## Hydrodynamics of a one-dimensional nearest neighbor model.(English)Zbl 0572.60095

Particle systems, random media and large deviations, Proc. Conf., Bowdoin Coll. 1984, Contemp. Math. 41, 329-342 (1985).
[For the entire collection see Zbl 0562.00006.]
The authors consider one-dimensional systems which satisfy the following equation: $dX_ i=dt(\phi '(X_{i+1}-X_ i)-\phi '(X_ i-X_{i- 1}))+dW_ i,\quad i\in {\mathbb{Z}},$ where $$X_ i(t)$$ denotes the position of particle i at time t; the $$(W_ i)$$ are independent Brownian motions. The labelling of particles is so that the $$X_ i(0)$$ form an increasing doubly infinite sequence. Then, the ordering of particles is preserved for all times t under following assumptions on $$\phi$$ : $$\phi$$ is a positive, decreasing, convex and twice differentiable function on $${\mathbb{R}}_+$$; $$\phi (0)=\infty$$, $$\phi (\infty)=0$$ and $$\int^{1}_{0}(\phi '(x))^ 2 \exp (-\phi (x))dx<\infty.$$
Denote by $$\mu$$ the independent product measure on $${\mathbb{R}}_+^{{\mathbb{Z}}}$$ of the measure exp(-$$\phi$$ (x)-$$\alpha$$ x)dx $$(\alpha >0$$ is fixed). Set $$\rho =\int u_ jd\mu$$, $$\chi =\int (u_ j- \rho)^ 2d\mu$$ and define the rescaling process $$\tilde N^{\epsilon}$$ by $\tilde N^{\epsilon}(g,t)=\sqrt{\epsilon}[\sum_{j}g(\epsilon X_ j(t\epsilon^{-2}))-(\epsilon \rho)^{-1}\int gdx]$ where $$g\in H^ d$$ $$=$$ the completion of $$C_ 0^{\infty}({\mathbb{R}})$$ under the norm $$f\mapsto [\int f(x)(-\Delta +x)^ df(x)dx]^{1/2}$$. The main result of the paper claims that the finite dimensional distributions of $$\tilde N^{\epsilon}(g,t)$$ $$(g\in H^ 5$$, $$t\geq 0)$$ converge to those of an Ornstein-Uhlenbeck process $$\tilde N$$ with covariance $${\mathbb{E}} \tilde N(g,0)\tilde N(h,t)=\chi \int gR_ thdx/\rho^ 3$$ where $$R_ t$$ is the kernel of Brownian motion, formally written as $$\exp [-(\rho^ 2/\chi)t\Delta /2]$$. Thus, the bulk diffusion coefficient of the process $$(X_ i(t):$$ $$i\in {\mathbb{Z}}$$, $$t\geq 0)$$ is given by $$\rho^ 2/\chi.$$
The construction of the process $$(X_ i(t))$$ and the discussion on the fluctuation process are based on the study of a related process, which is a solution to the equation: $dU_ j=dt[\phi '(U_{j+1})-2\phi '(U_ j)+\phi '(U_{j-1})]+dW_{j+1}-dW_ j,\quad j\in {\mathbb{Z}}.$
Reviewer: M.Chen

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J65 Brownian motion 80A20 Heat and mass transfer, heat flow (MSC2010)

Zbl 0562.00006