Fluctuation-dissipation equation of asymmetric simple exclusion processes. (English) Zbl 0884.60092

Landim, C.; Yau, H. T.

Probab. Theory Relat. Fields 108, No. 3, 321-356 (1997).

Summary: We consider asymmetric simple exclusion processes on the lattice \(\mathbb{Z}^d\) in dimension \(d\geq 3\). We denote by \(L\) the generator of the process, \(\nabla\) the lattice gradient, \(\eta\) the configuration, and \(w\) the current of the dynamics associated to the conserved quantity. We prove that the fluctuation-dissipation equation \(w= Lu+ D\nabla\eta\) has a solution for some function \(u\) and some constant \(D\) identified to be the diffusion coefficient. Intuitively, \(Lu\) represents rapid fluctuation and this equation describes a decomposition of the current into fluctuation and gradient of the density field, representing the dissipation. Using this result, we prove rigorously that the Green-Kubo formula converges and it can be identified as the diffusion coefficient.

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

asymmetric simple exclusion processes; fluctuation-dissipation equation; Green-Kubo formula