Stochastic approach to a space-time scaling limit for Hamiltonian systems
The deduction of the heat equation or the Fourier law for the macroscopic evolution of the energy through a diffusive space-time scaling limit from a microscopic dynamics given by Hamilton equations, is one of the most important problem in non-equilibrium statistical mechanics. The hydrodynamic limit is a kind of limiting procedure that changes scales in space and time for interacting systems with a large degree of freedom, which leads to nonlinear partial differential equations. The idea of the singular limit which derives the hydrodynamic equations goes back to Boltzmann and Maxwell. Despite many efforts, this limiting procedure has not been rigorously justified for Hamiltonian systems where a system evolves deterministically. Therefore, instead of Hamiltonian systems, stochastic microscopic models are studied intensively. In this talk, I will report some recent progress in the study of hydrodynamic limits for stochastic models originated form Hamiltonian systems.