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arXiv:hep-lat/9202004 · Physical Review Letters

Multicanonical ensemble: A new approach to simulate first-order phase transitions

Bernd A. Berg, Thomas Neuhaus

Published Jan 6, 1992• 1,582 citations
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Abstract

Relying on the recent proposed multicanonical algorithm, we present a numerical simulation of the first-order phase transition in the 2D 10-state Potts model on lattices up to sizes 100\ifmmode\times\else\texttimes\fi{}100. It is demonstrated that the new algorithm lacks an exponentially fast increase of the tunneling time between metastable states as a function of the linear size L of the system. Instead, the tunneling time diverges approximately proportional to ${\mathit{L}}^{2.65}$. On our largest lattice we gain more than 2 orders of magnitude as compared to a standard heat-bath algorithm. As a first physical application we report a high-precision computation of the interfacial free energy per unit area.

DOI
10.1103/physrevlett.68.9

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Relying on the recent proposed multicanonical algorithm, we present a numerical simulation of the first-order phase transition in the 2D 10-state Potts model on lattices up to sizes 100\ifmmode\times\else\texttimes\fi{}100. It is demonstrated that the new algorithm lacks an exponentially fast increase of the tunneling time between metastable states as a function of the linear size L of the system. Instead, the tunneling time diverges approximately proportional to L2.65{\mathit{L}}^{2.65}L2.65. On our largest lattice we gain more than 2 orders of magnitude as compared to a standard heat-bath algorithm. As a first physical application we report a high-precision computation of the interfacial free energy per unit area.
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Using multicanonical Monte Carlo simulations, we have calculated the interface free energy between the coexisting ordered and disordered phase at the temperature driven first-order phase transition of the two-dimensional 7-state Potts model. Our result 2f3= 0.0241±0.0010 is about an order of magnitude smaller than other estimates in the recent literature. We discuss the differences in analysis and give a possible explanation for the discrepancy. Finally, we briefly mention similar investigations at the deconfining phase transition of SU(3) lattice gauge theory.
We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.
Monte Carlo simulations are performed for first-order phase-transition models. The three-dimensional three-state Potts model has a weak first-order transition. For this model we calculate the density of states on L3{\mathit{L}}^{3}L3 block lattices (L up to size 36) and obtain high-precision estimates for the leading partition-function zeros. The finite-size-scaling analysis of the first zero exhibits the expected convergence of the critical exponent \ensuremath{\nu} toward 1/D for large L; in particular, we find \ensuremath{\nu}=2.955(26) from our two largest lattices. Analysis of our specific-heat Cv{\mathit{C}}_{\mathit{v}}Cv​ data yields l=0.080 31(26) for the latent heat. Along another line of approach, we calculate the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} is the correlation length) for cylindrical L2{\mathit{L}}^{2}L2Lz{\mathit{L}}_{\mathit{z}}Lz​ lattices (L up to 24 and Lz{\mathit{L}}_{\mathit{z}}Lz​=256). The finite size-scaling analysis of these results is also consistent with the convergence of \ensuremath{\nu} toward 1/D, but that the limiting value is 1/D is not yet conclusively established. Some theoretical arguments favor \ensuremath{\nu}\ensuremath{\rightarrow}0 in case of a first-order transition in a cylindrical ${\mathit{L}}^{\mathit{D}\mathrm{\ensuremath{-}}1}$\ensuremath{\infty} geometry. Therefore, we also applied our approach to the 2D ten-state Potts model, which is known to have a strong first-order transition. In this case we find unambiguous evidence in favor of 1/D as the limiting value.