Microcanonical Approach to the Simulation of First-Order Phase Transitions

Abstract

A generalization of the microcanonical ensemble suggests a simple strategy for the simulation of first order phase transitions. At variance with flat-histogram methods, there is no iterative parameters optimization, nor long waits for tunneling between the ordered and the disordered phases. We test the method in the standard benchmark: the Q-states Potts model (Q = 10 in 2 dimensions and Q = 4 in 3 dimensions), where we develop a cluster algorithm. We obtain accurate results for systems with 10 6 spins, outperforming flat-histogram methods that handle up to 10 4 spins. No cure is known for ECSD in canonical simulations (cluster methods [3, 4] do not help), which motivated the invention of the multicanonical ensemble [5]. The multicanonical probability for the energy density is constant, at least in the energy gap e o < e < e d (e o and e d : energy densities of the coexisting low-temperature ordered phase and high-temperature disordered phase), hence the name flat-histogram methods [5,6,7,8]. The canonical probability minimum in the energy gap (∝ exp[−ΣL D−1 ]) is filled by means of an iterative parameter optimization.In flat-histogram methods the system performs an energy random walk in the energy gap. The elementary step being of order L −D (a single spin-flip), one naively expects a tunneling time from e o to e d of order L 2D spinflips. But the (one-dimensional) energy random walk is not Markovian, and these methods suffer ECSD [10]. In fact, for the standard benchmark (the Q = 10 Potts model [9] in D = 2), the barrier of 10 4 spins was reached in 1992 [5], while the largest simulated system (to our knowledge) had 4 × 10 4 spins [6].ECSD in flat histogram simulations is probably understood [10]: on its way from e d to e o , the system undergoes several (four in D = 2) "transitions". First comes the condensation transition [10,11], at a distance of order L −D/(D+1) from e d , where a macroscopic droplet of the ordered phase is nucleated. Decreasing e, the droplet grows to the point that, for periodic boundary conditions, it reduces its surface energy by becoming a strip [12], see Fig. 2 (in D = 3, the droplet becomes a cylinder, then a slab [13]). At lower e the strip becomes a droplet of disordered phase. Finally, at the condensation transition close to e o , we encounter the homogeneous ordered phase.Here we present a method to simulate first order transitions without iterative parameter optimization nor energy random walk. We extend the configuration space as in Hybrid Monte Carlo [14]: to our N variables, σ i (named spins here, but they could be atomic positions) we add N real momenta, p i . The microcanonical ensemble for the {σ i , p i } offers two advantages. First, microcanonical simulations [15] are feasible at any value of e within the gap. Second, we obtain FluctuationDissipation Eqs. (5-8) where the (inverse) temperaturê β, a function of e and the spins, plays a role dual to that of e in the canonical ensemble. The e dependence of the mean value β e , interpolated from a grid as it is almost cons...

Numerical method for determining the interface free energy

“…4. All those quantities are always less than two standard deviations from the corresponding determinations of [15], which have been obtained on larger lattices.…”

“…[15,16]. Following [16], we use three different estimators for the transition temperature on lattices with finite extension.…”

Erratum to “Tethered Monte Carlo: Computing the effective potential without critical slowing down” [Nucl. Phys. B 807 (2009) 424–454]

“…Notice as well that we have constructed the effective potential from the starting point of a canonical ensemble. It would be elementary to retrace our steps for a microcanonical Ω N (we would just have to change the exponential of the energy in the previous equations to the appropriate microcanonical weight, see [17]). …”

“…The TMC method is an extension of the strategy introduced in [17]: there the configuration space was extended to work in a microcanonical ensemble, with entropy as the main physical variable. The motivation in [17] was handling first order transitions without the need for tunnelling between two coexisting phases.…”

“…The motivation in [17] was handling first order transitions without the need for tunnelling between two coexisting phases. The microcanonical method was first applied in [17] to a pure system and was later, in [18], instrumental in simulating a phase transition which remained first order in the presence of quenched disorder. Both the TMC and microcanonical methods have deep links to Creutz's microcanonical demon [19].…”