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Entropy production in random billiards
1. | Department of Mathematics and Statistics, Mount Holyoke College, 50 College St, South Hadley, MA 01075, USA |
2. | Department of Mathematics and Statistics, Washington University, Campus Box 1146, St. Louis, MO 63130, USA |
We consider a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smoluchowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters.
References:
[1] |
C. Cercignani and D. H. Sattinger, Scaling Limits and Models in Physical Processes, DMV Seminar, 28, Birkhäuser Verlag, Basel, 1998.
doi: 10.1007/978-3-0348-8810-3. |
[2] |
N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/127. |
[3] |
P. Collet and J.-P. Eckmann,
A model of heat conduction, Comm. Math. Phys., 287 (2009), 1015-1038.
doi: 10.1007/s00220-008-0691-2. |
[4] |
F. Comets, S. Popov, G. M. Schütz and M. Vachkovskaia,
Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal., 191 (2009), 497-537.
doi: 10.1007/s00205-008-0120-x. |
[5] |
S. Cook and R. Feres,
Random billiards with wall temperature and associated Markov chains, Nonlinearity, 25 (2012), 2503-2541.
doi: 10.1088/0951-7715/25/9/2503. |
[6] |
M. F. Demers, L. Rey-Bellet and H.-K. Zhang,
Fluctuation of the entropy production for the Lorentz gas under small external Forces, Comm. Math. Phys., 363 (2018), 699-740.
doi: 10.1007/s00220-018-3228-3. |
[7] |
J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet,
Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys., 201 (1999), 657-697.
doi: 10.1007/s002200050572. |
[8] |
J.-P. Eckmann and L.-S. Young,
Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys., 262 (2006), 237-267.
doi: 10.1007/s00220-005-1462-y. |
[9] |
J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet,
Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys., 95 (1999), 305-331.
doi: 10.1023/A:1004537730090. |
[10] |
S. N. Evans,
Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.
doi: 10.1214/aoap/1015345298. |
[11] |
R. Feres, Random walks derived from billiards, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 179-222.
doi: 10.1017/CBO9780511755187.008. |
[12] |
R. Feres and H.-K. Zhang,
The spectrum of the billiard Laplacian of a family of random billiards, J. Stat. Phys., 141 (2010), 1039-1054.
doi: 10.1007/s10955-010-0079-5. |
[13] |
R. Feres and H.-K. Zhang,
Spectral gap for a class of random billiards, Comm. Math. Phys., 313 (2012), 479-515.
doi: 10.1007/s00220-012-1469-0. |
[14] |
P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series, 9, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9780511628856.![]() ![]() |
[15] |
V. Jakšić, C.-A. Pillet and L. Rey-Bellet,
Entropic fluctuations in statistical mechanics: I. Classical dynamical systems, Nonlinearity, 24 (2011), 699-763.
doi: 10.1088/0951-7715/24/3/003. |
[16] |
V. Jakšić, C.-A. Pillet and A. Shirikyan,
Entropic fluctuations in Gaussian dynamical systems, Rep. Math. Phys., 77 (2016), 335-376.
doi: 10.1016/S0034-4877(16)30034-9. |
[17] |
D.-Q. Jiang, M. Qian and M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States, Lecture Notes in Mathematics, 1833, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b94615. |
[18] |
K. Khanin and T. Yarmola,
Ergodic properties of random billiards driven by thermostats, Comm. Math. Phys., 320 (2013), 121-147.
doi: 10.1007/s00220-013-1715-0. |
[19] |
H. Larralde, F. Leyvraz and C. Mejía-Monasterio,
Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197-231.
doi: 10.1023/A:1025726905782. |
[20] |
J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[21] |
Y. Li and L.-S. Young,
Existence of nonequilibrium steady state for a simple model of heat conduction, J. Stat. Phys., 152 (2013), 1170-1193.
doi: 10.1007/s10955-013-0801-1. |
[22] |
K. K. Lin and L.-S. Young,
Nonequilibrium steady states for certain Hamiltonian models, J. Stat. Phys., 139 (2010), 630-657.
doi: 10.1007/s10955-010-9958-z. |
[23] |
S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630.![]() ![]() |
[24] |
H. Qian, S. Kjelstrup, A. B. Kolomeisky and D. Bedeaux, Entropy production in mesoscopic stochastic thermodynamics: Nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces, J. Phys. Condensed Matter, 28 (2016).
doi: 10.1088/0953-8984/28/15/153004. |
[25] |
L. Rey-Bellet and L. E. Thomas,
Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré, 3 (2002), 483-502.
doi: 10.1007/s00023-002-8625-6. |
[26] |
L. Rey-Bellet, Nonequilibrium statistical mechanics of open classical systems, in XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005,447–454.
doi: 10.1142/9789812704016_0043. |
[27] |
L. Rey-Bellet and L. E. Thomas,
Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys., 225 (2002), 305-329.
doi: 10.1007/s002200100583. |
[28] |
U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, preprint, arXiv: 1205.4176. Google Scholar |
[29] |
T. Yarmola,
Sub-exponential mixing of open systems with particle-disk interactions, J. Stat. Phys., 156 (2014), 473-492.
doi: 10.1007/s10955-014-1014-y. |
show all references
References:
[1] |
C. Cercignani and D. H. Sattinger, Scaling Limits and Models in Physical Processes, DMV Seminar, 28, Birkhäuser Verlag, Basel, 1998.
doi: 10.1007/978-3-0348-8810-3. |
[2] |
N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/127. |
[3] |
P. Collet and J.-P. Eckmann,
A model of heat conduction, Comm. Math. Phys., 287 (2009), 1015-1038.
doi: 10.1007/s00220-008-0691-2. |
[4] |
F. Comets, S. Popov, G. M. Schütz and M. Vachkovskaia,
Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal., 191 (2009), 497-537.
doi: 10.1007/s00205-008-0120-x. |
[5] |
S. Cook and R. Feres,
Random billiards with wall temperature and associated Markov chains, Nonlinearity, 25 (2012), 2503-2541.
doi: 10.1088/0951-7715/25/9/2503. |
[6] |
M. F. Demers, L. Rey-Bellet and H.-K. Zhang,
Fluctuation of the entropy production for the Lorentz gas under small external Forces, Comm. Math. Phys., 363 (2018), 699-740.
doi: 10.1007/s00220-018-3228-3. |
[7] |
J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet,
Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys., 201 (1999), 657-697.
doi: 10.1007/s002200050572. |
[8] |
J.-P. Eckmann and L.-S. Young,
Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys., 262 (2006), 237-267.
doi: 10.1007/s00220-005-1462-y. |
[9] |
J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet,
Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys., 95 (1999), 305-331.
doi: 10.1023/A:1004537730090. |
[10] |
S. N. Evans,
Stochastic billiards on general tables, Ann. Appl. Probab., 11 (2001), 419-437.
doi: 10.1214/aoap/1015345298. |
[11] |
R. Feres, Random walks derived from billiards, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 179-222.
doi: 10.1017/CBO9780511755187.008. |
[12] |
R. Feres and H.-K. Zhang,
The spectrum of the billiard Laplacian of a family of random billiards, J. Stat. Phys., 141 (2010), 1039-1054.
doi: 10.1007/s10955-010-0079-5. |
[13] |
R. Feres and H.-K. Zhang,
Spectral gap for a class of random billiards, Comm. Math. Phys., 313 (2012), 479-515.
doi: 10.1007/s00220-012-1469-0. |
[14] |
P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series, 9, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9780511628856.![]() ![]() |
[15] |
V. Jakšić, C.-A. Pillet and L. Rey-Bellet,
Entropic fluctuations in statistical mechanics: I. Classical dynamical systems, Nonlinearity, 24 (2011), 699-763.
doi: 10.1088/0951-7715/24/3/003. |
[16] |
V. Jakšić, C.-A. Pillet and A. Shirikyan,
Entropic fluctuations in Gaussian dynamical systems, Rep. Math. Phys., 77 (2016), 335-376.
doi: 10.1016/S0034-4877(16)30034-9. |
[17] |
D.-Q. Jiang, M. Qian and M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States, Lecture Notes in Mathematics, 1833, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b94615. |
[18] |
K. Khanin and T. Yarmola,
Ergodic properties of random billiards driven by thermostats, Comm. Math. Phys., 320 (2013), 121-147.
doi: 10.1007/s00220-013-1715-0. |
[19] |
H. Larralde, F. Leyvraz and C. Mejía-Monasterio,
Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197-231.
doi: 10.1023/A:1025726905782. |
[20] |
J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[21] |
Y. Li and L.-S. Young,
Existence of nonequilibrium steady state for a simple model of heat conduction, J. Stat. Phys., 152 (2013), 1170-1193.
doi: 10.1007/s10955-013-0801-1. |
[22] |
K. K. Lin and L.-S. Young,
Nonequilibrium steady states for certain Hamiltonian models, J. Stat. Phys., 139 (2010), 630-657.
doi: 10.1007/s10955-010-9958-z. |
[23] |
S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630.![]() ![]() |
[24] |
H. Qian, S. Kjelstrup, A. B. Kolomeisky and D. Bedeaux, Entropy production in mesoscopic stochastic thermodynamics: Nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces, J. Phys. Condensed Matter, 28 (2016).
doi: 10.1088/0953-8984/28/15/153004. |
[25] |
L. Rey-Bellet and L. E. Thomas,
Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré, 3 (2002), 483-502.
doi: 10.1007/s00023-002-8625-6. |
[26] |
L. Rey-Bellet, Nonequilibrium statistical mechanics of open classical systems, in XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005,447–454.
doi: 10.1142/9789812704016_0043. |
[27] |
L. Rey-Bellet and L. E. Thomas,
Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys., 225 (2002), 305-329.
doi: 10.1007/s002200100583. |
[28] |
U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, preprint, arXiv: 1205.4176. Google Scholar |
[29] |
T. Yarmola,
Sub-exponential mixing of open systems with particle-disk interactions, J. Stat. Phys., 156 (2014), 473-492.
doi: 10.1007/s10955-014-1014-y. |









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