Classical one-dimensional Heisenberg model with an interaction of finite range

It is shown that the thermodynamic quantities and spin correlation functions of the classical Heisenberg model on a linear chain are expressed in terms of the eigenvalue with the smallest absolute value and the corresponding eigenfunction of a homogeneous linear integral equation, where the range of the interaction is assumed to be finite. The magnetization and susceptibility at nonzero external magnetic fields are given as a function of temperature, for the case of the nearest neighbour ferromagnetic and antiferromagnetic interaction. Efforts are paid to determine the properties near zero temperature.     

Kinetics of a finite one-dimensional mixture of hard rods with different masses

A one-dimensional binary mixture of impenetrable (hard core) particles with different mass ratios,m ₂/m ₁=1, 1.05, 1.2, 2, 3, and 4, was simulated to evolve in a computer by the molecular dynamics method. The systems withm ₂>m ₁ and initial velocity distribution ±v ₀ show a clear tendency to the equipartition of energy and relaxation toward a Maxwellian velocity distribution unlike the nonergodic system withm ₂=m ₁. Several quantities have been monitored during the evolution to investigate its dependence on the mass ratiom ₂/m ₁.     

Strong cluster properties for classical systems with finite range interaction

In a previous paper, “strong” decrease properties of the truncated correlation functions, taking into account the separation of all particles with respect to each other, have been presented and discussed. In this paper, we prove these properties for finite range interactions in various situations, in particular

1. at low activity for lattice and continuous systems,
2. at arbitrary activity and high temperature for lattice systems,
3. at ReH≠0, β arbitrary and atH=0 for appropriate temperatures in the case of ferromagnets.

We also give some general results, in particular an equivalence, on the links between analyticity and strong cluster properties of the truncated correlation functions.     

Exact renormalization transformation for the one-dimensional ising model with finite range interaction

The transfer matrix formalism is employed in order to construct the exact renormalization transformation for the one-dimensional Ising model with finite range interaction. The method is applied to the nearest-neighbour model and to the next nearest-neighbour model.     

Thermodynamic limit for the one-dimensional Ising systems with random interaction of finite range

The existence of the thermodynamic limit is proved for the random one-dimensional Ising systems under the assumption that the interaction energies are random variables taking on continuous values and the distribution of the random variables is given by a continuous function. It is assumed that the total number of possible configurations for each lattice site is finite and the range of interaction is finite.     

The classical fluid of associating hard rods in an external field

Two models of one-dimensional fluids of associating hard rods in an arbitrary external field are investigated. In the first model particles can only form dimers, while in the second model, which has been solved previously by Percus, aggregates of any size coexist. In both cases the grand canonical potential and the external potential are found exactly as functionals of the density. It is shown that Wertheim's thermodynamic perturbation theory of polymerization provides a straightforward route to the exact solution by expanding the functional space to include more density parameters. This suggests that Wertheim's theory should be used also for studying the structure (and not only the thermo-dynamics) of real associating fluids.     

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Perçus, and Sykes, removing some nontypical features of that system.Supported by NSF grant No. MCS 75-21684 A01 (M. A.), NSF grant No. MPS 75-20638 (J. L.), and USAFOSR grant No. 73-2430 B (J. M.)John Guggenheim Fellow on sabbatical leave from Belfer Graduate School of Science, Yeshiva University, New York.     

Ground-state properties of a one-dimensional system of hard rods

A quantum Monte Carlo simulation of a system of bosonic hard rods in one dimension is presented and discussed. The calculation is exact since the analytical form of the wave function is known and is in excellent agreement with predictions obtained from asymptotic expansions valid at large distances. The analysis of the static structure factor and the pair distribution function indicates that a solidlike and a gaslike phases exist at high and low densities, respectively. The one-body density matrix decays following a power law at large distances and produces a divergence in the low density momentum distribution at k=0 which can be identified as a quasicondensate.     

Equilibrium state of a classical fluid of hard rods in an external field

The external field required to produce a given density pattern is obtained explicitly for a classical fluid of hard rods. All direct correlation functions are shown to be of finite range in all pairs of variables. The one-sided factors of the pair direct correlation are also found to be of finite range.Supported in part by U.S. ERDA, Contract No. E(11-1)-3077.     

Phase transition for a one-dimensional lattice gas with hard core

Existence of a phase transition is proved for a one-dimensional lattice gas with long-range interaction and nearest neighbor exclusion.     

On the kinetic equations of a system of one-dimensional hard rods

The hydrodynamical approximation to an infinite system of one-dimensional identical hard rods interacting through elastic collisions, is shown to be an integrable system possessing a one-parameter family of nonlinear Hamiltonian structures.     

Infinite differentiability for one-dimensional spin system with long range random interaction

We consider one-dimensional spin systems with Hamiltonian: $$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$ , where ? _tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α>3/2 andE(? _tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ^∞ both inh and β. Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ? _tt′ 's, where the quenched magnetization isC ^∞ but fails to be analytic inh for suitableh and β.     

Microcanonical ensemble study of a classical fluid of hard rods under gravity

The study of a system of hard rods in a box of finite length in the presence of a uniform gravitational field is made by means of the microcanonical ensemble. Explicit expressions are derived for the phase volume and the density of states, the primary functions of this ensemble. Related statistical quantities are reported, such as the entropy, the temperature, the heat capacity and the forces exerted on the fluid by the bottom and top walls. The microcanonical number density and higher order molecular distribution functions are also derived. Received: 7 April 1998 / Received in final form: 20 July 1998 / Accepted: 28 July 1998     

Universality of the one-dimensional Bose gas with delta interaction

We consider several models of interacting bosons in a one-dimensional lattice. Some of them are not integrable like the Bose-Hubbard others are integrable. At low density all of these models can be described by the Bose gas with delta interaction. The lattice corrections corresponding to the different models are contrasted.     

Diffusion in a one-dimensional gas of hard point particles

We simulate the classical diffusion of a particle of massM in an infinite one-dimensional system of hard point particles of massm in equilibrium. Each computer run corresponds to about 10⁸ collisions of the diffusive particle. We find that (t) 1/t fort large enough, and a crossover from an M m regime where=2 to=3 forM=m. The diffusion constant has a sharp maximum atM=m. We study moments x(t)² and x(t)⁴, and examine the behavior ofq ² (t)=x(t)⁴/3x(t)²². We find thatq(t)1 (consistent with a normal distribution) in theM limit (for all timest) and in the t limit for allM. On sabbatical leave from IVIC-Instituto Venezolano de Investigaciones Cientificas.     

Dynamical effects of a one-dimensional multibarrier potential of finite range

We discuss the properties of a large number N of one-dimensional (bounded) locally periodic potential barriers in a finite interval. We show that the transmission coefficient, the scattering cross section σ, and the resonances of σ depend sensitively upon the ratio of the total spacing to the total barrier width. We also show that a time dependent wave packet passing through the system of potential barriers rapidly spreads and deforms, a criterion suggested by Zaslavsky for chaotic behaviour. Computing the spectrum by imposing (large) periodic boundary conditions we find a Wigner type distribution. We investigate also the S-matrix poles; many resonances occur for certain values of the relative spacing between the barriers in the potential. Received 1st August 2001 and Received in final form 18 November 2001     

Approach to field-induced stationary state in a gas of hard rods

The Boltzmann equation describing one-dimensional motion of a charged hard rod in a neutral hard rod gas at temperatureT = 0 is solved. Under the action of a constant and uniform field the charged particle attains a stationary state. In the long time limit the velocity autocorrelation function decays via damped oscillations. In the reference system moving with the mean particle velocity the decay of fluctuations in the position space is governed (in the hydrodynamic limit) by the diffusion equation. Both the stationary current and the diffusion coefficient are proportional to the square root of the field. It is conjectured that this result also holds forT > 0 in a strong field limit.On leave from the Institute of Theoretical Physics, University of Warsaw, Hoza 69, 00-081 Warsaw, Poland.     

Existence of phase transition for a lattice model with a repulsive hard core and an attractive short range interaction

We consider a lattice model with a repulsive hard core and an attractive short range interaction. We show that this model has at least three independent equilibrium states, when the temperature is sufficiently low and the chemical potential is suitably chosen.     

Analysis of a spatial Lotka-Volterra model with a finite range predator-prey interaction

We perform an analysis of a recent spatial version of the classical Lotka-Volterra model, where a finite scale controls individuals’ interaction. We study the behavior of the predator-prey dynamics in physical spaces higher than one, showing how spatial patterns can emerge for some values of the interaction range and of the diffusion parameter.