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Entropy production inequalities for the Kac Walk
Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation
1. | Institute of Mathematics, Polish Academy Of Sciences, ul. Śniadeckich 8, 00-656, Warsaw, Poland |
2. | Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland |
We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [
References:
[1] |
N. Ben Abdallah, A. Mellet and M. Puel,
Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262.
doi: 10.1142/S0218202511005738. |
[2] |
R. J. Adler,
Geometry of Random Fields, Wiley, 1981. |
[3] |
R. J. Adler,
An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Lecture Notes-Monograph Series, Vol. 12, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, 1990. |
[4] |
G. Bal, G. Papanicolaou and L. Ryzhik,
Radiative transport limit for the random Schrödinger equation, Nonlinearity, 15 (2002), 513-529.
doi: 10.1088/0951-7715/15/2/315. |
[5] |
G. Basile,
From a kinetic equation to a diffusion under an anomalous scaling, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 1301-1322.
doi: 10.1214/13-AIHP554. |
[6] |
G. Basile, C. Bernardin and S. Olla, A momentum conserving model with anomalous thermal conductivity in low dimension, Physical Review Letters, 96 (2006), 204303. Google Scholar |
[7] |
G. Basile, S. Olla and H. Spohn,
Wigner functions and stochastically perturbed lattice dynamics, Arch.Rat.Mech., 195 (2009), 171-203.
doi: 10.1007/s00205-008-0205-6. |
[8] |
A. Bensoussan, J. L. Lions and G. C. Papanicolaou,
Boundary Layers and Homogenization of transport Processes, Publ. RIMS, Kyoto Univ., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
[9] |
C. Bernardin, P. Gonçalves and M. Jara,
3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise, Arch. Ration. Mech. Anal., 220 (2016), 505-542.
doi: 10.1007/s00205-015-0936-0. |
[10] |
C. Gomez,
Wave decoherence for the random Schrödinger equation with long-range correlations, Comm. Math. Phys., 320 (2013), 37-71.
doi: 10.1007/s00220-013-1711-4. |
[11] |
H. J. Kushner,
Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press Series in Signal Processing, Optimization, and Control, 6. MIT Press, Cambridge, MA, 1984. |
[12] |
S. Janson,
Gaussian Hilbert Spaces, Cambridge University Press, 1997. |
[13] |
M. Jara and T. Komorowski,
Limit theorems for some continuous-time random walks, Advances in Applied Probability, 43 (2011), 782-813.
doi: 10.1017/S0001867800005140. |
[14] |
M. Jara, T. Komorowski and S. Olla,
Limit theorems for additive functionals of a Markov chain, Ann. of Appl. Prob., 19 (2009), 2270-2300.
doi: 10.1214/09-AAP610. |
[15] |
M. Jara, T. Komorowski and S. Olla,
Superdiffusion of energy in a chain of harmonic oscillators with noise, Commun. Math. Phys., 339 (2015), 407-453.
doi: 10.1007/s00220-015-2417-6. |
[16] |
T. Komorowski and S. Olla,
Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators, Nonlinearity, 29 (2016), 962-999.
doi: 10.1088/0951-7715/29/3/962. |
[17] |
T. Komorowski and L. Ryzhik,
Passive tracer in a slowly decorrelating random flow with a large mean, Nonlinearity, 20 (2007), 1215-1239.
doi: 10.1088/0951-7715/20/5/009. |
[18] |
T. Komorowski and Ł. Stȩpień,
Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension, Journ. Stat. Phys., 148 (2012), 1-37.
doi: 10.1007/s10955-012-0528-4. |
[19] |
T. Komorowski, S. Olla and L. Ryzhik,
Asymptotics of the solutions of the stochastic lattice wave equation, Arch. Rational Mech. Anal., 209 (2013), 455-494.
doi: 10.1007/s00205-013-0626-8. |
[20] |
T. Kurtz,
Semigroups of conditioned shifts and approximation of markov processes, Ann. Probab., 3 (1975), 618-642.
doi: 10.1214/aop/1176996305. |
[21] |
S. Lepri, R. Livi and A. Politi, Heat transport in low dimensions: Introduction and phenomenology,
Thermal Transport in Low Dimensions, edt S. Lepri, LNP, 921 (2016), 1-37. |
[22] |
J. Lukkarinen and H. Spohn,
Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.
|
[23] |
A. Mellet, S. Mischler and C. Mouhot,
Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.
doi: 10.1007/s00205-010-0354-2. |
[24] |
S. Peszat and Z. Zabczyk,
Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007. |
[25] |
M. A. Pinsky,
Introduction to Fourier Analysis and Wavelets, American Mathematical Society, 2009. |
[26] |
H. Spohn,
Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys., 154 (2014), 1191-1227.
doi: 10.1007/s10955-014-0933-y. |
[27] |
D. W. Stroock and S. R. S. Varadhan,
Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979. |
show all references
References:
[1] |
N. Ben Abdallah, A. Mellet and M. Puel,
Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262.
doi: 10.1142/S0218202511005738. |
[2] |
R. J. Adler,
Geometry of Random Fields, Wiley, 1981. |
[3] |
R. J. Adler,
An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Lecture Notes-Monograph Series, Vol. 12, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, 1990. |
[4] |
G. Bal, G. Papanicolaou and L. Ryzhik,
Radiative transport limit for the random Schrödinger equation, Nonlinearity, 15 (2002), 513-529.
doi: 10.1088/0951-7715/15/2/315. |
[5] |
G. Basile,
From a kinetic equation to a diffusion under an anomalous scaling, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 1301-1322.
doi: 10.1214/13-AIHP554. |
[6] |
G. Basile, C. Bernardin and S. Olla, A momentum conserving model with anomalous thermal conductivity in low dimension, Physical Review Letters, 96 (2006), 204303. Google Scholar |
[7] |
G. Basile, S. Olla and H. Spohn,
Wigner functions and stochastically perturbed lattice dynamics, Arch.Rat.Mech., 195 (2009), 171-203.
doi: 10.1007/s00205-008-0205-6. |
[8] |
A. Bensoussan, J. L. Lions and G. C. Papanicolaou,
Boundary Layers and Homogenization of transport Processes, Publ. RIMS, Kyoto Univ., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
[9] |
C. Bernardin, P. Gonçalves and M. Jara,
3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise, Arch. Ration. Mech. Anal., 220 (2016), 505-542.
doi: 10.1007/s00205-015-0936-0. |
[10] |
C. Gomez,
Wave decoherence for the random Schrödinger equation with long-range correlations, Comm. Math. Phys., 320 (2013), 37-71.
doi: 10.1007/s00220-013-1711-4. |
[11] |
H. J. Kushner,
Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press Series in Signal Processing, Optimization, and Control, 6. MIT Press, Cambridge, MA, 1984. |
[12] |
S. Janson,
Gaussian Hilbert Spaces, Cambridge University Press, 1997. |
[13] |
M. Jara and T. Komorowski,
Limit theorems for some continuous-time random walks, Advances in Applied Probability, 43 (2011), 782-813.
doi: 10.1017/S0001867800005140. |
[14] |
M. Jara, T. Komorowski and S. Olla,
Limit theorems for additive functionals of a Markov chain, Ann. of Appl. Prob., 19 (2009), 2270-2300.
doi: 10.1214/09-AAP610. |
[15] |
M. Jara, T. Komorowski and S. Olla,
Superdiffusion of energy in a chain of harmonic oscillators with noise, Commun. Math. Phys., 339 (2015), 407-453.
doi: 10.1007/s00220-015-2417-6. |
[16] |
T. Komorowski and S. Olla,
Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators, Nonlinearity, 29 (2016), 962-999.
doi: 10.1088/0951-7715/29/3/962. |
[17] |
T. Komorowski and L. Ryzhik,
Passive tracer in a slowly decorrelating random flow with a large mean, Nonlinearity, 20 (2007), 1215-1239.
doi: 10.1088/0951-7715/20/5/009. |
[18] |
T. Komorowski and Ł. Stȩpień,
Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension, Journ. Stat. Phys., 148 (2012), 1-37.
doi: 10.1007/s10955-012-0528-4. |
[19] |
T. Komorowski, S. Olla and L. Ryzhik,
Asymptotics of the solutions of the stochastic lattice wave equation, Arch. Rational Mech. Anal., 209 (2013), 455-494.
doi: 10.1007/s00205-013-0626-8. |
[20] |
T. Kurtz,
Semigroups of conditioned shifts and approximation of markov processes, Ann. Probab., 3 (1975), 618-642.
doi: 10.1214/aop/1176996305. |
[21] |
S. Lepri, R. Livi and A. Politi, Heat transport in low dimensions: Introduction and phenomenology,
Thermal Transport in Low Dimensions, edt S. Lepri, LNP, 921 (2016), 1-37. |
[22] |
J. Lukkarinen and H. Spohn,
Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.
|
[23] |
A. Mellet, S. Mischler and C. Mouhot,
Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.
doi: 10.1007/s00205-010-0354-2. |
[24] |
S. Peszat and Z. Zabczyk,
Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007. |
[25] |
M. A. Pinsky,
Introduction to Fourier Analysis and Wavelets, American Mathematical Society, 2009. |
[26] |
H. Spohn,
Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys., 154 (2014), 1191-1227.
doi: 10.1007/s10955-014-0933-y. |
[27] |
D. W. Stroock and S. R. S. Varadhan,
Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979. |
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