×

zbMATH — the first resource for mathematics

Differential entropy and dynamics of uncertainty. (English) Zbl 1124.82014
The author uses the notion of differential entropy for a continuous probability density in order to study time-dependent problems in both classical and quantum case. The paper begins introducing concepts related to the Shannon entropy and its relation to information and uncertainty. A connection is then set between Shannon entropy of a discrete probability measure and differential entropy of a related continuous probability density, which contrary to the von Neumann entropy does not vanish on pure states, quantifying the degree of probability localization or delocalization. The localization level of probability densities is then analyzed by means of entropy powers and Fisher information measure. Time dependent problems both in dissipative and quantum mechanical cases are dealt with, studying dynamics of differential entropy and of the Fisher information functional.

MSC:
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
94A17 Measures of information, entropy
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Adami, Physics of information, arXiv:quant-ph/040505 (2004).
[2] R. Alicki and M. Fannes, Quantum Dynamical Systems, Oxford University Press, Oxford (2001).
[3] V. Ambegaokar and A. Clerk, Entropy and time, Am. J. Phys. 67:1068–1073 (1999). · Zbl 1219.82106 · doi:10.1119/1.19084
[4] A. Arnold, et al., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Diff. Equations 26:43–100 (2001). · Zbl 0982.35113 · doi:10.1081/PDE-100002246
[5] B. C. Bag, Upper bound for the time derivative of entropy for nonequilibrium stochastic processes, Phys. Rev. E 65:046118 (2002).
[6] C. C. Bag, et al., Noise properties of stochastic processes and entropy production, Phys. Rev. E 64:026110 (2001). · Zbl 1229.01024
[7] R. Balian, Random matrices and information theory, Nuovo Cim. B 57:183–103 (1968). · Zbl 0209.21903 · doi:10.1007/BF02710326
[8] A. R. Barron, Entropy and the central limit theorem, Annals Probab. Theory 14:336–342 (1986). · Zbl 0599.60024 · doi:10.1214/aop/1176992632
[9] W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102:159–182 (1975). · Zbl 0338.42017 · doi:10.2307/1970980
[10] K. Berndl, et al., On the global existence of Bohmian mechanics, Commun. Math. Phys. 173:647–673 (1995). · Zbl 0845.34087 · doi:10.1007/BF02101660
[11] Białynicki-I. Birula and J. Madajczyk, Entropic uncertainty relations for angular distributions, Phys. Lett. A 108:384–386 (1985).
[12] Białynicki-I. Birula, and J. Mycielski, Uncertainty Relations for Information Entropy in Wave Mechanics, Commun. Math. Phys. 44:129–132 (1975). · doi:10.1007/BF01608825
[13] Ph. Blanchard and P. Garbaczewski, Non-negative Feynman-Kac kernels in Schr” odinger’s interpolation problem, J. Math. Phys. 38:1–15 (1997). · Zbl 0870.35046 · doi:10.1063/1.532004
[14] R. Blankenbecler and M. H. Partovi, Uncertainty, entropy, and the statistical mechanics of microscopic systems, Phys. Rev. Lett. 54:373–376 (1985). · doi:10.1103/PhysRevLett.54.373
[15] A. V. Bobylev and G. Toscani, On the generalization of the Boltzmann H-theorem for a spatially homogeneous Maxwell gas, J. Math. Phys. 33:2578–2586 (1992). · Zbl 0825.76713 · doi:10.1063/1.529578
[16] M. Bologna, et al., Trajectory versus probability density entropy, Phys. Rev. E 64:016223 (2001).
[17] L. Brillouin, Science and Information Theory, Academic Press, NY (1962). · Zbl 0098.32204
[18] Ĉ. Brukner and A. Zeilinger, Conceptual inadequacy of the Shannon information in quantum measurements, Phys. Rev. A 63:022113 (2002).
[19] V. Buyarov, et al., Computation of the entropy of polynomials orthogonal on an interval, SIAM J. Sci. Comp. to appear (2004), also math.NA/0310238. · Zbl 1082.33004
[20] E. A. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal. 101:194–211 (1991). · Zbl 0732.60020 · doi:10.1016/0022-1236(91)90155-X
[21] E. Carlen, Conservative diffusions, Commun. Math. Phys. 94:293–315 (1984). · Zbl 0558.60059 · doi:10.1007/BF01224827
[22] R. G. Catalan, J. Garay, and Lopez-R. Ruiz, Features of the extension of a statistical measure of complexity to continuous systems, Phys. Rev. E 66:011102 (2002).
[23] C. M. Caves, and C. Fuchs, Quantum information: how much information in a state vector ?, Ann. Israel Phys. Soc. 12:226–237 (1996).
[24] C. Cercignani, Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh (1975). · Zbl 0403.76065
[25] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15:1–89 (1943). · Zbl 0061.46403 · doi:10.1103/RevModPhys.15.1
[26] K. Ch. Chatzisavvas, Ch. C., Moustakidis and C. P. Panos, Information entropy, information distances and complexity of atoms, J. Chem. Phys. 123:174111 (2005).
[27] T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, NY (1991). · Zbl 0762.94001
[28] H. Cramér, Mathematical methods of statistics, Princeton University Press, Princeton (1946).
[29] R. Czopnik, and P. Garbaczewski, Frictionless Random Dynamics: Hydrodynamical Formalism, Physica A 317:449–471 (2003). · Zbl 1005.70014
[30] D. Daems and G. Nicolis, Entropy production and phase space volume contraction, Phys. Rev. E 59:4000–4006 (1999).
[31] G. Deco, et al: Determining the information flow of dynamical systems from continuous probability distributions, Phys. Rev. Lett. 78:2345–2348 (1997). · doi:10.1103/PhysRevLett.78.2345
[32] A. Dembo and T. Cover, Information theoretic inequalities, IEEE Trans. Inf. Th. 37:1501–1518 (1991). · Zbl 0741.94001 · doi:10.1109/18.104312
[33] D. Deutsch, Uncertainty in quantum measurements, Phys. Rev. Lett. 50:631–633 (1983). · doi:10.1103/PhysRevLett.50.631
[34] J. Dunkel and S. A. Trigger, Time-dependent entropy of simple quantum systems, Phys. Rev. A 71:052102 (2005).
[35] A. Eberle, Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators, LNM vol. 1718, Springer-Verlag, Berlin (2000). · Zbl 1016.47030
[36] R. Fortet, Résolution d’un systéme d’équations de M. Schrödingeer, J. Math. Pures Appl. 9:83 (1040).
[37] B. R. Frieden and B. H. Sofer, Lagrangians of physics and the game of Fisher-information transfer, Phys. Rev. E 52:2274–2286 (1995).
[38] S. R. Gadre, et al., Some novel characteristics of atomic information entropies, Phys. Rev. A 32:2602–2606 (1985).
[39] P. Garbaczewski and R. Olkiewicz, Feynman-Kac kernels in Markovian representations of the Schrödinger interpolating dynamics, J. Math. Phys. 37:732–751 (1996). · Zbl 0869.60101 · doi:10.1063/1.531412
[40] P. Garbaczewski and W. Karwowski, Impenetrable barrriers and canonical quantization, Am. J. Phys. 72:924–933 (2004). · doi:10.1119/1.1688784
[41] P. Garbaczewski, Perturbations of noise: Origins of isothermal flows, Phys. Rev. E 59:1498–1511 (1999).
[42] P. Garbaczewski, Signatures of randomness in quantum spectra, Acta Phys. Pol. A 33:1001–1024 (2002).
[43] P. Garbaczewski, Stochastic models of exotic transport, Physica A 285:187–198 (2000). · Zbl 1059.82529
[44] S. Goldstein and J. L. Lebowitz, On the (Boltzmann) entropy of non-equilibrium systems, Physica D 193:53–66 (2004). · Zbl 1076.82518
[45] M. J. W. Hall, Exact uncertainty relations, Phys. Rev. A 64:052103 (2001).
[46] M. J. W. Hall, Universal geometric approach to uncertainty, entropy and infromation, Phys. Rev. A 59: 2602–2615 (1999).
[47] J. J. Halliwell, Quantum-mechanical histories and the uncertainty principle: Information-theoretic inequalities, Phys. Rev. D 48:2739–2752 (1993).
[48] R. V. L. Hartley, Transmission of information, Bell Syst. Techn. J. 7:535–563 (1928).
[49] H. Hasegawa, Thermodynamic properties of non-equilibrium states subject to Fokker-Planck equations, Progr. Theor. Phys. 57:1523–1537 (1977). · doi:10.1143/PTP.57.1523
[50] T. Hatano and S. Sasa, Steady-State Thermodynamics of Langevin Systems, Phys. Rev. Lett. 86:3463–3466 (2001). · doi:10.1103/PhysRevLett.86.3463
[51] I. I. Hirschman, A note on entropy, Am. J. Math. 79:152–156 (1957). · Zbl 0079.35104 · doi:10.2307/2372390
[52] B. Hu, et al., Quantum chaos of a kicked particle in an infinite potential well, Phys. Rev. Lett. 82:4224–4227 (1999). · doi:10.1103/PhysRevLett.82.4224
[53] K. Huang, Statistical Mechanics, Wiley, New York (1987).
[54] R. S. Ingarden, A. Kossakowski, and M. Ohya, Information Dynamics and Open Systems, Kluwer, Dordrecht (1997). · Zbl 0891.94007
[55] E. T. Jaynes, Information theory and statistical mechanics.II., Phys. Rev. 108:171–190 (1957). · Zbl 0084.43701 · doi:10.1103/PhysRev.108.171
[56] E. T. Jaynes, Violations of Boltzmann’s H Theorem in Real Gases, Phys. Rev. A 4:747–750 (1971).
[57] D.-Q. Jiang, M. Qian, and M-P. Qian, Mathematical theory of nonequilibrium steady states, LNM vol. 1833, Springer-Verlag, Berlin (2004). · Zbl 1096.82002
[58] S. Kullback, Information Theory and Statistics, Wiley, NY (1959). · Zbl 0088.10406
[59] J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A: Math. Gen. 31:3719–3729 (1998). · Zbl 0910.60095 · doi:10.1088/0305-4470/31/16/003
[60] A. Lasota and M. C. Mackey, Chaos, Fractals and Noise, Springer-Verlag, Berlin (1994). · Zbl 0784.58005
[61] J. L. Lebowitz and Ch. Maes, Entropy - a Dialogue, pp. 269–273, in: On Entropy, Eds. A. Grevau, G. Keller, G. Warnecke, Princeton University Press, Princeton, (2003). · Zbl 1163.82307
[62] Y. V. Linnik, An information-theoretic proof of the central limit theorem, Theory Probab. App. 4:288–299 (1959). · Zbl 0097.13103 · doi:10.1137/1104028
[63] H. Maasen and J. B. M. Uffink, Generalized Entropic Uncertainty Relations, Phys. Rev. Lett. 60:1103–1106 (1988). · doi:10.1103/PhysRevLett.60.1103
[64] M. C. Mackey and M. Tyran-Kamińska, Effects of noise on entropy evolution, arXiv.org preprint cond-mat/0501092 (2005).
[65] M. C. Mackey and M. Tyran-Kamińska, Temporal behavior of the conditional and Gibbs entropies, arXiv.org preprint cond-mat/0509649 (2005).
[66] M. C. Mackey, The dynamic origin of increasing entropy, Rev. Mod. Phys. 61, 981–1015 (1989). · doi:10.1103/RevModPhys.61.981
[67] V. Majernik and T. Opatrný, Entropic uncertainty relations for a quantum oscillator, J. Phys. A: Math. Gen. 29:2187–2197 (1996). · Zbl 0901.60080 · doi:10.1088/0305-4470/29/9/029
[68] V. Majernik and L. Richterek, Entropic uncertainty relations for the infinite well, J. Phys. A: Math. Gen. 30: (1997), L49-L54. · Zbl 1001.81504 · doi:10.1088/0305-4470/30/4/002
[69] P. G. L. Mana, Consistency of the Shannon entropy in quantum experiments, Phys. Rev. A 69:062108 (2004).
[70] S. E. Massen and Panos C. P., Universal property of the information entropy in atoms, nuclei and atomic clusters, Phys. Lett. A 246:530–532 (1998).
[71] S. E. Massen, et al., Universal property of information entropy in fermionic and bosonic systems, Phys. Lett. A 299:131–135 (2002). · Zbl 0996.82028
[72] S. E. Massen, Application of information entropy to nuclei, Phys. Rev. C 67:014314 (2003).
[73] M. McClendon and H. Rabitz, Numerical simulations in stochastic mechanics, Phys. Rev. A 37:3479–3492 (1988).
[74] T. Munakata, A. Igarashi, and T. Shiotani, Entropy and entropy production in simple stochastic models, Phys. Rev. E 57:1403–1409 (1998).
[75] E. Nelson, Dynamical Theories of the Brownian Motion, Princeton University Press, Princeton, 1967. · Zbl 0165.58502
[76] R. G. Newton, What is a state in quantum mechanics?, Am. J. Phys. 72:348–350 (2004). · Zbl 1219.81027 · doi:10.1119/1.1636164
[77] M. Ohya and D. Petz, Quantum Entropy and Its use, Springer-Verlag, Berlin, 1993. · Zbl 0891.94008
[78] M. H. Partovi, Entropic formulation of uncertainty for quantum measurements, Phys. Rev. Lett. 50:1883–1885 (1983). · doi:10.1103/PhysRevLett.50.1883
[79] J. Pipek and I. Varga, Universal classification scheme for the spatial-localization properties of one-particle states in finite, d-dimensional systems, Phys. Rev. A 46:3148–3163 (1992).
[80] H. Qian, M. Qian, and X. Tang, Thermodynamics of the general diffusion process: time-reversibility and entropy production, J. Stat. Phys. 107:1129–1141 (2002). · Zbl 1029.82021 · doi:10.1023/A:1015109708454
[81] H. Qian, Mesoscopic nonequilibrium thermodynamics of single macromolecules and dynamic entropy-energy compensation, Phys. Rev. E 65:016102 (2001).
[82] H. Risken, The Fokker-Planck Equation, Springer-Verlag, Berlin, 1989. · Zbl 0665.60084
[83] D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Stat. Phys. 85:1–23 (1996). · Zbl 0973.37014 · doi:10.1007/BF02175553
[84] J. Sánchez-Ruiz, Asymptotic formula for the quantum entropy of position in energy eigenstates, Phys. Lett. A 226:7–13 (1997). · Zbl 0962.82502
[85] M. S. Santhanam, Entropic uncertainty relations for the ground state of a coupled sysytem, Phys. Rev. A 69:042301 (2004).
[86] C. E. Shannon, A mathematical theory of communication, Bell Syst. Techn. J. 27:379–423, 623–656 (1948). · Zbl 1154.94303
[87] J. D. H. Smith, Some observations on the concepts of information-theoretic entropy and randomness, Entropy, 3:1–11 (2001). · Zbl 1004.94009 · doi:10.3390/e3010001
[88] K. Sobczyk, Information Dynamics: Premises, Challenges and Results, Mechanical Systems and Signal Processing 15:475–498 (2001). · doi:10.1006/mssp.2000.1378
[89] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. and Control 2:101–112 (1959). · Zbl 0085.34701 · doi:10.1016/S0019-9958(59)90348-1
[90] A. Stotland, et al., The information entropy of quantum mechanical states, Europhys. Lett. 67:700–706 (2004). · doi:10.1209/epl/i2004-10110-1
[91] G. Toscani, Kinetic approach to the asymptotic behaviour of the solution to diffusion equation, Rend. di Matematica Serie VII 16:329–346 (1996). · Zbl 0904.35029
[92] J. Trebicki and K. Sobczyk, Maximum entropy principle and non-stationary distributions of stochastic systems, Probab. Eng. Mechanics 11:169–178 (1996). · doi:10.1016/0266-8920(96)00008-2
[93] M. Tribus and R. Rossi, On the Kullback information measure as a basis for information theory: Comments on a proposal by Hobson and Chang, J. Stat. Phys. 9:331–338 (1973). · doi:10.1007/BF01012165
[94] S. A. Trigger, Quantum nature of entropy increase for wave packets, Bull. Lebedev. Phys. Inst. 9:44–51 (2004).
[95] I. Varga and J. Pipek, Rényi entropies characterizing the shape and the extension of the phase-space representation of quantum wave functions in disordered systems, Phys. rev. E 68:026202 (2003).
[96] J. M. G. Vilar and J. M. Rubi, Thermodynamics beyond local equilibrium, Proc. Nat. Acad. Sci. (NY) 98:11081–11084 (2001).
[97] J. Voigt, Stochastic operators, Information and Entropy, Commun. Math. Phys. 81:31–38 (1981). · Zbl 0469.47030 · doi:10.1007/BF01941799
[98] J. Voigt, The H-Theorem for Boltzmann type equations, J. Reine Angew. Math 326:198–213 (1981). · Zbl 0451.34060 · doi:10.1515/crll.1981.326.198
[99] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50:221–260 (1978). · doi:10.1103/RevModPhys.50.221
[100] S. A. Werner, and H. Rauch, Neutron interferometry: Lessons in Experimental Quantum Physics, Oxford University Press, Oxford, 2000.
[101] R. J. Yañez, et al., Entropic integrals of hyperspherical harmonics and spatial entropy of D-dimensional central potentials, J. Math. Phys. 40:5675–5686 (1999). · Zbl 0968.81011 · doi:10.1063/1.533051
[102] R. J. Yañez, Van W. Assche, J. S. Dehesa, Position and information entropies of the D-dimensional harmonic oscillator and hydrogen atom, Phys. Rev. A 50:3065–3079 (1994).
[103] A. M. Yaglom and I. M. Yaglom, Probability and Information, D. Reidel, Dordrecht, 1983. · Zbl 0544.94001
[104] A. Zeilinger, et al., Single- and double-slit diffraction of neutrons, Rev. Mod. Phys. 60:1067–1073 (1988). · doi:10.1103/RevModPhys.60.1067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.