A Theorem on the origin of Phase Transitions

Details A Theorem on the origin of Phase Transitions A Theorem on the origin of Phase Transitions

2003-12-15

arxiv.org/abs/cond-mat/0312361v1

Phys.Rev.Lett., 92, 60601, (2004)

Physics

Condensed Matter

Statistical Mechanics

10 pages, Statistical Mechanics, Phase Transitions, General Theory. Phys. Rev. Lett., in press

Scientific paper

10.1103/PhysRevLett.92.060601

For physical systems described by smooth, finite-range and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that unless the equipotential hypersurfaces of configuration space \Sigma_v ={(q_1,...,q_N)\in R^N | V(q_1,...,q_N) = v}, v \in R, change topology at some v_c in a given interval [v_0, v_1] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (\beta(v_0), \beta(v_1)) also in the N -> \infty$ limit. Thus the occurrence of a phase transition at some \beta_c =\beta(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {\Sigma_v}_{v < v_c}$ and the {\Sigma_v}_{v > v_c}, which is the consequence of the existence of critical points of V on \Sigma_{v=v_c}, that is points where \nabla V=0.

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