By Tolman R. C.
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Extra resources for On the Energy and Entropy of Einsteins Closed Universe
Example text
Given , if M is sufficiently restrictions ξ , η, η L\M to large this means that either ξ ∨ η or η ∨ η L\M is not defined (we use the fact that F is locally finite). 18) by putting e−W ,M\ (ξ ∨η) = 0 if ξ ∨ η is not defined, e−B (η) = 0 if η ∨ η L\M is not defined, and (ξ ∨ η ∨ η L\M |X ) = 0 in B if ξ ∨ η ∨ η L\M is not defined. 19) to the limit η → exp[−U (ξ ) − W ,L\ (ξ ∨ η)] is then easily checked. In conclusion a thermodynamic limit of (μ( )ηL\ ) is necessarily a Gibbs state. We now derive an important result from the above estimates.
These appendices recall some well-known facts to establish terminology, and also provide access to less standard results. In general the reader is assumed to be familiar with basic facts of functional analysis, but no knowledge of physics is presupposed. A few open problems are collected in Appendix B. Appendix C contains a brief introduction to flows. Concerning notation and terminology we note the following points. We shall often write |X | for the cardinality of a finite set X . We shall use in Chapters 5–7 the notation Z> , Z , Z< , Z for the sets of integers which are respectively >0, 0, <0, 0.
E. given > 0 there exists finite such that M (A) − c < if ⊃ ). Let then σ, σ ∈ K ; we have σ (A) = lim σ (M (A)) = lim σ (M (A)) = σ (A) →L →L and therefore K consists of a single point. Suppose now that M A does not tend to a constant limit. We assume, as we may, that A = B ◦ α ∈ C for some . We can find sequences (Mn ), (Mn ) and ξn , ξn ∈ such that Mn → L , Mn → L and lim (M Mn (A))(ξn ) = lim (M Mn (A))(ξn ). n→∞ n→∞ But this means lim μ(Mn )ηn (B ◦ α n→∞ Mn ) = lim μ(Mn )ηn (B ◦ α n→∞ Mn ), where ηn = ξn |(L\Mn ), ηn = ξn |(L\Mn ).