Simulations Of First-order Phase Transitions Research Articles

Gravitational Waves from the Sound of a First Order Phase Transition

We report on the first three-dimensional numerical simulations of first-order phase transitions in the early Universe to include the cosmic fluid as well as the scalar field order parameter. We calculate the gravitational wave (GW) spectrum resulting from the nucleation, expansion, and collision of bubbles of the low-temperature phase, for phase transition strengths and bubble wall velocities covering many cases of interest. We find that the compression waves in the fluid continue to be a source of GWs long after the bubbles have merged, a new effect not taken properly into account in previous modeling of the GW source. For a wide range of models, the main source of the GWs produced by a phase transition is, therefore, the sound the bubbles make.

Order parameter free enhanced sampling of the vapor-liquid transition using the generalized replica exchange method

The generalized Replica Exchange Method (gREM) is extended into the isobaric-isothermal ensemble, and applied to simulate a vapor-liquid phase transition in Lennard-Jones fluids. Merging an optimally designed generalized ensemble sampling with replica exchange, gREM is particularly well suited for the effective simulation of first-order phase transitions characterized by "backbending" in the statistical temperature. While the metastable and unstable states in the vicinity of the first-order phase transition are masked by the enthalpy gap in temperature replica exchange method simulations, they are transformed into stable states through the parameterized effective sampling weights in gREM simulations, and join vapor and liquid phases with a succession of unimodal enthalpy distributions. The enhanced sampling across metastable and unstable states is achieved without the need to identify a "good" order parameter for biased sampling. We performed gREM simulations at various pressures below and near the critical pressure to examine the change in behavior of the vapor-liquid phase transition at different pressures. We observed a crossover from the first-order phase transition at low pressure, characterized by the backbending in the statistical temperature and the "kink" in the Gibbs free energy, to a continuous second-order phase transition near the critical pressure. The controlling mechanisms of nucleation and continuous phase transition are evident and the coexistence properties and phase diagram are found in agreement with literature results.

The Widom–Rowlinson mixture on a sphere: elimination of exponential slowing down atfirst-order phase transitions

Computer simulations of first-order phase transitions using ‘standard’ toroidal boundaryconditions are generally hampered by exponential slowing down. This is partly due tointerface formation, and partly due to shape transitions. The latter occur when dropletsbecome large such that they self-interact through the periodic boundaries. On a sphericalsimulation topology, however, shape transitions are absent. We expect that by using anappropriate bias function, exponential slowing down can be largely eliminated. In thiswork, these ideas are applied to the two-dimensional Widom–Rowlinson mixture confinedto the surface of a sphere. Indeed, on the sphere, we find that the number ofMonte Carlo steps needed to sample a first-order phase transition does not increaseexponentially with system size, but rather as a power law , with α≈2.5, and V the system area. This is remarkably close to a random walk for whichαRW = 2. The benefit of this improved scaling behavior for biased sampling methods, such as theWang–Landau algorithm, is investigated in detail.

Stefan Problem through Extended Finite Elements: Review and Further Investigations

A general review of the extended finite element method and its application to the simulation of first-order phase transitions is provided. Detailed numerical investigations are then performed by focusing on the one-dimensional case and studying: (i) spatial and temporal discretisations, (ii) different numerical techniques for the interface-condition enforcement, and (iii) different treatments for the blending elements. An embeddeddiscontinuity finite element approach is also developed and compared with the extended finite element method, so that a clearer insight of the latter can be given. Numerical examples for melting/solidification in planar, cylindrical, and spherical symmetry are presented and the results are compared with analytical solutions.

Microcanonical Approach to the Simulation of First-Order Phase Transitions

A simple microcanonical strategy for the simulation of first-order phase transitions is proposed. 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 two dimensions and Q=4 in D=3). We develop a cluster algorithm for this model, obtaining accurate results for systems with more than 10(6) spins.

2D Crystal Shapes, Droplet Condensation, and Exponential Slowing Down in Simulations of First-Order Phase Transitions

Multicanonical ensemble simulations for the simulation of first-order phase transitions suffer from exponential slowing down. Monte Carlo autocorrelation times diverge exponentially with free energy barriers ΔF, which in L d boxes grow as L d−1. We exemplify the situation in a study of the 2D Ising-model at temperature T/T c =0.63 for two different lattice manifolds, toroidal lattices, and surfaces of cubes. For both geometries the effect is caused by discontinuous droplet shape transitions between various classical crystal shapes obeying geometrical constraints. We use classical droplet theory and numerical simulations to calculate transition points and barrier heights. On toroidal lattices we determine finite size corrections to the droplet free energy, which are given by a linear combination of Gibbs–Thomson corrections, capillary wave fluctuation corrections, constant terms, and logarithmic terms in the droplet volume. Tolman corrections are absent. In addition, we study the finite size effects on the condensation phase transition, which occurs in infinite systems at the Onsager value of the magnetization. We find that this transition is of discontinuous order also. A combination of classical droplet theory and Gibbs–Thomson corrections yields a fair description for the transition point and for the droplet size discontinuity for large droplets. We also estimate the nucleation barrier that has to be surmounted in the formation of the stable droplet at coexistence.

Multibondic cluster algorithm

Inspired by the multicanonical approach to simulations of first-order phase transitions we propose for $q$-state Potts models a combination of cluster updates with reweighting of the bond configurations in the Fortuin-Kastelein-Swendsen-Wang representation of this model. Numerical tests for the two-dimensional models with $q=7, 10$ and $20$ show that the autocorrelation times of this algorithm grow with the system size $V$ as $\tau \propto V^\alpha$, where the exponent takes the optimal random walk value of $\alpha \approx 1$.

Multibondic cluster algorithm for Monte Carlo simulations of first-order phase transitions.

Inspired by the multicanonical approach to simulations of first-order phase transitions we propose for $q$-state Potts models a combination of cluster updates with reweighting of the bond configurations in the Fortuin-Kastelein-Swendsen-Wang representation of this model. Numerical tests for the two-dimensional models with $q=7, 10$ and $20$ show that the autocorrelation times of this algorithm grow with the system size $V$ as $\tau \propto V^\alpha$, where the exponent takes the optimal random walk value of $\alpha \approx 1$.

First-order phase transition with global Z3 and Z4 symmetry

Mechanism behind the weakness of first-order phase transition with global Z 3 symmetry in three spatial dimensions is discussed based on large-scale numerical simulations of first-order phase transitions in the three- and four-state Potts models.

Multicanonical multigrid Monte Carlo method.

To further improve the performance of Monte Carlo simulations of first-order phase transitions we propose to combine the multicanonical approach with multigrid techniques. We report tests of this proposition for the d-dimensional ${\mathrm{\ensuremath{\Phi}}}^{4}$ field theory in two different situations. First, we study quantum tunneling for d=1 in the continuum limit, and second, we investigate first-order phase transitions for d=2 in the infinite volume limit. Compared with standard multicanonical simulations we obtain improvement factors of several, and of about one order of magnitude, respectively.

Microcanonical simulation of first-order phase transitions in finite volumes

Microcanonical studies of first-order phase transitions provide essentially the same information as canonical ensemble calculations, and offer no significant advantage over the more common canonical approach. Interpretation of the S-shaped energy-temperature curve that signals a first-order transition as a “van der Waals” curve whose extrema denote “limits of metastability” and bracket an accesible “unstable phase” is unjustified; a simple model of finite-volume effects reproduces such S-shaped curves with good accuracy, as illustrated by the three-dimensional three-state Potts model.

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

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.

Kinetics of first-order phase transitions in adiabatic systems

Based on thermodynamic investigations a general scenario and a kinetic description of the process of first-order phase transitions in adiabatically closed systems, starting from metastable initial states, is developed. It is shown that, in analogy to isothermal constraints, three main stages of the transition may be distinguished: a first stage of dominating nucleation and simultaneous growth of the already formed supercritical clusters, a second stage of independent growth of the clusters, their number being nearly constant, and a third stage of competitive growth, of Ostwald ripening. The change of the temperature of the system due to the latent heat of the transition can be considered hereby as an additional depletion effect. It leads to an increase of the critical size of the clusters and thus to a significant decrease of the nucleation rate, compared with isothermal conditions, especially for relatively large initial supersaturations. Further, it may result also in variations of the stable heterogeneous equilibrium state—that is, configurations of stable clusters in the otherwise homogeneous medium. In particular, for a one-component system under a constant external pressure it makes the existence of such a state possible and results therefore in a qualitative change of the whole course of the phase transition from independent nucleation and growth to the three-stage scenario as characterized above. A theoretical description of the independent growth of the drops and of Ostwald ripening under adiabatic conditions is developed. The results are compared with growth processes in isothermal systems and both quantitative and possible qualitative differences are discussed. Further, they are applied to an interpretation of molecular-dynamics simulations of first-order phase transitions in adsorbed layers.

Accelerated dynamics in simulations of first-order phase transitions.

Accelerated dynamics in simulations of first-order phase transitions.