Entropy Methods in Hydrodynamical Scaling

Abstract

We shall describe these methods by examining very closely an explicit model. Consider N lattice sites arranged periodically in one dimension with a lattice spacing of 1/N. We have spin variables x _j attached to each site j/N, the sites being viewed as equally spaced points on the circle of unit circumference. The spins x _j vary in time in such a manner that they undergo a diffusion on IR^N denoted by {x ₁(t),..., x _N (t)}. The diffusion process is described by

$$d{x_{i}}(t) = s{z_{{i - 1,i}}}(t) - d{z_{{i,i + 1}}}(t),\,d{z_{{i,i + 1}}}(t) = \frac{{{N^{2}}}}{2}\left[ {\phi '({x_{i}}(t)) - \phi '({x_{{i + 1}}}(t))} \right]dt + Nd{\beta _{{i,i + 1}}}(t)$$

.