First-order Phase Transitions in Finite Systems I: Periodic Boundary Conditions

Igor Medveď

Published Jan 1, 2005• 0 citations

Abstract

We briefly review rigorous results on the finite-size effect near first-order phase transitions at which a two-phase coexistence takes place. We consider a large class of statistical mechanical models in (hyper)cubic volumes with periodic boundary conditions at low temperatures. The results show a universal behavior of the asymptotic smoothing of the phase transition discontinuities. The determination of the transition point from

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First-order Phase Transitions in Finite Systems I: Periodic Boundary Conditions

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Published Jan 1, 2005• 0 citations

OpenAlex

Abstract

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{\mathit{L}}^{\mathit{d}}

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.

p

distinct phases is possible, then

p

distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions

L

, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.

{\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.

{\mathit{U}}_{0}

(

{\mathit{tL}}^{\mathit{d}}

)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$

{\mathit{U}}_{1}

(

{\mathit{tL}}^{\mathit{d}}

) with t=1-

{\mathit{T}}_{\mathit{c}}

/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(

{\mathit{e}}_{1}

{\mathit{e}}_{2}

{\mathit{e}}_{2}

{\mathit{e}}_{1}

{)}^{2}

/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature

{\mathit{T}}_{\mathit{c}}^{(2)}

(L)-

{\mathit{T}}_{\mathit{c}}

=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

{\mathit{T}}_{\mathit{c}}

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.