Rigorous treatment of metastable states in the van der Waals-Maxwell theory

O. Penrose, J. L. Lebowitz

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We consider a classical system, in a ν-dimensional cube Ω, with pair potential of the form q(r) + γvφ(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 density ρ, with ρ and the temperature satisfying {Mathematical expression} and {Mathematical expression} where f(ρ, 0+) is the Helmholz free energy density (HFED) in the limit γ→ 0; α = ∫ φ(r)dvr and f0(ρ) is the HFED for the case φ = 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 time t > 0, given that it satisfied them at time 0, is at most λt, where λ is a quantity going to 0 in the limit {Mathematical expression} Here, r0 is a length characterizing the potential q, and x ≫ y means x/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.

Original languageEnglish (US)
Pages (from-to)211-236
Number of pages26
JournalJournal of Statistical Physics
Issue number2
StatePublished - Jun 1971
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Fluids
  • constraints
  • metastability
  • phase coexistence
  • statistical mechanics


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