Showing papers in "Journal of Statistical Physics in 1979"

The universal metric properties of nonlinear transformations

Abstract: The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx n+1 =λf(x n ) independent of a specificf is formally developed. A hierarchy of universal functionsg r (x) exists, each descriptive of the same local structure but at levels of a cluster of 2 r points. The hierarchy obeysg r−1 (x)=−αg r(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticg r ∼ g − δ−r h * where δ > 1 andh are determined as the associated eigenvalue and eigenvector of the operator ℒ: $$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right]$$ We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλ rather than just discreteλ r and bifurcation valuesΛ r and dynamics at suchλ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of theg r limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate tog r 's, thereby providing some elementary approximation schemes for obtainingλ r for a chosenf.

Translation of J. D. van der Waals' “The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density”

Abstract: Van der Waals justifies the choice of minimization of the (Helmholtz) free energy as the criterion of equilibrium in a liquid-gas system (Sections 1–4). If densityρ is a function of heighth then the local free energy density differs from that of a homogeneous fluid by a term proportional to (d 2 ρ/dh 2); the extra term arises from the energy not from the entropy (Section 5). He uses this result to show howρ varies withh (Section 6), how this variation leads to a stable minimum free energy (Section 7), and to calculate the capillary energy or surface tensionσ (Section 9). Near the critical pointσ varies as (τ k -τ)3/2, whereτ k is the critical temperature (Section 11). The paper closes with short discussions of the thickness of the surface layer (Section 12), of the difficulty of assuming thatρ varies discontinuously with height (Section 14), and of the possible effect of derivatives of higher order than (d 2 ρ/dh 2) on the free energy and surface tension (Section 15).

Randomly transitional phenomena in the system governed by Duffing's equation

Abstract: This paper deals with turbulent or chaotic phenomena which occur in the system governed by Duffing's equation, a special type of two-dimensional periodic system. By using analog and digital computers, experiments are carried out with special reference to the change of attractors and of average power spectra of the random processes under the variation of the system parameters. On the basis of the experimental results, an outline of the random process is made clear. The results obtained in this paper will be applied to various physical problems and will also serve as material for the development of a proper mathematics of this phenomenon.

Metastable Chaos: The Transition to Sustained Chaotic Behavior in the Lorenz Model

Abstract: The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbersr somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence uponr is studied both numerically and (very close to the criticalr) analytically.

“Thought-experiments” by molecular dynamics

Abstract: A method for studying the dynamical properties of liquids by molecular dynamics simulation is described. Its basis is the measurement of the response to a weak applied field of appropriate character. The explicit form of the mechanical perturbation is worked out in several cases, and details are given of the numerical techniques used in implementing the method.

Diffusion in a Bistable Potential: A Systematic WKB Treatment

Abstract: We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrodinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second “ half,” whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.

Mathematical properties of position-space renormalization-group transformations

Abstract: Properties of “position-space” or “cell-type” renormalization-group transformations from an Ising model object system onto an Ising model image system, of the type introduced by Niemeijer, van Leeuwen, and Kadanoff, are studied in the thermodynamic limit of an infinite lattice. In the case of a KadanofF transformation with finitep, we prove that if the magnetic field in the object system is sufficiently large (i.e., the lattice-gas activity is sufficiently small), the transformation leads to a well-defined set of image interactions with finite norm, in the thermodynamic limit, and these interactions are analytic functions of the object interactions. Under the same conditions the image interactions decay exponentially rapidly with the geometrical size of the clusters with which they are associated if the object interactions are suitably short-ranged. We also present compelling evidence (not, however, a completely rigorous proof) that under other conditions both the finite- and infinite-p (“majority rule”) transformations exhibit peculiarities, suggesting either that the image interactions are undefined (i.e., the transformation does not possess a thermodynamic limit) or that they fail to be smooth functions of the object interactions. These peculiarities are associated (in terms of their mathematical origin) with phase transitions in the object system governed not by the object interactions themselves, but by a modified set of interactions.

Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations

Abstract: Two infinite sequences of orbits leading to turbulence in a five-mode truncation of the Navier-Stokes equations for a 2-dimensional incompressible fluid on a torus are studied in detail. Their compatibility with Feigenbaum's theory of universality in certain infinite sequences of bifurcations is verified and some considerations on their asymptotic behavior are inferred. An analysis of the Poincare map is performed, showing how the turbulent behavior is approached gradually when, with increasing Reynolds number, no stable fixed point or periodic orbit is present and all the unstable ones become more and more unstable, in close analogy with the Lorenz model.

On the Hénon-Pomeau attractor

Abstract: The behavior of the iterates of the mapT(x, y) = (1+y−ax2,bx) can be useful for the understanding of turbulence. In this study we fix the value ofb at 0.3 and allowa to take values in a certain range. We begin with the study of the casea=1.4, for which we determine the existence of a strange attractor, whose region of attraction and Hausdorff dimension are obtained. As we changea, we study numerically the existence of periodic orbits (POs) and strange attractors (SAs), and the way in which they evolve and bifurcate, including the computation of the associated Lyapunov numbers. Several mechanisms are proposed to explain the creation and disappearance of SAs, the basin of attraction of POs, and the cascades of bifurcations of POs and of SAs for increasing and decreasing values ofa. The role of homoclinic and heteroclinic points is stressed.

Kinetic theory of hard spheres

Abstract: Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.

Chaotic response of a limit cycle

Abstract: External periodic modulation of a nonlinear oscillator may lead to chaotic behavior. This phenomenon is attributed to the existence of a strange attractor, which embodies essentially a folding motion as is met within the Bernoulli shift or the baker's transformation. The results obtained for the Brussels model are discussed from this viewpoint.

Percolation and cluster distribution. III. Algorithms for the site-bond problem

Abstract: Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.

The infinite-volume ground state of the Lennard-Jones potential

Abstract: We consider a finite chain of particles in one dimension, interacting through the Lennard-Jones potential. We prove the ground state is unique, and approaches uniform spacing in the infinite-particle limit.

Estimates of critical lengths and critical temperatures for classical and quantum lattice systems

Abstract: Local Ward identities are derived which lead to the mean-field upper bound for the critical temperature for certain multicomponent classical lattice systems (improving by a factor of two an estimate of Brascamp-Lieb). We develop a method for accurately estimating lattice Green's functionsId yielding 0.3069

Stationary measures for the periodic Euler flow in two dimensions

Abstract: We construct for the Euler flow in two dimensions with periodic boundary conditions the Gibbsian measures given by the energy and the enstrophy integrals. We show that they are inflnitesimally invariant under the Euler flow.

Series expansions from corner transfer matrices: The square lattice Ising model

Abstract: The corner transfer matrix formalism is used to obtain low-temperature series expansions for the square lattice Ising model in a field. This algebraic technique appears to be far more efficient than conventional methods based on combinatorial enumeration.

Existence of free energy for models with long-range random Hamiltonians

Abstract: Classical lattice systems with random Hamiltonians $$\frac{1}{2}\sum\limits_{x_1 e x_2 } {\frac{{\varepsilon (x_1 ,x_2 )\varphi (x_1 )\varphi (x_2 )}}{{\left| {x_1 - x_2 } \right|^{\alpha d} }}}$$ are considered, whered is the dimension, ande(x1,x2) are independent random variables for different pairs (x1,x2),Ee(x1,x2) = 0. It is shown that the free energy for such a system exiists with probability 1 and does not depend on the boundary conditions, providedα > 1/2.

Lattice Models for Liquid Crystals

Abstract: A problem in the theory of liquid crystals is to construct a model system which at low temperatures displays long-range orientational order, but not translational order in all directions. We present five lattice models (two two-dimensional and three three-dimensional) of hard-core particles with attractive interactions and prove (using reflection positivity and the Peierls argument) that they have orientational order at low temperatures; the two-dimensional models have no such ordering if the attractive interaction is not present. We cannot prove that these models do not have complete translational order, but their zero-temperature states are such that we are led to conjecture that complete translational order is always absent.

Diffusion-Controlled Processes Among Partially Absorbing Stationary Sinks

Abstract: A general linear response theory is presented to calculate the zero-wavevector and zero-frequency reaction rate coefficient for particles diffusing into absorbing spheres. Allowance is made for possible incomplete particle absorption. A Faxen-like theorem for chemical reactions is derived. The problem is solved completely for a simple regular array of sinks. Exact analytic expressions for the rate coefficient as a function of sink volume fraction are obtained for the sc and fcc lattices. The case of a disordered array of sinks is also considered and the leading order nonanalytic density dependence of the rate coefficient is calculated. In both cases an increase in the rate coefficient with sink density in a local region of the system is found. The general formalism is extended to examine the modification to the particle diffusion coefficient due to the presence of the spheres. For regular arrays of spheres, the mean field result is reproduced.

On the equivalence of boundary conditions

Abstract: We show that ifb andb′ are two boundary conditions (b.c.) for general spin systems on ℤ d such that the difference in the energies of a spin configuration σΛ in Λ ⊂ ℤ d is uniformly bounded, |H Λ,b (σΛ)−H Λ,b′(σΛ)|⩽C < ∞, then any infinite-volume Gibbs statesρ and ρ′ obtained with these b.c. have the same measure-zero sets. This implies that the decompositions ofρ and ρ′ into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, ifρ is extremal,ρ=ρ′. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.

The kinetic Ising model: Exact susceptibilities of two simple examples

Abstract: The susceptibility of a modified version of the one-dimensional kinetic Ising model is obtained and compared with the susceptibility of the Glauber version of this model. Spin-flip rates in the new model are picked so no spin-flip rate vanishes as the temperature vanishes. Despite the more rapid spin flips, the new model exhibits an infinitely slow approach to equilibrium in the low-temperature limit which is similar to the slowing down exhibited in the Glauber model. The new model also exhibits two different decay rates toward equilibrium, which are called the transient and slow decay rates. The Glauber model is characterized by only a single decay rate toward equilibrium.

The nonsymmetric pressure tensor in polyatomic fluids

Abstract: Nonequilibrium molecular dynamics calculations are used to show that polyatomic fluids can support antisymmetric stress. In a homogeneous system where the time dependence of vorticity is a step function, it is shown that the rate at which intrinsic angular velocity approaches its steady-state value (ω = 1/2▽ × u) is determined by the magnitude of the antisymmetric part of the pressure tensor.

Invariant subspaces of clustering operators

Abstract: A class of clustering operators is defined which is a generalization of a transfer matrix of a Gibbs lattice field with an exponential decay of correlations. It is proved that for small values ofβ the clustering operator has invariant subspaces which are similar tok-particle subspaces of the Fock space. The restriction of the clustering operator onto these subspaces resembles the operator exp(-H k, whereH k is thek- particle Schrodinger Hamiltonian in nonrelativistic quantum mechanics. The spectrum of eachH k,k⩾1, is contained in the interval (C 1βk,C 2βk). These intervals do not intersect with each other.

Nonlinear transport in the Boltzmann limit

Abstract: Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.

The symmetry of ground states under perturbation

Abstract: We consider a two-body potential which has only periodic ground states and prove that it can be perturbed, by an arbitrarily small perturbation, so as to have only aperiodic ground states.

Renormalized kinetic theory of nonequilibrium many-particle classical systems

Abstract: The far-from-equilibrium statistical dynamics of classical particle systems is formulated in terms of self-consistently determined phase-space density response, fluctuation, and vertex functions Collective and single-particle effects are treated on an equal footing Two approximations are discussed, one of which reduces to the Vlasov equation direct interaction approximation of Orszag and Kraichnan when terms that are explicitly due to particles are removed

Adsorption of dipolar hard spheres onto a smooth, hard wall in the presence of an electric field

Abstract: Integral equations have been solved for the density profile of dipolar hard spheres against a hard, smooth wall in the presence of an electric field. This density profile was examined as a function of the bulk medium's temperature and density with different field strengths and field directions. It was found to depend primarily upon the competitive interactions of the field with the monolayer particles and the first outer shell with the monolayer particles.

Low-temperature expansion for lattice systems with many ground states

Abstract: Low-temperature expansion for systems with many ground states is discussed. It is pointed out that, in general, different ground states may yield different formal perturbation expansions, and that the right expansion of the free energy is provided by ground states called here dominant.

Zeros of the Partition Function and Gaussian Inequalities for the Plane Rotator Model

Abstract: The partition function for ferromagnetic plane rotators in a complex external field μ, with ¦Im μ¦ ⩽ ¦Re μ ¦, is bounded below in modulus by its value at μ=0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and μ=0), the associated “Gaussian inequalities” imply that all truncated correlation functions decay at least as the two-point function.