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Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. (English) Zbl 0927.60060
Carmona, Rene A. (ed.) et al., Stochastic partial differential equations: six perspectives. Providence, RI: American Mathematical Society. Math. Surv. Monogr. 64, 107-152 (1999).
The main aim of the paper is to discuss stochastic improvements to the deterministic description provided by the equations \[ \frac{\partial}{\partial t} {\overline M}({\overline r},t) = {\overline F}({\overline M}({\overline r},t)),\tag{1} \] where \({\overline M}({\overline r},t)\) denotes an appropriate set of macroscopic variables depending on space and time and the structure of \({\overline F}\) depends in general only on the phenomena considered and not on the nature of the microscopic constituents on the macroscopic. An improved version of (1) is often written in the form \[ \frac{\partial}{\partial t} {\overline M}({\overline r},t) = {\overline F}^{\varepsilon}({\overline M}({\overline r},t)),\tag{2} \] in which \({\overline F}^{\varepsilon}\) depends on a small positive parameter \(\varepsilon\), representing the contribution of the finer scale, and \({\overline F}^{\varepsilon}\) approaches \({\overline F}\) as \(\varepsilon\) goes to zero. The authors review the origin of such SPDE’s in simple models in which the transition from microscopic to macroscopic scales and the corrections of (1) can be investigated with some mathematical rigor. More precisely they consider lattice systems with stochastic microscopic dynamics, the so-called interacting partical systems (IPS), where the configurations are updated at random (Poisson) times according to some local rules.
The paper consists of two parts. The first one deals with asymmetric non-reversible models. First, a microscopic model which is itself a deterministic PDE is studied. Then two IPS are considered, namely, the symmetric simple exclusion process (ASEP) and the independent particle process (IPP). The special case of ASEP, the so called weakly asymmetric simple exclusion process is on the discussion, too. The microscopic analogy between the ASEP and an interface model is reflected by the macroscopic analogy between the stochastic Burgers equation and the Kardar-Parisi-Zhang equation. The second part deals with IPS with long range potentials. These give rise to nontrivial translation invariant states of the dynamics, which is not necessarily conservative. The authors start with the hydrodynamics, the small fluctuation and the relation between Ornstein-Uhlenbeck processes and fluctuating hydrodynamics theory for these systems. The critical fluctuations are described. The fluctuations in unstable situation and the macroscopic effects that they generate are also studied. At last, a brief look at large deviation phenomena is given.
For the entire collection see [Zbl 0904.00017].

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)