Topology and phase transitions I. Preliminary results
2007
30 pages
Published in:
- Nucl.Phys.B 782 (2007) 189-218
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Abstract: (Elsevier)
In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first- or second-order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets { Σ v : = V N −1 ( v ) } v ∈ R of the potential function V N and the configuration space submanifolds enclosed by the Σ v defined by { M v : = V N −1 ( ( − ∞ , v ] ) } v ∈ R , which are labeled by the potential energy value v , and where N is the number of degrees of freedom. The proof of the theorem proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces { Σ v } v ∈ R , as well as of the { M v } v ∈ R , in an arbitrary interval of values for v ¯ = v / N , the Helmholtz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature. This preliminary theorem is essential to prove another theorem—in paper II —which makes a stronger statement about the relevance of topology for phase transitions.- 05.70.Fh
- 02.40.-k
- 05.20.-y
- Statistical mechanics
- Phase transitions
- Topology
- potential: confinement
- critical phenomena
- topology
- temperature
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