Showing papers in "Physica A-statistical Mechanics and Its Applications in 1981"

Liquid-crystalline ordering in the solution of long persistent chains

Abstract: The orientational ordering in the solution of long persistent macromolecules is studied by means of the Onsager method. It is shown that the liquid-crystalline transition occurs at significantly higher concentrations and that the order parameter at the transition point is much smaller than for the model of freely-jointed segments. It is concluded that the orientational ordering in the solutions of semiflexible chains depends essentially on the flexibility distribution along the chain contour.

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

Abstract: The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.

Phase diagram of the two-dimensional Lennard-Jones system; Evidence for first-order transitions

Abstract: There has recently been extensive interest in the nature of the melting/freezing transition for a two-dimensional system of molecules interacting with Lennard-Jones 6:12 potentials, which is a prototype for physisorbed systems. We have therefore made a detailed study of the thermodynamics of the phase diagram for this system. We first made calculations using liquid-state perturbation theory for the fluid state and a self-consistent cell theory for the solid state to determine thermodynamic functions; these results led to ordinary first-order phase transitions between solid/fluid and liquid/gas phases, in agreement with the constant-pressure Monte Carlo results of Abraham. We refined the calculations by using constant-pressure and constant-density Monte Carlo results to improve the accuracy of the calculated free energies, and we determined the two-phase equilibria by making direct Monte Carlo calculations for two-phase systems. The results are internally consistent and lead to a phase diagram qualitatively similar to the three-dimensional Lennard-Jones system.

Dissipative contributions of internal multiplicative noise: I. mechanical oscillator

Abstract: Langevin equations for closed systems with multiplicative fluctuations must also include appropriate dissipative terms that ensure eventual equilibration of the system. We consider an oscillator coupled to a heat bath and show that a particular nonlinear coupling to a harmonic heat bath leads to a fluctuating frequency and to nonlinear dissipative terms . We also analyze the effects of the multiplicative fluctuations and of the corresponding nonlinear dissipation on the temporal evolution of the average oscillator energy. We find that the rate of equilibration of this system can be significantly different from that of an oscillator with only additive fluctuations and linear dissipation.

Pinning of a rough interface by an external potential

Abstract: An analysis is given of the behavior of an interface between two phases in the presence of an external pinning potential in the solid-on-solid limit of the two-dimensional Ising model. It is found that the potential turns a rough interface into a smooth one, except in the case of a boundary potential, where a minimum potential strength is required. The connection with the roughening transition found by Abraham is discussed. The interface width is calculated as a function of the potential parameters in the limit of a weak pining potential.

Interpretation of the unusual behavior of H2O and D2O at low temperature: Are concepts of percolation relevant to the “puzzle of liquid water”?

Abstract: This talk will summarize the present status of an ongoing research program designed to answer the question posed in the title. Since a snapshot of liquid water with a subpicosecond shutter speed reveals that this system (a hydrogen-bonded liquid) is above its percolation threshold, it is tempting to imagine that connectivity concepts of the sort encompassed in percolation theory may prove useful. We find that the traditional approach of random-bond percolation theory-developed to describe the onset of gelation - is not sufficient, since water is well above its gelation threshold. Hence we develop a new correlated-site percolation model, whose predictions are found to be in quantitative agreement with molecular dynamics calculations and in qualitative agreement with a wide range of experimental data on low-temperature water. The picture that emerges is that of an “infinite” hydrogen-bonded network subject to continuous restructuring. At any instant of time, there are many strained and broken bonds. Tiny patches of this network have a local density and local entropy lower than the global density and global entropy of the network. These patches — described by correlated-site percolation theory — are all possible sizes and are characterized by highly ramified (“tree-like”) shapes, just as in random-site percolation. In particular, this model explains the paradoxical facts that at sufficiently low temperature, the isothermal compressibility KT ∝ <(δV)2TPN and the constant-pressure specific heat CP ∝ <(δS)2TPN increase as T decreases, while the thermal expansivity αP ∝ <δVδSTPN is negative. Finally, we discuss some ongoing calculations and experiments designed to provide additional tests of the overall picture.

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

Abstract: On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.

Analogs of renormalization group transformations in random processes

Abstract: We review the properties of a real-space renormalization group transformation of the free energy, including the existence of oscillatory terms multiplying the non-analytic part of the free energy. We then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation. (The essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary.) These random processes are shown to be generated directly from Cantor sets. In a spatial representation, the ensuing random process exhibits a transition between Gaussian and fractal behavior. In the fractal regime, the trajectories will, in an average sense, form self-similar clusters. In a temporal representation, the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior. The fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses.

Sensitivity of nonequilibrium systems

Abstract: Simple mechanisms through which nonequilibrium structures can be influenced by external fields are discussed. It is shown that a very weak gravitational or electric field can have a large influence on selection or creation of structures. In the absence of cooperativity, the influence of a weak field, to the leading order, is characterized by the ratio (Eint/kT), where Eint is the energy of interaction; however, when there is far-from-equilibrium cooperativity, it is shown that the influence of the field is characterized by (EintkT)13.

Rhythms and turbulence in populations of chemical oscillators

Abstract: Some phenomena of a statistical kind arise from populations of coupled nonlinear oscillators. In particular, we focus our attention on (A) turbulence in oscillating chemical reactions obeying deterministic kinetics, and (B) emergence and subsidence of collective rhythmicity in systems of coupled oscillators under stochastic driving forces; this phenomenon may have some relevance to physiological clocks in living organisms. If we assume weak interactions between the oscillators, and also weak stochastic forces in case (B), a simple perturbation method is applicable, which enables us to cast both problems into a common framework. Some numerical as well as analytical results will also be presented.

The hard sphere gas in the Boltzmann-Grad limit

Abstract: This talk will review what has been rigorously proved about the time-dependent behaviour of a gas of classical hard spheres in the limiting regime where the number n of particles per unit volume becomes infinitely large while the particle diameter ϵ goes to zero in such a way that nϵ2 approaches a finite non-zero limit. (It is in this limiting regime that the Boltzmann equation is expected to become exact.) The review will emphasize general conceptual issues including: (1) How one constructs, in a mathematically precise way, a reduced macroscopic description appropriate to this limiting regime. (2) How probabilistic considerations enter the analysis. (3) At what point irreversibility appears.

Quantum kinetic equations in systems with bound states

Abstract: Starting from the quantum mechanical BBGKY-hierarchy kinetic equations in systems with two particles bound states are given in this paper. With this equation it is possible to consider transport properties in nonideal gases with three particles reactions.

On vectorial internal variables and dielectric and magnetic relaxation phenomena

Abstract: In a previous paper it has been shown by the author that a vectorial internal variable may give rise to dielectric relaxation phenomena and that if such a variable occurs the polarization P may be written in the form P = P (0) + P (1) , where changes in P (0) are reversible processes and changes in P (1) are irreversible. In this paper we introduce a somewhat more general assumption concerning the entropy. This generalization leads to the possibility that both changes in P (0) and in P (1) are irreversible phenomena. In this way a formalism is obtained with two relaxation times for dielectric relaxation. In particular we investigate the linearized form of the theory. It is seen that in the linear case the relation between the electric field E and the polarization P has the form of a linear relation among E , P , the first derivatives with respect to time of E and P , and the second derivative with respect to time of P . Debye's equation for dielectric relaxation in polar liquids and the equation derived by De Groot and Mazur are special cases of the equation which has been obtained in this paper. Analogous results can be derived for magnetic relaxation phenomena. Snoek's equation and the equation obtained by De Groot and Mazur are special cases of the equation for magnetic relaxation which is derived in this paper.

Corner transfer matrices

Abstract: For two-dimensional lattice models with interactions only between nearest (and diagonally nearest) neighbour spins, a well-known concept is the row-to-row transfer matrix. Less well-known is the “corner” transfer matrix (CTM). This has some very useful properties. If it is normalized so that its largest eigenvalue is unity, and the eigenvalues are arranged in numerically decreasing order, then each eigenvalue tends to a limit as the lattice becomes large. For those models which have been solved exactly (notably the Ising, eight-vertex and hard hexagon models), this limiting eigenvalue distribution is very simple, being basically that of a direct product of two-by-two matrices. From it the order parameter can easily be obtained. For all models one can write down formally exact matrix relations for the CTM, but the matrices are of infinite size. If one uses a representation in which the CTM is diagonal, and then truncates these relations to finite size, then one obtains a quite accurate approximation. The larger the size the greater the accuracy. I.G. Enting and I have thereby obtained comparatively long series expansions for the Ising model in a field, and for the hard squares model.

Two-dimensional melting

Abstract: Recent theoretical work1) on the Kosterlitz-Thouless2) model of dislocation-mediated melting in two dimensions suggests the existence of a new “hexatic” phase of matter, intermediate between a solid and a liquid. Basic theoretical ideas are reviewed, together with experiments and computer simulations capable of testing these results. We also discuss the dynamics of two-dimensional melting3), phase transitions in smectic liquid crystal films4), anisotropic melting5), and melting of layered materials like smectics, cholesterics, and the Rayleigh-Benard convection cells6). Reviews of the material covered in this talk may be found in refs. 7 and 8 below.

Thermodynamics of relativistic systems

Abstract: This paper will review recent progress in the covariant formulation of thermodynamics and statistical mechanics, with emphasis on two topics: (i) a consistent treatment of transient effects which avoids the paradox of an infinite speed of heat, and (ii) the thermodynamics of black holes.

Static dielectric properties of polarizable Stockmayer fluids

Abstract: The effect on the thermodynamics and distribution functions in including point polarizability in the Stockmayer fluid is evaluated. A large effect on the dielectric constant is found. As in the non-polarizable case, the single super chain integral equation gives dielectric constants in excess of the computed values except at low dielectric constants.

On the evolution of higher dimensional Heisenberg continuum spin systems

Abstract: We consider the evolution of a classical Heisenberg ferromagnetic spin chain in its continuum limit in higher spatial dimensions. It is shown that the evolution of a radially symmetric chain could be identified with the motion of a helical space curve as in the linear case. The resulting invariant equations for the curvature (radial energy density) and torsion (related to current density) are shown to be equivalent to a generalized nonlinear Schrodinger equation, similar to the one derived by Ruijgrok and Jurkiewicz recently. Equivalent linear equations as well as special static solutions of point singular type are obtained. Similarity solutions, a class of which belong to Riccati type, are discussed in detail. For general higher dimensions, a potentially useful formulation is presented: Under stereographic projection of the unit sphere of spin, the equation of motion takes a neater form even with the inclusion of anisotropic interactions. Classes of explicit solutions are reported in higher dimensions. Propagating spin waves, static spin waves of point singular nature and of finite energy in some cases are also discussed.

Computer simulation of ionic systems. Influence of boundary conditions

Abstract: The results of molecular dynamics calculations for systems of charged particles under periodic boundary conditions are reported for the case in which the periodic array of particles makes a macroscopically large sphere surrounded by a continuum of dielectric constant ϵ′. It is shown that thermodynamic and most dynamic properties are independent of the nature of the surrounding medium. The conductivity σ(ω) of the system depends strongly on the dielectric properties of the surrounding medium.

Brownian motion of harmonic systems with fluctuating parameters: III Scaling and moment instabilities

Abstract: We examine the stability properties of equilibrium moments of all orders for the damped mechanical oscillator with a delta correlated fluctuating frequency. A Markovian master equation is derived startimg from a frequency fluctuation process with finite correlation time τc and the limit τc→0 is taken. To approach this limit systematically, the oscillator and frequency fluctuation parameters are expressed in terms of a dimensionless scaling parameter. We derive exact integer moment transport equations in the limit of vanishing correlation time. These equations, and hence the moments, depend only on the second cumulant of the frequency fluctuations and not on the higher cumulants. The conjecture of Bourret et al.1) that for given frequency fluctuations, however weak, all moments beyond a certain order diverge is proved. We therefore conclude that the equilibrium distribution of the oscillator displacement and momentum cannot be Gaussian. A simple algebraic relation is established between the order of the lowest unstable moments and the system parameters.

Statistical mechanics of two-dimensional coulomb systems: I. Lattice sums and simulation methodology

Abstract: The statistical mechanics of systems acting via two-dimensional charge-charge and dipole-dipole interactions is studied in periodic boundary conditions. The two-dimensional lattice theta-function transformation is obtained and the sum shown to contain a polarization correction to the Ewald sum, analogous to that found in three dimensions. Using a different approach, these sums are evaluated in closed form, for bodies of arbitrary shape. The methodology for the computer simulation of two dimensional dielectrics is discussed and the two-dimensional analogue of a generalized Kirkwood-Clausius-Mosotti-Debye formula derived.

Thermal diffusion factors at 300 K for seven binary noble gas systems containing helium or neon

Abstract: Thermal diffusion factors, α T , have been measured as a function of concentration at 300 K for seven binary noble gas systems containing helium or neon. The results, obtained with a two-bulb cell, are in agreement with those of Saviron et al. who used a thermal diffusion column, and in general greater than those of Taylor et al. who used a 20-tube trennschaukel. The experimental results agree well with values predicted by the Chapmen-Cowling theory and the recent potential functions reported in the literature.

The thermal conductivity of mixtures of nitrogen with four noble gases at room temperature

Abstract: The paper presents new, absolute measurements of the thermal conductivity of binary mixtures of nitrogen with four monatomic gases, He, Ne, Ar and Kr. The measurements have been performed at 27.5°C as a function of density within the pressure range 0.9–17 MPa. The experimental data have an estimated accuracy of ±0.3%. The experimental results are interpreted with the aid of the kinetic theory expressions of Monchick, Pereira and Mason for the thermal conductivity of polyatomic gas mixtures. These first-order formulae prove to be adequate to describe the experimental data within their uncertainty provided that empirical adjustments are made to the rotational relaxation collision numbers and collision integral ratios occuring in them. It is suggested that more accurate kinetic theory formulae would allow such quantities to be derived with greater precision and physical signifincance.

Critical point scaling in the Percus-Yevick equation

Abstract: It is pointed out that the solution of the Percus-Yevick equation for sticky hard spheres, obtained by Baxter, scales exactly in the critical region with classical, Van der Waals exponents. However, even though the correlation functions achieve the Ornstein-Zernike forms, the scaling functions for the equation of state are not classical: in particular, the usual asymptotic gas-liquid symmetry is strongly violated; there is a spinodal curve only for the liquid phase; the specific heat, C v , on the critical isochore is logarithmically divergent, and, quite unphysically, C v is also logarithmically divergent on the critical isotherm above the critical density. Finally, comparison with numerical calculations for Lennard-Jones interactions suggests that the scaling functions are, in general, nonuniversal for the PY equation.

General properties of gausson-conserving descriptions of quantal damped motion

Abstract: We investigate the time-evolution of correlated critical states (gaussons) for a time-dependent harmonic oscillator coupled to a loss mechanism. A general dissipativity condition can be formulated and a derivation of the structure of gausson-conserving, dissipative as well as nondissipative, Hamiltonians, is presented. The already known frictional Hamiltonians are found to be particular cases of this description of damped motion.

Electric and magnetic susceptibilities for a fluid-fluid interface; The ellipsometric coefficient

Abstract: Formulae for the electro-magnetic constitutive coefficients are derived in terms of the interfacial position autocorrelation function. Explicit expressions for these coefficients are then found in terms of e.g. the surface tension and the capillary length. The ellipsometric coefficient is expressed in these constitutive coefficients and a comparison with recent experiments by Beaglehole is made.

Self-diffusion in gaseous and liquid ethylene

Abstract: Proton spin echoes at 24 MHz have been used in an accurate study of the self-diffusion coefficient in gaseous and liquid ethylene over a large range of density and temperature. The high temperature data are well described by the corrected Enskog theory. As the temperature is lowered, increasing deviations from the hard sphere behaviour are observed, which are in qualitative agreement with computer simulations on square-well and Lennard-Jones particles.

Statistical theory of heterophase fluctuations

Abstract: We present the physical basis and mathematical foundations for constructing a microscopic statistical theory of systems with heterophase fluctuations. Averaging over these fluctuations allows us to find an effective Hamiltonian, corresponding to a quasi-equilibrium situation. Phase probabilities are defined and new classifications of phase transitions are suggested.

Light scattering from a fluid with a stationary temperature gradient

Abstract: The spectrum of light, scattered from a fluid with a stationary temperature gradient, is calculated on the basis of fluctuating hydrodynamics. Explicit expressions are obtained for the spectrum of the scattered light which is no longer symmetric around the frequency of the incident light. In particular the difference in height and intensity of the Brillouin lines is given. Furthermore the shift in the position of the maximum of the Rayleigh line is calculated.

Mathematical properties of renormalization-group transformations

Abstract: Renormalization-group transformations of the type introduced into statistical physics by K. G. Wilson have been widely used with great success for a variety of many-body calculations. The basic approach consists in integrating out certain degrees of freedom and using a new “image” Hamiltonian to describe the statistical properties of those that remain. The mathematical foundations of the procedure, however, remain somewhat obscure. P.A. Pearce and the author have pointed out that there are serious questions about the existence and properties of a thermodynamic limit for renormalization-group transformations. In particular, certain real-space transformations show peculiarities in this limit. These suggest that there may be serious mathematical problems with renormalization-group procedures, despite their considerable practical sucess.