Thermodynamics of structure-forming systems

Abstract

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations the approach is equivalent to the grand-canonical ensemble; for small systems we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition.

The Typical Set and Entropy in Stochastic Systems with Arbitrary Phase Space Growth

“…Questions like the definition of free energies or the possible need of extensivity, which have to be answered in order to progress towards a complete and consistent thermodynamic picture remain, however, open. For tentative answers to those questions, connections to early proposals could be drawn, both on the thermodynamic grounds; see, e.g., [36,65,66], and from the perspective of entropy characterization; see, e.g., [37,44,46,53,54]. In this paper we meet an equivalence relation underlying compact scales, which allows us to transform between compact scales that do not differ too strongly, i.e., not more than by a power, without essentially changing the structure of the typical set sequence, a fact that one can utilize to give the entropic functional particular properties.…”

“…random variables, the Gibbs-Shannon entropic functional arises naturally in the characterization of the typical set [24,25], establishing a clear connection between thermodynamics and phase space occupation. In systems/processes with collapsing or exploding phase spaces, path dependence or strong internal correlations [4,[18][19][20][21][32][33][34][35][36][37], the phase space may grow super-or sub-exponentially, and the emergence of the Shannon-Gibbs entropic functional derived from phase space volume occupancy considerations is no longer guaranteed. The same situation may arise in cases dealing with non-stationary information sources [38][39][40][41][42].…”

“…Generalized forms for entropies have been proposed to encompass these more general scenarios [43][44][45][46][47][48][49][50][51], some of them explicitly linking the entropic functional to the expected evolution of the phase space volumes [36,46,[52][53][54][55][56]. Despite the notable advances reported also for systems with physical significance [37,57,58], the concept of typicality has not been yet explored for systems/processes with exploding or shrinking phase spaces, which display path dependent dynamics or are subject to emergent internal constraints and correlations.…”

Potts model with invisible states on a scale-free network

“…This model was proposed to study the structure-forming systems ranging from the atoms that build molecules to the self-assembly of colloidal amphibolic particles. It was shown that the model exhibits a first-order phase transition when the number of "molecule states" is high [25]. An interesting is the case of a hybrid Potts model [26].…”

Equivalence of information production and generalised entropies in complex processes

“…Some examples for different process classes include i.i.d. processes, exchangeable and polynomial mixture processes [ 20 ], Polya processes [ 21 ], sample space reducing processes [ 6 ], and processes describing structure forming systems [ 22 ].…”