Microcanonical thermodynamics of first order phase transitions studied in the potts model

Abstract

Abstract. Phase transitions of first and second order can easily be distinguished in small systems in the microcanonical ensemble. Configurations of phase coexistence, which are suppressed in the conventional canonical formulation, carry important information about the main characteristics of first order phase transitions like the transition temperature, the latent heat, and the interphase surface tension. The characteristic backbending of the microcanonical caloric equation of state T ( E ) (not to be confused with the well known Van der Waals loops in ordinary thermodynamics or mean field approximations) leading to a negative specific heat is intimately linked to the interphase surface entropy.Keywords: Microcanonical thermodynamics; Phase transitions; Surface tension.Microcanonical thermodynamics describes the dependence of the volume RN of the N-body phase-space of an interacting many-body system on the globally conserved quantities like the total energy, momentum, angular momentum, mass, charge etc. This is the fundamental starting point of any statistical definition of thermodynamics. By Laplace transform of &(E) from the extensive quantities like the energy to intensive ones like inverse temperature Cp = l / T ) one obtains the more familiar Gibb's canonical partition function Z @ ) . For systems interacting by short range two-body forces (with hard cores) both formulations are identical in the thermodynamic limit of infinitely many particles at the same particle density [l]. However, the two formulations are different for finite systems and more essentially are different even in their physical content for systems with long range forces like gravity [2] or Coulomb dominated systems see for example [3]. This will be discussed in a forthcoming paper.One of the most interesting phenomena in thermodynamics are phase transitions. Computer simulations of simple models give a deep insight into the mechanism. Naturally these calculations can only be performed for small systems. The main characteristics like the transition temperature TI,, the specific latent heat qIar, and the interphase surface tension aSwrf are extrapolated to the infinite system. In the thermodynamic limit it should not matter whether the calculations for the finite systems are performed canonically or microcanonically. However, we will show for the two-dimensional, q = 10 states, Potts model, which has a clear phase transition of first order from ordered to disordered spins, the extrapolations converge faster when started from the microcanonical-ensemble. Moreover, our investigation will clarify

Complex pole approach in thermodynamic description of fluid mixtures with small number of molecules

“…This result is in good correspondence with published data obtained in experiments and in simulations by molecular dynamics [18]. The developed theory was applied to obtain numerical solutions for the binary mixture with droplet potential energy taking account of both surface energy and miscibility energy (13). The cluster energy model contains the miscibility term, which can be positive or negative…”

“…The non-classical thermodynamic effects were identified and studied in different areas of science, like chemical physics, biology, and nuclear physics [3][4][5]. Small systems can deviate from classical thermodynamics in adsorption and droplet formation [6][7][8], anomalous fluctuations [9,10], breakdown of equivalence of statistical ensembles [11][12][13], and loss of additive property of thermodynamic potentials [14]. These examples demonstrate various phenomena, which require extension of thermodynamics beyond thermodynamic limit.…”

Nonequilibrium Fluctuations in Small Systems: From Physics to Biology

“…Because macroscopic systems are collections of many molecules we expect that the same laws that have been found to be applicable in macroscopic systems are also valid in small systems containing a few number of molecules [1,2]. Yet, the phenomena that we will observe in the two regimes will be different.…”

On the dispute between Boltzmann and Gibbs entropy

“…In summary the Boltzmann entropy for a system with these features is not a concave function, consequently it cannot be the Legendre transformation of the Helmholtz free-energy density, and β B (ǫ) is a not-invertible function. In cases like this, the canonical ensemble has not foundation since it cannot be derived from the microcanonical ensemble, unlike the case of the extensive systems [57], where it can. Therefore, the case of the long-range interactions are outside the class of systems to which our proof applies, although we consider Boltzmann entropy the correct definition also for this class of systems.…”