# Showing papers in "Physica A-statistical Mechanics and Its Applications in 1989"

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TL;DR: In this article, the authors consider dynamic light scattering by non-ergodic media, such as glasses or gels, in which the scattering elements are able only to make limited Brownian excursions about fixed average positions.

Abstract: We consider dynamic light scattering (DLS) by non-ergodic media, such as glasses or gels, in which the scattering elements are able only to make limited Brownian excursions about fixed average positions. We point out that, for such media, the time-averaged correlation function of the intensity of scattered light, the quantity obtained from a single DLS measurement, is different from the ensemble-averaged function. An expression for this time-averaged intensity correlation function is derived and its properties and experimental analysis are discussed. Some of the literature on DLS by polymer gels is re-evaluated in the light of these new theoretical predictions.

510 citations

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TL;DR: In this paper, the authors show how statistical mechanics applies to powders, starting from the observations that powders have a large number of particles, and reproducible properties, and show that the volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable.

Abstract: Starting from the observations that powders have a large number of particles, and reproducible properties, we show how statistical mechanics applies to powders. The volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable. We introduce a variable X-the compactivity - which is analogous to the temperature in thermodynamics. Some simple models are considered which demonstrate how the problems involved can be tackled using the concept of compactivity. There is an increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic, nor in equilibrium, but which still fulfil a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way. Powders fall into this category. If a powder consists for example of uniform cubes of salt, and is poured into a container, falling at low density uniformly from a great height, one expects a salt powder of a certain density. Repeating the preparation reproduces the same density. A treatment such as shaking the powder by a definite routine produces a new density and the identical routine applied to another sample of the initial powder will result in the same final density. Clearly a Maxwell demon could arrange the little cubes of NaCl to make a material of different properties to that of our experiment, but if such demonics are ignored, and we restrict ourselves to extensive operations such as stirring, shaking, compressing - all actions which do not act on grains individually - then well defined states of the powder result. In this paper we will set up a framework for describing the state of the powder, basing our development on anologies with statistical mechanics. Some attempts have been made to apply information theory ideas directly to powders

510 citations

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TL;DR: In this paper, the authors studied universal amplitude ratios for three-dimensional nearest-neighbor Ising models and found that C + C - = 4.95±0.15, ƒ 1 + Ɣ 1 - = 1.960±0.01, A + A - = 0.523±0., αA + C + B + = 0,0581± 0.0010, while αA+ (ƒ1+)3 verifies hyperuniversality to within ± 0.8%.

Abstract: Several basic universal amplitude ratios are studied afresh for three-dimensional nearest-neighbor Ising models. In revising earlier work, modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes: C± for the susceptibility χ; ƒ1± for the correlation length ζ1; A± for the specific heat C(T); and B for the spontaneous magnetization M0. We find C + C - = 4.95±0.15 , ƒ 1 + ƒ 1 - = 1.960±0.01 , A + A - = 0.523±0.009 , αA + C + B + = 0.0581±0.0010 , while αA+ (ƒ1+)3 verifies hyperuniversality to within ±0.8%. A method for calculating amplitude ratios which allows for corrections to scaling yields estimates for C + C - and ƒ 1 + ƒ 1 - in excellent agreement with those derived from the individual amplitudes. Finally, explicit formulae are given for the numerical evaluation of χ(T), ζ1(T), C(T) and M0(T) over the full temperature range from criticality to T=0 and ∞; corresponding plots and convenient near-critical representations are also presented.

260 citations

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General Electric

^{1}TL;DR: The graphite/diamond/vapor/liquid triple point (or points) for graphite (or carbynes)/vapor-liquid remain controversial as mentioned in this paper, and the latest static and shock compression experiments on diamond indicate that it melts to a conducting liquid at about 5000 K at pressures of 15 to 30 GPa, but does not melt at about 6000 K at 125 GPa.

Abstract: Carbon atoms form very strong bonds to each other yielding solid crystalline materials like graphite and diamond. Because of the high bonding energies, the vaporization and melting temperatures are very high. Different kinds of atom-to-atom bonding make many solid forms possible, ranging from pure graphite to pure diamond, as well as many types of molecules in liquid or gaseous carbon. Rigorous conditions of high temperature, high pressure, or both, are required to change a given elemental phase of carbon to another. Currently the vapor-pressure line of graphite, the P, T equilibrium line between graphite and diamond, and the graphite/diamond/liquid triple point are fairly well established. The triple point (or points) for graphite (or carbynes)/vapor/liquid remain controversial. At pressures less than 0.1 GPa liquid carbon seems to be a poor electric conductor while at higher pressures it is a good one. Current experimental and theoretical evidence indicate that diamond is stable against collapse to metallic forms (unlike Si and Ge) up to pressures over 350 GPa, and possibly as high as 2300 GPa. The latest static and shock compression experiments on diamond indicate that it melts to a conducting liquid at about 5000 K at pressures of 15 to 30 GPa, but does not melt at about 6000 K at 125 GPa. This suggests that the melting temperature of diamond increases with pressure, and that at the melting temperature liquid carbon is slightly less dense than diamond.

209 citations

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TL;DR: In this paper, a new formulation of the statistical mechanics of powders is used to develop a theory for a mixture of grains of two different sizes. But the analysis is restricted to the spin formulation of an eight-vertex model.

Abstract: In this paper we use a new formulation of the statistical mechanics of powders to develop a theory for a mixture of grains of two different sizes. We map this problem onto the spin formulation of the eight-vertex model and reproduce the features of the phase separation diagram of the powder mixture that we would intuitively be led to expect. Finally, we discuss the insight afforded by this solution on the “thermodynamic” quantities of interest in the powder mixture.

182 citations

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TL;DR: In this article, the electrostatic contribution to the bending moduli of an amphiphilic monolayer is calculated on the basis of the Poisson Boltzmann equation, and the electrical free energy for a spherical and cylindrical surface is expanded in inverse powers of the radius of curvature a.

Abstract: The electrostatic contribution to the bending moduli of an amphiphilic monolayer is calculated on the basis of the Poisson-Boltzmann equation. The electrical free energy for a spherical and cylindrical surface is expanded in inverse powers of the radius of curvature a. The coefficient (1/a)2 in the electrical free energy gives the bending elastic moduli.

179 citations

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TL;DR: In this article, two independent approaches, the box counting and the sand box methods are used for the determination of the generalized dimensions associated with the geometrical structure of growing deterministic fractals.

Abstract: Two independent approaches, the box counting and the sand box methods are used for the determination of the generalized dimensions (Dq) associated with the geometrical structure of growing deterministic fractals. We find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated Dq's to their true values. Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres. In this case the latter approach provides better estimates of the generalized dimensions.

155 citations

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TL;DR: In this article, the electrical properties of polymers filled with conducting particles of various sizes and shapes are reviewed and compared with the predictions of percolation theory in the neighborhood of the familiar insulator-to-conductor transition.

Abstract: We review the electrical properties of polymers filled with conducting particles of various sizes and shapes. dc and ac experimental behaviors in the neighborhood of the familiar insulator to conductor transition are generally in fair agreement with the predictions of percolation theory. Observed discrepancies with the universal values of the critical exponents have been attributed to effects of percolation over a continuum and/or interparticle tunneling of electrons, although the latter is not the only possible conduction mechanism. The location of the transition appears to involve, aside from geometrical effects, the physico-chemistry of the colloid which is formed during material processing.

123 citations

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TL;DR: The tri-hamiltonian nature of Lax-equations is revealed in this article, starting with an R-matrix on an associative algebra g equipped with a trace form, there are g compatible Poisson brackets with linear, quadratic and cubic dependence on the coordinates.

Abstract: The tri-hamiltonian nature of Lax-equations is revealed: starting with an R-matrix on an associative algebra g equipped with a trace form there are g compatible Poisson brackets with linear, quadratic and cubic dependence on the coordinates. The invariant functions (Casimir functions) on g* are in involution relative to these brackets, they yield a hierarchy of integrable tri-hamiltonian Lax-equations. The results can be applied to solvable PDE’s such as the Korteweg-de Vries equation as well as to finite integrable systems such as the Toda lattice. In these cases the Poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well-known recursion operators.

108 citations

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Fermilab

^{1}TL;DR: In this paper, the spectral representation of the renormalized gauge propagator is calculated perturbatively in the limit of high temperature, and the interaction of an abelian field with an external current is also determined.

Abstract: The spectral representation of the renormalized gauge propagator is calculated perturbatively in the limit of high temperature. The interaction of an abelian field with an external current is also determined.

104 citations

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TL;DR: The Yukawa potential has three important mathematical properties that are apparently unrelated but, in fact, closely linked as discussed by the authors, and these properties have led to the appearance of the potential in a great variety of different physical problems.

Abstract: The Yukawa potential, φ(r)=A(λr)-1eλr, has three important mathematical properties that are apparently unrelated but, in fact, closely linked These properties have led to the appearance of the potential in a great variety of different physical problems These ramifications are discussed in roughly chronological order The potential is generalized to spaces of dimensionality other than 3, and the properties of this generalized potential are explored

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TL;DR: In this article, the authors present an algorithm for computing the self-consistent local fields on each atom in a liquid-vapour simulation, which can be converted from slowly converging sums to rapidly convergent sums with only a few terms.

Abstract: Computer simulations of liquid-vapour interfaces produce configurations of some finite number of atoms or molecules within a central cell. Periodic boundary conditions repeat the central cell to infinity. To extract local fields and optical properties from these configurations it is necessary to sum over dipolar fields to infinity. For N polarizable atoms in the central simulation cell, 1 2 N(N-1) dipolar sums are required, and N simultaneous linear equations are to be solved to find the self-consistent local fields on each atom. A realistic simulation of a liquid-vapour interface requires a thousand particles or more. Hence an efficient algorithm for the dipolar sums is required. Formulae are given which convert these slowly converging sums to rapidly convergent sums requiring only a few terms. These formulae enable local fields to be computed, for each atom, as a byproduct of the computer simulation. Of particular interest are the fluctuations in the local fields.

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TL;DR: In this article, it was shown that there are weak singularities in the droplet distribution corresponding to the disappearance of the surface tension of exponentially rare droplets off the coexistence curve.

Abstract: It has been known for more than a hundred years that one can drive a system around the liquid-gas critical point without any singular changes in the thermodynamic quantities. We argue that there are weak singularities in the droplet distribution corresponding to the disappearance of the surface tension of exponentially rare droplets off the coexistence curve. In the Coniglio-Klein-Swendsen-Wang droplet description of the Ising model such singularities occur in a natural way.

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TL;DR: In this article, a manifestly covariant relativistic Boltzmann equation is derived in a framework in which the fundamental dynamical constituents of the system are a family of N events with motion in space-time parametrized by an invariant historical time.

Abstract: A quantum mechanical derivation of a manifestly covariant relativistic Boltzmann equation is given in a framework in which the fundamental dynamical constituents of the system are a family of N events with motion in space-time parametrized by an invariant “historical time” τ. The relativistic analog of the BBGKY hierarchy is generated. Approximating the effect of correlations of second and higher order by two event collision terms, one obtains a manifestly covariant Boltzmann equation. The Boltzmann equation is used to prove the H-theorem for evolution in τ. For ensembles containing only positive energy (or only negative energy) states, a precise H-theorem is also proved for increasing t. In the nonrelativistic limit, the usual H-theorem is recovered. It is shown that a covariant form of the Maxwell-Boltzmann distribution is obtained in the equilibrium limit; an examination of the energy momentum tensor for the free gas yields the ideal gas law in this limit. It is shown that in local equilibrium the internal energy is defined by a Lorentz transformation to a local rest frame. We study the conserved quantities in this theory. There is a new nontrivially conserved quantity corresponding to the particle mass. We obtain continuity equations for these quantities.

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TL;DR: In this article, the authors developed the formalism for a continuous-time generalization of the persistent random walk, by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject.

Abstract: We develop the formalism for a continuous-time generalization of the persistent random walk, by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject. This generalization leads to evolution equations, in the time domain, that differ and are of higher order than the telegrapher's equation which is found in the case of the Markovian persistent random walk.

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TL;DR: In this article, the exact equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques, which reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath.

Abstract: The equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques. The exact equation of motion reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath. The equation separates into streaming or systematic terms and dissipative terms which are separately equal to zero at equilibrium. We find both statistical and dynamical system frequency shifts; the statistical shifts are present in equilibrium but the dynamical shifts affect the time-dependence, only. The general results are applied to the two-level system model for tunneling in condensed phases.

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TL;DR: In this paper, a theory is presented which relates the colloidal interactions to the microstructure of a Brownian suspension under weak shear and then to the bulk stresses via a new technique for renormalizing the thermodynamic contribution.

Abstract: A theory is presented which relates the colloidal interactions to the microstructure of a Brownian suspension under weak shear and then to the bulk stresses via a new technique for renormalizing the thermodynamic contribution. Further derivations of the interparticle stress provide an independent test of the accuracy of requisite closures. The results are very sensitive to the coupling between equilibrium and nonequilibrium distribution functions in the three-body closures; a closure in the spirit of the Percus-Yevick equation provides the most consistent results while superposition predicts aphysical results. Comparison with the available measurements on hard-sphere systems indicates that the Brownian stresses, renormalized into a hydrodynamic function, are responsible for the divergence in the low shear limiting viscosity in dense suspensions. However, pairwise additive hydrodynamics adequately predict neither the high frequency limiting complex viscosity nor the steady shear viscosity in dense suspensions.

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TL;DR: In this paper, the second and third virial coefficients and densities of five mixtures of carbon dioxide + nitrogen (xCO2=0.10560, 0.25147, 0.,50365, 0,71105, 0.90921), pure nitrogen, pure carbon dioxide, four mixtures, carbon dioxide plus methane (xco2= 0.09990,0.29858, 0,67607, 0), pure methane, and five mixture, carbon gas, ethane (x CO2=1.10043, 0 0.

Abstract: Densities of five mixtures of carbon dioxide + nitrogen (xCO2=0.10560, 0.25147, 0.50365, 0.71105, 0.90921), pure nitrogen, four mixtures of carbon dioxide + methane (xco2=0.09990, 0.29858, 0.67607, 0.90112), pure methane, and five mixtures of carbon dioxide + ethane (xCO2=0.10043, 0.25166, 0.49245, 0.73978, 0.90367) were measured at 300 and 320 K using the Burnett technique. Second and third virial coefficients and densities were derived for each mixture from the measurements. Cross second and third virial coefficients were determined for each system.

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TL;DR: In this paper, an experimental investigation has been made of the self-diffusion coefficient in xenon, using the NMR spin-echo technique, and the experimental results are compared with the results of computer simulations on systems of hard spheres and on system of Lennard-Jones molecules.

Abstract: An experimental investigation has been made of the self-diffusion coefficient in xenon, using the NMR spin-echo technique. This experiment in xenon is the first systematic investigation of self-diffusion in a monatomic gas over a wide range of density and temperature. The data have been obtained using an NMR high-pressure probe in a superconducting magnet. The experimental results are compared with the results of computer simulations on systems of hard spheres and on systems of Lennard-Jones molecules.

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TL;DR: The scaling behavior of the number of clusters near the critical point is confirmed in this paper, where it is shown that infinite size appears whenever the system has a nonzero magnetization.

Abstract: We study the clusters generated in the Swendsen-Wang algorithm in a magnetic field. It is shown that the number of clusters is related to that of Coniglio and Klein by simple factors. With this definition of clusters, infinite size appears whenever the system has a nonzero magnetization. Scaling behavior of the number of clusters near the critical point is confirmed. The number of clusters away from the critical point for large cluster size s is consistent with ln n ≌ |h|s − Γ s 2 3 on the low temperature side of the Coniglio-Klein cluster percolation transition line, and is consistent with ln n≌−(|h| + c)s on the high temperature side. We also argue that this transition line is given by h = ± h (T)×(T-T c ) 1 near Tc.

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TL;DR: In this article, the theory of subdynamics is presented in a new self-contained way, starting from the commutation relation П(ν) LH=LH В(ν), where LH is the Liouvillian.

Abstract: Work on the application of Poincare's theorem to large classical or quantum systems with a continuous spectrum is continued. In situations where it is applicable, Poincare's theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant. In contrast, the theory of subdynamics as developed by the Brussels group permits the construction of a unique set of projectors П(ν)(for t > 0), giving up the requirement of hermiticity which is replaced by “star-hermiticity”.
The theory of subdynamics is presented in a new self-contained way, starting from the commutation relation П(ν) LH=LH П(ν), where LH is the Liouvillian. This presentation is far more direct, and avoids some of the lengthy discussions associated with previous presentations (based mainly on the resolvent of the Liouvillian).
Subdynamics appears to be of interest from many points of view. It generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed Hamiltonian. In contrast, degrees of freedom are lost when going to the spectral representation (e.g. in the Friedrichs model). Subdynamics permits us to solve the initial value problem associated with the Liouville equation retaining the “non-Markovian” contributions which appear in the standard presentation. Finally, it introduces a classification of large dynamical systems, classical or quantum, into integrable and nonintegrable ones. It is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems, such as the interaction of matter with light. The applications of this technique to these problems will be worked out in subsequent papers.

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TL;DR: In this paper, a liquid-vapor interface deforms to a cylindrical surface which makes a definite contact angle with the walls, showing the presence of both a wetting and a drying transition.

Abstract: We have simulated a liquid-vapor interface which is confined by two parallel walls. Initially, a liquid slab is positioned perpendicular to the walls. The liquid-vapor interface deforms to a cylindrical surface which makes a definite contact angle with the walls. The angle could be varied between 0 and π by changing the solid-fluid interaction strength, showing the presence of both a wetting and a drying transition. The angle is related to the strengths of the wall-liquid, wall-vapor and liquid-vapor surface tensions via Young's law. A previous simulation positioned the liquid parallel to the walls and measured the surface tensions. Compared with the contact angles deduced from those measurements, the visually measured angles in this experiment give a different location of the drying transition.

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TL;DR: In this paper, a theoretical analysis of the Kerr effect of an oil continuous water/AOT/isooctane microemulsion in the limit of a low volume fraction of water is given.

Abstract: A theoretical analysis of the Kerr effect of an oil continuous water/AOT/isooctane microemulsion in the limit of a low volume fraction of water is given. The microemulsion consists of small nanometer sized waterdroplets, covered by a monomolecular layer of AOT. In the absence of a constant electric field these droplets are spherical. Because of the field their shape changes and becomes, if the field is not too strong, spheroidal. The eccentricity of the spheroids is found by minimizing the total energy of deformation which is a sum of an electromagnetic contribution and a contribution due to the curvature-dependent energy of the droplet surface. The Kerr constant per droplet is then found by deriving explicit expressions for the optical and dielectric polarizabilities along and orthogonal to the symmetry axis of the spheroid. We find that the negative birefringence for small radii of the droplets is a consequence of the optical anisotropy of the AOT molecules, which are all “lined up” orthogonal to the water surface. To test the theory, we repeated and extended measurements of Hilfiker et al. [1] on changes in the effect as a function of the radius of the droplets. We reproduce their results and are able to explain all data on the basis of the theoretical expressions given. Our results confirm the observation made by Hilfiker et al. that the negative birefringence is due to the AOT layer.

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TL;DR: In this paper, the theoretical and physical meaning of dissipation of background fields due to particle creation and statistical effects in interacting quantum field theories and in semiclassical gravitational theories is discussed.

Abstract: We discuss the theoretical and physical meaning of dissipation of background fields due to particle creation and statistical effects in interacting quantum field theories and in semiclassical gravitational theories. We indicate the possible existence of a fluctuation-dissipation relation for non-equilibrium quantum fields as occuring in cosmological particle creation and backreaction processes. We also conjecture that all effective theories, including quantum gravity, could manifest dissipative behavior.

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TL;DR: In this paper, the authors used the Burnett distribution and the Chapman-Enskog distribution to obtain asymptotic results for mass and heat fluxes corresponding to planar thermal transpiration and mechanocaloric effects.

Abstract: In Poiseuille flow between two parallel plates, the bulk flow is characterized by the Burnett distribution. We report explicit results for this distribution by solving numerically the relevant integral equations for a rigid sphere gas in the context of the linearized Boltzmann equation. Then, we use this distribution together with the Chapman-Enskog distribution to obtain asymptotic results (near-continuum) for mass and heat fluxes corresponding to planar thermal transpiration and mechanocaloric effects.

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TL;DR: In this article, a macroscopic static theory of gels upon swelling or volume phase transition was developed, and the stability criteria for uniaxially strained bulk gels were derived for spherically symmetric geometry with the outer boundary of the gel having a finite radius.

Abstract: We develop a macroscopic static theory of gels upon swelling or volume phase transition. After deriving the linear elasticity theory of strained systems from the general nonlinear formalism of deformation, we show that the surface modulational instability of a gel plate occurs as the result of softening of (generalized) Rayleigh surface waves. We also derive the stability criteria for uniaxially strained bulk gels. In addition to these linear analyses we developed, for the first time to our knowledge, a theory of three-dimensional phase coexistence of gels exhibiting a volume phase transition. Our study is limited mostly to the case of spherically symmetric geometry with the outer boundary of the gel having a finite radius. Various features of the two phase coexistence are found, some of which have no counterparts in the usual phase separation in binary fluids or gas-liquid systems.

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Fermilab

^{1}TL;DR: In this article, a general and consistent procedure for computing scattering amplitudes in hot gauge theories is presented, which is based on the same approach as the one described in this paper.

Abstract: A general and consistent procedure for computing scattering amplitudes in hot gauge theories is outlined.

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TL;DR: In this article, the authors consider the motion of a Brownian particle in a medium with inhomogeneous temperature in the presence of an external potential and derive the correct form for the Smoluchowski equation, which reduces to van Kampen's result in the absence of thermophoretic forces.

Abstract: We consider the motion of a Brownian particle in a medium with inhomogeneous temperature in the presence of an external potential. We start from the Klein-Kramers equation; in this equation a thermophoretic force, proportional to the temperature gradient, should in general be included to obtain a correct description of thermodiffusion effects in the hydrodynamic stage of the evolution. With the Chapman-Enskog method we derive the correct form for the Smoluchowski equation, which reduces to van Kampen's recent result in the absence of thermophoretic forces. We also give the first correction to this equation caused by deviations from local thermal equilibrium. For the system considered, such deviations persist even in the steady state.

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TL;DR: In this paper, the relaxation properties of a spin system weakly coupled to lattice degrees of freedom are described using an equation of motion for the spin density matrix, derived using a general weak coupling theory which has been previously developed.

Abstract: The relaxation properties of a spin system weakly coupled to lattice degrees of freedom are described using an equation of motion for the spin density matrix. This equation is derived using a general weak coupling theory which has been previously developed. To second order in the weak coupling parameter, the results are in agreement with those obtained by Bloch, Wangsness and Redfield but the derivation does not make use of second order perturbation theory for short times. The derivation can be extended beyond second order and ensures that the spin density matrix relaxes to its exact equilibrium form to the appropriate order in the weak coupling parameter.

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TL;DR: In this article, the spectrum of critical indices is found to be contained in the interval (α min, 1), where α min tends to 1 for increasing sets of numbers, and the scaling with respect to the sizes of subsets of natural numbers is also considered.

Abstract: The multifractal formalism is applied to prime numbers. The spectrum of critical indices is found to be contained in the interval ( α min , 1), where α min tends to 1 for increasing sets of numbers. Besides the scaling of moments with respect to the length of intervals the scaling with respect to the sizes of subsets of natural numbers is also considered. We have found the cusps in the plots of the functions ƒ(α) and we claim that they are not caused by numerical roundings but they are a real effect. Besides the computer method, some analytical calculations are presented.