Showing papers in "Physica A-statistical Mechanics and Its Applications in 1984"
TL;DR: A general formalism for the description of configurational cluster functions in multicomponent systems is developed and a new interpretation of the Cluster Variation Method as a self-consistency relation on the renormalized cluster energies is presented.
Abstract: A general formalism for the description of configurational cluster functions in multicomponent systems is developed. The approach is based on the description of configurational cluster functions in terms of an orthogonal basis in the multidimensional space of discrete spin variables. The formalism is used to characterize the reduced density matrices (or cluster probability densities) and the free energy functional obtained in the Cluster Variation Method approximation. For the particular representation chosen, the expectation values of the base functions are the commonly used multisite correlation functions. The latter form an independent set of variational parameters for the free energy which, in general, facilitates the minimization procedure. A new interpretation of the Cluster Variation Method as a self-consistency relation on the renormalized cluster energies is also presented.
1,341 citations
TL;DR: In this article, the wavevector dependent (short-time) diffusion coefficient for spherical particles in suspension is evaluated up to high concentrations and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres, as well as the resummed contributions from a special class of correlations.
Abstract: We evaluate the wavevector dependent (short-time) diffusion coefficient D(k) for spherical particles in suspension, by extending a previous study of selfdiffusion (which corresponds to the case of large k). Our analysis is valid up to high concentrations and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres, as well as the resummed contributions from a special class of correlations. Results obtained which agree well with available experimental data.
237 citations
TL;DR: In this article, a method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred is presented, which can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations.
Abstract: In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.
207 citations
TL;DR: In this article, a theory for the concentration and wave vector dependence of the effective viscosity of a suspension of spherical particles is given, up to high concentrations and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres.
Abstract: A theory is given for the concentration and wave vector dependence of the effective viscosity of a suspension of spherical particles. The analysis is valid up to high concentrations and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres. The relation to the diffusion coefficient of the spheres is discussed.
191 citations
TL;DR: In this article, the internal motion of a small element of fluid in inviscid and incompressible flows by neglecting the actions of the other elements which constitutes the whole fluid is studied.
Abstract: We look at the internal motion of a small element of fluid in inviscid and incompressible flows by neglecting the actions of the other elements which constitutes the whole fluid. This free motion of the elements leads, in a finite time, to the divergence of the velocity field in the element and to its flattening in a plane.
172 citations
TL;DR: In this article, the scaling exponent of the mobility and the correlation function of the chemical potential with an order paramater were examined for phase-separating unmixing mixtures.
Abstract: A dynamic scaling idea is examined for phase-separating unmixing mixtures. The scaling exponent of the mobility and that correlation function of the chemical potential with an order paramater determine the droplet growth rates. The results with those of existing phenomenological approaches. The scaling properties of the auto-correlation functions are discussed. If the correlation function associated with the chemical potential has a strong dependence on the length scale as shown in the text, then the auto-correlation function exhibits an anomalous scaling, under which the damping coefficient has a (negative) gap. Simplified scaling structure functions, both for critical and off-critical concentration mixtures, are given. They are compared with those of the computer simulation by Lebowitz et al.
161 citations
TL;DR: In this paper, a new statistical-mechanical theory of diffusion-controlled droplet coarsening is presented, where a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction φ to obtain kinetic equations for a single distribution function of droplets.
Abstract: A new statistical-mechanical theory of diffusion-controlled droplet coarsening is presented. With the aid of a scaling expansion method, a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction φ to obtain kinetic equations for a single distribution function of droplets. It is shown that there two characteristic stages of coarsening, depending on their space-time scales; an intermediate stage and a late stage. In both stages, new kinetic equations are systematically derived to order φ . These equations have two terms at order φ ; a collisionless drift term and a collision term. The collision term is shown to be different from the conventional encounter integral discussed by Lifshitz and Slyozov since the former is of order φ and describes distant (soft) collisions, while the latter is of order φ and describes close (hard) collisions. It is shown in both stages that the mean droplet radius increases as the cube root of the time (t 1 3 ). A scaling behavior of the distribution function is also found in both stages. In particular, in the late stage this scaling behavior is shown to coincide with that obtained by Lifshitz and Slysov in the limit φ→O. It is also pointed out that a naive expansion in powers of φ breaks down due to the long -range nature of interactions among droplets through diffusions. Fluctuations around the kinetic equations are also explicitly explored. They are shown to be nonthermal fluctuations generated by the soft collision process and to be small. Especially, in the late stage they are shown to obey a linear Gaussian Markov process, satisfying a fluctuation-dissipation relation of the second kind.
157 citations
TL;DR: In this article, the dynamics of inviscid and viscous Taylor-Green (TG) vortex flows are investigated by both direct spectral numerical solution of the Navier-Stokes equations (with up to 2563 modes) and by power series analysis in time.
Abstract: The dynamics of both the inviscid and viscous Taylor-Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier-Stokes equations (with up to 2563 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance cf(ct) of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that b(t) decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity (6 = 0) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag Rr. Frisch (1980) analysis from order t4* to order Po. Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t,,, at which the maximum energy dissipation is achieved. Early-time, high-R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t,,, the small scales of the flow are nearly isotropic provided that R 1000. Various features characteristic of fully developed turbulence are observed near t,,, when R = 3000 and R, = 110: (i) a k-n inertial range in the energy spectrum is obtained with n z 1.G2.2 (in contrast with a much steeper spectrum at earlier times) ;
155 citations
TL;DR: In this paper, it was shown that systems that break chiral symmetry are so sensitive to small chiral interaction that an interaction energy Eint > 10−15 -10−17kT is sufficient to have strong chiral selectivity.
Abstract: Nonequilibrium symmetry breaking and sensitivity are discussed in the context of the chemistry of chiral molecules. It is shown that systems that break chiral symmetry are so sensitive to small chiral interaction that an interaction energy Eint > 10−15 -10−17kT is sufficient to have strong chiral selectivity. Such energies are within the range of parity violating weak-neutral-current interactions. Chiral interaction energies due to a combination of electric, magnetic, gravitational and centrifugal fields, even when high field strengths are considered, are found to be less than 10−19kT. Plausible chemical systems that involve rhodium catalysts in which weak-neutral-current energies should be high are also briefly discussed.
119 citations
TL;DR: In this paper, the authors apply singularity analysis to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model.
Abstract: The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painleve property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painleve transcendents. Relaxing the Painleve property we find many partially integrable cases whose movable singularities are poles at leading order, with In( t - t 0 ) terms entering at higher orders. In an N th order, generalized Rossler model a precise relation is established between the partial fulfillment of the Painleve conditions and the existence of N - 2 integrals of the motion.
100 citations
TL;DR: In this article, the authors reported results for quantized Hall conductance in a two dimensional periodic potential, obtained in collaboration with David Thouless, Mahito Kohmoto, and Peter Nightingale.
Abstract: The Hall conductance αH of a two dimensional electron gas has been studied in a uniform magnetic field and a periodic potential. The periodic potential splits each Landau level in a nested devil's staircase like subband structure. The Kubo formula is written in the form of a topological invariant which makes apparent the quantization of σH in multiples of e2/h when the Fermi energy lies in a subgap. Explicit expressions for σH in the subgaps have been obtained. For increasing resolution of the nested subband structure σH makes increasingly larger jumps. Also a less rigorous but intuitively appealing explanation of this result is presented. Moreover it is shown how the devil's staircase structure describes phase diagrams of incommensurate monolayers adsorbed on surfaces with two competing periods. During this talk I will report results for the quantized Hall conductance in a two dimensional periodic potential, obtained in collaboration with David Thouless, Mahito Kohmoto, and Peter Nightingale. 1
TL;DR: In this paper, a simple derivation of the Atiyah-Singer index theorem for classical complexes and its G-index generalization using elementary properties of quantum mechanical supersymmetric systems is presented.
Abstract: Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.
TL;DR: In this paper, the authors describe a service they can provide: the proof of strict bounds on the results of (possibly very complicated) computations on real numbers, and illustrate, in a concrete example, how the availability of this kind of help in proving numerical bounds opens up a new way of approaching some qualitative questions which have proved hard to treat by more standard methods.
Abstract: Computers are useful in many ways in mathematical research. They make it possible, for example, to perform experiments on a wide variety of mathematical objects and to carry out or check complicated algebraic manipulations. In this talk I will describe another service they can provide: The proof of strict bounds on the results of (possibly very complicated) computations on real numbers. I will try to illustrate, in a concrete example, how the availability of this kind of help in proving numerical bounds opens up a new way of approaching some qualitative questions which have proved hard to treat by more standard methods.
TL;DR: The concept of distance between states in equilibrium thermodynamics introduced by Weinhold and Ruppeiner was based on the stability conditions and theory of fluctuations, respectively as discussed by the authors, and it has been shown that both these Riemannian matrices are equivalent up to a multiplicative function.
Abstract: The concepts of distance between states in equilibrium thermodynamics introduced by Weinhold and Ruppeiner were based on the stability conditions and theory of fluctuations, respectively. It has been shown that both these Riemannian matrices are equivalent up to a multiplicative function. They are connected by a contact transformation acting on the thermodynamic phase space. The family of contact transformation produces a set of new metrics from some reference metric. Such transformations can be also considered as a practical method for evaluating partial derivatives of thermodynamic functions.
TL;DR: In this article, it was shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions.
Abstract: It is shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard-core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions. The nonlinear integral equation is reduced to an algebraic equation of order d−3 for odd dimensions d greater than three. As an example the direct correlation function in five dimensions is derived explicitly.
TL;DR: In this article, two sets of nonlinear differential equations are derived and discussed for the time-dependent correlations between x-components of spins (S = 1 2 ) in an Ising chain in the presence of a transverse magnetic field.
Abstract: Two sets of nonlinear differential equations are derived and discussed for the time-dependent correlations between x-components of spins (S = 1 2 ) in an Ising chain in the presence of a transverse magnetic field. The equations are independent of temperature which enters only through the initial conditions for the correlations. The equations are valid for the (general) inhomogeneous case in which the exchange coupling as well as the magnetic field depend on the sites in the chain. In the derivation use is made of a general formulation of the thermodynamic Wick theorem. For the homogeneous case a nonlinear differential-difference equation is derived, generalizing the Painleve III equation found previously at zero temperature in the scaling limit. The finite-temperature field theory limit is discussed also.
TL;DR: In this paper, the authors present results related to an alternative mapping formulation (to the well-known Wigner-Weyl transform) which makes use of the Gaussian Wave Packet or coherent states representation | pq ↩.
Abstract: We present here some results related to an alternative mapping formulation (to the well-known Wigner-Weyl transform) which makes use of the Gaussian Wave Packet or coherent states representation | pq ↩. The pair p-q which labels this state defines a phase-space in which the abstract operators P, Q , of momentum and position, are represented as differential operators. In the mapped expression of an operator A ( P, Q ), the quantum effects appear when they are absent in the corresponding Wigner-Weyl transform.
TL;DR: In this paper, an equation of state is derived for degenerate plasmas, which connects the density with the chemical potentials of the species, and the formation of bound states is accounted for by a screened ladder approximation.
Abstract: In the framework of the Green's function technique an equation of state is derived for degenerate plasmas, which connects the density with the chemical potentials of the species. The formation of bound states is accounted for by a screened ladder approximation. A mass action law is given which includes several improvements as compared with the Saha equation. The influence of the latter on the ionization equilibrium is discussed.
TL;DR: In this paper, an extended Zubarev approach was used to express electrical conductivity, thermopower and thermal conductivity for a partially ionized plasma by equilibrium correlation functions.
Abstract: Electrical conductivity, thermopower and thermal conductivity for a partially ionized plasma are expressed within an extended Zubarev approach by equilibrium correlation functions. The Green function technique is used to evaluate the correlation functions in different approximations. Improvements of the Lenard-Balescu approximation are considered, which account for dynamical screening effects and higher Born approximations for the electron-electron, electron-ion and electron-atom interaction.
TL;DR: In this paper, an approximate renormalized equation of evolution for an arbitrary nonlinear single-degree-of-freedom system externally driven by Gaussian parametric fluctuations of finite correlation time was determined.
Abstract: We determine an approximate renormalized equation of evolution for an arbitrary nonlinear single-degree-of-freedom system externally driven by Gaussian parametric fluctuations of finite correlation time . The renormalization scheme used here gives a second order equation with a time-and-state-dependent “diffusion coefficient”. We are able to calculate the diffusion coefficient in closed form. The steady-state distribution can easily be obtained from the evolution equation. We are thus able to determine the parameter dependence of the steady-state distribution and, in particular, the influence of a correlation time of the fluctuations, which does not vanish, on the steady-state distribution.
TL;DR: In this article, the liquid-vapour coexistence densities and the vapour pressure are obtained for several potentials related to the Lennard-Jones (LJ) 12-6 potential; the shifted potential, the shifted force, the LJ m − n and the Stockmayer potentials.
Abstract: The liquid-vapour coexistence densities and the vapour pressure are obtained for several potentials related to the Lennard-Jones (LJ) 12-6 potential; the shifted potential, the shifted force, the LJ m − n and the Stockmayer potentials These are compared with the values obtained by other means such as by direct simulation of the liquid-vapour surface and by perturbation theory It would seem that the correction of truncated potentials to full potentials for inhomogenous systems is hazardous This technique and these results are expected to be generally useful in the analysis of the properties of liquids It is suggested that the LJ 12-6 sf3 potential should be adopted as a model potential replacing the usual hypothetical complete LJ 12-6 potential
TL;DR: In this article, a finite size renormalization technique is studied by examining a test model, which is exactly solvable, and the problem of lattice animals and self-avoiding walks.
Abstract: Finite size renormalization technique is studied by examining a test model, which is exactly solvable, and the problem of lattice animals and self-avoiding walks. Bulk estimates of the leading irrelevant RG eigenvalue are found and compared with independent estimates from finite-size data. We introduce an extrapolation technique which is applicable to non-monotonic sequences of exponent approximants.
TL;DR: In this paper, the Klimontovich form of the distribution function is used to derive the basic Poisson bracket relations for the canonical variables α, B, and E, which provide an explicit canonical realization of the Lie algebra of the Poincare group.
Abstract: Following an earlier work we derive a gauge-independent canonical structure for a fully relativistic multicomponent plasma theory. The Klimontovich form of the distribution function is used to derive the basic Poisson bracket relations for the canonical variables ⨍ α , B , and E . The Poisson bracket relations provide an explicit canonical realization of the Lie algebra of the Poincare group and they lead to the correct transformation properties for the canonical variables. We stress the importance of a canonical realization of the full symmetry group of the evolution equations. The covariance of the theory under the symmetry group can be used as a criterion to discriminate among different canonical structures for the evolution equations.
TL;DR: In this article, the authors consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates and derive an equation of evolution for the system variables alone.
Abstract: We consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates . Performing an explicit elimination of the bath variables we arrive at an equation of evolution for the system variables alone. In the weak coupling limit we show that the equation is of the generalized Langevin form, with fluctuations that are Gaussian and that obey a fluctuation-dissipation relation. If the system-bath coupling is linear in the system coordinates the resulting fluctuations are additive and the dissipation is linear. If the coupling is nonlinear in the system coordinates, the resulting fluctuations are multiplicative and the dissipation is nonlinear.
TL;DR: In this article, the authors extended the mobility tensors for an arbitrary number of spheres in an unbounded fluid to include the presence of a plane wall and obtained general expressions for the friction tensors and the fluid velocity field.
Abstract: A previously developed scheme—to evaluate the (translational and rotational) mobility tensors for an arbitrary number of spheres in an unbounded fluid—is extended to include the presence of a plane wall. General expressions for the friction tensors and the fluid velocity field are also obtained.
TL;DR: In this paper, the axial three-state chiral clock model on a d-dimensional lattice is analyzed near its multiphase point for d > 2 by using systematic low-temperature expansions carried to all orders where necessary.
Abstract: The phase diagram of the axial three-state chiral (or asymmetric) clock model on a d- dimensional lattice is analyzed near its multiphase point for d > 2 by using systematic low- temperature expansions carried to all orders where necessary. A matrix method simplifies the configurational analysis. Two infinitely long sequences of commensurate phases appear, each terminating in a triple point at T > 0 and having (mean) wavevectors q = 2πj 3(2j ± 1)a with j = 2, 3, …, jmax where, at fixed T, j max ≈ √2 ln(1 + √2)exp( 3J 2k B T) so that jmax → ∞ as T → 0.
TL;DR: In this article, it is shown that from irreducible representations of space groups, the superspace groups can be determined and vice versa, but there are some differences between the two approaches.
Abstract: Incommensurate crystal phases may be described either with irreducible representations of space groups or with so-called superspace groups which are space groups in more than three dimensions. It is shown that from those representations the superspace groups can be determined and vice versa. However, there are some differences between the two approaches.
TL;DR: In this article, the authors construct the evolution equation of the size distribution function of the unswitched domains of a model of a 1-D system in a switching field, and solve it as an initial value problem.
Abstract: We construct the evolution equation of the size distribution function of the unswitched domains of a model of a 1-D system in a switching field, and solve it as an initial value problem. As the time goes on, the distribution approaches to the “fixed point”, which is the exponential distribution. This feature is unaffected by the finiteness of the critical radius of nucleation.
TL;DR: In this paper, it was shown that a forced oscillation problem for a Hamiltonian equation on a torus possesses a periodic solution having minimal period T, for every sufficiently large prime number T. The proof uses the classical variational approach.
Abstract: : A forced oscillation problem for a Hamiltonian equation on a torus is studied, If the dimension of the torus is equal to 2n, and if the period of the time dependent Hamiltonian equation is equal to 1, there are at least (2n+1) periodic solutions having period 1. In this paper it is shown, that, under an additional, necessary nondegeneracy condition such an equation possesses a periodic solution having minimal period T, for every sufficiently large prime number T. The proof uses the classical variational approach. It is based on the Morse theory for periodic solution to its Morse index and on an iteration formula for the winding number. Originator-supplied keywords included: Hamiltonian systems, Periodic solutions, Variational principles, Morse-type index theory, Winding number of a periodic solution, and Reprints.