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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)

Citations:

Zbl 0562.00006