Physical review. B, Condensed matter

Comment on ‘‘Finite-size effects at temperature-driven first-order transitions’’

Alain H. Billoire, R. Lacaze, A. Morel, Sourendu Gupta A, Anders Irbäck, Bengt Petersson

Published Oct 1, 1990• 27 citations

Abstract

We reexamine the behavior near a first-order phase-transition point of the fourth-order energy cumulant ${\mathit{V}}_{\mathit{L}}$ introduced by Binder, Challa, and Landau. Our prediction is tested on the ten-states Potts model in two dimensions by Monte Carlo simulation using the Ferrenberg-Swendsen technique. We extract a latent heat in agreement with the exact result of Baxter.

DOI: 10.1103/physrevb.42.6743

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Physical review. B, Condensed matter

Comment on ‘‘Finite-size effects at temperature-driven first-order transitions’’

Alain H. Billoire, R. Lacaze, A. Morel, Sourendu Gupta A, Anders Irbäck, Bengt Petersson

Published Oct 1, 1990• 27 citations

Publisher

Abstract

We reexamine the behavior near a first-order phase-transition point of the fourth-order energy cumulant

{\mathit{V}}_{\mathit{L}}

introduced by Binder, Challa, and Landau. Our prediction is tested on the ten-states Potts model in two dimensions by Monte Carlo simulation using the Ferrenberg-Swendsen technique. We extract a latent heat in agreement with the exact result of Baxter.

DOI: 10.1103/physrevb.42.6743

Related papers

Finite-size scaling and Monte Carlo simulations of first-order phase transitions

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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)= ${\mathit{U}}_{0}$ ( ${\mathit{tL}}^{\mathit{d}}$ )+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$ ${\mathit{U}}_{1}$ ( ${\mathit{tL}}^{\mathit{d}}$ ) 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.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)=${\mathit{U}}_{0}$(${\mathit{tL}}^{\mathit{d}}$)+${\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.

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Wolfhard Janke, Stefan Kappler

We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point $β_t$ . By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point $β_t$. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.

arXiv:hep-lat/9701011Journal de Physique IMay 1, 19978 citations

We reexamine the behavior near a first-order phase-transition point of the fourth-order energy cumulant

{\mathit{V}}_{\mathit{L}}

{\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.

{\mathit{T}}_{0}

(V) have recently been proposed which involve only exponential corrections with respect to

{\mathit{T}}_{0}

\ensuremath{\equiv}

{\mathit{T}}_{0}

(\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.

First-order phase transition in

1d

Potts model with long-range interactions

The first-order phase transition in the one-dimensional

q

-state Potts model with long-range interactions decaying with distance as

1/r^{1+σ}

has been studied by Monte Carlo numerical simulations for

0 < σ\le 1

and integer values of

q > 2

. On the basis of finite-size scaling analysis of interface free energy

ΔF_L

, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for

σ

below a threshold value

σ_c(q)

β_t

. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.