After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our definitions are consistent with usual understanding of ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings are introduced. They are computed for right triangular billiards and the results indicate a very close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue that, besides ergodicity and mixing, there may exist a third class of quantum dynamics which is characterized by a maximized entropy.
Investigations on diffusion in systems with memory [I.V.L. Costa, R. Morgado, M.V.B.T. Lima, F.A. Oliveira, Europhys. Lett. 63 (2003) 173] have established a hierarchical connection between mixing, ergodicity, and the fluctuation-dissipation theorem (FDT). This hierarchy means that ergodicity is a necessary condition for the validity of the FDT, and mixing is a necessary condition for ergodicity. In this work, we compare those results with recent investigations using the Lee recurrence relations method [M.H. Lee, Phys. Rev. B 26 (1982) 2547; M.H. Lee, Phys. Rev. Lett. 87 (2001) 250601; M.H. Le…
The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out- of-equilibrium behavior. In particular we prove a form of (local) exponential ergodicity when the initial distribution is far from the stationary one and we prove that the mixing time in a finite box grows linearly in the side of the box.
In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier work [Krishnaswami and Senapati, arXiv:1810.01317, Oct. 2018, arXiv:1811.05807, Nov. 2018], an order-chaos-order transition was discovered in this system along with a band of global chaos for 5.33g < E < 5.6g. Here, we provide numerical evidence for ergodicity and mixing in this band. The distributions of relative angles and angular momenta along generic trajectories are show…
In this manuscript we analyse the behaviour of the probability density function of the sum of $N$ deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. When these properties are modified the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables $N$ increases the distance to Gaussian distribution increases. This behaviour goes against…
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…
Fading ergodicity and quantum dynamics in random matrix ensembles
Georg-August-Universität Göttingen（哥廷根大学）；J. Stefan Institute（约韦·斯特凡研究所）；University of Ljubljana（卢布尔雅那大学）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
Numerical evidence is given for non-ergodic (non-mixing) behavior, exhibiting ideal transport, of a simple non-integrable many-body quantum system in the thermodynamic limit, namely kicked $t-V$ model of spinless fermions on a ring. However, for sufficiently large kick parameters $t$ and $V$ we recover quantum ergodicity, and normal transport, which can be described by random matrix theory.
We study the unitary dynamics of the bosonic quantum East model, a kinetically constrained lattice model which generalises the quantum East model to arbitrary occupation per site. We consider the semiclassical limit of large (but finite) site occupancy, so that the dynamics are approximated by an evolution equation of the Gross-Pitaevskii kind. This allows us to numerically study in detail system sizes of hundreds of sites. Like in the spin-$1/2$ case, we find two dynamical phases, an active one of fast thermalisation, and an inactive one of slow relaxation and absence of ergodicity on numeric…
Unlike classical system, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entan…
Probing the ergodicity breaking transition via violations of random matrix theoretic predictions for local observables
Center for Quantum Technologies, Department of Physics, St. Kliment Ohridski University of Sofia（索非亚圣克利门特奥赫里德斯基大学量子技术中心和物理系）；Institute of Fundamental Physics IFF-CSIC（CSIC基础物理研究所）；Department of Physics and Astronomy, University of Exeter（埃克塞特大学物理与天文学系）
We show that the relaxation dynamics near a glass transition with continuous ergodicity breaking can be endowed with a geometric interpretation based on percolation theory. At mean-field level this approach is consistent with the mode-coupling theory (MCT) of type-A liquid-glass transitions and allows to disentangle the universal and nonuniversal contributions to MCT relaxation exponents. Scaling predictions for the time correlation function are successfully tested in the F12 schematic model and facilitated spin systems on a Bethe lattice. Our approach immediately suggests the extension of MCT…
A quantitative measure of convergence to effective ergodicity, the Thirumalai-Mountain (TM) metric, is applied to Metropolis and Glauber single-spin-flip dynamics. In computing this measure, finite lattice ensemble averages are obtained using the exact solution for a one dimensional Ising model, whereas the time averages are computed with Monte Carlo simulations. The time evolution of the effective ergodic convergence of Ising magnetization is monitored. By this approach, diffusion regimes of the effective ergodic convergence of magnetization are identified for different lattice sizes, nonzero…
At the present paper we have computed non-ergodicity paramater from Molecular Dynamics (MD) Simulation data after the mode-coupling theory (MCT) for a glass transition. MCT of dense liquids marks the dynamic glass-transition through a critical temperature $T_c$ that is reflected in the temperature-dependence of various physical quantities. Here, molecular dynamics simulations data of a model adapted to Ni$_{0.2}$Zr$_{0.8}$ are analyzed to deduce $T_c$ from the temperature-dependence of corresponding quantities and to check the consistency of the statements. Analyzed is the diffusion coefficien…
A growing body of empirical evidence suggests that the dynamics of wealth within a population tends to be non-ergodic, even after rescaling the individual wealth with the population average. Despite these discoveries, the way in which non-ergodicity manifests itself in models of economic interactions remains an open issue. Here, we shed valuable insight on these properties by studying the non-ergodicity of the population average wealth in a simple model for wealth dynamics in a growing and reallocating economy called Reallocating geometric Brownian motion (RGBM). When the effective wealth real…
We revisit the dynamics of the one-dimensional self-gravitating sheets models. We show that homogeneous and non-homogeneous states have different ergodic properties. The former is non-ergodic and the one-particle distribution function has a zero collision term if a proper limit is taken for the periodic boundary conditions. Non-homogeneous states are ergodic in a time window of the order of the relaxation time to equilibrium, as similarly observe in other systems with a long range interaction. For the sheets model this relaxation time is much larger than other systems with long range interacti…
We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement delta**2 around the ensemble value 3-alpha (1<alpha<2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the trajectory-to-trajectory average of delta**2 as function of the finite measurement time. This so-called finite-time amplitude depression and the scat…
In ergodic physical systems, time-averaged quantities converge (for large times) to their ensemble-averaged values. Large deviation theory describes rare events where these time averages differ significantly from the corresponding ensemble averages. It allows estimation of the probabilities of these events, and their mechanisms. This theory has been applied to a range of physical systems, where it has yielded new insights into entropy production, current fluctuations, metastability, transport processes, and glassy behaviour. We review some of these developments, identifying general principles.…
Multistability, i.e. the coexistence of several attractors for a given set of system parameters is one of the most important phenomena occurring in dynamical systems. We consider it in velocity dynamics of a Brownian particle driven by thermal fluctuations and moving in a biased periodic potential. In doing so we focus on the impact of ergodicity - a concept which lies at the core of statistical mechanics. The latter implies that a single trajectory of the system is representative for the whole ensemble and as a consequence the initial conditions of the dynamics are fully forgotten. The ergodi…
The dynamics of iterates at the transition to chaos in one-dimensional unimodal maps is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation and aging. The properties of the bifurcation gap induced by external noise, including a relationship between relaxation time and entropy, are seen to be comparable to those of a supercooled liquid above a glass transition temperature. Universal time evolution obtained from the Feigenbaum RG transformation is expressed analytically via $q$-exponentials, and interpreted in terms of nonextensive statistics.
We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and mobility, the feasibility of an individual to change their position in the income distribution. For this purpose, we explore the properties of an established model for income growth that includes "resetting" as a stabilising force which ensures stationary dynamics. We find that the dynamics of inequality is regime-dependent and may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequa…
We study a generic but simple non-integrable quantum {\em many-body} system of {\em locally} interacting particles, namely a kicked $t-V$ model of spinless fermions on 1-dim lattice (equivalent to a kicked Heisenberg XX-Z chain of 1/2 spins). Statistical properties of dynamics (quantum ergodicity and quantum mixing) and the nature of quantum transport in {\em thermodynamic limit} are considered as the kick parameters (which control the degree of non-integrability) are varied. We find and demonstrate {\em ballistic} transport and non-ergodic, non-mixing dynamics (implying infinite conductivity …
