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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 and 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.

[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. 

[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. 

[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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