Competing interactions in double-chains: Ising and spherical models

We investigate the thermodynamic properties of double chains of Ising and spherical spins with different first and a crossed second neighbour interaction in zero field. The interest is focussed on the region where different ground states are nearly degenerate due to competing interaction constants. The Ising system shows quasi-singular behaviour of the susceptibility for certain ratios of parameters. Moreover the nearest neighbour correlation function exhibits a sharp crossover from high-temperature “compensation-point” to low temperature ferro- or antiferromagnetic behaviour. An analogy is found between compensation points and tricritical points of higher dimensional systems.     

Nucleation near the spinodal in long-range Ising models

Properties of metastable long-range Ising models (LRIMs) are studied for deep quenches near the mean-field spinodal with Monte Carlo simulations using Glauber dynamics. The theory of spinodal-assisted nucleation is found to agree well with the data. Nucleating droplets are shown to have the same structure as large clusters in random long-range bond percolation.     

Random cascading models and the link between long-range and short-range interactions in multiparticle production

A general recursive relation for the total multiplicity distribution is derived from a random cascading model with intermittency. The conditions for the existence of a fixed point solution are formulated. An explicit example of such a solution, corresponding to the negative binomial multiplicity distribution is presented. The intimate connection between random cascading models and models of disordered systems is explained and explored. Long-distance properties of the interaction are related to the spectrum of states describing the interaction at short range.     

Weakly nonextensive thermostatistics and the Ising model with long-range interactions

We introduce a new nonextensive entropic measure that grows like , where N is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some N-body systems endowed with long-range interactions described by interparticle potentials. The power law (weakly nonextensive) behavior exhibited by is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized q-entropies. The functional is parametrized by the real number in such a way that the standard logarithmic entropy is recovered when . We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since is nonextensive. For , the entropy becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions. Received 24 May 2000     

Enhanced superdiffusion and finite velocity of Levy flights

A fractional differential equation is derived that describes the transformation of a stochastic transport from fast spreading \((\bar x \propto t^\alpha ,\alpha > 1)\) to a pseudowave regime α=1) due to the finiteness of the velocities of individual particles. Qualitative features of the new regime are discussed.     

Ferroelectric square lattice described by the transverse Ising model with long-range interactions

A ferroelectric square lattice described by the transverse Ising model is studied by taking into account the long-range interactions. The size dependence of the Curie temperature as well as the polarization of the lattice is studied. Dielectric peaks and pyroelectric peaks are found in the edge area of the lattice which vary in position with the interaction range. It is found that the interaction range has a strong influence on the ferroelectric properties of the lattice.     

Monte Carlo simulation of very large kinetic Ising models

We study the relaxation of Ising models in three and four dimensions aboveT _c , using multi-spin coding for lattices up to 360³ and 40⁴. The nonlinear relaxation time diverges as (T–T _c )^{–1.05±0.04} in three dimensions, where corrections to scaling are taken into account. In four dimensions the effective exponent is about 0.72; logarithmic correction factors make the analysis difficult here. The linear relaxation time for the asymptotic exponential decay is found to be larger, with effective exponents 1.31 (d=2) and 0.97 (d=4). The difference in the linear and nonlinear relaxation exponents is compatible in three dimensions with the orderparameter exponent , as predicted by Racz.Work supported by SFB 125 Aachen-Jülich-KölnWork started at Department de Physique des Systemes Desordonnes, Universite de Provence, Centre St-Jerome, F-13397 Marseille Cedex 13, France     

Short-time behavior of the kinetic spherical model with long-ranged interactions

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m₀ quenched from a very high temperature to T is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order is found in the intermediate time-regime. The short-time scaling relations of small m₀ are generalized to an arbitrary m₀ and all the time larger than . The characteristic functions for the scaling of m₀ and for are obtained. The crossover between scaling regimes is discussed in detail. Received 17 September 1999     

Statistical mechanics and dynamics of solvable models with long-range interactions

For systems with long-range interactions, the two-body potential decays at large distances as V(r)1/r^α, with α≤d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.     

Solution of the long-range gaussian-random Ising model

The spin glass model of Edwards and Anderson is solved for Ising spins starting from a renormalized diagrammatic expansion. One gets two qualitatively distinct phases for arbitrary external field. The high temperature phase is identical with the solution of Sherrington and Kirkpatrick. The low temperature phase does not have unphysical properties forT0, in contrast to previous investigations.     

Dynamical renormalization of kinetic van der Waals spin models

An exact dynamical renormalization approach in differential form is proposed for kinetic van der Waals spin systems with general many-body interactions. The problem of restoring covariance in the evolution equation after renormalization of the model is solved by introducing a suitable renormalized time parameter, which depends also on the magnetization of the spin configuration. The study of the behavior of this renormalized time near criticality leads to a scaling relation for the linear relaxation time. This relation can be shown to imply the exact results for the dynamical critical behavior of the system.On leave of absence from Instituto di Fisica e Unità G.N.S.M. del C.N.R., Università di Padova, Padova, Italy.     

Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems.     

Correlation-function identities and inequalities for Ising models with pair interactions

For Ising models with pair interactions in zero magnetic field a class of linear combinations of products of two correlation functions is studied. We derive sufficient and necessary conditions under which a function in this class is (a) zero for all values of the coupling parameters, or (b) nonnegative for all nonnegative values of the coupling parameters. Examples of correlation-function identities and inequalities of this type are given.     

Ising models with two- and three-spin interactions: Mean field equation of state

The Ising model with pair and triplet interactions on the triangular lattice is solved in the mean-field approximation. With a sufficiently strong triplet interaction two first-order transitions take place at low temperature, and at intermediate temperatures one transition, terminating in a critical point. For J₂ > 0.75J₃ only the latter transition remains.     

Calculation of spin correlations in two-dimensional Ising systems from one-dimensional kinetic models

Kinetic spin models of the type first introduced by Glauber are considered with the most general choice of transition rates. It is shown that their time evolution operator can be related to the transfer matrices of certain two-dimensional Ising lattices. This allows an exact calculation of spin correlation functions at certain temperatures. Specifically we show that for the triangular antiferromagnet this special temperature corresponds to the disorder point. For the Hamiltonian version of the ANNNI model the corresponding result restricts in an important way the possible structure of the phase diagram.     

Statistical mechanics of one-dimensional Ising and Potts models with exponential interactions

This paper continues the authors' work on a new method for discussing one-dimensional systems in statistical mechanics with exponentially decreasing interactions. It is shown how in the case of the S-spin Ising and the N-state Potts model the results in the classic paper of Kac et al. for these models emerge also from our method. It is the aim of the present paper to compare these two mathematically completely different methods and prepare the extension of our method to two-dimensional systems.     

Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach

The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.