Rigorous treatment of metastable states in the van der Waals-Maxwell theory |
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Authors: | O Penrose J L Lebowitz |
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Institution: | 1. The Open University, Walton Hall, Bletchley, Buckinghamshire, England 2. Belfer Graduate School of Science, Yeshiva University, New York, New York
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Abstract: | We consider a classical system, in a ν-dimensional cube Ω, with pair potential of the formq(r) + γ v φ(γr). Dividing Ω into a network of cells ω1, ω2,..., we regard the system as in a metastable state if the mean density of particles in each cell lies in a suitable neighborhood of the overall mean densityρ, withρ and the temperature satisfying $$f_0 (\rho ) + \tfrac{1}{2}\alpha \rho ^2 > f(\rho ,0 + )$$ and $$f''_0 (\rho ) + 2\alpha > 0$$ wheref(ρ, 0+) is the Helmholz free energy density (HFED) in the limit γ→ 0; α = ∫ φ(r)d v r andf 0 (ρ) is the HFED for the caseφ = 0. It is shown rigorously that, for periodic boundary conditions, the conditional probability for a system in the grand canonical ensemble to violate the constraints at timet > 0, given that it satisfied them at time 0, is at mostλt, whereλ is a quantity going to 0 in the limit $$|\Omega | \gg \gamma ^{ - v} \gg |\omega | \gg r_0 \ln |\Omega |$$ Here,r 0 is a length characterizing the potentialq, andx ? y meansx/y → +∞. For rigid walls, the same result is proved under somewhat more restrictive conditions. It is argued that a system started in the metastable state will behave (over times ?λ ?1) like a uniform thermodynamic phase with HFED f0(ρ) + 1/2αρ2, but that having once left this metastable state, the system is unlikely to return. |
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