First order phase transitions studied in the dynamical ensemble

A knowledge of the density of states () of a many particle system ( is the energy per particle) contains the complete information about its thermal behaviour. In this paper it will be demonstrated that () can very efficiently and precisely be determined by a Monte Carlo calculation in an ensemble where the system of interest is coupled to another system of comparable size. This is in contrast to the canonical approach where the system of interest is coupled to an infinite bath at a temperatureT. The usefulness of the approach will be demonstrated for theq-states Potts model withq=4, 5, and 8 where the numerical data can be compared with exact results. In our numerical data we can clearly identify a first order phase transition forq5 and a second order transition forq=4.     

Monte Carlo studies of finite-size effects at first-order transitions

First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.     

First order phase transitions in the canonical and the microcanonical ensemble

In the canonical ensemble any singularity of a thermodynamic function at a temperatureT _c is smeared over a temperature range of orderT _T /N. Therefore it is rather difficult to distinguish between a discontinuous and a continuous phase transition on the basis of numerical data obtained for finite systems in the canonical ensemble. It is demonstrated for four model systems that this problem cannot be circumvented by considering higher cumulants of the energy distribution or cumulant ratios. On the other hand, the distinction between first and a second order phase transition is rather direct if based on the microcanonical density of states which is readily obtainable in the dynamical ensemble.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Canonical Analysis of Condensation in Factorised Steady States

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L ^1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities. PACS numbers: 05.40.-a, 02.50.Ey, 64.60.-i.     

Elastic interaction and the phase transition in coherent metal-hydrogen systems

We study the statistical mechanics of hydrogen dissolved in metals. The underlying model is based on the assumption that the dominant attractive interaction between the protons in the metal is of an elastic nature.

In the first part of the paper we review some general properties of the elastic interaction. We then discuss the importance of boundary conditions for the form of the elastic interaction, which turns out to be of the Curie-Weiss type with macroscopic range.

In the second part we investigate the a-a' (‘gas-liquid’) phase transition in the hydrogen lattice fluid. The long-range part of the elastic interaction is treated in mean field approximation. In the canonical ensemble as opposed to the grand canonical ensemble one finds no co-existing phases near the critical point. Instead there is a continuous transition which changes into a first-order transition at tricritical points. In the temperature-density region which normally corresponds to the two-phase co-existence region the hydrogen density is inhomogeneous and varies on a macroscopic scale.

The peculiar nature of the a-a' phase transition is due to the long-range character of the elastic interaction, which ultimately results from the requirement of coherency of the host crystal. We argue that coherent metal-hydrogen systems offer examples of real systems where the classical theory of phase transitions applies.     

Some solvable models of nonuniform classical fluids

A model classical fluid is constructed by assuming that the direct correlation functionc(r – r) is independent of any applied external field. Thermodynamic consistency requires thatc(r – r) 0, and permits explicit representation of the model by a many-body interaction potential. In the canonical ensemble, the model shows a phase transition to an infinite density condensed phase, but in the grand canonical ensemble only an anomalous transition to zero density vapor is found to stably exist.     

Nonuniform van der Waals theory

The liquid-vapor interface of a confined fluid at the condensation phase transition is studied in a combined hydrostatic/mean-field limit of classical statistical mechanics. Rigorous and numerical results are presented. The limit accounts for strongly repulsive short-range forces in terms of local thermodynamics. Weak attractive longer-range ones, like gravitational or van der Waals forces, contribute a self-consistent mean potential. Although the limit is fluctuationfree, the interface is not a sharp Gibbs interface, but its structure is resolved over the range of the attractive potential. For a fluid of hard balls with –r ^–6 interactions the traditional condensation phase transition with critical point is exhibited in the grand ensemble: A vapor state coexists with a liquid state. Both states are quasiuniform well inside the container, but wall-induced inhomogeneities show up close to the boundary of the container. The condensation phase transition of the grand ensemble bridges a region of negative total compressibility in the canonical ensemble which contains canonically stable proper liquid-vapor interface solutions. Embedded in this region is a new, strictly canonical phase transition between a quasiuniform vapor state and a small droplet with extended vapor atmosphere. This canonical transition, in turn, bridges a region of negative total specific heat in the microanonical ensemble. That region contains subcooled vapor states as well as superheated very small droplets which are microcanonically stable.     

A new analytic solution for the Zwanzig-Lauritzen model of polymer chain folding

A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L₂ functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.     

Canonical typicality

It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schr?dinger in 1952.     

Nucleation on a sphere: the roles of curvature,confinement and ensemble

ABSTRACT

By combining Monte Carlo simulations and analytical models, we demonstrate and explain how the gas-to-liquid phase transition of colloidal systems confined to a spherical surface depends on the curvature and size of the surface, and on the choice of thermodynamic ensemble. We find that the geometry of the surface affects the shape of the free energy profile and the size of the critical nucleus by altering the perimeter–area ratio of isotropic clusters. Confinement to a smaller spherical surface results in both a lower nucleation barrier and a smaller critical nucleus size. Furthermore, the liquid domain does not grow indefinitely on a sphere. Saturation of the liquid density in the grand canonical ensemble and the depletion of the gas phase in the canonical ensemble lead to a minimum in the free energy profile, with a sharp increase in free energy for additional growth beyond this minimum.     

Thermodynamics of a 4‐site Hubbard model by analytical diagonalization

By use of the conservation laws a four‐site Hubbard model coupled to a particle bath within an external magnetic field in z‐direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many‐particle states differing in electron number and spin if the electron density is away from half‐filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high‐temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.     

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 forces on a single interacting Bose–Einstein condensate

Using double parabola approximation for a single Bose–Einstein condensate confined between double slabs we proved that in grand canonical ensemble (GCE) the ground state with Robin boundary condition (BC) is favored, whereas in canonical ensemble (CE) our system undergoes from ground state with Robin BC to the one with Dirichlet BC in small-L region and vice versa for large-L region and phase transition in space of the ground state is the first order. The surface tension force and Casimir force are also considered in both CE and GCE in detail.     

The canonical and grand canonical models for nuclear multifragmentation

Many observables seen in intermediate energy heavy-ion collisions can be explained on the basis of statistical equilibrium. Calculations based on statistical equilibrium can be implemented in microcanonical ensemble, canonical ensemble or grand canonical ensemble. This paper deals with calculations with canonical and grand canonical ensembles. A recursive relation developed recently allows calculations with arbitrary precision for many nuclear problems. Calculations are done to study the nature of phase transition in nuclear matter.     

Influence of atomic size mismatch on binary alloy phase diagrams

Abstract

The aim of this paper is to investigate the consequences of atomic size mismatch on the thermodynamics and the topology of binary phase diagrams of face centred cubic alloys. Simple pairwise interatomic potentials with few controlling parameters are used to identify general tendencies. Thermodynamic states are computed by Monte Carlo simulations on a non-rigid lattice. A special attention has been paid to the comparison between calculations in the canonical ensemble, where composition–temperature phase diagrams are determined through van der Waals loops, and in the grand canonical ensemble, where phase diagrams are computed using an interface migration technique. It is shown that these two procedures lead essentially to the same incoherent phase diagram. In the case of phase separating systems, we argue that the introduction of a size mismatch leads to a shrinkage of the solid solution domain and that the asymmetry of the miscibility gap is essentially controlled by the anharmonicity of the heteroatomic potential. Finally, in the case of ordering systems, we show that the asymmetry of the phase diagram may be due to the anharmonicity of the pair potentials or to the differences between their curvatures, the former effect being dominant if the atomic size mismatch is large.     

Phase structure of self-gravitating systems

The equilibrium properties of classical self-gravitating systems in the grand canonical ensemble are studied by using the correspondence with an euclidean field theory with infrared and ultraviolet cutoffs. It is shown that the system develops a first order phase transition between a low and a high density regime. In addition, due to the long range of the gravitational potential, the system is close to criticality within each phase, with the exponents of mean field theory. The coexistence of a sharp first order transition and critical behavior can explain both the presence of voids in large regions of the universe as well as the self-similar density correlations in terms of self-gravity alone.     

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.     

Nuclear multifragmentation, its relation to general physics

Heat can flow from cold to hot at any phase separation even in macroscopic systems. Therefore also Lynden-Bell's famous gravo-thermal catastrophe must be reconsidered. In contrast to traditional canonical Boltzmann-Gibbs statistics this is correctly described only by microcanonical statistics. Systems studied in chemical thermodynamics (ChTh) by using canonical statistics consist of several homogeneous macroscopic phases. Evidently, macroscopic statistics as in chemistry cannot and should not be applied to non-extensive or inhomogeneous systems like nuclei or galaxies. Nuclei are small and inhomogeneous. Multifragmented nuclei are even more inhomogeneous and the fragments even smaller. Phase transitions of first order and especially phase separations therefore cannot be described by a (homogeneous) canonical ensemble. Taking this serious, fascinating perspectives open for statistical nuclear fragmentation as test ground for the basic principles of statistical mechanics, especially of phase transitions, without the use of the thermodynamic limit. Moreover, there is also a lot of similarity between the accessible phase space of fragmenting nuclei and inhomogeneous multistellar systems. This underlines the fundamental significance for statistical physics in general.