From a kinetic equation to a diffusion under an anomalous scaling

Basile, Giada

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1301-1322.

Une équation de Boltzmann linéaire est interprétée comme équation de Fokker-Planck associée à la densité de probabilité d’un processus de Markov $\left(K\right(t),i(t),Y(t\left)\right)$ sur $({𝕋}^2\times \{1,2\}\times {ℝ}^2)$, où ${𝕋}^2$ est le tore bidimensionnel. Le processus Markovien $\left(K\right(t),i(t\left)\right)$ est ici un processus de sauts réversible avec des temps d’attente entre deux sauts à moyenne finie mais variance infinie. $Y\left(t\right)$ est une fonctionnelle additive de $K$, définie par $Y\left(t\right)={\int }_0^{t}v\left(K\left(s\right)\right)\mathrm{d}s$, où $\left|v\right|\sim 1$ pour $k$ petit. Nous prouvons que le processus ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converge en distribution vers un mouvement brownien bidimensionnel. En conséquence, et moyennant un changement d’échelle approprié, la solution de l’équation de Boltzmann converge vers celle d’ une équation de diffusion.

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $\left(K\right(t),i(t),Y(t\left)\right)$ on $({𝕋}^2\times \{1,2\}\times {ℝ}^2)$, where ${𝕋}^2$ is the two-dimensional torus. Here $\left(K\right(t),i(t\left)\right)$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y\left(t\right)$ is an additive functional of $K$, defined as ${\int }_0^{t}v\left(K\left(s\right)\right)\mathrm{d}s$, where $\left|v\right|\sim 1$ for small $k$. We prove that the rescaled process ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.

[1] O. Aalen. Weak convergence of stochastic integrals related to counting processes. Z. Wahrsch. Verw. Gebiete 38 (1977) 261-277. | MR | Zbl

[2] K. Aoki, J. Lukkarinen and H. Spohn. Energy transport in weakly anharmonic chain. J. Stat. Phys. 124 (2006) 1105-1129. | MR | Zbl

[3] G. Basile and A. Bovier. Convergence of a kinetic equation to a fractional diffusion equation. Markov Process. Related Fields 16 (2010) 15-44. | MR | Zbl

[4] G. Basile, C. Bernardin and S. Olla. Thermal conductivity for a momentum conserving model. Comm. Math. Phys. 287 (1) (2009) 67-98. | MR | Zbl

[5] G. Basile, S. Olla and H. Spohn. Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. 195 (1) (2009) 171-203. | MR | Zbl

[6] L. Bertini and B. Zegarlinsky. Coercive inequalities for Kawasaki dynamics. The product case. Markov Process. Related Fields 5 (1999) 125-162. | MR | Zbl

[7] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley-Interscience, New York, 1999. | MR | Zbl

[8] A. Dhar. Heat transport in low-dimensional systems. Adv. Phys. 57 (2008) 457-537. DOI:10.1080/00018730802538522.

[9] R. Durrett and S. Resnick. Functional limit theorems for dependent variables. Ann. Probab. 6 (1978) 829-849. | MR | Zbl

[10] A. Dvoretzky. Central limit theorems for dependent random variables. In Actes, Congrès int. Math. Tome 2 565-570. Gauthier-Villars, Paris, 1970. | MR | Zbl

[11] A. Dvoretzky. Asymptotic normality for sums of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probability 513-535. Univ. California Press, Berkeley, 1972. | MR | Zbl

[12] D. Freedman. Brownian Motion and Diffusion. Holden-Day, San Francisco, 1971. | MR | Zbl

[13] D. Freedman. On tail probabilities for martingales. Ann. Probab. 3 (1975) 100-118. | MR | Zbl

[14] B. V. Gnedenko and A. N. Kolmogorov. Predel'nye raspredeleniya dlya summ nezavisimyh slučaĭnyh veličin. (Russian) [Limit Distributions for Sums of Independent Random Variables]. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1949. | MR

[15] M. Jara, T. Komorowski and S. Olla. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (6) (2009) 2270-2300. | MR | Zbl

[16] S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau and A. A. Balandin. Dimensional crossover of thermal transport in few-layer graphene materials. Nature Materials 9 (2010) 555-558.

[17] I. S. Helland. Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9 (1982) 79-94. | MR | Zbl

[18] R. Lefevere and A. Schenkel. Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech. 2006 (2006) L02001. DOI:10.1088/1742-5468/2006/02/L02001.

[19] S. Lepri, R. Livi and A. Politi. Thermal conduction in classical low-dimensional lattice. Phys. Rep. 377 (2003) 1-80. | MR

[20] T. M. Liggett. $L^2$ rates of convergence for attractive reversible nearest particle systems. Ann. Probab. 19 (1991) 935-959. | MR | Zbl

[21] J. Lukkarinen and H. Spohn. Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal. 183 (2007) 93-162. | MR | Zbl

[22] J. Lukkarinen and H. Spohn. Anomalous energy transport in the FPU-$\beta$ chain. Comm. Pure Appl. Math. 61 (2008) 1753-1789. | MR | Zbl

[23] D. L. Mcleish. Dependent central limit theorems and invariance principles. Ann. Probab. 2 (4) (1974) 620-628. | MR | Zbl

[24] A. Mellet, S. Mischler and C. Mouhot. Fractional diffusion limit for collitional kinetic equations. Arch. Ration. Mech. Anal. 199 (2) (2011) 493-525. | MR | Zbl

[25] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability, 2nd edition. Cambridge Univ. Press, Cambridge, 2009. | MR | Zbl

[26] R. E. Peierls. Zur kinetischen Theorie der Waermeleitung in Kristallen. Ann. Phys. 3 (1929) 1055-1101. | JFM

[27] A. Pereverzev. Fermi-Pasta-Ulam $\beta$ lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E 68 (2003) 056124. | MR

[28] M. Röckner and F.-Y. Wang. Weak Poincaré inequalities and $L^2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001) 564-603. | MR | Zbl

[29] S. J. Sepanski. Some invariance principles for random vectors in the generalized domain of attraction of the multivariate normal law. J. Theoret. Probab. 10 (4) (1997) 153-1063. | MR | Zbl

[30] H. Spohn. The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124 (2-4) (2006) 1041-1104. | MR | Zbl