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April  2018, 11(2): 239-278. doi: 10.3934/krm.2018013

Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation

Tomasz Komorowski 1,, and  Łukasz Stȩpień 2,

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

* Corresponding author: Tomasz Komorowski

Received  May 2016 Revised  December 2016 Published  January 2018

Fund Project: Both authors acknowledge the support of the Polish National Science Center grant UMO-2012/07/B/ST1/03320.

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 [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

Keywords: Harmonic oscillators, kinetic limit, Ornstein-Uhlenbeck process, Wigner functions, correctors.
Mathematics Subject Classification: Primary: 82C44; Secondary: 82C20, 82C70.
Citation: Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
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.  Google Scholar

[2]

R. J. Adler, Geometry of Random Fields, Wiley, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

T. Kurtz, Semigroups of conditioned shifts and approximation of markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.  Google Scholar

[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.  Google Scholar

[22]

J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.   Google Scholar

[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.  Google Scholar

[24]

S. Peszat and Z. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007.  Google Scholar

[25]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, American Mathematical Society, 2009.  Google Scholar

[26]

H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys., 154 (2014), 1191-1227.  doi: 10.1007/s10955-014-0933-y.  Google Scholar

[27]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

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.  Google Scholar

[2]

R. J. Adler, Geometry of Random Fields, Wiley, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

T. Kurtz, Semigroups of conditioned shifts and approximation of markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.  Google Scholar

[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.  Google Scholar

[22]

J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, Arch. Ration. Mech. Anal., 183 (2007), 93-162.   Google Scholar

[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.  Google Scholar

[24]

S. Peszat and Z. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007.  Google Scholar

[25]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, American Mathematical Society, 2009.  Google Scholar

[26]

H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys., 154 (2014), 1191-1227.  doi: 10.1007/s10955-014-0933-y.  Google Scholar

[27]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

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