Non-equilibrium phase transitions in the two-temperature Ising model with Kawasaki dynamics

Phase transitions of the two-finite temperature Ising model on a square lattice are investigated by using a position space renormalization group (PSRG) transformation. Different finite temperatures, T _x ?and?T _y , and also different time-scale constants, ?? _x and ?? _y for spin exchanges in the x and y directions define the dynamics of the non-equilibrium system. The critical surface of the system is determined by RG flows as a function of these exchange parameters. The Onsager critical point (when the two temperatures are equal) and the critical temperature for the limit when the other temperature is infinite, previously studied by the Monte Carlo method, are obtained. In addition, two steady-state fixed points which correspond to the non-equilibrium phase transition are presented. These fixed points yield the different universality class properties of the non-equilibrium phase transitions.     

Exploiting the flexibility of a family of models for taxation and redistribution

In this paper we present and discuss results of Monte Carlo numerical simulations of the two-dimensional Ising ferromagnet in contact with a heat bath that intrinsically has a thermal gradient. The extremes of the magnet are at temperatures T ₁?<?T _c ?<?T ₂, where T _c is the Onsager critical temperature. In this way one can observe a phase transition between an ordered phase (T?<?T _c ) and a disordered one (T?>?T _c ) by means of a single simulation. By starting the simulations with fully disordered initial configurations with magnetization m????0 corresponding to T?=???, which are then suddenly annealed to a preset thermal gradient, we study the short-time critical dynamic behavior of the system. Also, by setting a small initial magnetization m?=?m ₀, we study the critical initial increase of the order parameter. Furthermore, by starting the simulations from fully ordered configurations, which correspond to the ground state at T?=?0 and are subsequently quenched to a preset gradient, we study the critical relaxation dynamics of the system. Additionally, we perform stationary measurements (t??????) that are discussed in terms of the standard finite-size scaling theory. We conclude that our numerical simulation results of the Ising magnet in a thermal gradient, which are rationalized in terms of both dynamic and standard scaling arguments, are fully consistent with well established results obtained under equilibrium conditions.     

Application of the Monte Carlo method to the problem of surface segregation simulation

A generalization of the Monte Carlo method to the case of grand canonical ensemble allowing the elimination of the problem of determination of the chemical potential of alloy components was proposed. The method is particularly convenient for the calculations of surface segregations because it excludes time-consuming calculation of the temperature-dependent bulk chemical potential μ(T). The new method was used for calculating segregations at the (100), (110), and (111) surfaces of the Ni₅₀Pd₅₀ alloy using the Ising model with ab initio effective interatomic interaction potentials.     

Negative heat capacity of Ar55 cluster

The solid–liquid phase transitions of Ar₅₅ cluster was simulated by the microcanonical molecular dynamics and microcanonical parallel tempering methods using Lennard–Jones potential, and thermodynamic quantities were calculated. The caloric curve of cluster has S-bend. To understand this behaviour, configurational and total entropies were evaluated, and the dents on the entropy curves were noticed as the sign of negative heat capacity. The heat capacities were evaluated by using configurational entropy data. The potential energy distributions have bimodal behaviour in the given range at the melting temperature. At the same time by using configurational entropy canonical caloric curve and canonical heat capacity were calculated. To obtain entropy change upon melting, total entropy were calculated from the caloric curve. The microcanonical results melting temperature, latent heat and entropy change upon melting values were reported and compared with the values reported in the literature and the values calculated from the thermodynamic relations offered for bulk matter, consistent values were found.     

Application of large deviation theory to the mean-field $\boldsymbol{\varphi}^\mathsf{4}$-model

A large deviation technique is used to calculate the microcanonical entropy function s(v,m) of the mean-field ϕ⁴-model as a function of the potential energy v and the magnetization m. As in the canonical ensemble, a continuous phase transition is found. An analytical expression is obtained for the critical energy v_c(J) as a function of the coupling parameter J.     

Microcanonical ensemble study of a gas column under gravity

The study of a classical ideal gas column of finite height H in a uniform gravitational field g is made by the microcanonical ensemble at energy E. The primary functions of this ensemble, the phase volume and the density of states, are derived. Related statistical quantities, such as the entropy, the temperature and the heat capacity, are also reported. The equivalence in the thermodynamic limit between the calculated microcanonical expressions and those obtained from the canonical ensemble is shown numerically. The expression for the temperature is used to analyze the temperature change when the gas is permitted to expand into an evacuated region increasing the height of the column from H ₁ to H ₂. The microcanonical single-particle momentum and height distributions are also reported.     

Finite size scaling of spin-spin correlations on the Ising square lattice

Using Monte Carlo data for the Ising square lattice, we show that the row spin-spin correlation functions scale as a function of both lattice size and ? = ∣1 ? T/T_c∣ forT >T_c.     

Phase behavior of unsymmetrical dimers on a square lattice

Monte Carlo simulations in the grand canonical ensemble have been carried out to study phase transitions in monolayers formed by heterogeneous dimers, composed of segments A and B, on a square lattice. The unsymmetrical segment-segment interactions are assumed (u_AA ≠ u_BB). The systems with u_AB = u_BB = −1 and different AA-interactions studied in our previous work [W. R?ysko, M. Borówko, Surf. Sci. 520 (2002) 151] are reconsidered. The structural phase transitions at high temperatures are investigated. It is shown that topology of a phase diagram depends on the energy u_AA. For repulsive AA-interactions a triple point is found and the system belongs to the universality class of a tricritical point. When u_AA < 0 the critical line of the order-disorder transition terminates at a critical end point and the system belongs to the universality class of the 2D Ising model.     

On the Mean-Field Spherical Model

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci/ of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor σ-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509?C542, 1988) model. Fix n??1 and ??>0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate ??(n?k)/n, where k is the distance from the node to the root. Denote by Z _n (t) the number of nodes with no descendants at time t and let T _n =?? ^?1 nln(n/ln4)+(ln2)/(2??). We prove that 2^?n Z _n (T _n +n??), ?????, converges to the Gompertz curve exp(?(ln2)?e ^??|? ). We also prove a central limit theorem for the martingale associated to Z _n (t).     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.     

Dynamics of the infinite-range ising spin glass

The Glauber dynamics of an Ising spin glass with infinite-range interactions and additional static field, h, is investigated near the freezing temperature, T_f. We obtain critical slowing down at and below the de Almeida-Thouless instability line, h_c(T), to order (1?T/T_f)³ with algebraic decay of the spin correlations ～t^?ν, where $\text{ν=}\text{1}\text{2}$ at T_f and $\text{ν≤}\text{1}\text{2}$ for T<T_f.     

Properties of the Ising magnet confined in a corner geometry

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ±h are applied, i.e., at the two boundary rows ending at the lower left corner a field +h acts, while at the two boundary rows ending at the upper right corner a field −h acts. For temperatures T less than the critical temperature T_c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T_f(h) runs from the upper left corner to the lower right corner, while for T<T_f(h) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T=T_f(h) the magnetization profile m(z) in the z-direction normal to the interface simply is linear and the interfacial width scales as w∝L, while for T>T_f(h) it scales as . The distribution P(?) of the interface position ? (measured along the z-direction from the corners) decays exponentially for T<T_f(h) from either corner, is essentially flat for T=T_f(h), and is a Gaussian centered at the middle of the diagonal for T>T_f(h). Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

On the susceptibility exponent of random anisotropy magnets

The temperature dependence of the non-linear susceptibility ≈₂(T) of random anisotropy magnets in the Ising limit (speromagnets) is calculated for temperatures above the freezing temperature T_f within the framework of the correlated molecular field theory. For the effective susceptibility exponent λ_s(T) = (T?T_f)≈₂d^-1_≈2/dT a non-monotonic temperature dependence is found as for the case of spin glasses. This must be taken into account in order to obtain reliable values for the critical susceptibility exponent from experimental data.     

Ratio of canonical and microcanonical temperatures of a vibratory antiferromagnetic ising chain

The ratio of canonical and microcanonical temperatures T(c)/T(&mgr;) of a vibratory antiferromagnetic Ising chain with N spins is given by analytical calculation. The result is T(c)/T(&mgr;)=1+O(N-1), which is consistent with the natural assumption given by Rugh.     

The equivalence of ensembles for classical systems of particles

For systems of particles in classical phase space with standard Hamiltonian, we consider (spatially averaged) microcanonical Gibbs distributions in finite boxes. We show that infinite-volume limits along suitable subsequences exist and are grand canonical Gibbs measures. On the way, we establish a variational formula for the thermodynamic entropy density, as well as a variational characterization of grand canonical Gibbs measures.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate correlations parallel to the layering in the diagonally layered model with periodv=2, the so-called “general square lattice” model (GS). If the model has a finite critical temperature,T _c>0, we have a spontaneous magnetization belowT _c vanishing atT _c with the Ising exponent β=1/8. AtT _c correlations decay algebraically with critical exponnet η=1/4 and exponentially forT>T _c. In the frustrated case we have oscillatory behaviour superposed on the exponential decay where the wavevector of the oscillations changes at some “disorder temperature”T _D(>T _c) from commensurate to temperature-dependent in commensurate periods. If the critical temperature vanishes,T _c=0 we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index η=1/2, i.e.T=0 is thus a critical point.