High-temperature Approximation Research Articles

Modelling of pressure-driven membrane separation of electrolytes.: `High temperature' approximation

A model of pressure-driven membrane process of electrolyte separation is presented. The electric field potential assumed as being known, exact solution for permeate composition is readily obtained. All species are assumed to have the same convection velocity. Local electroneutrality condition is not used. The electric potential has been taken into account under high temperature approximation, thus reducing the problem to algebraic equation in exp(Ψ), where Ψ is dimensionless flow potential, and making it possible to calculate concentrations of ions in permeate. Negative retention is shown to be possible for one-component electrolyte solution. For electrolyte mixtures, concentration of ion with high charge is shown to “govern” the membrane selectivity in respect to low-charge ions. Results obtained are in qualitative accordance with the earlier experimental data on membrane separation of reaction mixtures in homogeneous catalysis.

Unsupervised segmentation of Markov random field modeled textured images using selectionist relaxation

Among the existing texture segmentation methods, those relying on Markov random fields have retained substantial interest and have proved to be very efficient in supervised mode. The use of Markov random fields in unsupervised mode is, however, hampered by the parameter estimation problem. The recent solutions proposed to overcome this difficulty rely on assumptions about the shapes of the textured regions or about the number of textures in the input image that may not be satisfied in practice. In this paper, an evolutionary approach, selectionist relaxation, is proposed as a solution to the problem of segmenting Markov random field modeled textures in unsupervised mode. In selectionist relaxation, the computation is distributed among a population of units that iteratively evolves according to simple and local evolutionary rules. A unit is an association between a label and a texture parameter vector. The units whose likelihood is high are allowed to spread over the image and to replace the units that receive lower support from the data. Consequently, some labels are growing while others are eliminated. Starting with an initial random population, this evolutionary process eventually results in a stable labelization of the image, which is taken as the segmentation. In this work, the generalized Ising model is used to represent textured data. Because of the awkward nature of the partition function in this model, a high-temperature approximation is introduced to allow the evaluation of unit likelihoods. Experimental results on images containing various synthetic and natural textures are reported.

Evaluation of coherent-state path integrals in statistical mechanics by matrix multiplication

The numerical evaluation of coherent-state path-integral representations for partition functions and other quantities in equilibrium quantum statistical mechanics is discussed. Several coherent-state path-integral schemes are introduced, which differ from each other by the order of approximation and by the operator ordering employed in the high-temperature approximation of the density operator. Simple one-dimensional systems are used to test these schemes. For the harmonic oscillator, finite-dimensional approximations to the coherent-state path integral are calculated analytically and compared to each other and to the real-space path integral. For anharmonic systems, integrations must be approximated by quadrature formulas. This leads to a matrix multiplication scheme which is tested for the double-well potential. The results obtained are accurate from zero temperature way up into the high-temperature regime where quantum effects become negligible. This is a significant advantage over traditional real-space path integral schemes which break down at low temperatures.

Detection of low magnetic field by means of nuclear resonance of beats

The possibilities of detection of low magnetic fields by means of nuclear resonance of beats are discussed. It is shown that when low-frequency coherence is introduced into a system consisting of an arbitrary number of hyperfine sublevels of the atom's excited states in presence of an external magnetic field, in the high-temperature approximation and taking into account the dipole and quadrupole effects only, a complicated energy structure of hyperfine multiplets doesn't influence the dynamics of transverse components of nuclear magnetization. These occur to be just the same as in the case of two-level system. The multilevel feature of the spin system is not responsible for the fact of absence of the saturation phenomena on nuclear resonance of beats, which is characteristic of all the types of resonance of beats.

Nitrogen diffusion in lattice: a trapping diffusion process

X-ray diffraction, thermoconductivity and nuclear magnetic resonance experiments have been carried out in order to study the diffusion mechanism of N atoms in the lattice. These experiments reveal that a small number of N atoms which enter the lattice, but do not occupy the octahedral interstitial sites, are mobile at the nitrogenation temperature, while most of the N atoms, which enter the octahedral sites, are immobilized. Based on the characteristics of nitrogenation observed in (were R is the rare-earth element) systems, a tripping diffusion model has been proposed and discussed in detail. The previously proposed free diffusion model is the high-temperature approximation of the trapping diffusion model. The experiments and theoretical analysis show that it is the N - lattice interaction, instead of N - N interaction, that leads to the formation of the observed nitrided - unnitrided configuration, causes a tremendous decrease of the apparent diffusion frequency factor and is responsible for the N uptake stability and irreversibility. Also, this work shows how the N - lattice interaction affects the N uptake in various nitrogenation conditions.

An improved perturbation theory and van der Waals one-fluid theory of binary fluid mixtures: Part 1. Total and excess thermodynamic properties

We have examined the accuracy of a modified first-order perturbation theory of nonideal binary mixtures and van der Waals one-fluid theory, which uses an accurate equation of state for pure fluids. Perturbation theory consists of the first-order perturbation theory of high temperature approximation and the random phase approximation. Inclusion of random phase approximation enhances the applicability of perturbation theory to much wider ranges of temperatures and densities. In this part of the paper, results are reported for pressure, residual chemical potentials at finite concentration and in infinite dilution, total and excess properties of several fluid mixtures under highly nonideal conditions. In these mixtures, size and energy parameters of like as well as unlike components differ significantly. Comparisons of theoretical predictions with accurate simulation results show, in general, a very good performance of the perturbation theory in varying nonideal conditions. Van der Waals one-fluid theory is successful in predicting equation of state, residual chemical potentials at finite concentration and in infinite dilution in moderately nonideal mixtures only. It becomes less reliable in describing residual chemical potentials in infinite dilution in highly nonideal mixtures, and becomes unreliable in predicting excess properties of even moderately nonideal mixtures. In general, the present form of perturbation theory offers a significant improvement over the previous forms based on high temperature approximation, and is essentially more accurate than the van der Waals one-fluid theory.

An Improved First-Order Perturbation Theory of Simple Fluids Using High Temperature Approximation and Random Phase Approximation

In this paper, we have examined accuracy of an improved form of the first-order perturbation theory of simple fluids. Theory consists of the first-order perturbation theory of high temperature approximation and the random phase approximation. Inclusion of the random phase approximation enhances applicability of the perturbation theory to much wider ranges of temperatures and densities, especially to low densities. Comparisons are made between theoretical predictions and accurate computer simulation results for pressure, residual Helmholtz free energy, residual internal energy and vapor/liquid phase equilibria of Lennard-Jones fluids. In general, theoretical predictions are in very good agreement with simulation results. The random phase approximation is responsible for predicting accurate Gibbs ensemble Monte Carlo simulation results for vapor/liquid phase equilibria using the same theory in both vapor and liquid phases.

Melting of the Higgs vacuum: conserved numbers at high temperature

We discuss the computation of the grand canonical partition sum describing hot matter in systems with the Higgs mechanism in the presence of non-zero conserved global charges. We formulate a set of simple rules for that computation in the high-temperature approximation in the limit of small chemical potentials. As an illustration of the use of these rules, we calculate the leading term in the free energy of the standard model as a function of baryon number B. We show that this quantity depends continuously on the Higgs expectation value Φ, with a crossover at Φ ∼ T where Debye screening overtakes the Higgs mechanism—the Higgs vacuum “melts”. A number of confusions that exist in the literature regarding the B dependence of the free energy is clarified.

Cumulant calculations of thermodynamic quantities for the Hubbard and the emery model

The calculations of the first paper suggesting phase separation in the Hubbard model [2] are performed under modern computer facilties for the one and three-band Hubbard model. High-temperature approximations for specific heat and static correlation functions are obtained. The latter show, at these high temperatures, no tendency to phase separate in the same way as in the cited paper because our spin parallel nearest-neighbor charge correlation function is negative for all doping values. The spin antiparallel correlation function has positive and negative values for different doping regimes.

Quantum treatment of the effects of dipole–dipole interactions in liquid nuclear magnetic resonance

Experimental observation of anomalous intermolecular cross-peaks in two-dimensional solution NMR spectra have attracted significant recent attention. Extremely simple pulse sequences on extremely simple samples with large equilibrium magnetization give resonances in the indirectly detected dimension which are simply impossible in the conventional density matrix framework of NMR. Here we extend a recently proposed density matrix treatment [Science 262, 2005 (1993)] to calculate the exact time evolution for a variety of pulse sequences. This density matrix treatment explicitly removes two fundamental assumptions of the standard theory—it includes the dipolar interaction between spins in solution (which is only partially averaged away by diffusion) and completely removes the high temperature approximation to the equilibrium density matrix [exp(−βℋ)≊1−βℋ]. We compare this quantum mechanical treatment to a corrected classical model, which modifies the dipolar demagnetizing field formulation to account for the effects of residual magnetization, and show that the quantum picture can be reduced to this corrected classical model when certain assumptions about the retained dipolar couplings are valid. The combination of quantum and classical pictures provides enormously better predictive power and computational convenience than either technique alone.

A high-temperature approximation for the path-integral quantum Monte Carlo method

A high-temperature approximation for the discretized path-integral quantum Monte Carlo (PIQMC) method is formulated. At higher temperatures, all P fictitious classical particles representing a single quantum particle stay close together, and an efficient approximation is obtained when, in the primitive short-time propagator, an essentially local harmonic approximation is used for the external potential at the common centre of mass of P fictitious particles - the integration over P - 1 dimensions can then be carried out analytically, and a classical formula of the effective-potential type is obtained for the partition function.

An extensive study of the Helmholtz free energy of Lennard-Jones fluids using WCA theory

On the basis of results from molecular dynamics and using Weeks-Chandler-Andersen theory, a simple and exact analytical expression for the Helmholtz free energy of a system of particles interacting through the Lennard-Jones potential is obtained for a wide range of densities and temperatures. The values obtained from this expression and its derivative, the potential energy, are compared with existing published values, and are in reasonable agreement. The combination of accuracy with simplicity—only five parameters are used — could be very useful in theoretical and chemical engineering applications, where straightforward mathematical handling of thermodynamic properties is an important requirement. The high temperature approximation for the radial distribution function and for the Helmholtz free energy are compared with results from exact calculation. The results show this approximation to work well for the radial distribution function at high densities only, but to be good even at low densities for the Helmholtz free energy.

Perturbative treatments and learning techniques.

The replica trick ~RT! of statistical mechanics has become a very useful tool for the investigation of complex systems in general and, in particular, for studying learning and generalization processes in neural networks ~NN! @1,2#. The trick overcomes the difficulty of performing an ensemble average of the logarithm of the partition function Z by that of averaging Z over n replicas of the original network, with n a very large integer. For feedforward, single layered NN @3,4# the RT ansatz has been proved to be a valuable tool for the learning of a rule on the basis of a suitable set of examples. In particular, the full quenched theory has been carefully studied within the framework of a replica symmetry approximation @5#. However, as the temperature drops, symmetry breakings may invalidate this approximation, so that an approach to the full quenched theory that incorporates disorder effects due to ‘‘improper’’ examples may be of some interest. In the present effort we wish to study, in the spirit of a second-order approximation, noise effects in the training set, unavoidable in any realistic setting. The noise will here be the result of letting just part of the examples to be produced by the perceptron teacher ~PT!. The rest are to be randomly selected ~‘‘bad’’ examples!. Two types of situations are to be confronted: learnable and unlearnable rules @6#. For the former, there is at least a vector in the concomitant weight space that can learn the rule in an exact fashion. The latter arises mostly in cases of architectural mismatch. In such a situation the training error can never vanish. The question to be answered is, can a perceptron trained under these circumstances correctly respond to queries posed by a PT? In other words, is the rule underlying the ‘‘good’’ examples a learnable one? We will show here that these questions can be adequately dealt with. The paper is organized as follows. Section II is devoted to a brief recapitulation of basic concepts concerning the RT, while Sec. III deals with the thermodynamics of the situation that interests us here. A second order, high temperature approximation is derived. Boolean perceptrons with Ising weights on the learning curves are the subject of Sec. IV. Finally, some conclusions are drawn in Sec. V. II. THE REPLICA METHOD

The high temperature approximation and linearity of the thermodynamic properties on the WCA perturbation parameter

Abstract We carried out extensive computer simulations to obtain the radial distribution function, potential energy per particle and pressure for three-dimensional Lennard-Jones fluids. Each thermodynamic state (265 in total) was simulated for five values of the perturbative parameter of the Weeks-Chandler-Andersen theory. The results show that, for the Helmholtz free energy, the high temperature approximation works very well for all the temperatures and densities studied here, for the potential energy it is good at moderate and high densities for all temperatures, whereas for the pressure it is generally not good. In accordance with other works, for the radial distribution function the high temperature approximation can be used at high densities only. The linearity of the thermodynamic properties, with respect to the perturbative parameter of this theory was also analysed. The results show that the Helmholtz free energy, potential energy per particle and pressure are linear in the mentioned parameter for all densities and temperatures.

Electrostatic Protein–Protein Interactions: Comparison of Point-Dipole and Finite-Length Dipole Potentials of Mean Force

Based on summation of Coulombic interactions, a model is developed for finite-length dipole potentials of mean force in a salt-free dielectric continuum. Point-dipole and finite-length dipole potentials of mean force are compared for protein–protein interactions using parameters for bovine α-chymotrypsin. The two approximations made in the commonly used analytical point-dipole potentials of mean force are not valid at distances near contact. Relative to the finite-length dipole model, the high-temperature approximation overpredicts, and the point-dipole approximation underpredicts charge–dipole and dipole–dipole attractions.

Thermodynamics of quantum fields in black hole backgrounds.

We discuss the relation between the microcanonical and the canonical ensemble for black holes, and highlight some problems associated with extreme black holes already at the classical level. Then we discuss the contribution of quantum fields and demonstrate that the partition functions for scalar and Dirac (Majorana) fields in static space-time backgrounds can be expressed as functional integrals in the corresponding optical space, and point out that the difference between this and the functional integrals in the original metric is a Liouville-type action. The optical method gives both the correction to the black hole entropy and the bulk contribution to the entropy due to the radiation, while the conical singularity method just gives the divergent contribution to the black hole entropy unless one takes into account nonlocal terms. A simple derivation of a general formula for the free energy in the high-temperature approximation is given and applied to various cases. We conclude with a discussion of the second law.

Homogeneous versus inhomogeneous quantum-mechanical master equations

The relations between inhomogeneous and homogeneous quantum-mechanical master equations with and without high-temperature approximation are discussed. A simplified, general procedure is given for the conversion of the standard (high-temperature) inhomogeneous master equation into a homogeneous equation.

Coulomb pair density matrix. II

The Coulomb pair density matrixGβ(r, r′) for attractive and repulsive potentials is not only interesting for determining the two-particle effective potentials, but it is also essential in numerical studies of quantum systems. A high-temperature approximation is obtained for logGβ(r, r′), in the form of simple integrals or series expansions; large-distance expansions are also given.

Single-ion magnetic anisotropy of rare-earth-transition-metal compounds and its description by means of analytical expressions.

The single-ion magnetic anisotropy energy of 3d-4f compounds is shown to be generally irrepresentable in the form of analytical expressions involving anisotropy constants. Particular cases permitting analytical representation are analyzed and expressions for the anisotropy constants in terms of the crystal field and 3d-4f exchange parameters are derived. In particular, the high-temperature approximation appears useful for analysis of vital properties of hard magnetic materials. Limits of validity of the obtained formulas are established.

Equilibrium magnetization of concentrated ferrocolloids

A number of experiments were performed to investigate the temperature and concentration dependences of the initial susceptibility of magnetite ferrocolloids. Measurements were taken at infralow frequency of 0.05 Hz in a wide range of temperatures and concentrations. The experimental results provide verification of different theoretical models dealing with the equilibrium magnetization of ferrocolloids and taking into account magneto-dipole interparticle interactions. The limits of application for the models were qualified. It was shown that mean-spherical and high-temperature approximations provide adequate description of the real ferrocolloids. The Weiss model has been found to give satisfactory results only for diluted and moderately concentrated ferrocolloids, and, as well, in the region of strong magnetic fields.