Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.     

A new simulation method for the determination of phase equilibria in mixtures in the grand canonical ensemble

A grand canonical Monte Carlo (GCMC) simulation method is presented for the determination of the phase equilibria of mixtures. The coexistence is derived by expanding the pressure into a Taylor series as a function of the temperature and the chemical potentials that are the independent intensive variables of the grand canonical ensemble. The coefficients of the Taylor series can be calculated from ensemble averages and fluctuation formulae that are obtained from GCMC simulations in both phases. The method is able to produce the equilibrium data in a certain domain of the (T, p) plane from two GCMC simulations. The vapour-liquid equilibrium results obtained for a Lennard-Jones mixture agree well with the corresponding Gibbs ensemble Monte Carlo data.     

The thermodynamic model for nuclear multifragmentation

A great 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 (energy and number of particles in the system are kept fixed), canonical ensemble (temperature and number of particles are kept fixed) or grand canonical ensemble (fixed temperature and a variable number of particles but with an assigned average). This paper deals with calculations with 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 intermediate energy heavy ion collision, to study the caloric curves for nuclei and to explore the possibility of negative specific heat because of the finiteness of nuclear systems. The model can also be used for detailed calculations of other observables not connected with phase transitions, such as populations of selected isotopes in a heavy ion collision.The model also serves a pedagogical purpose. For the problems at hand, both the canonical and grand canonical solutions are obtainable with arbitrary accuracy hence we can compare the values of observables obtained from the canonical calculations with those from the grand canonical. Sometimes, very interesting discrepancies are found.To illustrate the predictive power of the model, calculated observables are compared with data from the central collisions of Sn isotopes.     

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.     

Microcanonical simulation of Ising systems

Numerical simulations of the microcanonical ensemble for Ising systems are described. We explain how to write very fast algorithms for such simulations, relate correlations measured in the microcanonical ensemble to those in the canonical ensemble and discuss criteria for convergence and ergodicity.     

Thermodynamics of one-dimensional nonlinear lattices

Analytic equations were obtained for the thermodynamic parameters of one-dimensional lattices of particles with the Toda and Morse interaction potentials in a canonical Gibbs ensemble. For the same systems, equations were derived for molecular dynamics simulations of thermodynamic processes. Stochastic differential equations were solved with simulating the thermostat by Langevin sources with random forced. Analytic equations for thermodynamic parameters (energy, temperature, and pressure) excellently coincided with molecular dynamics simulation results. The kinetics of system relaxation to the thermodynamic equilibrium state was analyzed. The advantages of simulating the physical properties of systems in a canonical compared with microcanonical ensemble were demonstrated.     

Lennard-Jones fluids confined in nanoscopic slits: Evidence for reentrant filling transitions

By using grand canonical and canonical ensemble Monte Carlo simulations, the structure and phase behavior of a Lennard-Jones (LJ) fluid confined between the parallel (100) planes of a face-centered cubic crystal are studied. Slit pores with a width which allows three adsorbate layers to form are used. It is shown that the filled pore consists of three commensurate layers over a wide range of the surface potential strength, while the pore-filling mechanism and the topology of the phase diagram change when the strength of this fluid-wall potential is varied. Condensation may occur in one step or via two layering-like transitions. The structure of monolayer films depends on the strength and corrugation of the surface potential, and the condensation of the middle layer may induce a reentrant first-order transition.Received: 16 January 2004, Published online: 23 March 2004PACS: 64.70.Nd Structural transitions in nanoscale materials - 64.60.Cn Order-disorder transformations; statistical mechanics of model systems - 68.35.Rh Phase transitions and critical phenomena     

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.     

Microcanonical equilibrium properties of chiral liquid crystals—a Monte Carlo study

Chiral liquid crystals have been investigated by means of a multicanonical Monte Carlo approach in order to characterize their phase behaviour by microcanonical equilibrium properties. The liquid crystals were described by three-dimensional lattice systems with intermolecular interactions given by the chiral Lebwohl-Lasher potential. Self-determined boundary conditions have been applied in order to enable the formation of chiral phases with equilibrium pitch. Selected thermodynamic properties, e.g. microcanonical entropy, temperature, heat capacity and a set of order parameters have been determined with dependence on microcanonical total energy. A cholesteric phase with temperature-induced helix inversion could be proven where the helical superstructure of the single component system studied changed its handedness through an infinite-pitch system. The thermodynamical behaviour in the microcanonical ensemble was found to be very similar to the behaviour in the canonical ensemble. The study of microcanonical equilibrium properties by means of multicanonical Monte Carlo simulations was shown to be a powerful tool for the study of the phase behaviour of model liquid crystals.     

Temperature and density extrapolations in canonical ensemble monte carlo simulations

We show how to use the multiple histogram method to combine canonical ensemble Monte Carlo simulations made at different temperatures and densities. The method can be applied to study systems of particles with arbitrary interaction potential and to compute the thermodynamic properties over a range of temperatures and densities. The calculation of the Helmholtz free energy relative to some thermodynamic reference state enables us to study phase coexistence properties. We test the method on the Lennard-Jones fluids for which many results are available.     

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.     

Thermodynamic and transport properties of simple fluids using lattice sums: bulk phases and liquid-vapour interface

Molecular dynamics simulations in the canonical ensemble have been performed to obtain the thermodynamic and transport properties of the Lennard-Jones fluid. The dispersion interactions were calculated using lattice sums. This method makes it possible to simulate the full potential avoiding the inclusion of the long range corrections (LRC) during or at the end of simulations. In the calculation of dynamic properties in bulk phases and thermodynamic quantities of inhomogeneous systems where the interface is physically present, in general the LRC cannot easily be included. By using the lattice sums method, the results are independent of the truncation of the potential. In the liquid-vapour interface simulations it is not necessary to make any pre-judgments about the form of the LRC formula to calculate coexisting properties such as the surface tension. The lattice sums method has been applied to evaluate how well the full interaction can be calculated in the liquid phase and in the liquid-vapour interface. In the liquid phase the pressure, configurational energy, diffusion coefficient and shear viscosity were obtained. The results of the thermodynamic properties are compared with those obtained using the spherically truncated and shifted (STS) potential with the LRC added at the end of simulations, and excellent agreement is found. The transport properties are calculated on different system sizes for a state near the triple point. The diffusion coefficient using the lattice sums method increases with the number of molecules, and the results are higher than those of the STS model truncated at 2.5σ (STS2.5). The shear viscosity does not show any system size dependence for systems with more than 256 molecules, and the lattice sums results are essentially the same as those for the STS2.5. In the liquid-vapour equilibria the coexisting densities and vapour pressures for the full potential agree well with those obtained using the Gibbs ensemble and the NPT + test particle methods. The surface tension using lattice sums and truncation of forces at 2.5σ agrees well with STS results using large system sizes and cutoff distances.     

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.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

正则系综的变电荷分子动力学方法

在迭代变电荷方法的基础上加以改进得到适于正则系综的变电荷方法.利用正则系综的热浴方法补偿模拟过程中动能的衰减.分子动力学模拟的结果表明,改进的变电荷方法能够避免能量漂移问题,在相同的电荷精度条件下,所需的迭代次数减少,可提高计算效率.     

Nuclear liquid-gas phase transition within the lattice gas model

We study the nuclear liquid-gas phase transition on the basis of a two-component lattice gas model. A Metropolis type of sampling method is used to generate microscopic states in the canonical ensemble. The effective equation of state and fragment mass distributions are evaluated in a wide range of temperatures and densities. A definition of the phase coexistence region appropriate for small systems is proposed. The caloric curve resulting from different types of freeze-out conditions are presented.     

Extended gaussian ensemble solution and tricritical points of a system with long-range interactions

The gaussian ensemble and its extended version theoretically play the important role of interpolating ensembles between the microcanonical and the canonical ensembles. Here, the thermodynamic properties yielded by the extended gaussian ensemble (EGE) for the Blume-Capel (BC) model with infinite-range interactions are analyzed. This model presents different predictions for the first-order phase transition line according to the microcanonical and canonical ensembles. From the EGE approach, we explicitly work out the analytical microcanonical solution. Moreover, the general EGE solution allows one to illustrate in details how the stable microcanonical states are continuously recovered as the gaussian parameter γ is increased. We found out that it is not necessary to take the theoretically expected limit γ → ∞ to recover the microcanonical states in the region between the canonical and microcanonical tricritical points of the phase diagram. By analyzing the entropy as a function of the magnetization we realize the existence of unaccessible magnetic states as the energy is lowered, leading to a breaking of ergodicity.     

The chemical potential of Lennard-Jones associating fluids from osmotic Monte Carlo simulations

The chemical potential for a two-component Lennard-Jones fluid with associative interaction between opposite species promoting the formation of dimers is calculated using osmotic Monte Carlo (OMC) canonical ensemble simulations. Grand canonical Monte Carlo simulations also are performed to verify the accuracy of the OMC approach. The data from both methods agree very well for thermodynamic states with different degrees of dimerization. It follows that the OMC is a promising approach for the determination of the thermodynamics of and equilibria between associating and non-associating fluids and associating fluid mixtures.     

Mesoscopic systems with fixed number of electrons

In this paper, we study the physics of mesoscopic systems with noninteracting electrons of fixed number. From a technical point of view, this means a discussion of the differences between the canonical and the grand canonical ensemble (fixed versus fluctuating number of particles). Such a discussion is not trivial since the grand canonical ensemble is the most convenient basis for the statistics of identical particles and one has to spend labour in order to retrieve the canonical ensemble. Specifically, we are considering ensembles of mesoscopic systems with disorder, either by atomic defects or by fluctuations in their geometric definitions and we discuss various forms of disorder averages.