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Physical Review Letters

New method to determine first-order transition points from finite-size data

Christian Borgs, Wolfhard Janke E

Published Mar 16, 1992• 81 citations
Publisher

Abstract

We consider a temperature-driven first-order phase transition describing the coexistence of q ordered low-temperature phases and one disordered high-temperature phase at the infinite-volume transition temperature ${\mathit{T}}_{0}$. Analyzing the exponential corrections to the periodic partition function in a box of volume V, ${\mathit{Z}}_{\mathrm{per}}$=${\mathit{tsum}}_{\mathit{m}}$ exp(-\ensuremath{\beta}${\mathit{f}}_{\mathit{m}}$V), where ${\mathit{f}}_{\mathit{m}}$ is the (metastable) free energy of the phase m, we propose several definitions of a finite-volume transition temperature ${\mathit{T}}_{0}$(V) which involve only exponential corrections with respect to ${\mathit{T}}_{0}$. We test our propositions in the d=2 Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update procedure.

DOI
10.1103/physrevlett.68.1738

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We consider a temperature-driven first-order phase transition describing the coexistence of q ordered low-temperature phases and one disordered high-temperature phase at the infinite-volume transition temperature T0{\mathit{T}}_{0}T0​. Analyzing the exponential corrections to the periodic partition function in a box of volume V, Zper{\mathit{Z}}_{\mathrm{per}}Zper​=tsumm{\mathit{tsum}}_{\mathit{m}}tsumm​ exp(-\ensuremath{\beta}fm{\mathit{f}}_{\mathit{m}}fm​V), where fm{\mathit{f}}_{\mathit{m}}fm​ is the (metastable) free energy of the phase m, we propose several definitions of a finite-volume transition temperature T0{\mathit{T}}_{0}T0​(V) which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​. We test our propositions in the d=2 Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update procedure.
Traditional observables used to locate first-order phase-transition points from Monte Carlo simulations on finite lattices usually exhibit powerlike finite-size scaling behavior. For systems in a box of volume V with periodic boundary conditions, several definitions of finite volume (pseudo-) transition temperatures T0{\mathit{T}}_{0}T0​(V) have recently been proposed which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​\ensuremath{\equiv}T0{\mathit{T}}_{0}T0​(\ensuremath{\infty}). One of these approaches suggests also a different and efficient way to compute the latent heat. These propositions are tested for the temperature-driven first-order phase transitions in the two-dimensional q-state Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update algorithm and histogram reweighting techniques.
We present a rigorous description of finite-size effects for a large class of models with an asymmetric first-order transition, assuming that all phases contributing to the transition have a finite correlation length. If the model describes the coexistence of two phases, it is shown that, at sufficiently low temperatures, the shift of the transition point due to finite-size effects in a volume Ld{\mathit{L}}^{\mathit{d}}Ld with periodic boundary conditions is O(${\mathit{L}}^{\mathrm{\ensuremath{-}}2\mathit{d}}$), in contrast to certain claims in the literature. We also discuss different ways to determine the transition point from finite-size data, which involve only exponentially small systematic errors in L.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.
We study the finite-size effects at a temperature-driven first-order transition by analyzing various moments of the energy distribution. The distribution function for the energy is approximated by the superposition of two weighted Gaussian functions yielding quantitative estimates for various quantities and scaling form for the specific heat. The rounding of the singularities and the shifts in the location of the specific-heat maximum are analyzed and the characteristic features of a first-order transition are identified. The predictions are tested on the ten-state Potts model in two dimensions by carrying out extensive Monte Carlo calculations. The results are found to be in good agreement with theory. Comparison is made with the second-order transitions in the two- and three-state Potts models.