Microcanonical entropy inflection points: Key to systematic understanding of transitions in finite systems

Abstract

We introduce a systematic classification method for the analogs of phase transitions in finite systems. This completely general analysis, which is applicable to any physical system and extends towards the thermodynamic limit, is based on the microcanonical entropy and its energetic derivative, the inverse caloric temperature. Inflection points of this quantity signal cooperative activity and thus serve as distinct indicators of transitions. We demonstrate the power of this method through application to the long-standing problem of liquid-solid transitions in elastic, flexible homopolymers.PACS numbers: 05.20. Gg,36.40.Ei,82.60.Nh Structure formation processes are typically accompanied by nucleation transitions, where crystalline shapes form out of a liquid or vapor phase. Thus, nucleation is governed by finite-size and surface effects. For small physical systems, it is difficult to understand thermodynamic transitions of this type, as they strongly depend on system size.Cooperativity refers to collective changes in a statistically significant fraction of the degrees of freedom in a system, which transforms the system into a new macrostate. In the thermodynamic limit of an infinitely large system, the ensemble of macrostates sharing similar thermodynamic properties would be called a "phase" and the transformation a "phase transition". The description of such a transformation in a finite system is more subtle, as it cannot be described in the traditional Ehrenfest scheme of singularities in response quantities. However, statistical physics and thus thermodynamics are also valid for systems with no thermodynamic limit. Examples include the structure formation in small atomic clusters and all biomolecules. This is particularly striking for proteins, i.e., heterogeneous linear chains of amino acids. The fact that the individual biological function is connected with the geometrical shape of the molecule makes it necessary to discriminate unfolded (non-functional) and folded (functional) states. Although these systems are finite, they undergo a structural transition by passing a single (or more) free-energy barrier(s). Since these finite-system transitions exhibit strong similarities compared to phase transitions, we extend the terminology once defined in the thermodynamic limit to all systems exhibiting cooperative behavior.In this paper, we introduce a commonly applicable and simple method for the identification and classification of cooperative behavior in systems of arbitrary size by means of microcanonical thermodynamics [1]. It also includes the precise and straightforward analysis of the finite-size effects, which are important to a general understanding of the onset of phase transitions. This is in contrast to canonical approaches, where detailed information is lost by averaging out thermal fluctuations. Re-gaining information about finite-size effects in canonical schemes, e.g., by the investigation of the distribution of Lee-Yang zeros in the complex temperature plane [2] or by inverse Laplace transform [3...

Entropic Description of Gas Hydrate Ice-Liquid Equilibrium via Enhanced Sampling of Coexisting Phases

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Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: A comparison for elastic flexible polymers

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