerry Z."https://www.zbmath.org/authors/?q=ai:yang.jerry-zhijian"Yin, Xiaobo"https://www.zbmath.org/authors/?q=ai:yin.xiaobo"Zhang, Jiwei"https://www.zbmath.org/authors/?q=ai:zhang.jiweiSummary: In this paper we focus on uniform convergence rates from nonlocal diffusion and subdiffusion solutions to the corresponding local limit with respect to a nonlocal effect parameter without extra assumptions on the regularity of nonlocal solutions, and present sufficient conditions to guarantee first- and second-order convergence rates, respectively. To do so, we first revisit the maximum principle for nonlocal models using the idea in [\textit{Y. Luchko}, J. Math. Anal. Appl. 351, No. 1, 218--223 (2009; Zbl 1172.35341)], and present the uniqueness of the nonlocal solutions. After that, we extend the methodology developed in [\textit{Q. Du} et al., Commun. Math. Sci. 17, No. 6, 1737--1755 (2019; Zbl 1426.47013)] to address the truncated errors on the volume constraints, and then combine the resulting errors from the boundary domain with the maximum principle to obtain the uniform convergence rates. Our analysis shows that the constant value continuation of the boundary conditions of local problems only leads to a first-order convergence rate. If one expects a second-order convergence rate, the information of first-order derivatives for local problems on the boundaries is required. One- and two-dimensional numerical examples are provided to validate our theoretical analysis.Nested triacontahedral shells or how to grow a quasi-crystalhttps://www.zbmath.org/1483.520162022-05-16T20:40:13.078697Z"Longuet-Higgins, Michael S."https://www.zbmath.org/authors/?q=ai:longuet-higgins.michael-s(no abstract)Surface tension and \(\Gamma\)-convergence of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic mediahttps://www.zbmath.org/1483.531192022-05-16T20:40:13.078697Z"Morfe, Peter S."https://www.zbmath.org/authors/?q=ai:morfe.peter-sThe paper ``Surface Tension and \(\Gamma\)-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media'' is centered on a problem of \(\Gamma\)-convergence of certain functionals which emerge in phenomenological mesoscopic theory of phase transitions. The departure point of the whole analysis is the family of functionals of the form
\[
\mathcal{F}^{\omega}(u) = \int_{\mathbb{R}^d}\left( \frac{1}{2}\varphi^{\omega}(x,Du(x))^2 + W(u(x))\right) dx\,, \tag{1}
\]
where \(\phi^{\omega}(x,\cdot)\) is a stationary ergodic Finsler metric, \(W\) is a double-well potential with wells of equal depth, and \(u\) is a scalar function taking value in the interval \([-1,1]\). Then, the main goal of the paper is to study the role played by randomness in determining the macroscopic surface tension in the Van der Waals-Cahn-Hilliard problem, the randomness being represented by a probability space \(\left( \Omega, \mathscr{B}, \mathbb{P} \right)\) with an action \(\tau\) of the group \(\mathbb{R}^d\).
The result of the paper is expressed in the language of \(\Gamma\)-convergence. Firstly a rescaled energy functional, \(\mathcal{F}^{\omega}_{\epsilon}(u)\) is defined, where \(\epsilon >0\) represents the length scale of the mesoscopic description, and localized to each open subset \(A\subset \mathbb{R}^d\), obtaining the functional
\[
\mathcal{F}_{\epsilon}^{\omega}(u;A) = \int_{A}\left( \frac{\epsilon}{2}\varphi^{\omega}(\epsilon^{-1}x,Du(x))^2 + \epsilon^{-1} W(u(x))\right) dx\,. \tag{2}
\]
Then, if \(\mathscr{E}\) denotes the functional
\[
\mathscr{E}(u;A)=\begin{cases}
\int_{\partial ^*\left\lbrace u=1 \right\rbrace \cap A} \tilde{\varphi}(\nu_{\left\lbrace u=1 \right\rbrace}(\xi)) \mathcal{H}^{d-1}(d\xi), &u\in BV(A;\left\lbrace -1,1 \right\rbrace)\\
\infty, &\text{otherwise}
\end{cases}, \tag{3}
\]
the main result is contained in the following Theorem 1:
\textbf{Theorem 1.} There is a one-homogeneous convex function \(\tilde{\varphi}\,:\,\mathbb{R}^d \,\rightarrow \,(0,\infty)\) depending only on \(\mathbb{P}\), such that, with probability one, \(\mathcal{F}^{\omega}\xrightarrow{\Gamma} \,\mathscr{E}\). More specifically, there is an event \(\hat{\Omega}\in \Sigma\) such that \(\mathbb{P}(\hat{\Omega})=1\) and no matter the choice of Lipschitz, open, bounded \(A \subseteq \mathbb{R}^d\) or \(\omega\in \hat{\Omega}\), the following occurs:
\begin{itemize}
\item[1.] if \((u_{\epsilon})_{\epsilon >0}\subset H^1(A;[-1,1])\) satisfies
\[
\sup \left\lbrace \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\mid \epsilon >0 \right\rbrace <\infty \tag{4}
\]
then \((u_{\epsilon})_{\epsilon >0}\) is relatively compact in \(L^1(A)\) and all of its limit points are in \(BV(A;\left\lbrace -1,1 \right\rbrace)\).
\item[2.] If \(u\in L^1(A;[-1,1])\) and \((u_{\epsilon})_{\epsilon >0}\subseteq H^1(A;[-1,1])\) satisfies \(u_{\epsilon}\,\rightarrow\, u\) in \(L^1(A)\), then
\[
\mathscr{E}(u;A)\leq \liminf_{\epsilon \rightarrow 0^+} \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\,. \tag{5}
\]
\item[3.] If \(u\in L^1(A;[-1,1])\), then there is a family \((u_{\epsilon})_{\epsilon >0}\subseteq H^1(A;[-1,1])\) such that \(u_{\epsilon}\,\rightarrow\, u\) in \(L^1(A)\) and
\[
\limsup_{\epsilon \rightarrow 0^+} \mathcal{F}^{\omega}_{\epsilon}(u_{\epsilon};A)\leq \mathscr{E}(u;A)\,. \tag{6}
\]
\end{itemize}
In the above functional \(\mathscr{E}\), \(\partial ^*E\) is the reduced boundary of the Caccioppoli set \(E\), whereas \(\nu_E\) is its normal vector and \(\mathcal{H}^d\) denotes the d-dimensional Hausdorff measure. In the main theorem \(\Sigma\subset \mathscr{B}\) is the \(\sigma\)-algebra of subsets of \(\Omega\) invariant under the action \(\tau\). Therefore, the above theorem states that even in presence of randomness there exists a macroscopic surface tension, the functional \(\mathscr{E}\), obtained as the limit in the sense of \(\Gamma\)-convergence of a mesoscopic energy functional.
The above theorem is proven in the last section of the work. In order to get there, the author proves a series of intermediate results which are collected in Sections 3 and 4. The goal of these two sections is to prove that there is a continuous function \(\tilde{\varphi}\) which is, then, used in the surface tension functional. This function is obtained via a procedure which the author calls thermodynamic limit. Firstly, chosen a function \(q\,:\,\mathbb{R}\,\rightarrow\,[-1,1]\) which is used to impose boundary conditions, a random process, called finite-volume surface tension and denoted \(\tilde{\varphi}^{\omega}(e,G,h)\), is obtained as the minimum
\[
\tilde{\varphi}^{\omega}(e,G,h) = \min \left\lbrace \mathcal{F}^{\omega}(u;A)\mid u\in H^1(A;[-1,1]), \; u-q_e \in H^1_0(A) \right\rbrace\,, \tag{7}
\]
where \(A=G\oplus_{e}(-h,h)\) is the following set
\[
G\oplus_{e}(-h,h) = \left\lbrace O_e(y)+te\,\mid\,y\in G,\; t\in (-h,h) \right\rbrace \tag{8}
\]
and \(O_e\,:\,\mathbb{R}^{d-1}\,\rightarrow\,\mathbb{R}^d\) is a linear isometry onto the hyperplane orthogonal to \(e\). Here \(e\) is a unit vector in the unit sphere \(S^{d-1}\subset \mathbb{R}^d\). Then, theorem 2 states:
\textbf{Theorem 2.} For each \(e\in S^{d-1}\), there is an event \(\hat{\Omega}\in \Sigma_e\) satisfying \(\mathbb{P}(\hat{\Omega}_e)=1\) such that if \(\omega\in \hat{\Omega}_e\), then
\[
\begin{split} \tilde{\varphi}(e) &= \lim_{R\rightarrow \infty}R^{1-d}\tilde{\varphi}^{\omega}_{\infty}(e,Q(0,R))\\
&= \lim_{h\rightarrow \infty} \limsup_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),h)\\
&= \lim_{h\rightarrow \infty} \liminf_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),h)\\
&= \lim_{R\rightarrow \infty} R^{1-d}\tilde{\varphi}^{\omega}(e,Q(0,R),kR) \,. \end{split}
\tag{9}
\]
In the above theorem, \(\Sigma_e\) is the \(\sigma\)-algebra of subsets invariant under the action \(\tau_x\) with \(x\) orthogonal to \(e\), \(Q(0,R)\subset \mathbb{R}^{d-1}\) is the cube centered at the origin with side length \(R/2\), and \(\tilde{\varphi}^{\omega}_{\infty}\) is the minimum
\[
\tilde{\varphi}^{\omega}_{\infty}(e,G) = \min \left\lbrace \mathcal{F}^{\omega}(u;G\oplus_e\mathbb{R})\mid -1 \leq u \leq 1, \; u-q =0 \; \text{on} \: \partial G \oplus_e \mathbb{R} \right\rbrace\,.
\]
Therefore, the above theorem states the existence of the thermodynamic limit for cubes centered at the origin. Then, the last part of section 3 extends the previous thermodynamic limit to any cube in \(\mathbb{R}^d\) as stated in
\textbf{Proposition 10.} There is an event \(\hat{\Omega}\in \Sigma\) satisfying \(\mathbb{P}(\hat{\Omega})=1\) such that if \(\omega \in \hat{\Omega}\), \(e\in S^{d-1}\), \(x_0\in \mathbb{R}^d\), and \(\rho>0\), then
\[
\tilde{\varphi}(e)\rho^{d-1} = \lim_{R\rightarrow \infty} R^{1-d}\tilde{\Phi}^{\omega}(e,Rx_0,RQ^{e}(x_0,\rho))\,, \tag{10}
\]
where
\[
\tilde{\Phi}^{\omega}(e,x,A)= \min \left\lbrace \mathcal{F}^{\omega}(u;A)\mid u\in H^1(A;[-1,1]), \; u-T_x q_e \in H^1_0(A) \right\rbrace\,. \tag{11}
\]
In the above theorem \(Q^{e}(x_0,\rho)\) denotes the cube in \(\mathbb{R}^d\)
\[
Q^{e}(x_0,\rho)= x+Q(0,R)\oplus_e \left( -\frac{\rho}{2}, \frac{\rho}{2} \right)\,, \tag{12}
\]
and \(T_xq(y) = q(x+y)\). Once the surface tension \(\tilde{\varphi}\) is found the results of Theorem 1 are obtained using some techniques proven in [\textit{N. Ansini} et al., Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 2, 265--296 (2003; Zbl 1031.49021)] of the paper.
The paper is very well organized and the Introduction is a fundamental guide to understand the development of the whole work. Indeed, it summarizes the contents of the main theorems and gives a short sketch of their proofs. Moreover, it provides useful references where finding some pieces of missing information. However, it could be useful starting from Section 2 in order to make contact with the notation of the work. The final appendices provide further definitions making the paper self-consistent. The technical core of the paper is divided into several sections and subsections, which facilitate the reading of the contents, together with a short initial comment anticipating the main results of any section.
Reviewer: Fabio Di Cosmo (Madrid)Quenched invariance principle for random walks on dynamically averaging random conductanceshttps://www.zbmath.org/1483.600662022-05-16T20:40:13.078697Z"Bethuelsen, Stein Andreas"https://www.zbmath.org/authors/?q=ai:bethuelsen.stein-andreas"Hirsch, Christian"https://www.zbmath.org/authors/?q=ai:hirsch.christian"Mönch, Christian"https://www.zbmath.org/authors/?q=ai:monch.christianSummary: We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on \(\mathbb{Z}\). In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.Switching interacting particle systems: scaling limits, uphill diffusion and boundary layerhttps://www.zbmath.org/1483.600672022-05-16T20:40:13.078697Z"Floreani, Simone"https://www.zbmath.org/authors/?q=ai:floreani.simone"Giardinà, Cristian"https://www.zbmath.org/authors/?q=ai:giardina.cristian"den Hollander, Frank"https://www.zbmath.org/authors/?q=ai:den-hollander.frank"Nandan, Shubhamoy"https://www.zbmath.org/authors/?q=ai:nandan.shubhamoy"Redig, Frank"https://www.zbmath.org/authors/?q=ai:redig.frankSummary: This paper considers three classes of interacting particle systems on \(\mathbb{Z}\): independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the \textit{type} of particle) between 1 (\textit{fast particles}) and \(\epsilon\in[0,1]\) (\textit{slow particles}). The switch between the two jump rates happens at rate \(\gamma\in(0,\infty)\). In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by \(N^{-1}\), time by \(N^2\), the switching rate by \(N^{-2}\), and letting \(N\rightarrow \infty\). The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick's law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on \([N]=\{1,\dots,N\}\), adding boundary reservoirs at sites 1 and \(N\) of fast and slow particles, respectively. Inside \([N]\) particles move as before, but now particles are injected and absorbed at sites 1 and \(N\) with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick's law made possible by the switching between types. We rescale the microscopic steady-state density profile and steady-state current and obtain the steady-state solution of a boundary-value problem for the double diffusivity model.Moments of the 2D SHE at criticalityhttps://www.zbmath.org/1483.600932022-05-16T20:40:13.078697Z"Gu, Yu"https://www.zbmath.org/authors/?q=ai:gu.yu.1"Quastel, Jeremy"https://www.zbmath.org/authors/?q=ai:quastel.jeremy"Tsai, Li-Cheng"https://www.zbmath.org/authors/?q=ai:tsai.li-chengSummary: We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius \(\varepsilon\to 0 \), we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit nontrivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of \textit{J. Dimock} and \textit{S. G. Rajeev} [J. Phys. A, Math. Gen. 37, No. 39, 9157--9173 (2004; Zbl 1067.81024)] to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamicshttps://www.zbmath.org/1483.601032022-05-16T20:40:13.078697Z"Baldassarri, Simone"https://www.zbmath.org/authors/?q=ai:baldassarri.simone"Nardi, Francesca Romana"https://www.zbmath.org/authors/?q=ai:nardi.francesca-romanaSummary: In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let \(\Lambda = \{ 0,1,\ldots,L\}^2 \subset \mathbb{Z}^2\) be a finite box. Particles perform simple exclusion on \(\Lambda\), but when they occupy neighboring sites they feel a binding energy \(-U_1 < 0\) in the horizontal direction and \(-U_2 < 0\) in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume \(\Lambda\). Along each bond touching the boundary of \(\Lambda\) from the outside to the inside, particles are created with rate \(\rho =e^{-\Delta\beta}\), while along each bond from the inside to the outside, particles are annihilated with rate 1, where \(\beta\) is the inverse temperature and \(\Delta > 0\) is an activity parameter. Thus, the boundary of \(\Lambda\) plays the role of an infinite gas reservoir with density \(\rho\). We consider the parameter regime \(U_1 > 2U_2\) also known as the strongly anisotropic regime. We take \(\Delta \in (U_1,U_1 +U_2)\) and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature \(\beta\). We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \textit{critical droplets}, as well as some of their properties. For the weakly anisotropic model corresponding to the parameter regime \(U_1 < 2U_2\), analogous results have already been obtained. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \textit{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the critical configurations.Covariant Symanzik identitieshttps://www.zbmath.org/1483.601122022-05-16T20:40:13.078697Z"Kassel, Adrien"https://www.zbmath.org/authors/?q=ai:kassel.adrien"Lévy, Thierry"https://www.zbmath.org/authors/?q=ai:levy.thierrySummary: Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.The fractal cylinder process: existence and connectivity phase transitionshttps://www.zbmath.org/1483.601452022-05-16T20:40:13.078697Z"Broman, Erik I."https://www.zbmath.org/authors/?q=ai:broman.erik-ivar"Elias, Olof"https://www.zbmath.org/authors/?q=ai:elias.olof"Mussini, Filipe"https://www.zbmath.org/authors/?q=ai:mussini.filipe"Tykesson, Johan"https://www.zbmath.org/authors/?q=ai:tykesson.johan-haraldSummary: We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension \(d\ge 2\), and a connectivity phase transition whenever \(d\ge 4\). We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.
A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.
In addition we also determine the almost sure Hausdorff dimension of the fractal set.Slow convergence of Ising and spin glass models with well-separated frustrated verticeshttps://www.zbmath.org/1483.601462022-05-16T20:40:13.078697Z"Gillman, David"https://www.zbmath.org/authors/?q=ai:gillman.david-w|gillman.david-saul|gillman.david-w.1"Randall, Dana"https://www.zbmath.org/authors/?q=ai:randall.dana-jSummary: Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an \(n\times n\) square lattice region Lambda with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be \(+\) and the left and right sides to be \(-\), a standard Peierls argument based on energy shows that below some critical temperature \(t_c\), any local Markov chain \(\mathcal{M}\) requires time exponential in \(n\) to mix.\par Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of \(\Lambda\) and yields weaker results when they are on the boundary of \(\Lambda\) but not near the corners. We show that there is a universal temperature \(T\) below which \(\mathcal{M}\) will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.
For the entire collection see [Zbl 1390.68020].Excess deviations for points disconnected by random interlacementshttps://www.zbmath.org/1483.601492022-05-16T20:40:13.078697Z"Sznitman, Alain-Sol"https://www.zbmath.org/authors/?q=ai:sznitman.alain-solSummary: We consider random interlacements on \(\mathbb{Z}^d\), \(d \ge 3\), when their vacant set is in a strongly percolative regime. Given a large box centered at the origin, we establish an asymptotic upper bound on the exponential rate of decay of the probability that the box contains an excessive fraction \(\nu\) of points that are disconnected by random interlacements from the boundary of a concentric box of double size. As an application, we show that when \(\nu\) is not too large this asymptotic upper bound matches the asymptotic lower bound derived in our work [``On bulk deviations for the local behavior of random interlacements'', Preprint, \url{arXiv:1906.05809}], and the exponential rate of decay is governed by the variational problem in the continuum involving the percolation function of the vacant set of random interlacements that we studied in [Prog. Probab. 77, 775--796 (2021; Zbl 1469.60347)]. This is a further confirmation of the pertinence of this variational problem.Practical MATLAB modeling with Simulink. Programming and simulating ordinary and partial differential equationshttps://www.zbmath.org/1483.650022022-05-16T20:40:13.078697Z"Eshkabilov, Sulaymon L."https://www.zbmath.org/authors/?q=ai:eshkabilov.sulaymon-lIn the first part of the book solution methods for ordinary differential equations (ODEs) are discussed. In Chapter 1 it is explained how one can evaluate the analytical solution of ODEs by using the Symbolic Math toolbox of MATLAB. Hereby, first-order and second order ODEs are considered. The application of numerical methods (Euler, Runge-Kutta, Adams-Bashford, Adams-Moulton and others methods) for solving first-order initial value problems of ODEs is explained in Chapter 2. Hereby, three possibilities are discussed, writing own MATLAB script files, using MATLAB's built-in ODE solvers, and modeling with Simulink. Chapter 3 is devoted to the solution of second-order ODEs by numerical methods and by using the symbolic toolbox as well as by modeling with Simulink. The topics of Chapters 4, 5, and 6 are possibilities for solving stiff ODEs, higher-order ODEs, coupled ODEs, and implicit ODEs. Chapter 7 gives a comparative analysis of the different ways for solving ODEs which are presented in the previous chapters. The accuracy and efficiency of these approaches is compared by one example. The topic of the second part of the book is the solution of boundary value problems of ODEs by using MATLAB built-in solvers. In the third part (Chapters 9--12) the solution of real-life problems modeled by ODEs is discussed: spring-mass-damper systems, electromechanical and mechanical systems, trajectory problems, Lorenz systems, and Lotka-Volterra problems. In the fourth part of the book the solution of partial differential equations (PDEs) is discussed. Hereby, the one-dimensional heat transfer problem, two-dimensional heat transfer problems as well as one- and two-dimensional wave propagation problems are considered. All presented solution methods are demonstrated by numerous examples. Some of the chapters contain also exercises.
Reviewer: Michael Jung (Dresden)Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equationshttps://www.zbmath.org/1483.651292022-05-16T20:40:13.078697Z"Braukhoff, Marcel"https://www.zbmath.org/authors/?q=ai:braukhoff.marcel"Jüngel, Ansgar"https://www.zbmath.org/authors/?q=ai:jungel.ansgarSummary: Structure-preserving finite-difference schemes for general nonlinear fourth-order parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida-Lebowitz-Speer-Spohn equations. The schemes conserve the mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation. Numerical examples in one and two space dimensions illustrate the dissipation properties.PIFE-PIC: parallel immersed finite element particle-in-cell for 3-D kinetic simulations of plasma-material interactionshttps://www.zbmath.org/1483.651992022-05-16T20:40:13.078697Z"Han, Daoru"https://www.zbmath.org/authors/?q=ai:han.daoru"He, Xiaoming"https://www.zbmath.org/authors/?q=ai:he.xiaoming.1"Lund, David"https://www.zbmath.org/authors/?q=ai:lund.david"Zhang, Xu"https://www.zbmath.org/authors/?q=ai:zhang.xu.1In this paper, the authors show the performance of a software package, called PIFE-PIC, developed by the authors for solving three space dimensional kinetic simulations of plasma-material interactions. The key ideas are to apply domain decomposition method and parallel computations. In particular, both the field equations and the particle equations are solved parallelly in overlapping subdomains. This can certainly provide a significant speed-up. The authors provide a set of numerical experiment to validate the performance of their software. The test set includes a large-scale simulation of plasma charging at a lunar crater. For some test cases, parallel efficiency up to approximately 110\% superlinear speedup was achieved.
Reviewer: Eric Chung (Hong Kong)Generalized Swift-Hohenberg and phase-field-crystal equations based on a second-gradient phase-field theoryhttps://www.zbmath.org/1483.740052022-05-16T20:40:13.078697Z"Espath, Luis"https://www.zbmath.org/authors/?q=ai:espath.luis-f-r"Calo, Victor M."https://www.zbmath.org/authors/?q=ai:calo.victor-manuel"Fried, Eliot"https://www.zbmath.org/authors/?q=ai:fried.eliotSummary: The principle of virtual power is used derive a microforce balance for a second-gradient phase-field theory. In conjunction with constitutive relations consistent with a free-energy imbalance, this balance yields a broad generalization of the Swift-Hohenberg equation. When the phase field is identified with the volume fraction of a conserved constituent, a suitably augmented version of the free-energy imbalance yields constitutive relations which, in conjunction with the microforce balance and the constituent content balance, delivers a broad generalization of the phase-field-crystal equation. Thermodynamically consistent boundary conditions for situations in which the interface between the system and its environment is structureless and cannot support constituent transport are also developed, as are energy decay relations that ensue naturally from the thermodynamic structure of the theory.Emergent quantumness in neural networkshttps://www.zbmath.org/1483.810082022-05-16T20:40:13.078697Z"Katsnelson, Mikhail I."https://www.zbmath.org/authors/?q=ai:katsnelson.mikhail-i"Vanchurin, Vitaly"https://www.zbmath.org/authors/?q=ai:vanchurin.vitalySummary: It was recently shown that the Madelung equations, that is, a hydrodynamic form of the Schrödinger equation, can be derived from a canonical ensemble of neural networks where the quantum phase was identified with the free energy of hidden variables. We consider instead a grand canonical ensemble of neural networks, by allowing an exchange of neurons with an auxiliary subsystem, to show that the free energy must also be multivalued. By imposing the multivaluedness condition on the free energy we derive the Schrödinger equation with ``Planck's constant'' determined by the chemical potential of hidden variables. This shows that quantum mechanics provides a correct statistical description of the dynamics of the grand canonical ensemble of neural networks at the learning equilibrium. We also discuss implications of the results for machine learning, fundamental physics and, in a more speculative way, evolutionary biology.Prepare non-classical collective spin state by designing control fieldshttps://www.zbmath.org/1483.810162022-05-16T20:40:13.078697Z"Zhao, X. L."https://www.zbmath.org/authors/?q=ai:zhao.xianglian|zhao.xiaolin|zhao.xilai|zhao.xianglan|zhao.xiaolong|zhao.xin-li|zhao.xile|zhao.xuelei|zhao.xiaoli|zhao.xiaole|zhao.xiuli|zhao.xinlong|zhao.xiulan|zhao.xueliang|zhao.xianlin|zhao.xuelong|zhao.xilu|zhao.xiaoling|zhao.xiaolei|zhao.xiaolu|zhao.xinlei|zhao.xiaoliang|zhao.xule"Ma, Y. L."https://www.zbmath.org/authors/?q=ai:ma.yulin"Ma, H. Y."https://www.zbmath.org/authors/?q=ai:ma.hongyang"Qiu, T. H."https://www.zbmath.org/authors/?q=ai:qiu.tianhui"Yi, X. X."https://www.zbmath.org/authors/?q=ai:yi.xuexiSummary: We propose a control method inspired by Lyapunov control to prepare non-classical state for a collective spin model. The spin squeezed states with entanglement can be prepared effectively. Meanwhile, the spin squeezed direction is fixed and the coherence can be maintained largely. It costs less time to obtain the non-classical state for larger systems and the control method performs effectively in a range of amplitudes of the control fields. The control method not only can be optimized by reducing the fast oscillations in the control fields but also behaves robustly for a range of deviation in the initial state and imperfections in the control fields during evolution. The non-classical states can be prepared in the presence of decoherence due to interaction with the environment in a range. We confirm that this closed-loop method can be applied to open-loop procedure in order to avoid quantum collapse resulting from measurement. This method can be applied in precision metrology by atom interferometry with a Bose-Einstein condensate.Anyons in the operational formalismhttps://www.zbmath.org/1483.810172022-05-16T20:40:13.078697Z"Neori, Klil H."https://www.zbmath.org/authors/?q=ai:neori.klil-h"Goyal, Philip"https://www.zbmath.org/authors/?q=ai:goyal.philipSummary: The operational formalism to quantum mechanics seeks to base the theory on a firm foundation of physically well-motivated axioms. It has succeeded in deriving the Feynman rules for general quantum systems. Additional elaborations have applied the same logic to the question of identical particles, confirming the so-called Symmetrization Postulate: that the only two options available are fermions and bosons. However, this seems to run counter to results in two-dimensional systems, which allow for anyons, particles with statistics which interpolate between Fermi-Dirac and Bose-Einstein. In this talk we will show that the results in two dimensions can be made compatible with the operational results. That is, we will show that anyonic behavior is a result of the topology of the space in two dimensions, and does not depend on the particles being identical; but that nevertheless, if the particles are identical, the resulting system is still anyonic.
For the entire collection see [Zbl 1470.00021].Thermal entanglement in a Ising spin chain with Dzyaloshinski-Moriya anisotropic antisymmetric interaction in a nonuniform magnetic fieldhttps://www.zbmath.org/1483.810282022-05-16T20:40:13.078697Z"Huang, Li-Yuan"https://www.zbmath.org/authors/?q=ai:huang.liyuanSummary: The thermal entanglement in the two-qubit Ising spin chain in the presence of the Dzyaloshinski-Moriya(DM) anisotropic antisymmetric interaction in a nonuniform magnetic field is investigated. The influences of the DM coupling constant \(D\), the temperature \(T\), the uniform external magnetic field \(B\) and the nonuniform magnetic field \(h\) on the thermal entanglement measured by the concurrence \(C\) are studied in detail. The results show that both the increasing \(T\) and \(|B|\) decrease the \(C\), but the increasing \(D\) develops the \(C\), and \(D\) can also heighten the values of the threshold magnetic field |\(B_t\)| and the temperature \(T_t\) above which the thermal entanglement vanishes. And for a definite \(D\), the increasing \(T\) makes the \(|B_t|\) become bigger as well. By comparison, before and after the critical temperature \(T_c\), the \(h\) has different effects on \(C\). Within a certain temperature range, the increasing \(h\) makes the \(C\) rise firstly and then fall. What's more, as the \(h\) increases, the key temperature \(T_k\) at which the \(C\) reaches the maximum value increases. As a result, the thermal entanglement can be controlled by adjusting the values of \(B, h, D\) and \(T\) in various terrible environment, such as in strong external magnetic field, or high temperature environment, which will be useful in the research of quantum information in solid systems.BCS effect on quantum correlation and tripartite quantum entanglement in spinless gapless Tomonaga-Luttinger liquid and cuprate superconducting nanowirehttps://www.zbmath.org/1483.810302022-05-16T20:40:13.078697Z"Montazeri, M. R."https://www.zbmath.org/authors/?q=ai:montazeri.m-r"Afzali, R."https://www.zbmath.org/authors/?q=ai:afzali.rSummary: It has been known that quantum information offers powerful instruments to investigate the properties of many-body systems. In this framework, we touched two particular aspect of this activity, namely the quantum entanglement and discord to compare the properties of gapless Tomonaga-Luttinger Liquid (TLL) model and the effect of BCS coupling in spinless fermions of TLL as a cuprate superconducting nanowire. Using two-fermion space-spin density matrix, we investigate quantum correlation of these cases via bipartite and tripartite entanglement, as well as quantum discord. The relations of concurrence (as a measure of quantum entanglement), the lower bound of the generalized robustness of tripartite entanglement and quantum discord in terms of the relative distance between fermions and the coupling parameter were accordingly obtained. The relationship between the compressibility as a physical property of system and quantum correlations has also been studied.Thermal entanglement in \(2 \times 3\) Heisenberg chains via distance between stateshttps://www.zbmath.org/1483.810312022-05-16T20:40:13.078697Z"Silva, Saulo L. L."https://www.zbmath.org/authors/?q=ai:silva.saulo-l-lSummary: Most of the work involving entanglement measurement focuses on systems that can be modeled by two interacting qubits. This is due to the fact that there are few studies presenting entanglement analytical calculations in systems with spins \(s > 1/2\). In this paper, we present for the first time an analytical way of calculating thermal entanglement in a dimension \(2 \otimes 3\) Heisenberg chain through the distance between states. We use the Hilbert-Schmidt norm to obtain entanglement. The result obtained can be used to calculate entanglement in chains with spin-1/2 coupling with spin-1, such as ferrimagnetic compounds as well as compounds with dimer-trimer coupling.Symmetry enriched phases of quantum circuitshttps://www.zbmath.org/1483.810392022-05-16T20:40:13.078697Z"Bao, Yimu"https://www.zbmath.org/authors/?q=ai:bao.yimu"Choi, Soonwon"https://www.zbmath.org/authors/?q=ai:choi.soonwon"Altman, Ehud"https://www.zbmath.org/authors/?q=ai:altman.ehudSummary: Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry.
Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with \(\mathbb{Z}_2\) symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a \(U(1)\) critical phase at moderate measurement rates and a Kosterlitz-Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.Shortcut to adiabatic two-qubit state swap in a superconducting circuit QED via effective drivingshttps://www.zbmath.org/1483.810412022-05-16T20:40:13.078697Z"Li, Ming"https://www.zbmath.org/authors/?q=ai:li.ming|li.ming.3|li.ming.1|li.ming.8|li.ming.9|li.ming.2|li.ming.5|li.ming.4|li.ming.6|li.ming.7"Dong, Xin-Ping"https://www.zbmath.org/authors/?q=ai:dong.xin-ping"Yan, Run-Ying"https://www.zbmath.org/authors/?q=ai:yan.run-ying"Lu, Xiao-Jing"https://www.zbmath.org/authors/?q=ai:lu.xiaojing"Zhao, Zheng-Yin"https://www.zbmath.org/authors/?q=ai:zhao.zheng-yin"Feng, Zhi-Bo"https://www.zbmath.org/authors/?q=ai:feng.zhi-boSummary: Optimal two-qubit operation is of significance to quantum information processing. An efficient scheme is proposed for realizing the shortcut to adiabatic two-qubit state swap in a superconducting circuit quantum electrodynamics (QED) via effective drivings. Two superconducting qutrits are coupled to a common cavity field and individual classical drivings. Based on two Gaussian-type Rabi drivings, two-qubit state swap can be adiabatically implemented within a reduced three-state system. To speed up the operation, these two original Rabi drivings are modified in the framework of shortcuts to adiabaticity, instead of adding an extra counterdiabatic driving. Moreover, owing to a shorter duration time, the decoherence effects on the accelerated quantum operation can be mitigated significantly. The strategy could offer an optimized method to construct fast and robust quantum operations on superconducting qubits experimentally.Gibbs measures of nonlinear Schrödinger equations as limits of quantum many-body states in dimension \(d\leq 3\)https://www.zbmath.org/1483.810652022-05-16T20:40:13.078697Z"Sohinger, Vedran"https://www.zbmath.org/authors/?q=ai:sohinger.vedranSummary: Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. In [\textit{J. Fröhlich} et al., Commun. Math. Phys. 356, No. 3, 883--980 (2017; Zbl 1381.81177)], we prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. We consider bounded defocusing interaction potentials in dimensions \(d = 1, 2, 3\), and we work either on \(\mathbb{T}^d\) or on \(\mathbb{R}^d\) with a confining potential. The analogous problem for \(d = 1\) and in higher dimensions with smooth nontranslation-invariant interactions was recently studied by \textit{M. Lewin} et al. [J. Éc. Polytech., Math. 2, 65--115 (2015; Zbl 1322.81082)] by means of entropy methods.
In our work, we apply a perturbative expansion of the interaction, motivated by ideas from field theory. The terms of the expansion are analysed using a diagrammatic representation and their sum is controlled using Borel resummation techniques. When \(d = 2,3\), we apply a Wick ordering renormalisation procedure. The latter allows us to deal with the rapid growth of the number of particles. Moreover, in the one-dimensional setting our methods allow us to obtain a microscopic derivation of time-dependent correlation functions for the cubic nonlinear Schrödinger equation [\textit{J. Fröhlich} et al., Adv. Math. 353, 67--115 (2019; Zbl 1421.82022)]. All results presented in this chapter are based on joint work with Jürg Fröhlich (ETH Zürich), Antti Knowles (University of Geneva), and Benjamin Schlein (University of Zürich).
For the entire collection see [Zbl 1473.53004].Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonianshttps://www.zbmath.org/1483.810722022-05-16T20:40:13.078697Z"Cornean, Horia"https://www.zbmath.org/authors/?q=ai:cornean.horia-d"Monaco, Domenico"https://www.zbmath.org/authors/?q=ai:monaco.domenico"Moscolari, Massimo"https://www.zbmath.org/authors/?q=ai:moscolari.massimoSummary: It is well known that, given a \(2d\) purely magnetic Landau Hamiltonian with a constant magnetic field \(b\) which generates a magnetic flux \(\varphi\) per unit area, then any spectral island \(\sigma_b\) consisting of \(M\) infinitely degenerate Landau levels carries an integrated density of states \(\mathcal{I}_b=M \varphi \). Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any \(2d\) Bloch-Landau operator \(H_b\) which also has a bounded \(\mathbb{Z}^2\)-periodic electric potential. Assume that \(H_b\) has a spectral island \(\sigma_b\) which remains isolated from the rest of the spectrum as long as \(\varphi\) lies in a compact interval \([\varphi_1,\varphi_2]\). Then \(\mathcal{I}_b=c_0+c_1\varphi\) on such intervals, where the constant \(c_0\in\mathbb{Q}\) while \(c_1\in \mathbb{Z} \). The integer \(c_1\) is the Chern marker of the spectral projection onto the spectral island \(\sigma_b\). This result also implies that the Fermi projection on \(\sigma_b\), albeit continuous in \(b\) in the strong topology, is nowhere continuous in the norm topology if either \(c_1\ne 0\) or \(c_1=0\) and \(\varphi\) is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.Generalized multifractality at spin quantum Hall transitionhttps://www.zbmath.org/1483.810732022-05-16T20:40:13.078697Z"Karcher, Jonas F."https://www.zbmath.org/authors/?q=ai:karcher.jonas-f"Charles, Noah"https://www.zbmath.org/authors/?q=ai:charles.noah"Gruzberg, Ilya A."https://www.zbmath.org/authors/?q=ai:gruzberg.ilya-a"Mirlin, Alexander D."https://www.zbmath.org/authors/?q=ai:mirlin.aleksandr-davidovichSummary: Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C.
Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then ``translate'' them to the language of eigenstate observables. Performing numerical simulations on network models for SQH and IQH transitions, we confirm the analytical predictions for scaling observables and determine the corresponding exponents. Remarkably, the generalized-multifractality exponents at the SQH critical point strongly violate the generalized parabolicity of the spectrum, which implies violation of the local conformal invariance at this critical point.Band structures of hybrid graphene quantum dots with magnetic fluxhttps://www.zbmath.org/1483.810752022-05-16T20:40:13.078697Z"Lemaalem, Bouchaib"https://www.zbmath.org/authors/?q=ai:lemaalem.bouchaib"Zahidi, Youness"https://www.zbmath.org/authors/?q=ai:zahidi.youness"Jellal, Ahmed"https://www.zbmath.org/authors/?q=ai:jellal.ahmedSummary: We study the band structures of hybrid graphene quantum dots subject to a magnetic flux and electrostatic potential. The system is consisting of a circular single layer graphene surrounded by an infinite bilayer graphene. By solving the Dirac equation we obtain the solution of the energy spectrum in two regions. For the valley \(K\), it is found that the magnetic flux strongly acts by decreasing the gap and shifting energy levels away from zero radius with some oscillations, which are not observed for null flux case. As for the valley \(K^\prime\), the energy levels rapidly increase when the radius increases. A number of oscillations appeared that is strongly dependent on the values taken by the magnetic flux.Exciton stability and luminescence in InN/(In,Ga)N quantum dots under size and shell content effectshttps://www.zbmath.org/1483.810772022-05-16T20:40:13.078697Z"Benhaddou, Farid"https://www.zbmath.org/authors/?q=ai:benhaddou.farid"El Ghazi, Haddou"https://www.zbmath.org/authors/?q=ai:el-ghazi.haddou"Abboudi, Hassan"https://www.zbmath.org/authors/?q=ai:abboudi.hassan"Zorkani, Izeddine"https://www.zbmath.org/authors/?q=ai:zorkani.izeddine"Jorio, Anouar"https://www.zbmath.org/authors/?q=ai:jorio.anouarSummary: The electronic structure and associated excitonic properties of core/shell nanocrystals based on InN/(In,Ga)N quantum dots with InN-core and (In,Ga)N-shell are investigated numerically using perturbation theory. The shell In(Ga)N and ligand in this nanocrystal strengthen confinement, passivate the structure, and improve the InN optical characteristics. Such confinement is modelled by a finite potential considering the effective mass and dielectric constant mismatches between the core and shell. The impacts of Indium-content and quantum dot size on charge carrier position, excitonic states, excitonic binding energy, exciton stability, effective band-gap, and fundamental excitonic transition are investigated. Our numerical results reveal that the quantum dots size and Indium-content can be used to tailor nanocrystals and their luminescence properties for eventual opto-electronic applications. It is found that the exciton is more stable at room temperature in nanostrucure with lower Indium-content and thin core.Majorana representation for a composite systemhttps://www.zbmath.org/1483.810822022-05-16T20:40:13.078697Z"Yang, Jing"https://www.zbmath.org/authors/?q=ai:yang.jing"Zhang, Yong"https://www.zbmath.org/authors/?q=ai:zhang.yong.2|zhang.yong.10|zhang.yong.1|zhang.yong.13|zhang.yong.11|zhang.yong.15|zhang.yong.7|zhang.yong.14|zhang.yong.12|zhang.yong.9|zhang.yong.8|zhang.yong.5Summary: The Majorana representation, which provides an intuitive way to represent the quantum state by stars on the Bloch sphere, has drawn considerable attention in the context of many body systems or systems with high-dimensional Hilbert space. In this work, we study Majorana representation for an interacting spin-1/2 composite system and its relation with the quantum correlation between the subsystems. Similar with the Majorana representation, we can define a new intuitive representation for the composite system by defining three stars which represents the dynamics and correlation of the two subsystems. The results show that, these two representations are the same for the separable case. The Berry phase has been found to be directly related to the concurrence between the two subsystems. By study the cases of no correlation and large correlation, the quantum entanglement and the Berry phases of the subsystem and their two representations on the Bloch sphere are discussed. These representations by stars provides a intuitive way to study the dynamics, entanglement and Berry phase composite system and its subsystems.Boson-fermion correspondence, QQ-relations and Wronskian solutions of the T-systemhttps://www.zbmath.org/1483.810862022-05-16T20:40:13.078697Z"Tsuboi, Zengo"https://www.zbmath.org/authors/?q=ai:tsuboi.zengoSummary: It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra \(U_q(gl(2r | 1)^{(2)})\) and the non-twisted quantum affine algebra \(U_q(so(2r + 1)^{(1)})\), we proposed, in the previous paper [the author, ibid. 870, No. 1, 92--137 (2013; Zbl 1262.17017)], a Wronskian solution of the T-system for \(U_q(so(2r + 1)^{(1)})\) as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra \(U_q(gl(2r | 1)^{(1)})\). In this paper, we elaborate on this solution, and give a proof missing in [loc. cit.]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [\textit{A. Kuniba} et al., J. Phys. A, Math. Gen. 28, No. 21, 6211--6226 (1995; Zbl 0871.17015)]. We also propose Wronskian-type expressions of T-functions (eigenvalues of transfer matrices) labeled by non-rectangular Young diagrams, which are quantum affine algebra analogues of the Weyl character formula for \(so(2r + 1)\). We show that T-functions for spinorial representationsof \(U_q(so(2r + 1)^{(1)})\) are related to reductions of T-functions for asymptotic typical representations of \(U_q(gl( 2r | 1)^{(1)})\).Survival of entanglement in an optical parametric oscillation from decoherencehttps://www.zbmath.org/1483.810882022-05-16T20:40:13.078697Z"Ayehu, Desalegn"https://www.zbmath.org/authors/?q=ai:ayehu.desalegnSummary: We consider a parametric oscillation driven by coherent light beams and coupled to a two-mode thermal reservoir. Applying the solutions of the quantum Langevin equations, we investigate the effect of decoherence and the driving coherent light beams on the squeezing and entanglement as well as statistical properties of the two-mode cavity radiation. We show that the two-mode thermal noise and the driving coherent light beams lead to an increase in the intensity of the cavity radiation. However, the driving coherent light beams do not have any effect on the squeezing and entanglement properties of the two-mode cavity radiation while the two-mode thermal noise leads to a decrease in the degree of squeezing and entanglement of the two-mode cavity radiation.Wilson loop expectations in lattice gauge theories with finite gauge groupshttps://www.zbmath.org/1483.811142022-05-16T20:40:13.078697Z"Cao, Sky"https://www.zbmath.org/authors/?q=ai:cao.skyUsual computations in quantum field theory (QFT) are based on a perturbative expansion in the coupling constant which breaks down at strong coupling. This situation is particularly relevant for quantum chromodynamics (QCD) which is a SU(3) gauge theory describing the strong interaction of the Standard Model between particles possessing a charge called color. One of the most important question in this context is how confinement arises, that is, why only color-singlet particles can be observed. A Wilson loop operator is gauge-invariant observable obtained from the holonomy of the gauge field along a closed loop whose expectation value provides an order parameter for the confinement phase transition. A very successful approach is lattice QFT which consists in discretizing spacetime and reducing the path integral to usual integrals. As such, lattice QFT is particularly convenient for both analytical and numerical computations.
The current paper follows the first approach and aims at studying lattice QFT with discrete gauge groups. The main result is the computation of the Wilson loop at first order in a small coupling constant expansion \(\beta^{-1}\) for an infinite lattice \(\mathbb Z^4\). The idea of the proof is to define the theory on a lattice \(\Lambda\) of finite size, computes the Wilson loop, and takes a subsequential limit \(\Lambda \to \mathbb Z^4\). This is done by replacing edge variables by surfaces, similar to what happens for the Ising model (with spin variables replaced by loops); then, these surfaces are described by a Poisson random process whose generating function is the expectation value of the Wilson loop. This last point requires having a discrete group, which explains the focus of the author. The proof considers first Abelian groups before generalizing to non-Abelian groups.
Reviewer: Harold Erbin (Boston)The R-matrix formalism for two-particle scattering problemshttps://www.zbmath.org/1483.811322022-05-16T20:40:13.078697Z"Anghel, Dragoş-Victor"https://www.zbmath.org/authors/?q=ai:anghel.dragos-victor"Preda, Amanda Teodora"https://www.zbmath.org/authors/?q=ai:preda.amanda-teodora"Nemnes, George Alexandru"https://www.zbmath.org/authors/?q=ai:nemnes.george-alexandruSummary: The R-matrix formalism is extended to include two-particle scattering events. The system is the same as in the case of one-particle R-matrix formalism and consists of a number (\(\geq 2\)) of \textit{leads} connected to a \textit{scattering region}. When the particles are identical fermions, our approach conserves the anti-symmetry of the wavefunction in the entire system, which opens the possibility to account for asymptotic entangled states. The particles have a mutual interaction when they are both in the scattering region. Using a proper Ansatz for the two-particle wavefunctions, we obtain a consistent system of equations which gives the coefficients for the two-particle scattering functions expanded in terms of anti-symmetrized products of one-particle scattering functions, when at least one particle is in the leads, or of the two-particle eigenstates of the Hamiltonian, when both particles are in the scattering region. Our formalism applies also to bosons.Scattering matrix of elementary excitations in the antiperiodic \textit{XXZ} spin chain with \(\eta = \frac{i\pi}{3}\)https://www.zbmath.org/1483.811352022-05-16T20:40:13.078697Z"Sun, Pei"https://www.zbmath.org/authors/?q=ai:sun.pei"Yang, Jintao"https://www.zbmath.org/authors/?q=ai:yang.jintao"Qiao, Yi"https://www.zbmath.org/authors/?q=ai:qiao.yi"Cao, Junpeng"https://www.zbmath.org/authors/?q=ai:cao.junpeng"Yang, Wen-Li"https://www.zbmath.org/authors/?q=ai:yang.wenliSummary: We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter \(\eta = \frac{\pi i}{3}\). We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.QCD \(\theta\)-vacuum in a uniform magnetic fieldhttps://www.zbmath.org/1483.811402022-05-16T20:40:13.078697Z"Adhikari, Prabal"https://www.zbmath.org/authors/?q=ai:adhikari.prabalSummary: We study the \(\theta\)-vacuum of QCD using two-flavor chiral perturbation theory (\(\chi\)PT) in the presence of a uniform, background magnetic field calculating the magnetic field-dependent free energy density, the topological density, the topological susceptibility and the fourth cumulant at one-loop order. We find that the topological susceptibility is enhanced by the magnetic field while the fourth topological cumulant is diminished at weak fields and enhanced at larger fields when \(\theta = 0\). However, in the QCD vacuum with \(\theta \neq 0\), the topological susceptibility can be either monotonically enhanced or diminished relative to their \(\theta\)-vacuum values. The fourth cumulant also exhibits monotonic enhancement or suppression except for regions of \(\theta\) near 0 and \(2\pi\), where it is both diminished and enhanced. Finally, the topological density is enhanced for all magnetic fields with its relative shift being identical to the relative shift of the up and down quark condensates in the \(\theta \)-vacuum.Non-conformal attractor in boost-invariant plasmashttps://www.zbmath.org/1483.811432022-05-16T20:40:13.078697Z"Chattopadhyay, Chandrodoy"https://www.zbmath.org/authors/?q=ai:chattopadhyay.chandrodoy"Jaiswal, Sunil"https://www.zbmath.org/authors/?q=ai:jaiswal.sunil-prasad"Du, Lipei"https://www.zbmath.org/authors/?q=ai:du.lipei"Heinz, Ulrich"https://www.zbmath.org/authors/?q=ai:heinz.ulrich"Pal, Subrata"https://www.zbmath.org/authors/?q=ai:pal.subrataSummary: We study the dissipative evolution of (0+1)-dimensionally expanding media with Bjorken symmetry using the Boltzmann equation for massive particles in relaxation-time approximation. Breaking conformal symmetry by a mass induces a non-zero bulk viscous pressure in the medium. It is shown that even a small mass (in units of the local temperature) drastically modifies the well-known attractor for the shear Reynolds number previously observed in massless systems. For generic nonzero particle mass, neither the shear nor the bulk viscous pressure relax quickly to a non-equilibrium attractor; they approach the hydrodynamic limit only late, at small values of the inverse Reynolds numbers.
Only the longitudinal pressure, which is a combination of thermal, shear and bulk viscous pressures, continues to show early approach to a far-off-equilibrium attractor, driven by the rapid longitudinal expansion at early times. Second-order dissipative hydrodynamics based on a gradient expansion around locally isotropic thermal equilibrium fails to reproduce this attractor.Monopole hierarchy in transitions out of a Dirac spin liquidhttps://www.zbmath.org/1483.811642022-05-16T20:40:13.078697Z"Dupuis, Éric"https://www.zbmath.org/authors/?q=ai:dupuis.eric"Witczak-Krempa, William"https://www.zbmath.org/authors/?q=ai:witczak-krempa.williamSummary: Quantum spin liquids host novel emergent excitations, such as monopoles of an emergent gauge field. Here, we study the hierarchy of monopole operators that emerges at quantum critical points (QCPs) between a two-dimensional Dirac spin liquid and various ordered phases. This is described by a confinement transition of quantum electrodynamics in two spatial dimensions (QED\(_3\) Gross-Neveu theories) Focusing on a spin ordering transition, we get the scaling dimension of monopoles at leading order in a large-\(N\) expansion, where \(2 N\) is the number of Dirac fermions, as a function of the monopole's total magnetic spin. Monopoles with a maximal spin have the smallest scaling dimension while monopoles with a vanishing magnetic spin have the largest one, the same as in pure QED\(_3\). The organization of monopoles in multiplets of the QCP's symmetry group SU\((2)\times\)SU(N) is shown for general N.A new second-order upper bound for the ground state energy of dilute Bose gaseshttps://www.zbmath.org/1483.811652022-05-16T20:40:13.078697Z"Basti, Giulia"https://www.zbmath.org/authors/?q=ai:basti.giulia"Cenatiempo, Serena"https://www.zbmath.org/authors/?q=ai:cenatiempo.serena"Schlein, Benjamin"https://www.zbmath.org/authors/?q=ai:schlein.benjaminSummary: We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee-Huang-Yang formula. This result was first established in [J. Stat. Phys. 136, No. 3, 453--503 (2009; Zbl 1200.82002)] by \textit{H.-T. Yau} and \textit{J. Yin}. Our proof, which applies to repulsive and compactly supported \(V \in L^3 (\mathbb{R}^3)\), gives better rates and, in our opinion, is substantially simpler.How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: quantum Hall plateau transitions (almost) all the way downhttps://www.zbmath.org/1483.811662022-05-16T20:40:13.078697Z"Karcher, Jonas F."https://www.zbmath.org/authors/?q=ai:karcher.jonas-f"Foster, Matthew S."https://www.zbmath.org/authors/?q=ai:foster.matthew-sSummary: We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear \(\sigma\)-model picture imply that most TSC surface states should Anderson localize for arbitrarily weak disorder (CI, AIII), or exhibit weak antilocalizing behavior (DIII). The numerical studies reviewed here instead indicate \textit{spectrum-wide surface quantum criticality}, characterized by robust eigenstate multifractality throughout the surface-state energy spectrum. In other words, there is an ``energy stack'' of critical wave functions. For class AIII, multifractal eigenstate and conductance analysis reveals identical statistics for states throughout the stack, consistent with the class A integer quantum-Hall plateau transition (QHPT). Class CI TSCs exhibit surface stacks of class C spin QHPT states. Critical stacking of a third kind, possibly associated to the class D thermal QHPT, is identified for \textit{nematic} velocity disorder of a single Majorana cone in class DIII. The Dirac theories studied here can be represented as perturbed 2D Wess-Zumino-Novikov-Witten sigma models; the numerical results link these to Pruisken models with the topological angle \(\vartheta=\pi\). Beyond applications to TSCs, all three stacked Dirac theories (CI, AIII, DIII) naturally arise in the effective description of dirty \(d\)-wave quasiparticles, relevant to the high-\(T_c\) cuprates.Hilbert-space fragmentation, multifractality, and many-body localizationhttps://www.zbmath.org/1483.811682022-05-16T20:40:13.078697Z"Pietracaprina, Francesca"https://www.zbmath.org/authors/?q=ai:pietracaprina.francesca"Laflorencie, Nicolas"https://www.zbmath.org/authors/?q=ai:laflorencie.nicolasSummary: Investigating many-body localization (MBL) using exact numerical methods is limited by the exponential growth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space.
Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to a Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space.Green's functions on a renormalized lattice: an improved method for the integer quantum Hall transitionhttps://www.zbmath.org/1483.811692022-05-16T20:40:13.078697Z"Puschmann, Martin"https://www.zbmath.org/authors/?q=ai:puschmann.martin"Vojta, Thomas"https://www.zbmath.org/authors/?q=ai:vojta.thomasSummary: We introduce a performance-optimized method to simulate localization problems on bipartite tight-binding lattices. It combines an exact renormalization group step to reduce the sparseness of the original problem with the recursive Green's function method. We apply this framework to investigate the critical behavior of the integer quantum Hall transition of a tight-binding Hamiltonian defined on a simple square lattice. In addition, we employ an improved scaling analysis that includes two irrelevant exponents to characterize the shift of the critical energy as well as the corrections to the dimensionless Lyapunov exponent. We compare our findings with the results of a conventional implementation of the recursive Green's function method, and we put them into broader perspective in view of recent development in this field.The microscopic picture of the integer quantum Hall regimehttps://www.zbmath.org/1483.811702022-05-16T20:40:13.078697Z"Römer, Rudolf A."https://www.zbmath.org/authors/?q=ai:romer.rudolf-a"Oswald, Josef"https://www.zbmath.org/authors/?q=ai:oswald.josefSummary: Computer modeling of the integer quantum Hall effect based on self-consistent Hartree-Fock (HF) calculations has now reached an astonishing level of maturity. Spatially-resolved studies of the electron density at near macroscopic system sizes of up to \(1\mu\text{m}^2\) reveal self-organized clusters of locally fully filled and locally fully depleted Landau levels depending on which spin polarization is favored. The behavior results, for strong disorders, in an exchange-interaction induced \(g\)-factor enhancement and, ultimately, gives rise to narrow transport channels, including the celebrated narrow edge channels. For weak disorder, we find that bubble and stripes phases emerge with characteristics that predict experimental results very well. Hence the HF approach has become a convenient numerical basis to \textit{quantitatively} study the quantum Hall effects, superseding previous more qualitative approaches.Moments of the ground state density for the \(d\)-dimensional Fermi gas in an harmonic traphttps://www.zbmath.org/1483.811722022-05-16T20:40:13.078697Z"Forrester, Peter J."https://www.zbmath.org/authors/?q=ai:forrester.peter-jInteracting electrons in a random medium: a simple one-dimensional modelhttps://www.zbmath.org/1483.811732022-05-16T20:40:13.078697Z"Klopp, Frédéric"https://www.zbmath.org/authors/?q=ai:klopp.frederic"Veniaminov, Nikolaj A."https://www.zbmath.org/authors/?q=ai:veniaminov.nikolaj-aSummary: The present paper is devoted to the study of a simple model of interacting electrons in a random background. In a large interval \(\Lambda\), we consider \(n\) one-dimensional particles whose evolution is driven by the Luttinger-Sy model, i.e., the interval \(\Lambda\) is split into pieces delimited by the points of a Poisson process of intensity \(\mu\) and, in each piece, the Hamiltonian is the Dirichlet Laplacian. The particles interact through a repulsive pair potential decaying polynomially fast at infinity. We assume that the particles have a positive density, i.e., n \(/|\Lambda|\rightarrow \rho > 0\) as \(|\Lambda|\rightarrow +\infty\). In the low density or large disorder regime, i.e., \(\rho / \mu\) small, we obtain a two-term asymptotic for the thermodynamic limit of the ground state energy per particle of the interacting system; the first order correction term to the non- interacting ground state energy per particle is controlled by pairs of particles living in the same piece. The ground state is described in terms of its one- and two-particle reduced density matrix. Comparing the interacting and the non-interacting ground states, one sees that the effect of the repulsive interactions is to move a certain number of particles living together with another particle in a single piece to a new piece that was free of particles in the non-interacting ground state.
For the entire collection see [Zbl 1473.53004].Photon interactions with superconducting topological defectshttps://www.zbmath.org/1483.811762022-05-16T20:40:13.078697Z"Battye, Richard A."https://www.zbmath.org/authors/?q=ai:battye.richard-a"Viatic, Dominic G."https://www.zbmath.org/authors/?q=ai:viatic.dominic-gSummary: Using a toy model for the interactions between a defect-forming field and the photon field where the photon becomes massive in the defect core (motivated by recent work on defects in the 2HDM), we study the impact on photon propagation in the background of the defect. We find that, when the photon frequency (in natural units) is much lower than the symmetry breaking scale, domain walls reflect most of an incoming photon signal leading to potential interesting astrophysical signals. We also adapt the calculations for vortices and monopoles. We find that the case of strings is very similar to the standard case for massive scalar particles, but in the case of monopoles the cross-section is proportional to the geometrical area of the monopole.Virial series for a system of classical particles interacting through a pair potential with negative minimumhttps://www.zbmath.org/1483.820012022-05-16T20:40:13.078697Z"Procacci, Aldo"https://www.zbmath.org/authors/?q=ai:procacci.aldoUniversality for critical lines in Ising, vertex and dimer modelshttps://www.zbmath.org/1483.820022022-05-16T20:40:13.078697Z"Mastropietro, Vieri"https://www.zbmath.org/authors/?q=ai:mastropietro.vieriThe present paper reviews the proof of universality in the case of fixed point of planar lattice models such as Ising model with quartic interactions, vertex and dimer models.
Reviewer: Farruh Mukhamedov (Kuantan)Introduction to the algebraic Bethe ansatzhttps://www.zbmath.org/1483.820032022-05-16T20:40:13.078697Z"Slavnov, N. A."https://www.zbmath.org/authors/?q=ai:slavnov.nikita-aThis note is a short introduction to the Algebraic Bethe Ansatz that is one of the essential achievements of the Quantum Inverse Scattering Method. This note is based on the lecture given in the Scientific and Educational Center of Steklov Mathematical Institute in Moscow. The symmetry is limited to \(\mathfrak{sl}_2\) in this note. In Section 2 the \(R\)-matrix \(R(u,v)\) and the monodromy matrix \({T}(u)\) are introduced by the Yang-Baxter relation and the \(RTT\)-relation. The transfer matrix \(\mathcal{T}(u)\) is introduced as the sum of diagonal elements of the monodromy matrix \(T(u)=\left(\begin{smallmatrix} A(u)&B(u)\\ C(u)&D(u) \end{smallmatrix}\right)\). The transfer matrix \(\mathcal{T}(u)\) commutes with the Hamiltonian, hence constructing eigenfunctions of the transfer matrix determines those of the Hamiltonian. In Section 3 the XXX Heisenberg magnet is introduced as an example. In Section 4 the monodromy matrix of the XXZ Heisenberg magnet is introduced. In Section 5 is devoted to construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The \(RTT\) relation and the requirements of the spectral parameters called the Bethe equations allow construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The spectrums and the Bethe equations of the XXX Heisenberg magnet are given as an example.
For the entire collection see [Zbl 1472.53006].
Reviewer: Takeo Kojima (Yonezawa)The maxent extension of a quantum Gibbs family, convex geometry and geodesicshttps://www.zbmath.org/1483.820042022-05-16T20:40:13.078697Z"Weis, Stephan"https://www.zbmath.org/authors/?q=ai:weis.stephan|weis.stephan.1Summary: We discuss methods to analyze a quantum Gibbs family in the ultra-cold regime where the norm closure of the Gibbs family fails due to discontinuities of the maximum-entropy inference. The current discussion of maximum-entropy inference and irreducible correlation in the area of quantum phase transitions is a major motivation for this research. We extend a representation of the irreducible correlation from finite temperatures to absolute zero.
For the entire collection see [Zbl 1470.00021].Solution of the Kadanoff-Baym equations by iterated expansionshttps://www.zbmath.org/1483.820052022-05-16T20:40:13.078697Z"Özel, Cenap"https://www.zbmath.org/authors/?q=ai:ozel.cenap"Linker, Patrick"https://www.zbmath.org/authors/?q=ai:linker.patrick"Nauman, Syed Khalid"https://www.zbmath.org/authors/?q=ai:nauman.syed-khalidSummary: In this paper, we will show how the general nonlinear Kadanoff-Baym equations can be solved with iterated series expansion. In this regard, we will neglect vertex corrections. Further, we will obtain a formal solution in terms of colored tree graphs. The iteration procedure will show that after proper discretization, well-structured matrix equations would be obtained which are easy to implement in numerical simulation.Complete characterization of flocking versus nonflocking of Cucker-Smale model with nonlinear velocity couplingshttps://www.zbmath.org/1483.820062022-05-16T20:40:13.078697Z"Kim, Jong-Ho"https://www.zbmath.org/authors/?q=ai:kim.jongho"Park, Jea-Hyun"https://www.zbmath.org/authors/?q=ai:park.jea-hyunSummary: We consider the Cucker-Smale model with a regular communication rate and nonlinear velocity couplings, which can be understood as the parabolic equations for the discrete \(p\)-Laplacian \((p\geq 1)\) with nonlinear weights involving a parameter \(\beta(>0)\). For this model, we study the initial data and the ranges of \(p\) and \(\beta\) to characterize when flocking and nonflocking occur. Specifically, we analyze the nonflocking case, subdividing it into \textit{semi-nonflocking} (only velocity alignment holds) and \textit{full nonflocking} (group formation and velocity alignment do not hold). More precisely, we show that if \(\beta\in(0, 1]\), \(p\in[1,3)\), then flocking occurs for any initial data. If \(\beta\in(0,1]\), \(p\in[3,\infty)\), then semi-nonflocking occurs for any initial data. If \(\beta\in(1,\infty)\), \(p\in[1,3)\), then flocking occurs for some initial data. In the case \(\beta\in(1,\infty)\) and \(p\in[3,\infty)\), we observe alternative states. Finally, we have numerically verified the conclusions obtained by analytical calculations.Analytical solutions to the boundary integral equation: a case of angled dendrites and paraboloidshttps://www.zbmath.org/1483.820072022-05-16T20:40:13.078697Z"Alexandrov, Dmitri V."https://www.zbmath.org/authors/?q=ai:aleksandrov.dmitrii-valerevich"Galenko, Peter K."https://www.zbmath.org/authors/?q=ai:galenko.peter-konstantinovichThe present paper, by means of the boundary integral equations, the authors found generalized analytic solutions for the parabolic/paraboloidal growing shapes and angle-like dendrites in two/three space dimensions.
Reviewer: Farruh Mukhamedov (Kuantan)Noise-induced dynamics in a Josephson junction driven by trichotomous noiseshttps://www.zbmath.org/1483.820082022-05-16T20:40:13.078697Z"Jin, Yanfei"https://www.zbmath.org/authors/?q=ai:jin.yanfei"Wang, Heqiang"https://www.zbmath.org/authors/?q=ai:wang.heqiangSummary: Noise-induced dynamics is explored in a Josephson junction system driven by multiplicative and additive trichotomous noises in this paper. Under the adiabatic approximation, the analytical expression of average output current for the Josephson junction is obtained, which can be used to characterize stochastic resonance (SR). If only the additive trichotomous noise is considered, the large correlation time of additive noise can induce the suppression and the SR in the curve of average output current. When the effects of both multiplicative and additive trichotomous noises are considered, two pronounced peaks exist in the curves of average output current for large multiplicative noise amplitude and optimal additive noise intensity. That is, the stochastic multi-resonance phenomenon is observed in this system. Moreover, the curve of average output current appears a single peak as a function of multiplicative noise intensity, which disappears for the case of small fixed additive noise amplitude. Especially, the mean first-passage time (MFPT) as the function of additive trichotomous noise intensity displays a non-monotonic behavior with a maximum for the large multiplicative noise amplitude, which is called the phenomenon of the noise enhanced stability (NES).A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particleshttps://www.zbmath.org/1483.820092022-05-16T20:40:13.078697Z"Ferreira, Luís Simão"https://www.zbmath.org/authors/?q=ai:ferreira.luis-simaoSummary: In this paper, we proceed as suggested in the final section of [\textit{E. Carlen} et al., Ann. Probab. 48, No. 6, 2807--2844 (2020; Zbl 1456.60252)] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around \(0.02 \), which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.A method for simulating the dynamics of rarefied gas based on lattice Boltzmann equations and the BGK equationhttps://www.zbmath.org/1483.820102022-05-16T20:40:13.078697Z"Ilyin, O. V."https://www.zbmath.org/authors/?q=ai:ilyin.o-vSummary: A hybrid method for solving boundary value problems for rarefied gas using the Bhatnagar-Gross-Krook (BGK) model and the lattice Boltzmann equation is studied. One-dimensional boundary value problems subject to membrane-type boundary conditions are considered. In strongly nonequilibrium regions, the BGK model should be used, and in the regions in which the distribution function is close to Maxwell's one, the lattice Boltzmann equations can be used. On the region boundaries, a matching procedure should be performed; such a procedure is proposed in this paper. Note that the standard lattice Boltzmann models distort the distribution function on the region boundaries, but this distortion has no physical meaning. It is shown that, in order to correctly join the solutions on the region boundaries, the semi-moments of Maxwell's distribution must be exactly reproduced. For this purpose, novel lattice models of the Boltzmann equation are constructed using the entropy method. Results of numerical computations of the temperature and density profiles for the Knudsen number equal to 0.1 are presented, and the numerically obtained distribution function at the matching point is compared with the theoretical distribution function. Computation of the matching point is discussed.Direct simulation of charge transport in graphene nanoribbonshttps://www.zbmath.org/1483.820112022-05-16T20:40:13.078697Z"Nastasi, Giovanni"https://www.zbmath.org/authors/?q=ai:nastasi.giovanni"Camiola, V. Dario"https://www.zbmath.org/authors/?q=ai:camiola.vito-dario"Romano, Vittorio"https://www.zbmath.org/authors/?q=ai:romano.vittorioSummary: Graphene nanoribbons are considered as one of the most promising ways to design electron devices where the active area is made of graphene. In fact, graphene nanoribbons present a gap between the valence and the conduction bands as in standard semiconductors such as Si or GaAs, at variance with large area graphene which is gapless, a feature that hampers a good performance of graphene field effect transistors.
To use graphene nanoribbons as a semiconductor, an accurate analysis of their electron properties is needed. Here, electron transport in graphene nanoribbons is investigated by solving the semiclassical Boltzmann equation with a discontinuous Galerkin method. All the electron-phonon scattering mechanisms are included. The adopted energy band structure is that devised in
[\textit{M. Bresciani} et al., ``Simple and efficient modeling of the E-k relationship and low-field mobility in graphene nano-ribbons'', Solid-State Eletron. 54, No. 9, 1015--1021 (2010; \url{doi:10.1016/j.sse.2010.04.038})] while according to [loc. cit.] the edge effects are described as an additional scattering stemming from the Berry-Mondragon model which is valid in presence of edge disorder. With this approach a spacial 1D transport problem has been solved, even if it remains two dimensional in the wave-vector space. A degradation of charge velocities, and consequently of the mobilities, is found by reducing the nanoribbon width due mainly to the edge scattering.Mass and horizon Dirac observables in effective models of quantum black-to-white hole transitionhttps://www.zbmath.org/1483.830222022-05-16T20:40:13.078697Z"Bodendorfer, Norbert"https://www.zbmath.org/authors/?q=ai:bodendorfer.norbert"Mele, Fabio M."https://www.zbmath.org/authors/?q=ai:mele.fabio-m"Münch, Johannes"https://www.zbmath.org/authors/?q=ai:munch.johannesThere have been several recent studies of the black hole interior in loop quantum gravity, exploiting the isomorphism between the Schwarzschild interior and the Kantowski-Sachs spacetime. This allows the use of loop quantum cosmology techniques. The singularity that appears in the classical theory is resolved, replacing it with a black hole to white hole transition. Dirac observables have been identified associated with the mass of the black hole and the white hole and relations between them found. This paper further analyzes the issue of Dirac observables in these and other ``polymerized'' models.
Reviewer: Jorge Pullin (Baton Rouge)Chiral and non-chiral spinning string dynamo instability from quantum torsion sourceshttps://www.zbmath.org/1483.830422022-05-16T20:40:13.078697Z"Garcia de Andrade, L. C."https://www.zbmath.org/authors/?q=ai:garcia-de-andrade.l-cSummary: In this paper dynamo instability is investigated by making use of a condensed matter screw dislocation model of spinning cosmic strings in the framework of teleparallel \(T_4\) spacetime, The magnetic field is encoded into the \(T_4\) metric tensor. By using quantum torsion from quantisation flux dependence on the spin angular momentum of strings, one is able to obtain dynamo instability where chiral currents grow exponentially as in Witten superconducting cosmic string. Our results are compared to [\textit{V. Galitsky} et al., Phys. Rev. Lett. 121, No. 17, Article ID 176603, 6 p. (2018; \url{doi:10.1103/PhysRevLett.121.176603})], who have investigated dynamo instability from Weyl semimetals, showing that chiral anomaly term reduces the Reynolds number for dynamo instability. String dynamos in the framework of Einstein-Cartan (EC) [the author, Cosmol. Astropart. Phys. 2014, No. 8, Paper No. 23, 9 p. (2014; \url{doi:10.1088/1475-7516/2014/08/023})], ER - [\textit{L. C. Garcia de Andrade}, Cosmic magnetism in modified theories of gravity. ChisinauÉditions universitaires europ'eennes (2017)] cosmology have been previously addressed. Moreover, it is shown that dynamo strings possess a monopole singularity at Big Bang. Topological defects dynamos, such as torsion spin-polarised fermions orthogonal to the wall [the author, Classical Quantum Gravity 38, No. 6, Article ID 065005, 9 p. (2021; Zbl 1479.83135)] have been obtained from torsional anomalies sources. Galitski et al. also discovered that chiral anomalies help dynamo effect in real magnetic fields. Non-chiral solutions where the current is along the string but the magnetic field is not, such as in pion string dynamo [\textit{R. Gwyn} et al., ``Magnetic fields from heterotic cosmic strings'', Phys. Rev. D (3) 79, No. 8, Article ID 083502, 13 p. (2009; \url{doi:10.1103/PhysRevD.79.083502})] are obtained from the Heaviside step function which tells us that the magnetic field grows, in the plane above \(z=0\) where the spinning string crosses the plane, and vanishes below the crossing plane.Thermodynamical critical properties of \(4D\) charged AdS massive black hole by thermo-shadow methodhttps://www.zbmath.org/1483.830542022-05-16T20:40:13.078697Z"Liu, Yun"https://www.zbmath.org/authors/?q=ai:liu.yun"Nie, Zhenxiong"https://www.zbmath.org/authors/?q=ai:nie.zhenxiong"Chen, Juhua"https://www.zbmath.org/authors/?q=ai:chen.juhua"Wang, Yongjiu"https://www.zbmath.org/authors/?q=ai:wang.yongjiuSummary: In this paper, we investigate the effect of the characteristic parameter \(m\) of the black hole solution in massive gravity on the thermodynamical critical properties of the black hole. We find that when the characteristic parameter \(m\) is less than the critical value \(m_c\), there is no the phase transition in the black hole system. On this basis, we use the black hole shadow radius to detect the phase transition of the \(4D\) charged AdS massive black hole in the extended phase space. The acquired result shows that the variation of thermodynamic quantities with shadow radius is very similar to that with event horizon radius. Using the shadow radius, we construct the black hole shadow thermal profile. By changing the temperature on the shadow contour, the Van der Waals-like phase transition can be clearly presented on the shadow contour.Polytropic anti-de Sitter black holehttps://www.zbmath.org/1483.830622022-05-16T20:40:13.078697Z"Salti, M."https://www.zbmath.org/authors/?q=ai:salti.mustafa"Aydogdu, O."https://www.zbmath.org/authors/?q=ai:aydogdu.oktay"Sogut, K."https://www.zbmath.org/authors/?q=ai:sogut.kenanSummary: In the present study, we mainly discuss the features of polytropic anti-de Sitter (PAdS) black hole solution, which can be taken into account also as a stable configuration for dark energy stars. A dark energy star is a hypothetical compact astrophysical object and it has gained astrophysical relevance in literature for several reasons, for instance, it may be another interpretation for observations of astronomical black hole candidates. The idea basically points out that falling matter is transformed into dark energy (or vacuum energy), as the matter falls through the event horizon. In the first step of our investigation, assuming thermodynamical parameters of the asymptotically AdS black hole are identical to those introduced for the generalized polytropic gas, we obtain an exact solution for the metric function describing interior domain of the PAdS black hole. Then, we investigate the physical features of intermediate (shell) region. Subsequently, we describe physical properties like energy conditions and hydrostatic equilibrium via mathematical calculations as well as graphical analyses. In the final step, because black holes can be naturally regarded as a thermal device, we focus on a heat engine process for the PAdS black hole and obtain an analytical expression for the efficiency in terms of entropy and temperature.Geometrothermodynamics of black holes with a nonlinear sourcehttps://www.zbmath.org/1483.830632022-05-16T20:40:13.078697Z"Sánchez, Alberto"https://www.zbmath.org/authors/?q=ai:rivadulla-sanchez.albertoSummary: We study thermodynamics and geometrothermodynamics of a particular black hole configuration with a nonlinear source. We use the mass as fundamental equation, from which it follows that the curvature radius must be considered as a thermodynamic variable, leading to an extended equilibrium space. Using the formalism of geometrothermodynamics, we show that the geometric properties of the thermodynamic equilibrium space can be used to obtain information about thermodynamic interaction, critical points and phase transitions. We show that these results are compatible with the results obtained from classical black hole thermodynamics.Thermodynamics of a static electric-magnetic black hole in Einstein-Born-Infeld-AdS theory with different horizon geometrieshttps://www.zbmath.org/1483.830652022-05-16T20:40:13.078697Z"Tataryn, M. B."https://www.zbmath.org/authors/?q=ai:tataryn.m-b"Stetsko, M. M."https://www.zbmath.org/authors/?q=ai:stetsko.mykola-mSummary: We consider black hole solutions with electric and magnetic sources in the four-dimensional Einstein-Born-Infeld-AdS theory with spherical, planar and hyperbolic horizon geometries. Exact analytical solutions for the metric function, electric and magnetic fields were obtained and they recover the RN-AdS black hole in the limit \(\beta \rightarrow +\infty\) for spherical horizon in the absence of the magnetic charge. Expressions for temperature, electric and magnetic potential were obtained and they satisfy the first law of the extended black hole thermodynamics, where a negative cosmological constant is associated with thermodynamic pressure. Also, the Born-Infeld vacuum polarization term \(Bd\beta\) was included into the first law in order to satisfy the Smarr relation. Critical behavior of the black hole was examined and condition on electric and magnetic charges were obtained when phase transition appears. Also, the critical ratio and capacity at constant pressure were calculated. Electric and magnetic charges enter into the metric function and thermodynamic quantities symmetrically and thus the presence of the magnetic charge does not bring very significant new features. Finally, we examine the Joule-Thomson expansion if the black hole mass is fixed. The inversion and isenthalpic curves were plotted and the cooling and heating regions were demonstrated. These results recover the Joule-Thomson expansion recently considered for the RN-AdS black hole in the corresponding limit.Lifshitz scaling effects on the holographic paramagnetic-ferromagnetic phase transitionhttps://www.zbmath.org/1483.830692022-05-16T20:40:13.078697Z"Ghotbabadi, B. Binaei"https://www.zbmath.org/authors/?q=ai:ghotbabadi.b-binaei"Sheykhi, A."https://www.zbmath.org/authors/?q=ai:sheykhi.ahmad"Bordbar, G. H."https://www.zbmath.org/authors/?q=ai:bordbar.g-hSummary: We disclose the effects of Lifshitz dynamical exponent \(z\) on the properties of holographic paramagnetic-ferromagnetic phase transition in the background of Lifshitz spacetime. To preserve the conformal invariance in higher dimensions, we consider the Power-Maxwell (PM) electrodynamics as our gauge field. We introduce a massive 2-form coupled to the PM field and perform the numerical shooting method in the probe limit by assuming the PM and the 2-form fields do not back-react on the background geometry. The obtained results indicate that the critical temperature decreases with increasing the strength of the power parameter \(q\) and dynamical exponent \(z\). Besides, the formation of the magnetic moment in the black hole background is harder in the absence of an external magnetic field. At low temperatures, and in the absence of an external magnetic field, our result show the spontaneous magnetization and the ferromagnetic phase transition. We find that the critical exponent takes the universal value \(\beta=1/2\) regardless of the parameters \(q, z, d\), which is in agreement with the mean field theory. In the presence of an external magnetic field, the magnetic susceptibility satisfies the Curie-Weiss law.Holographic superconductors in 4D Einstein-Gauss-Bonnet gravity with backreactionshttps://www.zbmath.org/1483.830722022-05-16T20:40:13.078697Z"Pan, Jie"https://www.zbmath.org/authors/?q=ai:pan.jie"Qiao, Xiongying"https://www.zbmath.org/authors/?q=ai:qiao.xiongying"Wang, Dong"https://www.zbmath.org/authors/?q=ai:wang.dong.7|wang.dong|wang.dong.3|wang.dong.8|wang.dong.2|wang.dong.4|wang.dong.1|wang.dong.6|wang.dong.5"Pan, Qiyuan"https://www.zbmath.org/authors/?q=ai:pan.qiyuan"Nie, Zhang-Yu"https://www.zbmath.org/authors/?q=ai:nie.zhang-yu"Jing, Jiliang"https://www.zbmath.org/authors/?q=ai:jing.jiliangSummary: We construct the holographic superconductors away from the probe limit in the consistent \(D \to 4\) Einstein-Gauss-Bonnet gravity. We observe that, both for the ground state and excited states, the critical temperature first decreases then increases as the curvature correction tends towards the Chern-Simons limit in a backreaction dependent fashion. However, the decrease of the backreaction, the increase of the scalar mass, or the increase of the number of nodes will weaken this subtle effect of the curvature correction. Moreover, for the curvature correction approaching the Chern-Simons limit, we find that the gap frequency \(\omega_g/T_c\) of the conductivity decreases first and then increases when the backreaction increases in a scalar mass dependent fashion, which is different from the finding in the \((3 + 1)\)-dimensional superconductors that increasing backreaction increases \(\omega_g/T_c\) in the full parameter space. The combination of the Gauss-Bonnet gravity and backreaction provides richer physics in the scalar condensates and conductivity in the \((2 + 1)\)-dimensional superconductors.The unified history of the viscous accelerating universe and phase transitionshttps://www.zbmath.org/1483.830822022-05-16T20:40:13.078697Z"Astashenok, A. V."https://www.zbmath.org/authors/?q=ai:astashenok.artyom-v"Odintsov, S. D."https://www.zbmath.org/authors/?q=ai:odintsov.sergei-d"Tepliakov, A. S."https://www.zbmath.org/authors/?q=ai:tepliakov.a-sSummary: We propose the unified description of the early acceleration (cosmological inflation) and the present epoch of so called ``dark energy''. The inflation can be described by cosmic fluid with van der Waals equation of state and with viscosity term. Viscosity leads to slow-roll inflation with the parameters such as the spectral index, and the tensor-to-scalar ratio in concordance with observational data. Our next step is to modify this equation of state (EoS) to describe the present accelerated expansion. One can add the term into EoS so that the contribution of which is small for inflation but crucial for late-time acceleration. The key point of the model is possible phase transition which leads to decrease of the viscosity. We show that proposed model describes observational data about standard ``candles'' and correct dependence of Hubble parameter from redshift. Moreover, we propose the possible scenario to resolve dark matter problem.Inflation, phase transitions and the cosmological constanthttps://www.zbmath.org/1483.830832022-05-16T20:40:13.078697Z"Bertolami, Orfeu"https://www.zbmath.org/authors/?q=ai:bertolami.orfeuSummary: Cosmological phase transitions are thought to have taken place at the early Universe imprinting their properties on the observable Universe. There is strong evidence that, through the dynamics of a scalar field that lead a second order phase transition, inflation shaped the Universe accounting for the most conspicuous features of the observed Universe. It is argued that inflation has also striking implications for the vacuum energy. Considerations for subsequent second order phase transitions are also discussed.Cosmological implications of the hydrodynamical phase of group field theoryhttps://www.zbmath.org/1483.830842022-05-16T20:40:13.078697Z"Gabbanelli, Luciano"https://www.zbmath.org/authors/?q=ai:gabbanelli.luciano"De Bianchi, Silvia"https://www.zbmath.org/authors/?q=ai:de-bianchi.silviaSummary: In this review we focus on the main cosmological implications of the Group Field Theory approach, according to which an effective continuum macroscopic dynamics can be extracted from the underlying formalism for quantum gravity. Within this picture what counts is the collective behaviour of a large number of quanta of geometry. The resulting state is a condensate-like structure made of ``pre-geometric'' excitations of the Group Field Theory field over a no-space vacuum. Starting from the kinematics and dynamics, we offer an overview of the way in which Group Field Theory condensate cosmology treats solutions for the homogeneous and isotropic universe. These solutions including a bounce, share with other quantum cosmological approaches the resolution of the singularity characterizing general relativity. Contrary to what is usually done in quantum cosmology, in Group Field Theory cosmology no preliminary symmetry reduction is needed for this purpose. We conclude with a discussion of the limits and future perspectives of the Group Field Theory approach.Magnetogenesis in Higgs inflationhttps://www.zbmath.org/1483.830872022-05-16T20:40:13.078697Z"Kamarpour, Mehran"https://www.zbmath.org/authors/?q=ai:kamarpour.mehranSummary: We study the generation of magnetic fields in the Higgs inflation model with the axial coupling in order to break the conformal invariance of the Maxwell action and produce strong enough magnetic fields for observed large-scale magnetic fields. This interaction breaks the parity and enables a production of only one of the polarization states of the electromagnetic field due to axion-like coupling of electromagnetic field to the inflation. Therefore, the produced magnetic fields are helical. In fact, calculations show the mode of one polarization undergoes amplification, while the other one diminishes. We consider radiatively corrected Higgs inflation potential. In comparison to the Starobinsky potential, we obtain an extra term as a one loop correction and determine the spectrum of generalized electromagnetic fields. The effect of quantum correction modifies potential so that in some certain conditions when
back reaction is weak the observed large-scale magnetic field can be explained by our modified potential. We should emphasize in this model we only consider linear approximation for electromagnetic field so that the theory does not contain higher-order derivatives and the so-called ghost degrees of freedom. Therefore, the theory is consistent with cosmology. In addition, the magnetic field generated in this model has very small correlation length. It is impossible to explain within this model both the strength of magnetic field and its large coherence length. Due to the nontrivial helicity, the produced magnetic fields undergo the inverse cascade process in the turbulent plasma which can strongly increase their correlation length. We find that, for two values of coupling parameter \(\chi_1=5\times 10^9M_p^{-2}\) and \(\chi_1=7.5\times 10^9M_p^{-2}\), the back-reaction is weak and our analysis is valid.Interface dynamics with heat and mass fluxeshttps://www.zbmath.org/1483.850042022-05-16T20:40:13.078697Z"Ilyin, Dan V."https://www.zbmath.org/authors/?q=ai:ilyin.dan-v"Abarzhi, Snezhana I."https://www.zbmath.org/authors/?q=ai:abarzhi.snezhana-iSummary: Unstable interfaces ubiquitously occur in fluids, plasmas and materials, from the celestial event of supernova to the atomic level of plasma fusion. Knowledge of their fundamentals is in demand in science, mathematics, and engineering. Our work considers the classical problem of stability of a phase boundary having heat and mass fluxes across it and yields a rigorous theory resolving challenges not addressed before. By employing self-consistent boundary conditions for thermal heat flux, by identifying the perturbation waves' structure, and by thoroughly investigating the dependence of waves' coupling on system parameters, we discover new fluid instabilities in the advection, diffusion and low Mach regimes. We find that the interface stability is set primarily by the interplay of macroscopic inertial stabilization with destabilizing acceleration, and the microscopic thermodynamics and thermal heat flux creates vortical fields in the bulk.Contribution to the angular momentum transport paradigm for accretion diskshttps://www.zbmath.org/1483.850092022-05-16T20:40:13.078697Z"Montani, Giovanni"https://www.zbmath.org/authors/?q=ai:montani.giovanni"Carlevaro, Nakia"https://www.zbmath.org/authors/?q=ai:carlevaro.nakiaSummary: We analyze the stationary configuration of a thin axisymmetric stellar accretion disk, neglecting non-linear terms in the plasma poloidal velocity components. We set up the Grad-Shafranov equation for the system, including the plasma differential rotation (according to the so-called co-rotation theorem). Then, we study the small scale backreaction of the disk to the central body magnetic field and we calculate the resulting radial infalling velocity. We show that the small scale radial oscillation of the perturbed magnetic surface is associated to the emergence of relevant toroidal current densities, able to balance the Ohm law even in the presence of quasi-ideal values of the plasma resistivity. The contribution to the infalling velocity of the averaged backreaction contrasts accretion, but it remains negligible as far as the induced magnetic field is small compared to that of the central body.Aspects of GRMHD in high-energy astrophysics: geometrically thick disks and tori agglomerates around spinning black holeshttps://www.zbmath.org/1483.850102022-05-16T20:40:13.078697Z"Pugliese, D."https://www.zbmath.org/authors/?q=ai:pugliese.daniela"Montani, G."https://www.zbmath.org/authors/?q=ai:montani.giovanniSummary: This work focuses on some key aspects of the general relativistic (GR) -- magneto-hydrodynamic (MHD) applications in high-energy astrophysics. We discuss the relevance of the GRHD counterparts formulation exploring the geometrically thick disk models and constraints of the GRMHD shaping the physics of accreting configurations. Models of clusters of tori orbiting a central super-massive black hole (\textbf{SMBH}) are described. These orbiting tori aggregates form sets of geometrically thick, pressure supported, perfect fluid tori, associated to complex instability processes including tori collision emergence and
empowering a wide range of activities related expectantly to the embedding matter environment of Active Galaxy Nuclei. Some notes are included on aggregates combined with proto-jets, represented by open cusped solutions associated to the geometrically thick tori.
This exploration of some key concepts of the GRMHD formulation in its applications to High-Energy Astrophysics starts with the discussion of the initial data problem for a most general Einstein-Euler-Maxwell system addressing the problem with a relativistic geometric background. The system is then set in quasi linear hyperbolic form, and the reduction procedure is argumented. Then, considerations follow on the analysis of the stability problem
for self-gravitating systems with determined symmetries considering the perturbations also of the geometry part on the quasi linear hyperbolic onset. Thus we focus on the ideal GRMHD and self-gravitating plasma ball. We conclude with the models of geometrically thick GRHD disks gravitating around a Kerr \textbf{SMBH} in their GRHD formulation and including in the force balance equation of the disks the influence of a toroidal magnetic field, determining its impact in tori topology and stability.Towards a mathematical theory of behavioral human crowdshttps://www.zbmath.org/1483.900422022-05-16T20:40:13.078697Z"Bellomo, Nicola"https://www.zbmath.org/authors/?q=ai:bellomo.nicola"Gibelli, Livio"https://www.zbmath.org/authors/?q=ai:gibelli.livio"Quaini, Annalisa"https://www.zbmath.org/authors/?q=ai:quaini.annalisa"Reali, Alessandro"https://www.zbmath.org/authors/?q=ai:reali.alessandro