# States of Equilibrium and Local Equilibrium

Chapter

## Abstract

Given the dynamics the initial state of the system has to be specified. From the point of view of statistical mechanics states are probability measures on the phase space *Γ*. The initial measure *μ* is transported along by the flow *T*_{ t } to the measure at time *t* as *μ*_{ t } = *μ°**T*_{ −t }. If *μ* has a density *f*, *μ*(*d*^{ Nd } q *d*^{ Nd } p)= *f*(q, p)*d*^{ Nd } q *d*^{ Nd } p, then *μ*_{ t }(*d*^{ Nd } q *d*^{ Nd } p) = (*f°* *T*_{ −t })(q, p)
*d*^{ Nd } q *d*^{ Nd } p. For stochastic boundary conditions the probability measure at time *t* is *μ*_{ t }(*d*^{ Nd } q *d*^{ Nd } p) *=* ∫*μ*(*d*^{ d } q’ *d*^{ Nd } p’) *P*_{ t }(q’, p’|*d*^{ Nd } q *d*^{ Nd } p), *t* ≧ 0, according to the rules for Markov processes.

## Keywords

Correlation Function Probability Measure Local Equilibrium Gibbs Measure Grand Canonical Ensemble
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1991