Abstract
The exclusion process, sometimes called Kawasaki dynamics or lattice gas model, describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each site. We mostly discuss the symmetric and reversible case. The weakly asymmetric case recently attracts attention related to KPZ equation; cf. Bertini and Giacomin (Commun Math Phys 183:571–607, 1995) for a simple exclusion case and Gonçalves and Jara (Arch Ration Mech Anal 212:597–644, 2014) for an exclusion process with speed change, see also Gonçalves et al. (Ann Probab 43:286–338, 2015), Gubinelli and Perkowski (J Am Math Soc 31:427–471, 2018). In Sect. 1, as a warm-up, we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time scaling. From this model, one can derive a linear heat equation and a stochastic partial differential equation (SPDE) in the limit, respectively. Section 2 is devoted to the entropy method originally invented by Guo et al. (Commun Math Phys 118:31–59, 1988). We consider the exclusion process with speed change, in which the jump rate of a particle depends on the configuration nearby the particle. This gives a non-trivial interaction among particles. We study only the case that the jump rate satisfies the so-called gradient condition. The hydrodynamic limit, which leads to a nonlinear diffusion equation, follows from the local ergodicity or the local equilibrium of the system, and this is shown by establishing one-block and two-block estimates. We also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs principle. Section 3 explains the relative entropy method originally due to Yau (Lett Math Phys 22:63–80, 1991). This is a variant of GPV method and gives another proof for the hydrodynamic limit. The difference between these two methods is as follows. Let \(N^d\) be the volume of the domain on which the system is defined (typically, d-dimensional discrete box with side length N) and denote the (relative) entropy by H. Then, H relative to a global equilibrium behaves as \(H=O(N^d)\) (or entropy per volume is O(1)) as \(N\rightarrow \infty .\) GPV method rather relies on the fact that the entropy production I, which is the time derivative of H, behaves as \(O(N^{d-2})\) so that I per volume is o(1), and this characterizes the limit measures. On the other hand, Yau’s method shows \(H=o(N^d)\) for H relative to local equilibria so that the entropy per volume is o(1) and this proves the hydrodynamic limit. In Sect. 4, we consider Kawasaki dynamics perturbed by relatively large Glauber effect, which allows creation and annihilation of particles. This leads to the reaction–diffusion equation in the hydrodynamic limit. We discuss especially the equation with reaction term of bistable type and the problem related to the fast reaction limit or the sharp interface limit leading to the motion by mean curvature. We apply the estimate on the relative entropy due to Jara and Menezes (Non-equilibrium fluctuations of interacting particle systems, 2017; Symmetric exclusion as a random environment: invariance principle, 2018), which is actually obtained as a combination of GPV and Yau’s estimates. This makes possible to study the hydrodynamic limit for microscopic systems with another diverging factors apart from that caused by the space–time scaling.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1995)
Chen, X., Hilhorst, D., Logak, E.: Asymptotic behavior of solutions of an Allen–Cahn equation with a nonlocal term. Nonlinear Anal. 28, 1283–1298 (1997)
De Masi, A., Ferrari, P., Lebowitz, J.: Reaction diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589–644 (1986)
De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. In: Lecture Notes in Mathematics, vol. 1501. Springer, Berlin, pp. x+196 (1991)
Dembo, A., Cover, T.M., Thomas, J.A.: Information theoretic inequalities. IEEE Trans. Inform. Theory 37, 1501–1518 (1991)
Deuschel, J.-D., Stroock, D.W.: Large Deviations. Pure and Applied Mathematics, vol. 137. Academic Press, Cambridge, pp. xiv+307 (1989)
Farfan, J., Landim, C., Tsunoda, K.: Static large deviations for a reaction–diffusion model. Probab. Theory Relat. Fields (2018)
Fleming, W.H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843 (1979)
Funaki, T.: Equilibrium fluctuations for lattice gas. In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds.) Itô’s Stochastic Calculus and Probability Theory, pp. 63–72. Springer, Berlin (1996)
Funaki, T.: Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin. 15, 407–438 (1999)
Funaki, T.: Hydrodynamic limit for \(\nabla \phi \) interface model on a wall. Probab. Theory Relat. Fields 126, 155–183 (2003)
Funaki, T.: Lectures on Random Interfaces. In: SpringerBriefs in Probability and Mathematical Statistics. Springer, Berlin, xii+138 pp (2016)
Funaki, T., Handa, K., Uchiyama, K.: Hydrodynamic limit of one-dimensional exclusion processes with speed change. Ann. Probab. 19, 245–265 (1991)
Funaki, T., Ishitani, K.: Integration by parts formulae for Wiener measures on a path space between two curves. Probab. Theory Relat. Fields 137, 289–321 (2007)
Funaki, T., Olla, S.: Fluctuations for \(\nabla \phi \) interface model on a wall. Stoch. Proc. Appl. 94, 1–27 (2001)
Funaki, T., Tsunoda, K.: Motion by mean curvature from Glauber–Kawasaki dynamics, preprint (2018)
Funaki, T., Uchiyama, K., Yau, H.-T.: Hydrodynamic limit for lattice gas reversible under Bernoulli measures. In: Funaki, T., Woyczynski, W. (eds.) Nonlinear Stochastic PDE’s: Hydrodynamic Limit and Burgers’ Turbulence, IMA volume (Univ. Minnesota) 77, Springer, Berlin, pp. 1–40 (1996)
Gonçalves, P., Jara, M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212, 597–644 (2014)
Gonçalves, P., Jara, M., Sethuraman, S.: A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43, 286–338 (2015)
Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. J. Am. Math. Soc. 31, 427–471 (2018)
Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)
Jara, M., Menezes, O.: Symmetric exclusion as a random environment: invariance principle, arXiv:1807.05414 (2018)
Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. In: Grundlehren der Mathematischen Wissenschaften, vol. 320, pp. xvi+442. Springer, Berlin (1999)
Liggett, T.M.: Interacting particle systems. In: Grundlehren der Mathematischen Wissenschaften, vol. 276. Springer, Berlin, pp. xv+488 (1985)
Liggett, T.M.: Stochastic interacting systems: contact, voter and exclusion processes. In: Grundlehren der Mathematischen Wissenschaften, vol. 324, pp. xii+332. Springer, Berlin (1999)
Menezes, O.: Non-equilibrium fluctuations of interacting particle systems. Thesis, IMPA, Brazil (2017)
Olla, S., Varadhan, S.R.S., Yau, H.-T.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993)
Sasada, M.: On the Green–Kubo formula and the gradient condition on currents. Ann. Appl. Probab. 28, 2727–2739 (2018)
Varadhan, S.R.S., Yau, H.-T.: Diffusive limit of lattice gas with mixing conditions. Asian J. Math. 1, 623–678 (1997)
Yau, H.-T.: Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22, 63–80 (1991)
Acknowledgements
This manuscript grew out of notes on lectures given at Beijing Jiaotong University, Gran Sasso Science Institute in L’Aquila and KAIST in Daejeon in January–February, March and July–August 2018, respectively. The author thanks Xiangchan Zhu, Errico Presutti and Paul Jung for their kind invitations. He also thanks Zhiming Ma and Tusheng Zhang for suggesting the publication of the notes and Makiko Sasada for her useful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Funaki, T. Hydrodynamic Limit for Exclusion Processes. Commun. Math. Stat. 6, 417–480 (2018). https://doi.org/10.1007/s40304-018-0161-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-018-0161-x
Keywords
- Exclusion process
- Kawasaki dynamics
- Hydrodynamic limit
- Scaling limit
- Relative entropy
- Nonlinear diffusion equation