Saddle index properties, singular topology, and its relation to thermodynamic singularities for aϕ4mean-field model

2004/05/31 by D. A. Garanin, R. Schilling, A. Scala +1 · 1 citation
Materials Science · Mathematics · Physics and Astronomy · #Atomic physics #Combinatorics #Energy (signal processing) #Geometry #Ground state #Material Dynamics and Properties #Mathematical physics #Mathematics #Physics #Quantum mechanics #Saddle point #Statistical Mechanics and Entropy #Theoretical and Computational Physics #Zero (linguistics) #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.70.036125

published as Phys.Rev. E70 (2004) 036125 · 9 PR pages, 6 figures

arxiv created 2004/08/04 · openalex publication_date 2004/09/30 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We investigate the potential energy surface of a \ensuremathφ⁴ model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers \ensuremathα₊,\ensuremathα₀,\ensuremathα_\ensuremath- with \ensuremathα₊+\ensuremathα₀+\ensuremathα_\ensuremath-=1, provided that the interaction strength \ensuremathμ is smaller than a critical value. The saddle index n_s is equal to \ensuremathα₀ and its distribution function has a maximum at n_s^max=1∕3. The density p(e) of stationary points with energy per particle e, as well as the Euler characteristic \ensuremathχ(e), are singular at a critical energy e_c(\ensuremathμ), if the external field H is zero. However, e_c(\ensuremathμ)\ensuremath≠\ensuremathυ_c(\ensuremathμ), where \ensuremathυ_c(\ensuremathμ) is the mean potential energy per particle at the thermodynamic phase transition point T_c. This proves that previous claims that the topological and thermodynamic transition points coincide is not valid, in general. Both types of singularities disappear for H\ensuremath≠0. The average saddle index n_s as function of e decreases monotonically with e and vanishes at the ground state energy, only. In contrast, the saddle index n_s as function of the average energy e(n_s) is given by n_s(e)=1+4e (for H=0) that vanishes at e=\ensuremath-1∕4>\ensuremathυ₀, the ground state energy.

Citations

• Geometry, Topology and Physics
• Theorem on the Origin of Phase Transitions
• Topology and Phase Transitions: Paradigmatic Evidence
• Geometry of dynamics and phase transitions in classical latticeφ4theories
• Dynamics and geometric properties of thek-trigonometric model
• Energy barriers and activated dynamics in a supercooled Lennard-Jones liquid
• Configuration Space Connectivity across the Fragile-to-Strong Transition in Silica

Cited by

• The mean-fieldφ4model: Entropy, analyticity, and configuration space topology