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Physics Research International

Microcanonical Entropy of the Infinite-State Potts Model

Jonas Johansson, Mats‐Erik Pistol

Published Nov 24, 2011• 2 citations
Publisher

Abstract

In this investigation we show that the entropy of the two-dimensional infinite-state Potts model is linear in configurational energy in the thermodynamic limit. This is a direct consequence of the local convexity of the microcanonical entropy, associated with a finite system undergoing a first-order transition. For a sufficiently large number of states , this convexity spans the entire energy range of the model. In the thermodynamic limit, the convexity becomes insignificant, and the microcanonical entropy (the logarithm of the density of states) tends to a straight line. In order to demonstrate the behaviour of the convexity, we use the Wang-Landau Monte-Carlo technique to numerically calculate the density of states for a few finite but high values of . Finally, we calculate the free energy and discuss the generality of our results.

DOI
10.1155/2011/437093

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In this investigation we show that the entropy of the two-dimensional infinite-state Potts model is linear in configurational energy in the thermodynamic limit. This is a direct consequence of the local convexity of the microcanonical entropy, associated with a finite system undergoing a first-order transition. For a sufficiently large number of states , this convexity spans the entire energy range of the model. In the thermodynamic limit, the convexity becomes insignificant, and the microcanonical entropy (the logarithm of the density of states) tends to a straight line. In order to demonstrate the behaviour of the convexity, we use the Wang-Landau Monte-Carlo technique to numerically calculate the density of states for a few finite but high values of . Finally, we calculate the free energy and discuss the generality of our results.
Traditional definition of phase transition involves an infinitely large system in thermodynamic limit. Finite systems such as biological proteins exhibit cooperative behavior similar to phase transitions. We employ recently discovered analysis of inflection points of microcanonical entropy to estimate the transition temperature of the phase transition in q state Potts model on a finite two dimensional square lattice for q=3 (second order) and q=8 (first order). The difference of energy density of states (DOS) Δ ln g(E) = ln g(E+ ΔE) −ln g(E) exhibits a point of inflexion at a value corresponding to inverse transition temperature. This feature is common to systems exhibiting both first as well as second order transitions. While the difference of DOS registers a monotonic variation around the point of inflexion for systems exhibiting second order transition, it has an S-shape with a minimum and maximum around the point of inflexion for the case of first order transition.
Boltzmann's microcanonical entropy is the link between statistical physics and thermodynamics, forasmuch as the behavior of any thermodynamic quantity is directly related to the number of microscopic configurations. Accordingly, in this work, we investigate the behavior of the logarithm of the density of states of the three-state Potts model with an external field applied to one of the states using joint entropic simulations based on the Wang-Landau algorithm. Our analysis reveals that the microcanonical entropy curve is detachable, and each resulting path is related to the formation of clusters. Such a description is consistent with the energy-entropy argument related to the inception of a phase transition. When the external field is reversed and strong, the observed phase transition is from an ordered configuration to cluster formations. The behavior of the microcanonical inverse temperature indicates both first and second-order phase transitions occurring at different temperatures for high values of the external field.
The appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition. However, a convex dip also shows up in some systems with a continuous transition as, for example, in the Baxter-Wu model and in the four-state Potts model in two dimensions. We demonstrate that the appearance of a convex dip in those cases can be traced back to a finite-size effect. The properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite-size scaling theory for continuous phase transitions. Results obtained from numerical simulations corroborate the predictions of the scaling theory.
It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface eS(E,N)e^{S(E,N)}eS(E,N) of the mechanical N-body phase space or more precisely of the curvature determinant D(E,N)=∂2S/∂E2∗∂2S/∂N2−(∂2S/∂E∂N)2D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2D(E,N)=∂2S/∂E2∗∂2S/∂N2−(∂2S/∂E∂N)2 without taking the thermodynamic limit. The first calculation of the entire entropy surface S(E,N)S(E,N)S(E,N) for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where S(E,N)S(E,N)S(E,N) has a maximum curvature ∼0\sim 0∼0. One is the border between the regions in \{E,NE,NE,N\} with D(E,N)>0D(E,N)>0D(E,N)>0 and with D(E,N)<0D(E,N)<0D(E,N)<0, the other line is critical starting as a valley in D(E,N)D(E,N)D(E,N) running from the continuous transition in the ordinary q=3-Potts model, converting at PmP_mPm​ into a flat ridge/plateau (maximum) deep inside the convex intruder of S(E,N)S(E,N)S(E,N) which characterizes the first order liquid-gas transition. The multi-critical point PmP_mPm​ is their crossing.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
Using a multicanonical Metropolis algorithm we have performed Monte Carlo simulations of the 3D 3-state Potts model on L3{L}^{3}L3 lattices with L=20L=20L=20, 30, 40, 50. Covering a range of inverse temperatures from ${\ensuremath{\beta}}_{\mathrm{min}}=0$ to ${\ensuremath{\beta}}_{\mathrm{max}}=0.33$ we calculated the infinite volume limit of the entropy density $s(\ensuremath{\beta})$ with its normalization obtained from s(0)=ln3s(0)=\mathrm{ln}3s(0)=ln3. At the transition temperature the entropy and energy endpoints in the ordered and disordered phase are estimated employing a novel reweighting procedure. We also evaluate the transition temperature and the order-disorder interface tension. The latter estimate increases when capillary waves are taken into account.