Investigation of Conditions for the Asymptotic Existence of the Configuration Integral of Gibbs’ Distribution

Let V be a cube of dimension ν, with volume $|V|$. Let |V| / N → ν, N → ∞. Let x = (x₁ , ⋯ ,xN ), xᵢ ∈ V, i = 1, ⋯ ,N, \[ Q(V,N) = ∫_V { ⋯ ∫_V {exp \{ { - β U({ x})} \}} } dx_1 ⋯ dx_N , \] where \[ U({ x}) = ∑1 i < j N {Φ ( {| {x_i - x_j } |} )} . \] The conditions on Φ (y), which are sufficient and in some sense necessary for the existence of the finite limit \[ {lim }N → ∞ 1/Nlog {1}{{N!}}Q(V,N) \] are given.