Classification of Phase Transitions by Microcanonical Inflection-Point Analysis

Abstract

By means of the principle of minimal sensitivity we generalize the microcanonical inflection-point analysis method by probing derivatives of the microcanonical entropy for signals of transitions in complex systems. A strategy of systematically identifying and locating independent and dependent phase transitions of any order is proposed. The power of the generalized method is demonstrated in applications to the ferromagnetic Ising model and a coarse-grained model for polymer adsorption onto a substrate. The results shed new light on the intrinsic phase structure of systems with cooperative behavior.

Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions

“…In the case of the model, the caloric curve displays an inflection point just at the critical energy density value identified by the bifurcation point of the order parameter—highlighted with the vertical dfashed line in Figure 2 —and this is in perfect agreement with the proposition put forward by Bachmann in Refs. [ 40 , 43 ].…”

“…The inequivalence of statistical ensembles in presence of long-range interactions, and Molecular Dynamics studies of energy conserving systems have motivated several investigations of the microcanonical description of phase transitions [ 35 , 36 , 37 , 38 , 39 , 40 ]. In particular, we emphasize a recent and very interesting proposal in Reference [ 40 ] which proves very effective to interpret the outcomes of numerical simulations, as it will be seen in the following. A complementary viewpoint à la Ehrenfest has been heuristically put forward in Reference [ 41 ].…”

Entropy Inflection and Invisible Low-Energy States: Defensive Alliance Example

“…Since the slope of s(ρ) defines the intrinsic (microcanonical) inverse temperature, as the temperature T decreases below 1/β x the system is no longer ca-pable of finding a matching stable equilibrium and will stay out-of-equilibrium if it has not fortuitously reached a ground state. Notice that entropy-inflection is qualitatively different from the temperature-inflection phenomenon of [35] (see also [36]) as the latter does not result in a non-concave entropy curve. Non-concave microcanonical entropy was also discussed earlier in the contexts of ferromagnetic metastable states [18,19] and constraint satisfiability problems [37].…”

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

“…If E c stands for the critical value of the Hamiltonian function at the transition point. Hence, the starting of a topological theory of phase transitions that goes beyond the existing theories on this topic [4]-namely, the Yang-Lee theory [15,16] and the Dobrushin-Lanford-Ruelle theory [17]-requires the limit N → ∞ to account for the loss of analyticity of thermodynamic observables; but the study of transitional phenomena in finite N systems (with N extremely smaller than the Avogadro number) is particularly relevant in many other contemporary problems [18], for instance, those related to polymers' thermodynamics and biophysics [19][20][21], Bose-Einstein condensation and Dicke's superradiance in microlasers, superconductive transitions in small metallic objects, just to quote some example.…”

“…. , n − 1; thus, we can write ∂ u 0 = f ν where f is an unknown function that, through the condition (20), produces f = χ. This means that the basis vector ∂ u 0 actually is not normalized and…”