Boltzmann-Gibbs Equilibrium Research Articles

Tsallis q-Statistics in Seismology.

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann-Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events.

Covariant momentum map thermodynamics for parametrized field theories

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

Relaxation to equilibrium in models of classical spins with long-range interactions

For a model long-range interacting system of classical Heisenberg spins, we study how fluctuations, such as those arising from having a finite system size or through interaction with the environment, affect the dynamical process of relaxation to Boltzmann–Gibbs equilibrium. Under deterministic spin precessional dynamics, we unveil the full range of quasistationary behavior observed during relaxation to equilibrium, whereby the system is trapped in nonequilibrium states for times that diverge with the system size. The corresponding stochastic dynamics, modeling interaction with the environment and constructed in the spirit of the stochastic Landau–Lifshitz–Gilbert equation, however shows a fast relaxation to equilibrium on a size-independent timescale and no signature of quasistationarity, provided the noise is strong enough. Similar fast relaxation is also seen in Glauber Monte Carlo dynamics of the model, thus establishing the ubiquity of what has been reported earlier in particle dynamics (hence distinct from the spin dynamics considered here) of long-range interacting systems, that quasistationarity observed in deterministic dynamics is washed away by fluctuations induced through contact with the environment.

Brownian dynamics of fully confined suspensions of rigid particles without Green's functions.

We introduce a Rigid-Body Fluctuating Immersed Boundary (RB-FIB) method to perform large-scale Brownian dynamics simulations of suspensions of rigid particles in fully confined domains, without any need to explicitly construct Green's functions or mobility operators. In the RB-FIB approach, discretized fluctuating Stokes equations are solved with prescribed boundary conditions in conjunction with a rigid-body immersed boundary method to discretize arbitrarily shaped colloidal particles with no-slip or active-slip prescribed on their surface. We design a specialized Split-Euler-Maruyama temporal integrator that uses a combination of random finite differences to capture the stochastic drift appearing in the overdamped Langevin equation. The RB-FIB method presented in this work only solves mobility problems in each time step using a preconditioned iterative solver and has a computational complexity that scales linearly in the number of particles and fluid grid cells. We demonstrate that the RB-FIB method correctly reproduces the Gibbs-Boltzmann equilibrium distribution and use the method to examine the time correlation functions for two spheres tightly confined in a cuboid. We model a quasi-two-dimensional colloidal crystal confined in a narrow microchannel and hydrodynamically driven across a commensurate periodic substrate potential mimicking the effect of a corrugated wall. We observe partial and full depinning of the colloidal monolayer from the substrate potential above a certain wall speed, consistent with a transition from static to kinetic friction through propagating kink solitons. Unexpectedly, we find that particles nearest to the boundaries of the domain are the first to be displaced, followed by particles in the middle of the domain.

The 5G Cellular Backhaul Management Dilemma: To Cache or to Serve

To reap the benefits of cache-enabled small cell networks, new backaul management mechanisms are needed to prevent the predicted files that are downloaded at the small base stations (SBSs) for caching purposes from jeopardizing the urgent requests that need to be served via the backhaul. Such mechanisms must account for the heterogeneity of the backhaul that will encompass both wireless backhaul links (at various frequency bands) and a wired backhaul component. In this paper, the heterogeneous backhaul management problem is formulated as a minority game in which each SBS has to define the number of predicted files to download, without affecting the required transmission rate of the current requests. For the formulated game, it is shown that a unique fair proper mixed Nash equilibrium (PMNE) exists. A self-organizing reinforcement learning algorithm is then proposed and shown to converge to a unique Boltzmann-Gibbs equilibrium, which approximates the desired PMNE. Simulation results show that the performance of the proposed approach can be close to that of the ideal optimal algorithm while it outperforms a centralized greedy approach in terms of the amount of data that is cached without jeopardizing the quality-of-service of current requests.

The world of long-range interactions: A bird’s eye view

In recent years, studies of long-range interacting (LRI) systems have taken center stage in the arena of statistical mechanics and dynamical system studies, due to new theoretical developments involving tools from as diverse a field as kinetic theory, non-equilibrium statistical mechanics, and large deviation theory, but also due to new and exciting experimental realizations of LRI systems. In the first, introductory, Section 1, we discuss the general features of long-range interactions, emphasizing in particular the main physical phenomenon of non-additivity, which leads to a plethora of distinct effects, both thermodynamic and dynamic, that are not observed with short-range interactions: Ensemble inequivalence, slow relaxation, broken ergodicity. In Section 2, we discuss several physical systems with long-range interactions: mean-field spin systems, self-gravitating systems, Euler equations in two dimensions, Coulomb systems, one-component electron plasma, dipolar systems, free-electron lasers. In Section 3, we discuss the general scenario of dynamical evolution of generic LRI systems. In Section 4, we discuss an illustrative example of LRI systems, the Kardar–Nagel spin system, which involves discrete degrees of freedom, while in Section 5, we discuss a paradigmatic example involving continuous degrees of freedom, the so-called Hamiltonian mean-field (HMF) model. For the former, we demonstrate the effects of ensemble inequivalence and slow relaxation, while for the HMF model, we emphasize in particular the occurrence of the so-called quasistationary states (QSSs) during relaxation towards the Boltzmann–Gibbs equilibrium state. The QSSs are non-equilibrium states with lifetimes that diverge with the system size, so that in the thermodynamic limit, the systems remain trapped in the QSSs, thereby making the latter the effective stationary states. In Section 5, we also discuss an experimental system involving atoms trapped in optical cavities, which may be modelled by the HMF system. In Section 6, we address the issue of ubiquity of the quasistationary behavior by considering a variety of models and dynamics, discussing in each case the conditions to observe QSSs. In Section 7, we investigate the issue of what happens when a long-range system is driven out of thermal equilibrium. Conclusions are drawn in Section 8.

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.

Temporal integrators for fluctuating hydrodynamics

Including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics. These Langevin equations possess a special fluctuation-dissipation structure that needs to be preserved by spatio-temporal discretizations in order for the computed solution to reproduce the correct long-time behavior. In particular, numerical solutions should approximate the Gibbs-Boltzmann equilibrium distribution, and ideally this will hold even for large time step sizes. We describe finite-volume spatial discretizations for the fluctuating Burgers and fluctuating incompressible Navier-Stokes equations that obey a discrete fluctuation-dissipation balance principle just like the continuum equations. We develop implicit-explicit predictor-corrector temporal integrators for the resulting stochastic method-of-lines discretization. These stochastic Runge-Kutta schemes treat diffusion implicitly and advection explicitly, are weakly second-order accurate for additive noise for small time steps, and give a good approximation to the equilibrium distribution even for very strong fluctuations. Numerical results demonstrate that a midpoint predictor-corrector scheme is very robust over a broad range of time step sizes.

Continuous- and discrete-time Glauber dynamics. First- and second-order phase transitions in mean-field Potts models

As is known, at the Gibbs-Boltzmann equilibrium, the mean-field q-state Potts model with a ferromagnetic coupling has only a first-order phase transition when q ⩾ 3, while there is no phase transition for an antiferromagnetic coupling. The same equilibrium is asymptotically reached when one considers the continuous time evolution according to a Glauber dynamics. In this paper we show that, when we consider instead the Potts model evolving according to a discrete-time dynamics, the Gibbs-Boltzmann equilibrium is reached only when the coupling is ferromagnetic while, when the coupling is anti-ferromagnetic, a period-2 orbit equilibrium is reached and a stable second-order phase transition in the Ising mean-field universality class sets in for each component of the orbit. We discuss the implications of this scenario in real-world problems.

QUANTUM FLUCTUATION DYNAMICS DURING THE TRANSITION OF A MESOSCOPIC BOSONIC GAS INTO A BOSE–EINSTEIN CONDENSATE

The condensate number distribution during the transition of a dilute, weakly interacting gas of N = 200 bosonic atoms into a Bose–Einstein condensate is modeled within number conserving master equation theory of Bose–Einstein condensation. Initial strong quantum fluctuations occuring during the exponential cycle of condensate growth reduce in a subsequent saturation stage, before the Bose gas finally relaxes towards the Gibbs–Boltzmann equilibrium.

An alternative to the conventional micro-canonical ensemble

The usual approach to the foundations of quantum statistical physics is based on the conventional micro-canonical ensemble as a starting point for deriving the Boltzmann–Gibbs (BG) equilibrium. It leaves, however, a number of conceptual and practical questions unanswered. Here we discuss these questions, thereby motivating the study of a natural alternative known as the quantum micro-canonical (QMC) ensemble. We present a detailed numerical study of the properties of the QMC ensemble for finite quantum systems, revealing good agreement with the existing analytical results for large quantum systems. We also propose the way to introduce analytical corrections accounting for finite-size effects. With the above corrections, the agreement between the analytical and the numerical results becomes very accurate. We demonstrate that the variance of energy fluctuations can be used to discriminate the QMC equilibrium from the BG equilibrium.

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.

Linear response theory for long-range interacting systems in quasistationary states

Long-range interacting systems, while relaxing to equilibrium, often get trapped in long-lived quasistationary states which have lifetimes that diverge with the system size. In this work, we address the question of how a long-range system in a quasistationary state (QSS) responds to an external perturbation. We consider a long-range system that evolves under deterministic Hamilton dynamics. The perturbation is taken to couple to the canonical coordinates of the individual constituents. Our study is based on analyzing the Vlasov equation for the single-particle phase-space distribution. The QSS represents a stable stationary solution of the Vlasov equation in the absence of the external perturbation. In the presence of small perturbation, we linearize the perturbed Vlasov equation about the QSS to obtain a formal expression for the response observed in a single-particle dynamical quantity. For a QSS that is homogeneous in the coordinate, we obtain an explicit formula for the response. We apply our analysis to a paradigmatic model, the Hamiltonian mean-field model, which involves particles moving on a circle under Hamiltonian dynamics. Our prediction for the response of three representative QSSs in this model (the water-bag QSS, the Fermi-Dirac QSS, and the Gaussian QSS) is found to be in good agreement with N-particle simulations for large N. We also show the long-time relaxation of the water-bag QSS to the Boltzmann-Gibbs equilibrium state.

Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates

This work describes the statistics for the occupation numbers of quantum levels in a large isolated quantum system, where all possible superpositions of eigenstates are allowed provided all these superpositions have the same fixed energy. Such a condition is not equivalent to the conventional microcanonical condition because the latter limits the participating eigenstates to a very narrow energy window. The statistics is obtained analytically for both the entire system and its small subsystem. In a significant departure from the Boltzmann-Gibbs statistics, the average occupation numbers of quantum states exhibit in the present case weak algebraic dependence on energy. In the macroscopic limit, this dependence is routinely accompanied by the condensation into the lowest-energy quantum state. This work contains initial numerical tests of the above statistics for finite systems and also reports the following numerical finding: when the basis states of large but finite random matrix Hamiltonians are expanded in terms of eigenstates, the participation of eigenstates in such an expansion obeys the newly obtained statistics. The above statistics might be observable in small quantum systems, but for the macroscopic systems, it rather re-enforces doubts about self-sufficiency of nonrelativistic quantum mechanics for justifying the Boltzmann-Gibbs equilibrium.

Boltzmann-Gibbs thermal equilibrium distribution for classical systems and Newton law: a computational discussion

We implement a general numerical calculation that allows for a direct comparison between nonlinear Hamiltonian dynamics and the Boltzmann-Gibbs canonical distribution in Gibbs $\Gamma$-space. Using paradigmatic first-neighbor models, namely, the inertial XY ferromagnet and the Fermi-Pasta-Ulam $\beta$-model, we show that at intermediate energies the Boltzmann-Gibbs equilibrium distribution is a consequence of Newton second law (${\mathbf F}=m{\mathbf a}$). At higher energies we discuss partial agreement between time and ensemble averages.

The Vlasov equation and the Hamiltonian mean-field model

We show that the quasi-stationary states of homogeneous (zero magnetization) states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite N effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann–Gibbs equilibrium.

Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N→∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N, dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via nonstationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann–Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N 1.7. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.

Dynamics and thermodynamics of a model with long-range interactions

The dynamics and the thermodynamics of particles/spins interacting via long-range forces display several unusual features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular the model shows a second order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact the numerical simulations show the presence of quasi-stationary state (QSS), i.e. metastable states which become stable if the thermodynamic limit is taken before the infinite time limit. The QSS differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heat, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly-decaying correlations and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS have been recently interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which is not alternative to the previous one.

Long-term evolution of stellar self-gravitating systems away from thermal equilibrium: connection with nonextensive statistics.

With particular attention to the recently postulated introduction of a nonextensive generalization of Boltzmann-Gibbs statistics, we study the long-term stellar dynamical evolution of self-gravitating systems on time scales much longer than the two-body relaxation time. In a self-gravitating N-body system confined in an adiabatic wall, we show that the quasiequilibrium sequence arising from the Tsallis entropy, so-called stellar polytropes, plays an important role in characterizing the transient states away from the Boltzmann-Gibbs equilibrium state.

Crossover in diffusion equation: anomalous and normal behaviors.

Ubiquitous phenomena exist in nature where, as time goes on, a crossover is observed between different diffusion regimes (e.g., anomalous diffusion at early times which becomes normal diffusion at long times, or the other way around). In order to focus on such situations we have analyzed particular relevant cases of the generalized Fokker-Planck equation integral dgamma(')tau(gamma('))[ partial differential (gamma('))rho(x,t)]/ partial differential t(gamma('))= integral dmu(')dnu'D(mu('),nu('))[ partial differential (mu('))[rho(x,t)](nu('))]/ partial differential x(mu(')), where tau(gamma(')) and D(mu('),nu(')) are kernels to be chosen; the choice tau(gamma('))=delta(gamma(')-1) and D(mu('),nu('))=delta(mu(')-2)delta(nu(')-1) recovers the normal diffusion equation. We discuss in detail the following cases: (i) a mixture of the porous medium equation, which is connected with nonextensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with Lévy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. In cases (i) and (iii), the less diffusive regime occurs for short times, while at long times the more diffusive regime emerges. The opposite occurs in case (ii). The present results could be easily extended to more complex situations (e.g., crossover between two, or even more, different anomalous regimes), and are expected to be useful in the analysis of phenomena where nonlinear and fractional diffusion equations play an important role. Such appears to be the case for isolated long-ranged interaction Hamiltonians, which along time can exhibit a crossover from a longstanding metastable anomalous state to the usual Boltzmann-Gibbs equilibrium one. Another illustration of such crossover occurs in active intracellular transport.