Unique Equilibrium State Research Articles

Equilibrium States of Grid Functions

It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.

A completely solvable model of the nonlinear Boltzmann equation

Abstract In one space and one time dimension, a model of the nonlinear Boltzmann equation is presented that is exactly solvable for all initial conditions. Furthermore, this model has the following desirable properties: (i) conservation of the number of particles; (ii) energy conservation; (iii) nonlinearity; (iv) positivity of distribution functions; and (v) unique equilibrium state (for any given density) which is approached as t → ∞ for most physically interesting initial conditions. Some of the simple properties of this model are studied.

The cluster expansion for classical and quantum lattice systems

We develop a high-temperature expansion for general lattice systems which can be applied to classical as well as quantum systems. Applying the expansion we prove analyticity of correlation functions, uniqueness of equilibrium states, and cluster properties for classical and quantum lattice systems in the high-temperature region.

Equilibrium states of grid functions

It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.

Equilibrium and Stability in a Periodic Marketing Ring

A dynamic economic model for a periodic marketing network of the following sort is constructed herein. Traders operating out of urban centers pass among rural markets during the marketing week buying up a commodity, which they transport back to their respective urban centers and sell to wholesalers. The traders follow fixed and in general different, rings of markets, which they repeat from week to week. A set of nonlinear difference equations is devised that determines the time-dependent prices and commodity flows throughout the network. when the traders’ rings are all the same, we prove that the system has a unique equilibrium state. This is established for two cases, one where the traders do not store goods and the second where they are allowed to do so. In the former case, we also establish conditions under which the equilibrium state is locally asymptotically stable. We finally show that, in the two-ring case, a peculiar price variation can occur, a result that may explain some of the reported “unpred...

Efficient integration of the time dependent collisional-radiative equations

The properties of a set of conservative positive rate equations, such as the collisional-radiative equations, are investigated. It is shown that the equations are positivity maintaining, and furthermore, converge uniformly onto a unique equilibrium state consistent with physical constraints. Two methods of integration are considered. The first involving eigenvector decomposition yields an exact solution, but is expensive in computer time, and cumbersome if the rate matrix is defective. A simple weighted finite difference approach may be integrated stiffly and unconditionally stably, and is therefore suitable for inclusion in multi-celled fluid codes. The improvement in accuracy obtained by weighting is investigated, and attention is drawn to the importance of a positivity-maintaining form.

The effect of positive assortative mating at one locus on a second linked locus

A model for positive assortative mating based on genotype for one locus is employed to investigate the effect of this mating system on the genotypic structure of a second linked locus as well as on the joint genotypic structure of these two loci. It is shown that the second locus does not attain a precise positive assortative mating structure, but yet it shares a property that is characteristic of positive assortative mating, namely an increase in the frequency of homozygotes over that typically found in panmictic structures. Given any arbitrary genotypic structure for the parental population, the resulting offspring generation possesses a structure at the second locus that does not depend on the recombination frequency, while the joint structure of course does. In case assortative mating as well as linkage are not complete, there exists a unique joint equilibrium state for the two loci, which is characterized by complete stochastic independence between the two loci as well as by Hardy-Weinberg proportions at the second locus. For the second locus alone, Hardy-Weinberg equilibrium is realized if and only if gametic linkage equilibrium and an additionally specified condition are realized.

Multiple Equilibrium States of Nonlinearly Elastic Strings

This paper presents a comprehensive treatment of the existence, multiplicity, and qualitative behavior of equilibrium states for nonlinearly elastic strings under three different loading systems. It is shown that vertically loaded strings have unique convex equilibrium states and have concave equilibrium states that come in pairs. All equilibrium states are of one of these two forms. Similar, though more complicated, results hold for strings under normal pressure.

A remark on Dobrushin's uniqueness theorem

Ten years ago, Dobrushin [1] proved a beautiful result showing that under suitable hypotheses, a statistical mechanical lattice system interaction has a unique equilibrium state. In particular, there is no long range order, etc. see [6,7] for related material, Israel [4] for analyticity results and Gross [3] for falloff of correlations. There does not appear to have been systematic attempts to obtain very good estimates on precisely when Dobrushin's hypotheses hold, except for certain spin models [6,4]. Our purpose here is to note that with one simple device one can obtain extremely good estimates which are fairly close to optimal. Let Ω be a fixed compact space (single spin configuration space), dμ0 a probability measure on Ω and for each α e Z, Ωα a copy of Ω. For X a finite subset of TD', let Ω — X Ωα. An interaction is an assignment of a continuous function, ΦpO, on Ω to each finite XcTL. While it is not necessary for Dobrushin's theorem, it is convient notationally to suppose Φ translation covariant in the obvious sense.

Asymptotic behavior of nonlinear compartmental systems: Nonoscillation and stability

This paper discusses properties related to the stability of a class of nonlinear compartmental systems. Specifically, mathematical conditions which guarantee the same qualitative behavior inherent in linear compartmental systems are considered. We first consider the nonoscillatory property of solutions and show that the system has no periodic oscillation under a mild condition. The result is then used to derive a necessary and sufficient condition for every solution to converge to a set of equlibrium points which may depend both on the input and the initial state. A sufficient condition is also given for an equilibrium state to depend only on the input. The asymptotic behavior of the free systems is also considered, and a sufficient condition is given for the origin to be globally asymptotically stable. Furthermore, for a closed compartmental system it is shown that for each given initial state, unique equilibrium state, if it exists, depends only on the total sum of the components of the initial state. Finally a sufficient condition is given for solutions to converge to the unique point.

On the uniqueness of the equilibrium state for plane rotators

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.

An algebraic condition for the approach to equilibrium of an open N-level system

We give an algebraic condition in order that a completely positive dynamical semigroup of an N-level system has a unique (invariant) equilibrium state and that every initial state approaches this equilibrium state as t→∞. We apply our result to a semigroup arising in the weak coupling limit.

Flows with Unique Equilibrium States

Flows with Unique Equilibrium States

Examples for the nonuniqueness of the equilibrium state

In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the functions of this class which satisfy Ruelle’s Perron-Frobenius condition, those which admit a measure determined by a homogeneity condition, and those which have unique equilibrium state. In particular, we get examples for the nonuniqueness of the equilibrium state.

Approach to equilibrium for completely positive dynamical semigroups of N-level systems

We give a simple sufficient condition for a completely positive dynamical semigroup of a N-level system to have a unique (invariant) equilibrium state and for every initial state to approach this equilibrium state as t → ∞. We estimate the rate of approach to equilibrium.

Some systems with unique equilibrium states

where h , ( f ) is the entropy of/z. We want conditions o n f a n d ~o which guarantee a unique equilibrium state. f i s called expansive if there is an ~ > 0 such that for any two points x ¢ y in X there is an n e Z so that d( f " (x ) , f " (y ) ) > E. fsat isf ies specification if for each 8 > 0 there is an integer p(8) for which the following is true: if I~,. •., I , are intervals o f integers contained in [a, b] with d(ll, I j) >_ p(8) for i ¢ j and x l , . . . , x , eX, then there is a point x e X with fb-"+PO)(X)= X and d(fk(x), fk(xi)) < 8 for k e Ii. This condition allows us to construct a lot of periodic points. For ~o e C(X) and n _ 1 let

Existence and uniqueness of equilibrium states for some spin and continuum systems

The one to one correspondence between the existence of a unique equilibrium state and the differentiability of the free energy density with respect to the external field previously shown for Ising ferromagnetis is extendend to higher valued spin systems as well as to continuum systems satisfying the Fortuin, Kasteleyn and Ginibre inequalities. In particular this is shown to hold for a mixture ofA –B particles in which there is no interaction between like particles and a repulsion between unlike particles. Where the derivative of the free energy is discontinuous there are at least two equilibrium states.

A SUFFICIENT CONDITION FOR THE EXISTENCE OF A UNIQUE EQUILIBRIUM STATE IN AN ADIABATIC TUBULAR REACTOR

This paper presents a sufficient condition for the existence of a unique equilibrium state in an adiabatic tubular reactor in which a single exothermic chemical reaction takes place and Peclet number for mass transfer is equal to that for heat transfer. The condition is obtained by analyzing the relations between properties of the heat generation curve for an adiabatic tubular reactor and those of the curve for an adiabatic complete mixing reactor with the same residence time, inlet temperature and inlet concentration as the tubular reactor, and is given in terms of the character of the heat generation curve for the complete mixing reactor, which can be analyzed much more easily than the tubular reactor. Compared with other conditions for a first-order reaction, the condition is shown to be stronger than the condition of Luss and Amundson for equilibrium states with small Peclet numbers and low temperature, but more conservative for equilibrium states with large Peclet numbers.

On the uniqueness of the equilibrium state for Ising spin systems

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a “periodic” external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T c and is non-unique forT<T c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.

Burgers turbulence model for a channel flow

The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case &amp; Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.