Ensemble inequivalence in systems with long-range interactions

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We illustrate our results showing an application to the Blume-Emery-Griffiths model. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble.Comment: 12 pages, no figure

Ensemble inequivalence: A formal approach

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble.Comment: 4 pages, no figures, given at the NEXT2001 conference on non-extensive thermodynamic

Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales

Motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter $\lambda$ suitably defined on the unfolded scale. Using results for the spectral two-point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For $\lambda$ << 1 the correlations are given by Dawson's integral while for $\lambda$ >> 1 we derive a novel analytical formula for the two-point function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticable can be identified. We also derive an exact expansion of the local level density for finite level number.Comment: 15 pages, Revtex, no figures, to be published in special issue of J. Math. Phys. (1997

Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation

The scaling theory of irreversible aggregation is discussed in some detail. First, we review the general theory in the simplest case of binary reactions. We then extend consideration to ternary reactions, multispecies aggregation, inhomogeneous situations with arbitrary size dependent diffusion constants as well as arbitrary production terms. A precise formulation of the scaling hypothesis is given as well as a general theory of crossover phenomena. The consequences of this definition are described at length. The specific issues arising in the case in which an infinite cluster forms at finite times (the so-called gelling case) are discussed, in order to address discrepancies between theory and recent numerical work. Finally, a large number of exactly solved models are reviewed extensively with a view to pointing out precisely in which sense the scaling hypothesis holds in these various models. It is shown that the specific definition given here will give good results for almost all cases. On the other hand, we show that it is usually possible to find counterexamples to stronger formulations of the scaling hypothesis.Comment: 160 pp. 1 figure, submitted to Physics Report

Transport properties of a modified Lorentz gas

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsager's reciprocity relations to within numerical accuracy as well as the Green-Kubo relations . We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.Comment: 16 pages, 16 figures, submitted to J. Stat. Phy

Scaling of Reaction Zones in the A+B->0 Diffusion-Limited Reaction

We study reaction zones in three different versions of the A+B->0 system. For a steady state formed by opposing currents of A and B particles we derive scaling behavior via renormalization group analysis. By use of a previously developed analogy, these results are extended to the time-dependent case of an initially segregated system. We also consider an initially mixed system, which forms reaction zones for dimension d<4. In this case an extension of the steady-state analogy gives scaling results characterized by new exponents.Comment: 4 pages, REVTeX 3.0 with epsf, 2 uuencoded postscript figures appended, OUTP-94-33