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\(\mathrm{L}^2\)-hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system. (English) Zbl 1486.82034

The purpose of this work is to study the behavior of the distribution function of charged particles \(f_{cp}(r_{cp},v_{cp},t)\) at large times. The study is carried out within the framework of the Vlasov-Poisson-Fokker-Planck system of equations: \[ \begin{gathered} \frac{\partial}{\partial t} f_{cp}+ \vec{v}_{cp}\cdot\nabla_r f_{cp}- \frac{1}{m_{cp}} \nabla_r[\Phi+ eZ\psi(\vec{r}_{cp})]\cdot\nabla_V f_{cp}= D_{\Delta V}f_{cp}+A\nabla_V\cdot(\vec{v}_{cp} f_{cp}),\\ \iiint f_{cp}d^3 v_{cp}=n_{cp}. \end{gathered}\tag{1} \] The Poisson equation for the electrical potential \(\Phi(r_{cp})\) are used in the next form: \[ \Delta_r\Phi=- eZ\iiint f_{cp} d^3v_{cp}/\varepsilon_0.\tag{2} \] Here \(D= \frac{k_BT}{m_{cp}} v\), \(A= v= \sqrt{\frac{k_B T}{m_{cp}}} n_{cp}\langle Q_{cp^0}\rangle\), \(\langle Q_{cp^0}\rangle\) is the average cross section of elastic scatterings of charged particles on background particles, \(\varepsilon_0\) is the vacuum permittivity, the rest of the designations are well known.
The system of equations (1), (2) describe the charged particles’ evolution under the influence of a self-consistent, nonlinear interaction through the mean field potential \(\Phi\) and collisions with a background. The collision integral, written in the Fokker-Planck approximation, takes into account diffusion in velocity space and friction. The nonlinear system of equations (1), (2) has a unique stationary solution: \[ \begin{gathered} f_s(r_{cp},v_{cp})= f_{s1}(r_{cp})f_{s2}(v_{cp}),\\ f_{s1}(\vec{r}_{cp})= \exp\biggl[-\frac{\Phi+eZ\psi_0}{k_BT}\biggr],\ f_{s2}(\vec{v}_{cp})= n_{cp} \exp\biggl[-\frac{m_{cp}\vec{v}^2_{cp}}{k_BT}\biggr].\\ \Delta\psi_0= eZn_{cp}\exp\biggl[-\frac{\Phi+ eZ\psi_0}{k_BT}\biggr]/\varepsilon_0. \end{gathered}\tag{3} \] When the non-stationary distribution function is represented as \[ f_{cp}(r_{cp}, v_{cp},t)= f_s(1+\eta f_1(r_{cp}, v_{cp},t)),\ \psi(r_{cp},t)= \psi_0+ \eta\psi(r_{cp}, t),\quad \eta<1, \] the large-time behavior of the charged particles distribution function is investigated using the linearized Vlasov-Poisson-Fokker-Planck system: \[ \begin{gathered} \frac{\partial}{\partial t} f_1+ \vec{v}_{cp}\cdot\nabla_r f_1- \frac{1}{m_{cp}} \nabla_r[\Phi+ eZ\psi_0]\cdot\nabla_V f_1-\vec{v}_{cp}\frac{eZ}{k_BT} \nabla_r\psi_1= D\Delta_V f_1+ A\nabla_V(\vec{v}_{cp} f_1),\\ \Delta_r\psi_1= eZ\iiint f_s f_1 d^3v_{cp}/\varepsilon_0. \end{gathered}\tag{4} \] This system of equations has one characteristic length equal to the Debye radius \(r_D=\sqrt{\frac{\varepsilon_0 k_B T}{n_{cp} e^2Z}}\) and three characteristic time scales: \[ t_1\equiv 1/v= 1/\sqrt{\frac{k_BT}{m_{cp}}} n_{cp}\langle Q_{cp0}\rangle\ll t_2= \mathrm{Max}[\Phi(r_D)= eZ\psi_0(r_D)]/r_D m_{cp} \sqrt{\frac{k_BT}{m_{cp}}}\ll t_3= \frac{t^2_2}{t_1}. \] By introducing dimensionless parameters \[ r'=r/r_D,\ \Phi'(r)= \Phi(r)/eZ\psi_0(r_D),\ v'=v/\sqrt{\frac{k_BT}{m_{cp}}},\ t'=t/t_3, \] it is possible to write the system of equations (4) in a dimensionless form containing a small parameter \(\varepsilon\ll 1\): \[ \begin{gathered} \varepsilon\frac{\partial}{\partial t'} f_1+\vec{v}'\cdot \nabla_r' f_1- \nabla_r'[\Phi'+ \psi_0']\cdot\nabla_v' f_1- \vec{v}'\cdot\nabla_r'\psi_1'= \frac{1}{\varepsilon}[\Delta_V'f_1+ \nabla_V'\cdot(\vec{v}'f_1)],\\ \Delta_r' \psi_1'=- \iiint f_s f_1 d^3v_{cp}/n_{cp}. \end{gathered}\tag{5} \] The authors prove the main result of this works (Theorem 1) using the diffusion limit of the linearized system (5). When \(d\ge 1\), \(\Phi(r)= |r|^\alpha\), \(\alpha>0\) the behavior of the distribution function of charged particles \(f_1(r_{cp}, v_{cp}, t)\) at large times is \[ \begin{gathered} \Vert f_1(\vec{r}_{cp}, \vec{v}_{cp},t)\Vert^2\le C\Vert f_1(\vec{r}_{cp}, \vec{v}_{cp}, t=0)\Vert^2\, e^{-\lambda t},\ t\ge 0,\\ \Vert f_1(\vec{r}_{cp}, \vec{v}_{cp}, t)\Vert^2= \iint f_1 f_s d\vec{r}\,d\vec{v}_{cp}+ \int|\nabla_r \psi_1|^2 d\vec{r},\ \lambda>0,\ C\approx 1. \end{gathered}\tag{6} \] Here another important result of this works is Theorem 2. It are proved, if conditions are met: \(d\ge 1\), \(\Phi(r)= |r|^\alpha\), \(\alpha>0\) an initial datum \(f_1(r_{cp}, v_{cp}, t=0)\) of zero average, and \(\Vert f_1(\vec{r}_{cp}, \vec{v}_{cp}, t=0)\Vert^2<\infty\). The theorem states: the behavior of the distribution function of charged particles \(f_1(r_{cp},v_{cp},t)\) at large times satisfies to inequality (6) for any \(\varepsilon>0\), and there exist two constants \(\lambda>0\) and \(C>1\), which do not depend on \(\varepsilon\). Unfortunately, the authors do not obtain a more specific estimate of the damping decrements in inequality (6) \(\lambda\). More accurate decrement estimates can be made based on heuristic and physical considerations.
If inequality \(\Phi(r_D)< eZ\psi_0(r_D)\) holds, then the damping decrements in inequality (6) in dimensional form have the form: \[ \lambda_1\approx v,\ \lambda_2\approx \sqrt{\frac{\pi}{2e^2_L}}\,\omega_{cp}. \] Here \(\omega_{cp}= \sqrt{\frac{n_{cp}(eZ)^2}{\varepsilon_0 m_{cp}}}\) is the plasma oscillation frequency, \(e_L\) is the base of natural logarithms.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
35H10 Hypoelliptic equations
35P15 Estimates of eigenvalues in context of PDEs
35Q84 Fokker-Planck equations
35Q83 Vlasov equations
35Q60 PDEs in connection with optics and electromagnetic theory
35R09 Integro-partial differential equations
47G20 Integro-differential operators
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
82D10 Statistical mechanics of plasmas
82D37 Statistical mechanics of semiconductors
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References:

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