Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics

Abstract

The elsewhere surmised topological origin of phase transitions is given here new important evidence through the analytic study of an exactly solvable model for which both topology and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second order phase transition for k = 2 and a first order one for k > 2. This opens a completely new perspective for the understanding of the deep origin of first and second order phase transitions, respectively. In particular, a remarkable theoretical result consists of a new mathematical characterization of first order transitions. Moreover, we show that a "reduced" configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical N -body system and its macroscopic thermodynamic behaviour. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic 2N -dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

Phase transitions in small systems: Microcanonical vs. canonical ensembles

“…Later on, Fisher [10] and Grossmann et al [11][12][13] employed a very similar approach to analyze the temperature dependence of phase transitions in the canonical ensemble. Recently, further significant progress in the understanding of critical phenomena has been achieved by studying the connection between PTs and phase (or configuration) space topology [14][15][16][17][18][19][20].…”

“…Hence, a singularity in the microcanonical TDFs may arise whenever A changes its geometry or 'shape' in an irregular manner during a small variation of the control parameters E and V . One possible origin for this may be a change in the topology of A (as discussed by Pettini et al [18][19][20]). For the models considered here, however, the topology of A remains unaffected and another general mechanism is at work.…”

Entropy: Form Follows Function

“…In fine print that we did not state or reproduce, Angelani et al (2004) use a complex number in the partition function exponential to derive phase transition and conventional entropy. However, they forget to address the imaginary part and it is left dangling in free or phase space, after they had no use remaining for it.…”

“…The homogenous interactions arise from topology of a problem and are related to entropy and information loss. Topology and entropy are intimately linked as shown by Angelani et al (2004). Homogenous interactions can also be multientity interactions and can be categorized again on the basis of number of entities involved in the interaction.…”

“…Since Occam's razor is a non-mathematical statement; it is untenable to accept it as the justification of Tsallis, C. (1988) entropy as the claimant that describes phase transitions. Phase transitions have topological origin as investigated by Angelani et al (2004). Thus for a statistical mechanics model without non-homogenous interactions, it could be unreasonable to accept that Tsallis, C. (1988) entropy contributes anything of substance over Shannon-Gibbs-Boltzman formulation.…”