2005/05/31 by Roberto Franzosi, Marco Pettini, Lionel Spinelli
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Combinatorics #Computer science #Configuration space #Entropy (arrow of time) #Material Dynamics and Properties #Mathematical Dynamics and Fractals #Mathematics #Morse code #Morse theory #Order (exchange) #Phase space #Phase transition #Physics #Pure mathematics #Quantum mechanics #Space (punctuation) #Topological and Geometric Data Analysis #Topological entropy #Topology (electrical circuits) #cond-mat.stat-mech #hep-th #math-ph #math.MP #msc:58E05 #msc:82B03 #msc:82B05 #msc:82B26
paper · pdf · doi:10.1016/j.nuclphysb.2007.04.035
published as Nucl.Phys.B782:219-240,2007 · 35 pages. This is an improved version of math-ph/0305032. Added minor chances: Title, Abstract, Introduction
arxiv created 2006/04/18 · openalex publication_date 2007/05/09 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Σv := VN-1 (v)v ∈ R of the potential function VN and the configuration space submanifolds enclosed by the Σv defined by Mv := VN-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 Σvv ∈ R, as well as of the Mvv ∈ R, in an arbitrary interval of values for \vb=v/N, the Helmoltz 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.