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arXiv:cond-mat/9904073

Phase transitions in finite systems = topological peculiarities of the microcanonical entropy surface

D. H. E. Gross, Evgeny V. Votyakov

Published Apr 6, 1999• 3 citations
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Abstract

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 $e^{S(E,N)}$ of the mechanical N-body phase space or more precisely of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ without taking the thermodynamic limit. The first calculation of the entire entropy surface $S(E,N)$ for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where $S(E,N)$ has a maximum curvature $\sim 0$. One is the border between the regions in \{$E,N$\} with $D(E,N)>0$ and with $D(E,N)<0$, the other line is critical starting as a valley in $D(E,N)$ running from the continuous transition in the ordinary q=3-Potts model, converting at $P_m$ into a flat ridge/plateau (maximum) deep inside the convex intruder of $S(E,N)$ which characterizes the first order liquid-gas transition. The multi-critical point $P_m$ is their crossing.

DOI
10.48550/arxiv.cond-mat/9904073

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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.
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.
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.
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.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
The basic quantity for the description of the statistical properties of physical systems is the density of states or equivalently the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are for example related to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions already in finite systems. In this paper we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities which announce a continuous phase transition in the infinite system are characterised by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analyzing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
Physical quantities obtained from the microcanonical entropy surfaces of classical spin systems show typical features of phase transitions already in finite systems. It is demonstrated that the singular behaviour of the microcanonically defined order parameter and susceptibility can be understood from a Taylor expansion of the entropy surface. The general form of the expansion is determined from the symmetry properties of the microcanonical entropy function with respect to the order parameter. The general findings are investigated for the four-state vector Potts model as an example of a classical spin system.