Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/r^d+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.     

Entropy of classical systems with long-range interactions

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.     

Nonequilibrium statistical mechanics of systems with long-range interactions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary states (qSSs), the lifetime of which diverges with the number of particles. Therefore, in the thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle–wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.     

Critical behavior in random-spin systems with long-range interactions

Effects of the potential range of the interaction to critical behaviors of quenched random-spin systems are investigated in the limit M → 0 of the MN-component models by means of the renormalization-group theories. As static critical phenomena the stability of the fixed points is investigated and the critical exponents (η, γ, , crossover index) and the equation of state are derived. These phenomena are different from those in pure systems, for the positive specific heat exponent of the pure Heisenberg system.     

Analyticity for one-dimensional systems with long-range superstable interactions

We consider unbounded spin systems and classical continuous particle systems in one dimension. We assume that the interaction is described by a superstable two-body potential with a decay at large distances at least asr ^?2(lnr)^?(2+ε), ε > 0. We prove the analyticity of the free energy and of the correlations as functions of the interaction parameters. This is done by using a “renormalization group technique” to transform the original model into another, physically equivalent, model which is in the high-temperature (small-coupling) region.     

Ergodicity breaking and parametric resonances in systems with long-range interactions

We explore the mechanism responsible for the ergodicity breaking in systems with long-range forces. In thermodynamic limit such systems do not evolve to the Boltzmann-Gibbs equilibrium, but become trapped in an out-of-equilibrium quasi-stationary-state. Nevertheless, we show that if the initial distribution satisfies a specific constraint-a generalized virial condition-the quasistationary state is very close to ergodic and can be described by Lynden-Bell statistics. On the other hand, if the generalized virial condition is violated, parametric resonances are excited, leading to chaos and ergodicity breaking.     

Optimized energy calculation in lattice systems with long-range interactions

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blote algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N2) problem for systems of size N. We show how this can be reduced to an O(N log N) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.     

Critical behavior in anisotropic cubic systems with long-range interactions

Effects of the potential range of interaction to critical behaviors of anisotropic cubic systems are investigated by means of the Callan-Symanzik equations. As the static critical behavior the stability of fixed points, the critical exponents η^C, γ^C, φ^C_s and φ^C_c, and the equation of state are also investigated. As the dynamic critical behavior the dynamic critical exponent z_φ is derived based on the time-dependent Ginzburg-Landau stochastic model. The two- and three-dimensional critical behaviors are discussed.     

Monte Carlo simulations of the equilibrium and non-equilibrium properties of low-dimensional magnetic systems with long-range dipolar interactions

The equilibrium and non-equilibrium properties of a two-dimensional Ising-like spin system with both ferromagnetic exchange and long-range dipolar interactions are studied. Implementing the Ewald Sums (ES) for the calculation of the dipolar interaction we reproduced the results of the literature concerning the equilibrium-phase diagram and the stability of the phases. We also investigated the aging regime and the Fluctuation–Dissipation Theorem (FDT) violations in the system and we showed that, despite usual claims in the literature, the dynamical behavior of the system is independent of the strength of the dipolar interaction, at least in the zone of the phase diagram under study.     

Equilibrium properties of classical systems with long-range forces. BBGKY equation,neutrality, screening,and sum rules

We introduce a generalization of the BBGKY equation to define the equilibrium states for systems with long-range forces and study the properties of such states. We show that there are properties typical of short-range forces (shape independence, normal fluctuations, asymptotic behavior of correlation functions) and others which are typical of long-range forces (possible shape dependence, neutrality, sum rules and screening, abnormal fluctuations, boundedness of the internal electric field). If the force decreases at infinity faster than the Coulomb force, the properties will be those typical of short-range forces; on the other hand, if the force decreases at infinity as the Coulomb force or slower, the properties will be those typical of long-range forces.     

Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions

We discuss the nature of quasi-stationary states (QSS) with non-Boltzmannian distribution in systems with long-range interactions in relation with a process of incomplete violent relaxation based on the Vlasov equation. We discuss several attempts to characterize these QSS. We show that their distribution is non-universal and explain why their prediction is difficult in general.     

Optical properties of nonequilibrium low-dimensional systems

The optical properties of low-dimensional carrier systems ("quantum wire" type) driven away from equilibrium are studied. The frequency and wave-vector-dependent dielectric function of a quasi-one-dimensional electron system under the action of an exciting external pumping source is derived. The optical responses of the system are obtained in terms of its nonequilibrium thermodynamic state, the latter characterized resorting to a nonequilibrium statistical ensemble formalism.     

Thermodynamic properties of the XXZ double chain with long-range interactions

Using the Jordan-Wigner transformation and cut-off approximation, the Letter studies the one-dimensional spin-1/2 XXZ double chain with z-directional uniform long-range interactions in the external magnetic field. Thermodynamic properties as the Helmholtz free energy, internal energy, specific heat and magnetization of the system are obtained. The first and second-order phase transition of the system at a finite temperature are also given. Our results agree with those of the other literatures.     

Breaking of ergodicity and long relaxation times in systems with long-range interactions

The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.