Takaaki Monnai
We evaluate the relaxation time to equilibrium, and especially show that it is almost independent from the system size for macroscopic isolated quantum systems. It at most polynomially depends on the system size. This estimation holds when the Hamiltonian is non-integrable, the initial deviation of the quantity of interest is of order its spectral norm, and the relaxation process is monotonic.
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A quantum mechanical explanation of the relaxation to equilibrium is shown for macroscopic systems for nonintegrable cases and numerically verified. The macroscopic system is initially in an equilibrium state, subsequently externally perturbed during a finite time, and then isolated for a sufficiently long time. We show a quantitative explanation that the initial microcanonical state typically reaches to a state whose expectation values are well-approximated by the average over another microcanonical ensemble.
We constructed a model that evolved from a non-equilibrium state to an equilibrium state. The model only needs two basic coefficients, including self-similar coefficients and non-equilibrium coefficients. The coefficients of the model can be obtained directly from real data. The model can well represent the phenomena of anomalous diffusion, non-Gaussian distribution and so on in non-equilibrium complex systems. It is important to mention that the model is a global iterative model that can exhibit statistical physical properties on multiple time scales, as well as exhibiting long-term memory pr…
We present the first analytic study of finite-size effects on critical diffusion above and below T_c of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below T_c. Two new universal dynamic amplitude ratios at T_c are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte-Carlo simulations.
Far from equilibrium relaxation in the weak coupling limit
本文以伊辛链为例，证明离散能量系统在无限弱耦合下仍可因能量景观起伏而出现远离平衡的异常弛豫，挑战了传统准静态弛豫假设。
We employ Monte Carlo simulations to study the non-equilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density auto-correlation function in the non-equilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time auto-correlations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice …
Nonequilibrium relaxation behaviors in the Ising model on a square lattice based on the Wolff algorithm are totally different from those based on local-update algorithms. In particular, the critical relaxation is described by the stretched-exponential decay. We propose a novel scaling procedure to connect nonequilibrium and equilibrium behaviors continuously, and find that the stretched-exponential scaling region in the Wolff algorithm is as wide as the power-law scaling region in local-update algorithms. We also find that relaxation to the spontaneous magnetization in the ordered phase is cha…
In this contribution we show that a suitably defined nonequilibrium entropy of an N-body isolated system is not a constant of the motion in general and its variation is bounded, the bounds determined by the thermodynamic entropy, i.e., the equilibrium entropy. We define the nonequilibrium entropy as a convex functional of the set of n-particle reduced distribution functions (n=0,......., N) generalizing the Gibbs fine-grained entropy formula. Additionally, as a consequence of our microscopic analysis we find that this nonequilibrium entropy behaves as a free entropic oscillator. In the approac…
The melting transition in the hard-disk system is considered. Non-equilibrium relaxation analysis of the six-fold bond-orientational order parameter has been carried out. The critical point between the hexatic and the fluid phase is determined on the basis of the dynamic scaling hypothesis. The value of the critical exponent $η$ is determined from the fluctuation of the order parameter at the criticality as $η= 0.25(2)$ which is consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.
Systems with long-range interactions often relax towards statistical equilibrium over timescales that diverge with $N$, the number of particles. A recent work [S. Gupta and D. Mukamel, J. Stat. Mech.: Theory Exp. P03015 (2011)] analyzed a model system comprising $N$ globally coupled classical Heisenberg spins and evolving under classical spin dynamics. It was numerically shown to relax to equilibrium over a time that scales superlinearly with $N$. Here, we present a detailed study of the Lenard-Balescu operator that accounts at leading order for the finite-$N$ effects driving this relaxation. …
The problem of relaxation to equilibrium
本文通过非严格边界条件研究经典气体系统趋向平衡的过程，证明在海森堡不确定性原理指导下可实现平衡态。
The Mpemba effect denotes an anomalous relaxation phenomenon where a system initially at a hot temperature cools faster than a system that starts at a less elevated temperature. We introduce an isothermal analog of this effect for a system prepared in a non-equilibrium steady state that then relaxes towards equilibrium. Here, the driving strength, which determines the initial non-equilibrium steady state, takes the role of the temperature in the original version. As a paradigm, we consider a particle initially driven by a non-conservative force along a one-dimensional periodic potential. We sh…
We consider the non-stationary quantum relaxation of the Ising spin chain in a transverse field of strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary state by a process possessing quasi long range correlations in time and space, independent of the value of $h$. In particular the system exhibits aging (or lack of time translational invariance on intermediate time scales) although no indications of coarsening are present.
In this work, the relaxation process of the spin-3/2 Blume-Capel model with quenched random crystal field on a two dimensional square lattice has been investigated by a method which combines the statistical equilibrium theory and the thermodynamic of linear irreversible processes. Generalized force and flux have been defined in irreversible thermodynamics limit. The kinetic equation for the magnetization has been obtained by using linear response theory. In order to understand effect of the first-order phase transition on the relaxation process, low temperature regions have been analyzed. And …
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 …
We investigate the evolution of a system composed of $N$ non-interacting point particles of mass $m$ in a container divided into two chambers by a movable adiabatic piston of mass $M\gg m$. Using a two-time-scale perturbation approach in terms of the small parameter $α=2m/(M+m)$, we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times ${\cal O}(M)$, is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A ve…
Paper in honour of Freeman Dyson on the occasion of his 80th birthday. Normal N-body systems relax to equilibrium distributions in which classical kinetic energy components are 1/2 kT, but, when inter-particle forces are an inverse cubic repulsion together with a linear (simple harmonic) attraction, the system pulsates for ever. In spite of this pulsation in scale, r(t), other degrees of freedom relax to an ever-changing Maxwellian distribution. With a new time, tau, defined so that r^2d/dt =d/d tau it is shown that the remaining degrees of freedom evolve with an unchanging reduced Hamiltonian…
Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually…
In the present paper the problem of the relaxation of magnetization to equilibrium (i.e., with no magnetization) is investigated numerically for a variant of the well-known model introduced by Bohr to study the diamagnetism of electrons in metals. Such a model is mathematically equivalent to a billiard with obstacles in a magnetic field. We show that it is not guaranteed that equilibrium is attained within the typical time scales of microscopic dynamics. Indeed, considering an out of equilibrium state produced by an adiabatic switching on of a magnetic field, we show that, depending on the val…
A classical long-range-interacting $N$-particle system relaxes to thermal equilibrium on time scales growing with $N$; in the limit $N\to \infty$ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of $N$ and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution fun…
Relaxation of the thermal Casimir or van der Waals force for a model dielectric medium is investigated. We start with a model of interacting polarization fields with a dynamics that leads to a frequency dependent dielectric constant of the Debye form. In the static limit the usual zero frequency Matsubara mode component of the Casimir force is recovered. We then consider the out of equilibrium relaxation of the van der Waals force to its equilibrium value when two initially uncorrelated dielectric bodies are brought into sudden proximity. It is found that the spatial dependence of the out of e…
We study the non-equilibrium relaxation of an elastic line described by the Edwards-Wilkinson equation. Although this model is the simplest representation of interface dynamics, we highlight that many (not though all) important aspects of the non-equilibrium relaxation of elastic manifolds are already present in such quadratic and clean systems. We analyze in detail the aging behaviour of several two-times averaged and fluctuating observables taking into account finite-size effects and the crossover to the stationary and equilibrium regimes. We start by investigating the structure factor and e…
We provide evidence of an extremely slow thermalization occurring in the Discrete NonLinear Schrödinger (DNLS) model. At variance with many similar processes encountered in statistical mechanics - typically ascribed to the presence of (free) energy barriers - here the slowness has a purely dynamical origin: it is due to the presence of an adiabatic invariant, which freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events, where energy is suddenly released towards the background. We conjecture that this exponentially slow relaxation is a key ingredient contribu…
We examine the question of the criteria of the relaxation to the equilibrium in the hard disk dynamics. In the Event-Chain Monte Carlo, we check the displacement distributions which follows to the exponential law.
Relaxation to equilibrium after strong and collective excitation is studied, by using a Hamiltonian dynamical system of one dimensional XY model. After an excitation of a domain of $K$ elements, the excitation is concentrated to fewer elements, which are made farther away from equilibrium, and the excitation intensity increases logarithmically with $K$. Equilibrium is reached only after taking this ``roundabout'' route, with the time for relaxation diverging asymptotically as $K^γ$ with $γ\approx 4.2$.
We elucidate a non-conserved relaxational nonequilibrium dynamics of a O(2) symmetric model. We drive the system out of equilibrium by introducing a non-zero noise cross-correlation of amplitude $D_\times$ in a stochastic Langevin description of the system, while maintaining the O(2) symmetry of the order parameter space. By performing dynamic renormalization group calculations in a field-theoretic set up, we analyze the ensuing nonequilibrium steady states and evaluate the scaling exponents near the critical point, which now depend explicitly on $D_\times$. Since the latter remains unrenormal…
Finite-time transitions in optimal control and non-equilibrium relaxation
研究有限时间最优控制中，粒子在非均匀环境中运动时，状态依赖惩罚与路径依赖耗散的竞争导致控制策略在临界时间出现突变，并与非平衡弛豫中的动态相变相关联。
Eigenvalue crossing as a phase transition in relaxation dynamics
本文研究了系统参数骤变后的弛豫动力学，发现弛豫矩阵第二和第三本征值交叉会导致类似一阶相变的轨迹奇点，并在4态系统和一维Ising模型中验证了该现象。
We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetisation that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterised by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase t…
Off-equilibrium dynamics of a three-dimensional lattice model with nearest- and next nearest-neighbors exclusions is studied. At equilibrium, the model undergoes a first-order fluid-solid transition. Non-equilibrium filling, through random sequential adsorption with diffusion, creates amorphous structures and terminates at a disordered state with random closest packing density that lies in the equilibrium solid regime. The approach towards random closest packing is characterized by two distinct power-law regimes, reflecting the formation of small densely packed grains in the long time regime o…
We employ Monte Carlo simulations to investigate the non-equilibrium relaxation properties of the two- and three-dimensional Coulomb glass with different long-range repulsive interactions. Specifically, we explore the aging scaling laws in the two-time density autocorrelation function. We find that in the time window and parameter range accessible to us, the scaling exponents are not universal, depending on the filling fraction and temperature: As either the temperature decreases or the filling fraction deviates more from half-filling, the exponents reflect markedly slower relaxation kinetics.…
The Hamiltonian Mean-Field model (HMF), an inertial XY ferromagnet with infinite-range interactions, has been extensively studied in the last few years, especially due to its long-lived meta-equilibrium states, which exhibit a series of anomalies, such as, breakdown of ergodicity, anomalous diffusion, aging, and non-Maxwell velocity distributions. The most widely investigated meta-equilibrium states of the HMF arise from special (fully magnetized) initial conditions that evolve to a spatially homogeneous state with well defined macroscopic characteristics and whose lifetime increases with the …
Starting from the second law of thermodynamics applied to an isolated system consisting of the system surrounded by an extremely large medium, we formulate a general non-equilibrium thermodynamic description of the system when it is out of equilibrium. We then apply it to study the structural relaxation in glasses and establish the phenomenology behind the concept of the fictive temperature and of the empirical Tool-Narayanaswamy equation on firmer theoretical foundation.
Ferromagnetic transition in double-exchange systems is studied by non-equilibrium relaxation technique combined with Monte Carlo calculations. Critical temperature and critical exponents are estimated from relaxation of the magnetic moment. The results are consistent with the previous Monte Carlo results in thermal equilibrium. The exponents estimated by these independent techniques suggest that the universality class of this transition is the same as that of short-range interaction models but is different from the mean-field one.
It is well known that ergodic theory can be used to formally prove a weak form of relaxation to equilibrium for finite, mixing, Hamiltonian systems. In this Letter we extend this proof to any dynamics that preserves a mixing equilibrium distribution. The proof uses an approach similar to that used in umbrella sampling, and demonstrates the need for a form of ergodic consistency of the initial and final distribution. This weak relaxation only applies to averages of physical properties. It says nothing about whether the distribution of states relaxes towards the equilibrium distribution or how l…
We present a dynamical description of slow relaxation processes based on the extension of Onsager's fluctuation theory to systems in local quasi-equilibrium. A non-Markovian Fokker-Planck equation for the conditional probability density is derived and from it we obtain the relaxation equation for the moments. We show that the fluctuation-dissipation theorem can be formulated in terms of the temperature of the system at local quasi-equilibrium. Our theory may be applied to a wide variety of systems undergoing slow relaxation. We discuss in particular slow dynamics in glassy systems and Brownian…
After a general overview of some features of the relaxation dynamics of the Hamiltonian Mean Field model, its equilibrium thermodynamic properties are used to rephrase the out-of-equilibrium regime for energies below the critical point $u_{c}=0.75$ in terms of an effective dynamical coexistence between a clustered and a gaseous phases, whose existence could be associated to the large relaxation times observed when $u_{1}<u<u_{c}$, with $u_{1}=0.5$. Starting from the hypothesis that the \textit{parametric resonance} is the microscopic mechanism allowing the energetic interchange between the par…
It has recently been shown that relaxation of the rotational energy of hot non-equlibrium photofragments (i) slows down significantly with the increase of their initial rotational temperature and (ii) differs dramatically from the relaxation of the equilibrium rotational energy correlation function, manifesting thereby breakdown of the linear response description [Science 311, 1907 (2006)]. We demonstrate that this phenomenon may be caused by the angular momentum dependence of rotational friction. We have developed the generalized Fokker-Planck equation whose rotational friction depends upon a…
We examine the problem of how excited populations of electrons relax after they have been excited by a pump. We include three of the most important relaxation processes: (i) impurity scattering; (ii) Coulomb scattering; and (iii) electron-phonon scattering. The relaxation of an excited population of electrons is one of the most fundamental processes measured in pump/probe experiments, but its interpretation remains under debate. We show how several common assumptions about non-equilibrium relaxation that are pervasive in the field may not hold under quite general conditions. The analysis shows…
Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSSs) whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation", predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (non zero magnetization) QSSs. When varying the initial condition within a family of "water-bags" with different initial magnetization…
Adding transitions to an equilibrium system increases the activity. Naively, one would expect this to hold also in out of equilibrium systems. This surprising effect is caused by adding heretofore forbidden transitions into less and less active states. We demonstrate, using relatively simple models, how adding transitions to an out of equilibrium system may in fact reduce the activity and even cause it to vanish. We investigate six related kinetically-constrained lattice gas models, some of which behave as naively expected while others exhibit this non-intuitive behavior. We introduce a semi-m…
The concept of a non-equilibrium interfacial tension, defined via the work required to deform the system such that the interfacial area is changed while the volume is conserved, is investigated theoretically in the context of the relaxation of an initial perturbation of a colloidal fluid towards the equilibrium state. The corresponding general formalism is derived for systems with planar symmetry and applied to fluid models of colloidal suspensions and polymer solutions. It is shown that the non-equilibrium interfacial tension is not necessarily positive, that negative non-equilibrium interfac…
Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures
本文针对随机图上的Ising p-spin模型，提出改进的动力学平均场闭合方案，通过重整化微观时间尺度，准确捕捉低温玻璃态的两步弛豫，并与蒙特卡洛模拟结果完美匹配。
We propose a new efficient scheme for the quantum Monte Carlo study of quantum critical phenomena in quantum spin systems. Rieger and Young's Trotter-number-dependent finite-size scaling in quantum spin systems and Ito {\it et al.}'s evaluation of the transition point with the non-equilibrium relaxation in classical spin systems are combined and generalized. That is, only one Trotter number and one inverse temperature proportional to system sizes are taken for each system size, and the transition point of the transformed classical spin model is estimated as the point at which the order paramet…
This letter reports experimental and numerical results on particle dynamics in an out-of-equilibrium granular medium. We observed two distinct types of grain motion: the well known cage motion, during which a grain is always surrounded by the same neighbors, and low probability "jumps", during which a grain moves significantly more relative to the others. These observations are similar to the results obtained for other out-of-equilibrium systems (glasses, colloidal systems, etc.). Although such jumps are extremely rare, by inhibiting them in numerical simulations we demonstrate that they play …
In these four lectures I describe basic ideas and methods applicable to both classical and quantum systems displaying slow relaxation and non-equilibrium dynamics. The first half of these notes considers classical systems, and the second half, quantum systems. In Lecture 1, I briefly review the glass transition problem as a paradigm of slow relaxation and dynamical arrest in classical many-body systems. I discuss theoretical perspectives on how to think about glasses, and in particular how to model them in terms of kinetically constrained dynamics. In Lecture 2, I describe how via large deviat…
Recent studies have experienced the acceleration of convergence in Markov chain Monte Carlo methods implemented by the systems without detailed balance condition (DBC). However, such advantage of the violation of DBC has not been confirmed in general. We investigate the effect of the absence of DBC on the convergence toward equilibrium. Surprisingly, it is shown that the DBC violation always makes the relaxation faster. Our result implies the existence of a kind of thermodynamic inequality that connects the nonequilibrium process relaxing toward steady state with the relaxation process which h…
Fluctuation theorems are a generalization of thermodynamics on small scales and provide the tools to characterize the fluctuations of thermodynamic quantities in non-equilibrium nanoscale systems. They are particularly important for understanding irreversibility and the second law in fundamental chemical and biological processes that are actively driven, thus operating far from thermal equilibrium. Here, we apply the framework of fluctuation theorems to investigate the important case of a system relaxing from a non-equilibrium state towards equilibrium. Using a vacuum-trapped nanoparticle, we …
It is common practice in molecular dynamics and Monte Carlo computer simulations to run multiple, separately-initialized simulations in order to improve the sampling of independent microstates. Here we examine the utility of an extreme case of this strategy, in which we run a large ensemble of $M$ independent simulations (a "swarm"), each of which is relaxed to equilibrium. We show that if $M$ is of order $10^3$, we can monitor the swarm's relaxation to equilibrium, and confirm its attainment, within $\sim 10\barτ$, where $\barτ$ is the equilibrium relaxation time. As soon as a swarm of this s…
Relaxation of Femtosecond Non-equilibrium Electrons in a Metallic Sample
针对飞秒时间尺度下双温模型不适用的问题，建立金属热电子弛豫动力学模型，考虑费米能级附近费米阻塞导致的弛豫减慢效应，预测热电子分布的费米能级峰值，可通过飞秒泵浦-探测实验验证。
We study thermal relaxation in ordered arrays of coupled nonlinear elements with external driving. We find, that our model exhibits dynamic self-organization manifested in a universal stretched-exponential form of relaxation. We identify two types of self-organization, cooperative and anti-cooperative, which lead to fast and slow relaxation, respectively. We give a qualitative explanation for the behavior of the stretched exponent in different parameter ranges. We emphasize that this is a system exhibiting stretched-exponential relaxation without explicit disorder or frustration.
Systems far from equilibrium approach stability slowly due to "anti-mixing" characterized by regions of the phase-space that remain disconnected after prolonged action of the flow. We introduce the Optimal Growth Mode (OGM) to capture this slow initial relaxation. The OGM is calculated from Markov matrices approximating the action of the Fokker-Planck operator onto the phase space. It is obtained as the mode having the largest growth in energy before decay. Important nuances between the OGM and the more traditional slowest decaying mode are detailed in the case of the Lorenz 63 model. The impl…
We study the non--equilibrium motion of an elastic string in a two dimensional pinning landscape using Langevin dynamics simulations. The relaxation of a line, initially flat, is characterized by a growing length, $L(t)$, separating the equilibrated short length scales from the flat long distance geometry that keep memory of the initial condition. We show that, in the long time limit, $L(t)$ has a non--algebraic growth with a universal distribution function. The distribution function of waiting times is also calculated, and related to the previous distribution. The barrier distribution is narr…
The theory of entropy production in nonequilibrium, Hamiltonian systems, previously described for steady states using partitions of phase space, is here extended to time dependent systems relaxing to equilibrium. We illustrate the main ideas by using a simple multibaker model, with some nonequilibrium initial state, and we study its progress toward equilibrium. The central results are (i) the entropy production is governed by an underlying, exponentially decaying fractal structure in phase space, (ii) the rate of entropy production is largely independent of the scale of resolution used in the …
We study instabilities and relaxation to equilibrium in a long-range extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting initially the lowest Fourier mode. Localization in mode space is stronger for the long-range FPU model. This allows us to uncover the sporadic nature of instabilities, i.e., by varying initially the excitation amplitude of the lowest mode, which is the control parameter, instabilities occur in narrow amplitude intervals. Only for sufficiently large values of the amplitude, the system enters a permanently unstable regime. These findings also clarify t…
We study the relaxation of a spin I that is weakly coupled to a quantum mechanical environment. Starting from the microscopic description, we derive a system of coupled relaxation equations within the adiabatic approximation. These are valid for arbitrary I and also for a general stationary non--equilibrium state of the environment. In the case of equilibrium, the stationary solution of the equations becomes the correct Boltzmannian equilibrium distribution for given spin I. The relaxation towards the stationary solution is characterized by a set of relaxation times, the longest of which can b…
We revisit the issue of relaxation to thermal equilibrium in the so-called "sheet model", i.e., particles in one dimension interacting by attractive forces independent of their separation. We show that this relaxation may be very clearly detected and characterized by following the evolution of order parameters defined by appropriately normalized moments of the phase space distribution which probe its entanglement in space and velocity coordinates. For a class of quasi-stationary states which result from the violent relaxation of rectangular waterbag initial conditions, characterized by their v…
The long-time behaviour of spin-spin correlators in the slow relaxation of systems undergoing phase-ordering kinetics is studied in geometries of finite size. A phenomenological finite-size scaling ansatz is formulated and tested through the exact solution of the kinetic spherical model, quenched to below the critical temperature, in $2<d<4$ dimensions.
The aging regime of the trap model, observed for a temperature T below the glass transition temperature T_g, is a prototypical example of non-stationary out-of-equilibrium state. We characterize this state by evaluating its "distance to equilibrium", defined as the Shannon entropy difference ΔS (in absolute value) between the non-equilibrium state and the equilibrium state with the same energy. We consider the time evolution of ΔS and show that, rather unexpectedly, ΔS(t) continuously increases in the aging regime, if the number of traps is infinite, meaning that the "distance to equilibrium" …
So far, only the energy thresholds of single discrete breathers in nonlinear Hamiltonian systems have been analytically obtained. In this work, the energy thresholds of discrete breathers in thermal equilibrium and the energy thresholds of long-lived discrete breathers which can remain after a long time relaxation are analytically estimated for nonlinear chains. These energy thresholds are size dependent. The energy thresholds of discrete breathers in thermal equilibrium are same as the previous analytical results for single discrete breathers. The energy thresholds of long-lived discrete brea…
Mesoscopic non-equilibrium thermodynamics is used to formulate a model describing non-homogeneous and non-Debye dielectric relaxation. The model is presented in terms of a Fokker-Planck equation for the probability distribution of non-interacting polar molecules in contact with a heat bath and in the presence of an external time-dependent electric field. Memory effects are introduced in the Fokker-Planck description through integral relations containing memory kernels, which in turn are used to establish a connection with fractional Fokker-Planck descriptions. The model is developed in terms o…
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