Off-equilibrium Dynamics Research Articles

Introduction to the dynamics of disordered systems: Equilibrium and gradient descent

This paper contains the lecture notes of the short courses given by one of us (F.Z.) at the summer school Fundamental Problems in Statistical Physics XV, held in Brunico, Italy, in July 2021, and, just before that, at the summer school Glassy Systems and Inter-Disciplinary Applications, held in Cargese, France, in June 2021. The course was a short introductory overview of the dynamics of disordered systems, focused in particular on the equilibrium dynamics (with the associated glass transition), and on the simplest case of off-equilibrium dynamics, namely gradient descent. A few selected topics (and references) are chosen, based on the authors’ own taste and competences, and on pedagogical reasons, without aiming at a complete review of the subject.

Temperature chaos is present in off-equilibrium spin-glass dynamics

Experiments featuring non-equilibrium glassy dynamics under temperature changes still await interpretation. There is a widespread feeling that temperature chaos (an extreme sensitivity of the glass to temperature changes) should play a major role but, up to now, this phenomenon has been investigated solely under equilibrium conditions. In fact, the very existence of a chaotic effect in the non-equilibrium dynamics is yet to be established. In this article, we tackle this problem through a large simulation of the 3D Edwards-Anderson model, carried out on the Janus II supercomputer. We find a dynamic effect that closely parallels equilibrium temperature chaos. This dynamic temperature-chaos effect is spatially heterogeneous to a large degree and turns out to be controlled by the spin-glass coherence length ξ. Indeed, an emerging length-scale ξ* rules the crossover from weak (at ξ ≪ ξ*) to strong chaos (ξ ≫ ξ*). Extrapolations of ξ* to relevant experimental conditions are provided.

Quantized aging mode in metallic glass-forming liquids

Physical aging exhibits complex off-equilibrium dynamics and its studies are of particular importance for understanding the nature of glassy state. We investigate the aging process of supercooled Cu50Zr50 glass-forming liquid numerically and theoretically. The relaxation time spectra based on molecular dynamics simulations show that the aging dynamics features the superposition of discrete homogeneous dynamic modes. The visiting probabilities of these modes decay toward slower relaxation regime, which is tuned by aging time from the compressed to the stretched exponential manner. Such multi-mode dynamics is found to be accompanied by the quantized hierarchy of mobile clusters with identical size increment. With this finding, we propose an aging model, which describes the aging dynamics as the aging-dependent weighted average over quantized dynamic modes. The two-time correlation functions associated with aging thus are separated into aging-time dependent factor and aging independent relaxation-time term. The model well predicts the heterogeneous dynamics in aging and equilibration of supercooled metallic liquids, and reveals the intrinsic aspects influencing aging process.

Quantized Aging Mode in Metallic Glass-Forming Liquids

Physical aging exhibits complex off-equilibrium dynamics and its studies are of particular importance for understanding the nature of glassy state. We investigate the aging process of supercooled Cu50Zr50 glass-forming liquid numerically and theoretically. The relaxation time spectra based on molecular dynamics simulations show that the aging dynamics features the superposition of discrete homogeneous dynamic modes. The visiting probabilities of these modes decay toward slower relaxation regime, which is tuned by aging time from the compressed to the stretched exponential manner. Such multi-mode dynamics is found to be accompanied by the quantized hierarchy of mobile clusters with identical size increment. With this finding, we propose an aging model, which describes the aging dynamics as the aging-dependent weighted average over quantized dynamic modes. The two-time correlation functions associated with aging thus are separated into aging-time dependent factor and aging independent relaxation-time term. The model predicts the heterogeneous dynamics in aging and equilibration of supercooled metallic liquids and reveals the intrinsic aspects influencing aging process.

Non-Markovian Methods in Glass Transition.

A model for the heterogeneity of local dynamics in polymer and other glass-forming materials is provided here. The fundamental characteristics of the glass transition phenomenology emerge when simulating a condensed matter open cluster that has a strong interaction with its heterogeneous environment. General glass transition features, such as non-exponential structural relaxations, the slowing down of relaxation times with temperature and specific off-equilibrium glassy dynamics can be reproduced by non-Markovian dynamics simulations with the minimum computer resources. Non-Markovian models are shown to be useful tools for obtaining insights into the complex dynamics involved in the glass transition phenomenon, including whether or not there is a need for a growing correlation length or the relationship between the non-exponentiality of structural relaxations and dynamic heterogeneity.

Dynamical off-equilibrium scaling across magnetic first-order phase transitions

We investigate the off-equilibrium dynamics of a classical spin system with symmetry in 2 < D < 4 spatial dimensions and in the limit . The system is set up in an ordered equilibrium state and is subsequently driven out of equilibrium by slowly varying the external magnetic field h across the transition line at fixed temperature . We distinguish the cases where the magnetic transition is continuous and where the transition is discontinuous. In the former case, we apply a standard Kibble–Zurek approach to describe the non-equilibrium scaling and formally compute the correlation functions and scaling relations. For the discontinuous transition we develop a scaling theory which builds on the coherence length rather than the correlation length since the latter remains finite for all times. Finally, we derive the off-equilibrium scaling relations for the hysteresis loop area during a round-trip protocol that takes the system across its phase transition and back. Remarkably, our results are valid beyond the large-n limit.

Dynamic scaling behavior at thermal first-order transitions in systems with disordered boundary conditions

We investigate the equilibrium and off-equilibrium behaviors of systems at thermal first-order transitions (FOTs) when the boundary conditions favor one of the two phases. As a theoretical laboratory we consider the two-dimensional Potts model. We show that an anomalous finite-size scaling emerges in systems with open boundary conditions favoring the disordered phase, associated with a mixed regime where the two phases are spatially separated. Correspondingly, if the system is slowly heated across the transition, the characteristic times of the off-equilibrium dynamics scale with a power of the size. We argue that these features generally apply to systems at FOTs, when boundary conditions favor one of the two phases. In particular, they should be relevant for the experimental search of FOTs of the quark-gluon plasma in heavy-ion collisions.

Off-equilibrium infrared structure of self-interacting scalar fields: Universal scaling, vortex-antivortex superfluid dynamics, and Bose-Einstein condensation

We map the infrared dynamics of a relativistic single component ($N=1$) interacting scalar field theory to that of nonrelativistic complex scalar fields. The Gross-Pitaevskii (GP) equation, describing the real time dynamics of single component ultracold Bose gases, is obtained at first nontrivial order in an expansion proportional to the powers of $\lambda \phi^2/m^2$ where $\lambda$, $\phi$ and $m$ are the coupling constant, the scalar field and the particle mass respectively. Our analytical studies are corroborated by numerical simulations of the spatial and momentum structure of overoccupied scalar fields in (2+1)-dimensions. Universal scaling of infrared modes, vortex-antivortex superfluid dynamics and the off-equilibrium formation of a Bose-Einstein condensate are observed. Our results for the universal scaling exponents are in agreement with those extracted in the numerical simulations of the GP equation. As in these simulations, we observe coarsening phase kinetics in the Bose superfluid with strongly anomalous scaling exponents relative to that of vertex resummed kinetic theory. Our relativistic field theory framework further allows one to study more closely the coupling between superfluid and normal fluid modes, specifically the turbulent momentum and spatial structure of the coupling between a quasi-particle cascade to the infrared and an energy cascade to the ultraviolet. We outline possible applications of the formalism to the dynamics of vortex-antivortex formation and to the off-equilibrium dynamics of the strongly interacting matter formed in heavy-ion collisions.

Spin-glass-like aging in colloidal and granular glasses.

Motivated by the mean field prediction of a Gardner phase transition between a "normal glass" and a "marginally stable glass", we investigate the off-equilibrium dynamics of three-dimensional polydisperse hard spheres, used as a model for colloidal or granular glasses. Deep inside the glass phase, we find that a sharp crossover pressure PG separates two distinct dynamical regimes. For pressure P < PG, the glass behaves as a normal solid, displaying fast dynamics that quickly equilibrate within the glass free energy basin. For P > PG, instead, the dynamics become strongly anomalous, displaying very large equilibration timescales, aging, and a constantly increasing dynamical susceptibility. The crossover at PG is strongly reminiscent of the one observed in three-dimensional spin-glasses in an external field, suggesting that the two systems could be in the same universality class, consistent with theoretical expectations.

Global view on coupled dynamics of heavy and light flavor observables from EPOSHQ

We present an analysis of the comparison between 2nd and 3rd flow harmonics of π, protons and D mesons produced in ultrarelativistic heavy ion collisions. We advocate that such an analysis could turn out to be a good strategy in order to quantify the off-equilibrium dynamics of heavy flavor at lower transverse momentum pT.

Fixed points and flow analysis on off-equilibrium dynamics in the boson Boltzmann equation

We consider fixed points of steady solutions and flow directions using the boson Boltzmann equation that is a one-dimensionally reduced kinetic equation after the angular integration. With an elastic collision integral of the two-to-two scattering process, in the dense (dilute) regime where the distribution function is large (small), the boson Boltzmann equation has approximate fixed points with a power-law spectrum in addition to the thermal distribution function. We argue that the power-law fixed point can be exact in special cases. We elaborate a graphical presentation to display evolving flow directions similarly to the renormalization group flow, which explicitly exhibits how fixed points are connected and parameter space is separated by critical lines. We discuss that such a flow diagram contains useful information on thermalization processes out of equilibrium.

Time reversibility from visibility graphs of nonstationary processes.

Visibility algorithms are a family of methods to map time series into networks, with the aim of describing the structure of time series and their underlying dynamical properties in graph-theoretical terms. Here we explore some properties of both natural and horizontal visibility graphs associated to several nonstationary processes, and we pay particular attention to their capacity to assess time irreversibility. Nonstationary signals are (infinitely) irreversible by definition (independently of whether the process is Markovian or producing entropy at a positive rate), and thus the link between entropy production and time series irreversibility has only been explored in nonequilibrium stationary states. Here we show that the visibility formalism naturally induces a new working definition of time irreversibility, which allows us to quantify several degrees of irreversibility for stationary and nonstationary series, yielding finite values that can be used to efficiently assess the presence of memory and off-equilibrium dynamics in nonstationary processes without the need to differentiate or detrend them. We provide rigorous results complemented by extensive numerical simulations on several classes of stochastic processes.

Dynamical transition in the D=3 Edwards-Anderson spin glass in an external magnetic field.

We study the off-equilibrium dynamics of the three-dimensional Ising spin glass in the presence of an external magnetic field. We have performed simulations both at fixed temperature and with an annealing protocol. Thanks to the Janus special-purpose computer, based on field-programmable gate array (FPGAs), we have been able to reach times equivalent to 0.01 s in experiments. We have studied the system relaxation both for high and for low temperatures, clearly identifying a dynamical transition point. This dynamical temperature is strictly positive and depends on the external applied magnetic field. We discuss different possibilities for the underlying physics, which include a thermodynamical spin-glass transition, a mode-coupling crossover, or an interpretation reminiscent of the random first-order picture of structural glasses.

Replica-symmetry-breaking transitions and off-equilibrium dynamics

I consider branches of replica-symmetry-breaking (RSB) solutions in glassy systems that display a dynamical transition at a temperature T_{d} characterized by a mode-coupling-theory dynamical behavior. Below T_{d} these branches of solutions are considered to be relevant to the system complexity and to off-equilibrium dynamics. Under general assumptions I argue that near T_{d} it is not possible to stabilize the one-step (1RSB) solution beyond the marginal point by making a full RSB (FRSB) ansatz. However, depending on the model, there may exist a temperature T strictly lower than T_{d} below which the 1RSB branch can be continued to a FRSB branch. Such a temperature certainly exists for models that display the so-called Gardner transition and in this case T_{G}<T_<T_{d}. An analytical study in the context of the truncated model reveals that the FRSB branch of solutions below T is characterized by a two-plateau structure and it ends where the first plateau disappears. These general features are confirmed in the context of the Ising p-spin model with p=3 by means of a numerical solution of the FRSB equations. The results are discussed in connection with off-equilibrium dynamics within Cugliandolo-Kurchan theory. In this context I assume that the RSB solution relevant for off-equilibrium dynamics is the 1RSB marginal solution in the whole range (T ,T_{d}) and it is the end point of the FRSB branch for T<T. Remarkably, under these assumptions it can be argued that T marks a qualitative change in off-equilibrium dynamics in the sense that the decay of various dynamical quantities changes from power law to logarithmic.

Critical off-equilibrium dynamics in glassy systems

We consider off-equilibrium dynamics at the critical temperature in a class of glassy system. The off-equilibrium correlation and response functions obey a precise scaling form in the aging regime. The structure of the {\it equilibrium} replicated Gibbs free energy fixes the corresponding {\it off-equilibrium} scaling functions implicitly through two functional equations. The details of the model enter these equations only through the ratio $w_2/w_1$ of the cubic coefficients (proper vertexes) of the replicated Gibbs free energy. Therefore the off-equilibrium dynamical exponents are controlled by the very same parameter exponent $\lambda=w_2/w_1$ that determines equilibrium dynamics. We find approximate solutions to the equations and validate the theory by means of analytical computations and numerical simulations.

Jamming, relaxation, and crystallization of a supercooled fluid in a three-dimensional lattice

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. Nonequilibrium 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 toward random closest packing is characterized by two distinct power-law regimes, reflecting the formation of small densely packed grains in the long-time regime of the filling process. We then study the fixed-density relaxation of these amorphous structures toward the solid phase. The route to crystallization for the high-density, deeply supercooled regime is shown to deviate from the simple grain growth proposed by classical nucleation theory. Our measurements suggest that relaxation in this regime is driven mainly by coalescence of neighboring crystallized grains which exist in the initial amorphous state.

Equilibrium and off-equilibrium trap-size scaling in one-dimensional ultracold bosonic gases

We study some aspects of equilibrium and off equilibrium quantum dynamics of dilute bosonic gases in the presence of a trapping potential. We consider systems with a fixed number of particles N and study their scaling behavior with increasing the trap size. We focus on one-dimensional (1D) bosonic systems, such as gases described by the Lieb-Liniger model and its Tonks-Girardeau limit of impenetrable bosons, and gases constrained in optical lattices as described by the Bose-Hubbard model. We study their quantum (zero-temperature) behavior at equilibrium and off equilibrium during the unitary time evolution arising from changes of the trapping potential, which may be instantaneous or described by a power-law time dependence, starting from the equilibrium ground state for a initial trap size. Renormalization-group scaling arguments, analytical and numerical calculations show that the trap-size dependence of the equilibrium and off-equilibrium dynamics can be cast in the form of a trap-size scaling in the low-density regime, characterized by universal power laws of the trap size, in dilute gases with repulsive contact interactions and lattice systems described by the Bose-Hubbard model. The scaling functions corresponding to several physically interesting observables are computed. Our results are of experimental relevance for systems of cold atomic gases trapped by tunable confining potentials.

Off-equilibrium relaxational dynamics with an improved Ising Hamiltonian

We study the off-equilibrium relaxational dynamics at criticality in the three-dimensionalBlume–Capel model whose static critical behaviour belongs to the 3D Ising universalityclass. Using an ‘improved’ Hamiltonian (the leading corrections to scaling have vanishingamplitude) we perform Monte Carlo simulations of the relaxational dynamics after aquench from to Tc. Analysing the off-equilibrium dynamics atTc we obtain an estimate of the dynamical critical exponentz = 2.020(8) that is perfectly consistent with the field theory predictions.

Quantum Quenches and Off-Equilibrium Dynamical Transition in the Infinite-Dimensional Bose-Hubbard Model

We study the off-equilibrium dynamics of the infinite-dimensional Bose-Hubbard model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.

Linear response subordination to intermittent energy release in off-equilibrium aging dynamics

The interpretation of experimental and numerical data describing off-equilibrium aging dynamics crucially depends on the connection between spontaneous and induced fluctuations. The hypothesis that linear response fluctuations are statistically subordinated to irreversible outbursts of energy, so-called quakes, leads to predictions for the average values and the fluctuation spectra of physical observables in reasonable agreement with experimental results (see e.g. Sibani et al 2006 Phys. Rev. B 74 224407). Using simulational data from a simple but representative Ising model with plaquette interactions, direct statistical evidence supporting the subordination hypothesis is presented and discussed in this work. Both energy and magnetic fluctuations are analyzed, with and without an external magnetic field present. In all cases, fluctuation spectra have a Gaussian zero centered component. For large negative values, the energy spectrum additionally features an intermittent tail describing the quakes. In the magnetization spectrum, two intermittent tails are present. These are symmetric around zero for zero-field, but asymmetric in other cases. The field has thus a biasing effect on the spontaneous intermittent magnetic fluctuations. Furthermore, the field has a negligible effect on the energy fluctuation spectra. From the observed strict temporal correlation between quakes and intermittent magnetization fluctuations, it is possible to conclude that the linear response is controlled by the quakes and inherits their temporal statistics. On this basis, the information culled from intermittent linear response data can be analyzed in the same way as spontaneous thermal energy fluctuations. The latter have a central rôle in thermally activated dynamics, but are harder to measure than linear response data.