Techniques Of Statistical Mechanics Research Articles

Scaling properties of three-dimensional foams

We obtain the scaling states of three-dimensional soap foams by using equilibrium statistical mechanics techniques. We consider a phase space where each bubble is characterized by its center of mass position, volume, surface and number of faces. A Gibbs entropy function is then defined and, as done in standard statistical mechanics, we obtain the probability density function defined in the phase space by maximizing the entropy subject to convenient constraints. The froth is supposed to completely fill the available volume and we consider energy terms accounting for surface tension and the thermal energy of the gas inside the bubbles. We find that the volume distribution presents an exponential decay with cell size and the correlation between cell volume and number of faces fits very well the available numerical data. Additional distribution functions and average values are also in good agreement with numerical simulations.

Statistical Inference, Occam's Razor, and Statistical Mechanics on the Space of Probability Distributions

The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how Occam's razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and I derive and discuss an interpretation of Jeffreys' prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.

Thermodynamics of trimethylindium adducts of N,N′,N ″-trialkyl-1,3,5-triazacyclohexanes †

Adducts of formula InMe3·C3H6N3R3, where R = Me, Pri or But, have been prepared. The adduct bonding of three lone pairs is shown to be only slightly stronger than a standard one lone-pair adduct bond, and estimated to be between 85 and 90 kJ mol–1. The vapour pressures of these adducts have been determined and interpreted by statistical mechanical techniques to show the presence of disorder within the molecules, both in the alkyl groups and in the adduct bonding. The former is shown to affect the solid phases; its use in modelling the disorder in a crystal structure and in predicting the entropy of phase change to disordered phases is outlined. The latter appears in the liquid and vapour phases and is used to explain melting points and entropies of vaporisation.

Mass and Energy Transport Through Slit Pores: Application to Planar Poiseuille Flow

Abstract We develop a simple, efficient and general statistical mechanical technique for calculating the pressure tensor and the heat flux vector in atomic fluids. The method is applied to the case of planar Poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with good computational efficiency. We also find that the commonly used approximation to the Irving-Kirkwood expression for the heat flux and the pressure tensor (where the Irving-Kirkwood Oij operator is set equal to unity-the so-called IK1 approximation), leads to incorrect results for highly inhomogeneous fluids. In such cases the pressure tensor and heat flux vector display spurious oscillations. We calculate the spatially dependent viscosity across a narrow pore and find that it exhibits real but weak oscillations, a consequence of ...

Oxygen vacancies in silica and germania glasses

A statistical mechanical technique with no adjustable parameters is used to calculate the density of oxygen vacancies in pure germania glasses. The calculated values are in fair agreement with the experimental values obtained by Schurman. A slightly different technique utilizing one adjustable parameter is used to calculate the density of oxygen vacancies in binary germano-silicate glasses. The results are also in fair agreement with experimental values.

Investigation of artifacts due to periodic boundary conditions

Periodic boundary conditions are routinely employed in molecular dynamics simulations to simulate bulk material and remove surface effects. However, the effect of periodic boundary conditions on simulation results is not well understood. In this paper three distinct approaches are explored in an effort to determine whether artifacts exist due to the introduction of artificial periodicity. The techniques of group theoretical statistical mechanics are applied to provide tests for artifacts due to the cubic symmetry introduced. Autocorrelation functions and generalized squared displacements for large and small system sizes are compared for artifacts and finally transport coefficients for very low density systems are compared with accepted theoretical results. None of these methods show signs of numerically significant artifacts.

Embedded-atom-method effective-pair-interaction study of the structural and thermodynamic properties of Cu-Ni, Cu-Ag, and Au-Ni solid solutions.

The structural and thermodynamic properties of Cu-Ni, Cu-Ag, and Au-Ni solid solutions have been studied using a computational approach which combines an embedded-atom-method (EAM) description of alloy energetics with a second-order-expansion (SOE) treatment of compositional and displacive disorder. It is discussed in detail how the SOE approach allows the EAM expression for the energy of a substitutional alloy to be cast in the form of a generalized lattice-gas Hamiltonian containing effective pair interactions with arbitrary range. Furthermore, we show how the SOE-EAM method can be combined with either mean-field or Monte Carlo statistical mechanics techniques in order to calculate short-range-order (SRO) parameters, average nearest-neighbor bond lengths, and alloy thermodynamic properties which include contributions from static displacive relaxations and dynamic atomic vibrations. We demonstrate that the contributions to alloy heats of mixing arising from displacive relaxations can be sizeable, and that the neglect of these terms can lead to large overestimations of calculated phase-transition temperatures. The effects of vibrational free-energy contributions on the results of composition-temperature phase diagram calculations are estimated to be relatively small for the phase-separating alloy systems considered in this study.It is shown that within the SOE approach displacive effects can act only to displace the peak in the Fourier-transformed SRO parameter away from Brillouin-zone-boundary special points and towards the origin. Consistent with this result, we show that the unusual SRO observed in diffuse scattering experiments for Au-Ni solid solutions can be understood as arising from a competition between chemical and displacive driving forces which favor ordering and clustering, respectively. \textcopyright{} 1996 The American Physical Society.

Pressure tensor for inhomogeneous fluids.

We develop a simple, efficient, and general statistical mechanical technique for calculating the pressure tensor of an atomic fluid. The method is applied to the case of planar Poiseuille flow through a narrow slit pore, and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with low statistical uncertainty. Using both approaches, the viscosity is calculated as a function of position across the pore, and is seen to oscillate because of a wall-induced local structure in the fluid. We discuss these methods in relation to the well-known ambiguity of the pressure tensor.

Heat flux vector in highly inhomogeneous nonequilibrium fluids.

We develop a simple, efficient, and general statistical mechanical technique, based upon the previously developed method of planes (MOP) technique of calculating the pressure tensor, in order to calculate the heat flux vector of an atomic fluid. The method is applied to the case of Poiseuille flow through a narrow channel and is compared to the corresponding Irving-Kirkwood heat flux vector, using the first approximation (IK1). Our exact MOP method is shown to be more efficient than the approximate IK1 approach. Additionally, for the special case of planar Poiseuille flow, we derive an alternative mesoscopic expression by integrating the energy continuity equation. This mesoscopic calculation is shown to be extremely efficient and is in excellent numerical agreement with the MOP.

Projection techniques in non-equilibrium statistical mechanics III. A microscopic theory of turbulence

Two novel procedures are applied to a reformulation of projection techniques in non-equilibrium statistical mechanics. The time evolution of a spatially uniform system is treated in detail. We obtain and solve a sequence of kinetic equations, which can exhibit extremely irregular behavior, strongly suggestive of turbulence. The same techniques are applied to a uniform but anisotropic system, substantiating the validity of the procedures. The techniques and results strongly suggest a microscopic theory of turbulence emphasizing the role of molecular structure. These same ideas more than hint a possible quantum basis of turbulence.

Theoretical investigation of the thermodynamic stability of nano-scale systems—II: Relaxation of a junction profile

Abstract Nonlinear relaxation of a sharp density profile typical of layered semiconductor junctions is studied using an irreversible statistical mechanical technique, the Path Probability Method, taking into account nearest neighbor correlations. The vacancy mechanism and the pair approximation are used. It is found that atoms near a sharp density profile diffuse up against the density gradient. Our numerical examples demonstrate that in this range there is a possibility that the atom flux can go either down along or up against the local chemical potential (μ) gradient. However, the calculations do not deny the possibility of modifying the definition of μ in such a way that the atoms always flow toward the direction of decreasing μ.

Empirical Risk Minimization versus Maximum-Likelihood Estimation: A Case Study

We study the interaction between input distributions, learning algorithms, and finite sample sizes in the case of learning classification tasks. Focusing on the case of normal input distributions, we use statistical mechanics techniques to calculate the empirical and expected (or generalization) errors for several well-known algorithms learning the weights of a single-layer perceptron. In the case of spherically symmetric distributions within each class we find that the simple Hebb rule, corresponding to maximum-likelihood parameter estimation, outperforms the other more complex algorithms, based on error minimization. Moreover, we show that in the regime where the overlap between the classes is large, algorithms with low empirical error do worse in terms of generalization, a phenomenon known as overtraining.

A replica analysis of the minimum weight solution to the linear equations problem

We investigate, through statistical mechanics techniques, the problem of finding N-tuples J=(Ji,J2,...,JN) with a fixed fraction k of non-vanishing entries that solve a set of P= alpha N linear equations. In particular, we determine the region in the plane ( alpha ,k) where these solutions exist with a probability of 1 in the limit of large N. We also obtain at insight into the microscopic structure of these solutions by calculating the distribution of the probability of an arbitrary non-vanishing entry Ji. Moreover we evaluate analytically the performances of several easy-to-implement heuristic algorithms and compare them with the optimal solution, a task that is suited very well to the statistical mechanics approach.

The classical N-body problem within a generalized statistical mechanics

The modifications that must be introduced within the framework of standard, classical N-body statistical mechanics techniques, in order to deal with the recently proposed generalized scenario of non-extensive statistical mechanics are discussed. A generalized version of the equipartition theorem is shown to hold. The ideal gas is revisited as a simple application.

LEARNING TEMPORAL SEQUENCES FROM EXAMPLES IN A LOCAL FEEDBACK NEURAL NETWORK

Statistical-mechanical techniques are used to study temporal sequence association in a local feedback neural network consisting of an input layer of context units and a single output unit. Each context unit has a feedback connection onto itself such that it accumulates an exponentially decaying moving average or trace of previous inputs to that unit. The formation of these traces allows the network to extract temporal information over an interval delta, where delta = 1/magnitude of ln gamma and gamma is the decay-rate of the moving average. The particular problem of learning a rule from examples is considered where the rule is generated by a teacher local feedback network. The replica method and other mean-field theory techniques are used to determine how the resulting generalization error of the network varies as a function of sequence length M and decay-rate gamma.

Turbulence renormalization group calculations using statistical mechanics methods

The renormalization group theory of fluid turbulence is developed from a statistical mechanical viewpoint using an exact expression for the functional probability distribution of the velocity field. The latter is similar in form to an equilibrium Gibbs distribution, and is derived by combining Lagrangian statistical mechanics with an Eulerian fluid description. It is shown that this distribution enables an RG transformation to be defined, evaluated, and analyzed using statistical mechanical techniques. The method of determining amplitudes used here also differs from previous work, in that the indeterminacy of the relevant stirring force parameters is resolved through the essential requirement that all coarse-grained distributions must yield the actual dissipation rate ℰ. Consequently, although the fixed point solutions of the dynamic RG approach are recovered, slightly different numerical results are obtained for amplitudes. The present approach yields, for example, for Kolmogorov’s constant, CK=1.44.

Information processing by a perceptron in an unsupervised learning task

We study the ability of a simple neural network (a perceptron architecture, no hidden units, binary outputs) to process information in the context of an unsupervised learning task. The network is asked to provide the best possible neural representation of a given input distribution, according to some criterion taken from information theory. The authors compare various optimization criteria that have been proposed: maximum information transmission, minimum redundancy and closeness to factorial code. They show that for the perceptron one can compute the maximum information that the code (the output neural representation) can convey about the input. They show that one can use statistical mechanics techniques, such as replica techniques, to compute the typical mutual information between input and output distributions. More precisely, for a Gaussian input source with a given correlation matrix, they compute the typical mutual information when the couplings are chosen randomly. They determine the correlations between the synaptic couplings that maximize the gain of information. They analyse the results in the case of a one-dimensional receptive field.

Radial distribution functions and their role in modeling of mixtures behavior

Radial distribution (Pair correlation) functions are the primary linkage between macroscopic thermodynamics properties and intermolecular interactions of fluids and fluid mixtures. Numerous theories of mixtures exist based on the definition of the radial distribution functions. Utilization of such theories require the knowledge about mixture radial distribution functions and their relations. Generally the exact radial distribution functions of real fluid mixtures are not available. In this paper a number of approximations for the mixture radial distribution functions are presented. Application of these approximation can result in analytic mixture theories and mixing rules. A uniform statistical mechanical technique is presented through which one can derive equation of state mixing rules based on different RDF approximations. The relative merits of different sets of mixing rules derived by this technique and their applications for modeling of the behavior of mixtures are discussed.

Thermodynamic properties of small zinc clusters based on atomistic simulations

Molecular-dynamics calculations were performed on zinc atom clusters to determine their equilibrium configurations using an embedded-atom method (EAM) potential developed for zinc. Calculation of the thermodynamic properties at different temperatures involved a Monte Carlo scheme in conjunction with statistical mechanical techniques. The harmonic approximation was used in the calculation of the vibrational contribution to the cluster partition function and the rigid-body approximation was used in the calculation of the rotational contribution. These calculations were used to examine the Helmholtz free energy of formation of the clusters as a function of cluster size, temperature and pressure with the aim of determining the nucleation rates and critical supersaturation pressures. Three cluster-growth patterns were considered in all the calculations and stability diagrams were plotted indicating the relative stability of clusters as a function of cluster size and temperature for these three growth patterns.

Information processing by a perceptron in an unsupervised learning task

We study the ability of a simple neural network (a perceptron architecture, no hidden units, binary outputs) to process information in the context of an unsupervised learning task. The network is asked to provide the best possible neural representation of a given input distribution, according to some criterion taken from information theory. The authors compare various optimization criteria that have been proposed: maximum information transmission, minimum redundancy and closeness to factorial code. They show that for the perceptron one can compute the maximum information that the code (the output neural representation) can convey about the input. They show that one can use statistical mechanics techniques, such as replica techniques, to compute the typical mutual information between input and output distributions. More precisely, for a Gaussian input source with a given correlation matrix, they compute the typical mutual information when the couplings are chosen randomly. They determine the correlations b...