A survey of the hydrodynamical behavior of many-particle systems.

*(English)*Zbl 0567.76006
Nonequilibrium phenomena II: From stochastics to hydrodynamics, Stud. Stat. Mech. 11, 123-294 (1984).

[For the entire collection see Zbl 0551.00016.]

It is the aim of statistical mechanics to derive the macroscopic properties of matter from its molecular structure. There is a rather satisfactory equilibrium theory of thermodynamics. The situation is quite different in the non-equilibrium case, where one aims at deriving the hydrodynamical equations. At present no physically realistic models can be treated. Only recently hydrodynamical behaviour of some idealized systems of interacting particles were studied rigorously. It is the main purpose of these models, that they show how hydrodynamical behaviour arises. The present article reviews these results. It describes both these models and the general structure. ”Local ergodicity” and ”hydrodynamical limit” turn out to provide the foundation, from which Euler-type equations of the conserved quantities can be derived. For some models these conditions are proven to hold. Further properties like the linearized theory, fluctuations and first-order corrections are studied under stronger assumptions.

It is the aim of statistical mechanics to derive the macroscopic properties of matter from its molecular structure. There is a rather satisfactory equilibrium theory of thermodynamics. The situation is quite different in the non-equilibrium case, where one aims at deriving the hydrodynamical equations. At present no physically realistic models can be treated. Only recently hydrodynamical behaviour of some idealized systems of interacting particles were studied rigorously. It is the main purpose of these models, that they show how hydrodynamical behaviour arises. The present article reviews these results. It describes both these models and the general structure. ”Local ergodicity” and ”hydrodynamical limit” turn out to provide the foundation, from which Euler-type equations of the conserved quantities can be derived. For some models these conditions are proven to hold. Further properties like the linearized theory, fluctuations and first-order corrections are studied under stronger assumptions.

Reviewer: M.Mürmann

##### MSC:

76A02 | Foundations of fluid mechanics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

82D15 | Statistical mechanics of liquids |