States of Equilibrium and Local Equilibrium
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.
KeywordsCorrelation Function Probability Measure Local Equilibrium Gibbs Measure Grand Canonical Ensemble
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