Broken Ergodicity Research Articles

Double Helix of Icosahedra Structure and Spin Glass Magnetism of the δ-Co2.5Zn17.5-xMnx (x = 0.4-3.5) Pseudo-Binary Alloys.

We have synthesized δ-Co2.5Zn17.5-xMnx (x = 0.4-3.5) pseudo-binary alloys of 10 different compositions by a high-temperature solid-state synthetic route, determined their crystal structures and the Mn substitution pattern, and estimated the existence range of the δ-phase. The alloys crystallize in two chiral enantiomorphic space groups P62 and P64, where the basic atomic polyhedron of the chiral structure is an icosahedron and the neighboring icosahedra share vertices to form an infinitely long double helix along the hexagonal axis (like in the δ-Co2.5Zn17.5 parent binary phase). The alloys are pure δ-phase up to the Mn content x ≈ 3.5. The Mn atoms partially substitute Zn atoms at particular crystallographic sites located on the icosahedra. The study of magnetism was performed on the Co2.5Zn17.1Mn0.4 alloy with the lowest Mn content. Contrary to the expectation that structural chirality may induce the formation of a nontrivial magnetic state, a spin glass state with no relation to the structural chirality was found. The magnetic sublattice contains all of the necessary ingredients (randomness and frustration) for the formation of a spin glass state. Typical out-of-equilibrium dynamic phenomena of a spin system with broken ergodicity were detected below the spin freezing temperature Tf ≈ 8 K.

Ergodicity breaking from Rydberg clusters in a driven-dissipative many-body system.

It is challenging to probe ergodicity breaking trends of a quantum many-body system when dissipation inevitably damages quantum coherence originated from coherent coupling and dispersive two-body interactions. Rydberg atoms provide a test bed to detect emergent exotic many-body phases and nonergodic dynamics where the strong Rydberg atom interaction competes with and overtakes dissipative effects even at room temperature. Here, we report experimental evidence of a transition from ergodic toward ergodic breaking dynamics in driven-dissipative Rydberg atomic gases. The broken ergodicity is featured by the long-time phase oscillation, which is attributed to the formation of Rydberg excitation clusters in limit cycle phases. The broken symmetry in the limit cycle is a direct manifestation of many-body collective effects, which is verified experimentally by tuning atomic densities. The reported result reveals that Rydberg many-body systems are a promising candidate to probe ergodicity breaking dynamics, such as limit cycles, and enable the benchmark of nonequilibrium phase transition.

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

We review and extend the theory of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence. The theory contains a solution to the ‘dynamo problem’, i.e., the problem of determining how a planetary or stellar body produces a global dipole magnetic field. We extend the theory to the case of ideal MHD turbulence with a mean magnetic field that is aligned with a rotation axis. The existing theory is also extended by developing the thermodynamics of ideal MHD turbulence based on entropy. A mathematical model is created by Fourier transforming the MHD equations and dynamical variables, resulting in a dynamical system consisting of the independent Fourier coefficients of the velocity and magnetic fields. This dynamical system has a large but finite-dimensional phase space in which the phase flow is divergenceless in the ideal case. There may be several constants of the motion, in addition to energy, which depend on the presence, or lack thereof, of a mean magnetic field or system rotation or both imposed on the magnetofluid; this leads to five different cases of MHD turbulence that must be considered. The constants of the motion (ideal invariants)—the most important being energy and magnetic helicity—are used to construct canonical probability densities and partition functions that enable ensemble predictions to be made. These predictions are compared with time averages from numerical simulations to test whether or not the system is ergodic. In the cases most pertinent to planets and stars, nonergodicity is observed at the largest length-scales and occurs when the components of the dipole field become quasi-stationary and dipole energy is directly proportional to magnetic helicity. This nonergodicity is evident in the thermodynamics, while dipole alignment with a rotation axis may be seen as the result of dynamical symmetry breaking, i.e., ‘broken ergodicity’. The relevance of ideal theoretical results to real (forced, dissipative) MHD turbulence is shown through numerical simulation. Again, an important result is a statistical solution of the ‘dynamo problem’.

Observing strain glass transition in Ti33Nb15Zr25Hf25O2 high entropy alloy with Elinvar effect

Exploring the phase transition of high entropy alloys (HEAs) with multiple major elements is of great importance for understanding the underlying physical mechanisms. Macroscopic martensitic phase transition has been frequently reported in HEAs, however, nanoscale microstructural phase evolution has not been investigated to the same extent. Herein, we have prepared the Ti33Nb15Zr25Hf25O2 HEA and investigated the strain glass transition and its associated properties using dynamic mechanical analysis and microstructure characterization. We have found that the elastic modulus in Ti33Nb15Zr25Hf25O2 HEA deviates from Wachtman's equation and observed the Elinvar effect in the form of temperature-independent modulus in the temperature range from 150 K to 450 K and frequency-dependence modulus around 220 K. The strain glass transition has been evidenced in Ti33Nb15Zr25Hf25O2 HEA by the formation and growth of nano-sized domains during in-situ transmission electron microscopy (TEM) cooling, and substantiated by the broken ergodicity during zero-field-cooling/field-cooling. The strain glass transition is believed to account for the Elinvar effect, where the modulus hardening of nano-sized domains compensates dynamically with the modulus softening of the transformable matrix.

Ergodic characterization of nonergodic anomalous diffusion processes

Anomalous diffusion in various complex systems abounds in nature and spans multiple space and time scales. Canonical characterization techniques that rely upon mean squared displacement break down for nonergodic processes, making it challenging to characterize anomalous diffusion from an individual time-series measurement. Nonergodicity reigns when the time-averaged mean square displacement differs from the ensemble-averaged mean squared displacement even in the limit of long measurement series. In these cases, the typical theoretical results for ensemble averages cannot be used to understand and interpret data acquired from time averages. The difficulty then lies in obtaining statistical descriptors of the measured diffusion process that are not nonergodic. We show that linear descriptors such as the standard deviation, coefficient of variation, and root mean square break ergodicity in proportion to nonergodicity in the diffusion process. In contrast, time series of descriptors addressing sequential structure and its potential nonlinearity: multifractality change in a time-independent way and fulfill the ergodic assumption, largely independent of the time series' nonergodicity. We show that these findings follow the multiplicative cascades underlying these diffusion processes. Adding fractal and multifractal descriptors to typical linear descriptors would improve the characterization of anomalous diffusion processes. Two particular points bear emphasis here. First, as an appropriate formalism for encoding the nonlinearity that might generate nonergodicity, multifractal modeling offers descriptors that can behave ergodically enough to meet the needs of linear modeling. Second, this capacity to describe nonergodic processes in ergodic terms offers the possibility that multifractal modeling could unify several disparate nonergodic diffusion processes into a common framework.

Prominent quantum many-body scars in a truncated Schwinger model

The high level of control and precision achievable in current synthetic quantum matter setups has enabled first attempts at quantum-simulating various intriguing phenomena in condensed matter physics, including those probing thermalization or its absence in closed quantum systems. In a recent work [Desaules \textit{et al.} [arXiv:2203.08830], we have shown that quantum many-body scars -- special low-entropy eigenstates that weakly break ergodicity in nonintegrable systems -- arise in spin-$S$ quantum link models that converge to $(1+1)-$D lattice quantum electrodynamics (Schwinger model) in the Kogut--Susskind limit $S\to\infty$. In this work, we further demonstrate that quantum many-body scars exist in a truncated version of the Schwinger model, and are qualitatively more prominent than their counterparts in spin-$S$ quantum link models. We illustrate this by, among other things, performing a finite-$S$ scaling analysis that strongly suggests that scarring persists in the truncated Schwinger model in the limit $S\to\infty$. Although it does not asymptotically converge to the Schwinger model, the truncated formulation is relevant to synthetic quantum matter experiments, and also provides fundamental insight into the nature of quantum many-body scars, their connection to lattice gauge theories, and the thermalization dynamics of the latter. Our conclusions can be readily tested in current cold-atom setups.

Scrambling in quantum cellular automata

Scrambling is the delocalization of quantum information over a many-body system and underlies all quantum-chaotic dynamics. We employ discrete quantum cellular automata as classically simulable toy models of scrambling. We observe that these automata break ergodicity, i.e., they exhibit quantum scarring. We also find that the time scale of scrambling rises with the local Hilbert-space dimension and obeys a specific combinatorial pattern. We then show that scarring is mostly suppressed in a semiclassical limit, demonstrating that semiclassical-chaotic systems are more ergodic.

Multifractal descriptors ergodically characterize non-ergodic multiplicative cascade processes

Biological and psychological processes routinely break ergodicity, meaning they fail to have stable means (Mean) and independent variation over time that we might find in additive white Gaussian noise (awGn). One possible reason for this failure of ergodicity is the failure of biological and psychological processes to exhibit independence across time. Multifractal evidence has long suggested that biological and psychological processes show strong signatures of nonlinear interactions across scales. These cross-scale interactions sooner befit a cascade-dynamical process than awGn. The present work thus compares awGn to simulations of multiplicative binomial cascades, submitting both types of series and shuffled versions of each to the Thirumalai-Mountain method for estimating ergodicity breaking. Estimating ergodicity breaking for original awGn and cascade series allows us to examine the sources of ergodicity breaking across the sequence, e.g., in temporal correlations specifying nonlinear interactions across scales, and examining ergodicity breaking of the shuffled series allows us to assess the raw, sequence-independent contribution of distributional properties (e.g., the heavy tails of a cascade) without the original temporal sequence. Raw cascade fluctuations and the standard deviation (SD) and root mean square (RMS) series describing those raw fluctuations break ergodicity, but nonlinear, cascade-dynamical descriptors: multifractal spectrum width (Δα) and multifractal nonlinearity (tMF), maintain ergodicity. Interestingly, the fundamentally linear descriptor, fractal Hurst exponent (HfGn) shows moderate ergodicity breaking when describing the fundamentally nonlinear cascade processes, but the linear descriptor coefficient of variation (CV) controls for multiplicative relationships between SD and Mean and maintains ergodicity. We conclude that the ergodicity of statistical descriptors depends on how well they can portray nonlinearity (Δα and tMF) or at least multiplicativity (CV) of the underlying cascade processes.

Phase field simulations for the crossover from sharp martensitic transformation into smooth strain glass transition by fine precipitates

Recent studies have shown an intriguing phenomenon that by introducing nano-precipitates, its sharp first order martensitic transformation (MT) can be changed into smooth strain glass transition which is characterized by sluggish evolution of nano-sized martensitic domains. However, it remains unclear how the precipitates can fundamentally change the nature of the normal MT. In the present study, we reveal the origin of this phenomenon and explore the novel properties associated with the precipitate-induced strain glass using phase field simulations. Our model and phase field simulations show that the shape and size of the Ni4Ti3 precipitates influence the geometrical confinement. The geometrical confinement caused by precipitates with disk shape and high density could produce strong resistance to the growth of large-scale martensitic domains, and can lead to the freezing of nano-sized martensitic domains. When the average size of the B2 pocket between adjacent Ni4Ti3 precipitates is below a critical value (∼16 nm), the avalanche-like behavior of the normal MT is altered to a smooth nanoscale MT with nearly zero transition hysteresis, which exhibits interesting hysteresis-free superelasticity over a wide temperature range. Our results further show that the nanoscale MT exhibits characteristic signatures of strain glass, such as broken ergodicity and Vogel-Fulcher frequency dispersion of internal friction peaks and suggest the non-ergodic nature of the geometrically confined MT. Through precipitate engineering, it is possible to regulate and modify the normal MT into the nanoscale MT and consequently achieve novel properties.

Fractal and multifractal descriptors restore ergodicity broken by non-Gaussianity in time series

Ergodicity breaking is a challenge for biological and psychological sciences. Ergodicity is a necessary condition for linear causal modeling. Long-range correlations and non-Gaussianity characterizing various biological and psychological measurements break ergodicity routinely, threatening our capacity for causal modeling. Long-range correlations (e.g., in fractional Gaussian noise, a.k.a. “pink noise”) break ergodicity—in raw Gaussian series, as well as in some but not all standard descriptors of variability, i.e., in coefficient of variation (CV) and root mean square (RMS) but not standard deviation (SD) for longer series. The present work demonstrates that progressive increases in non-Gaussianity conspire with long-range correlations to break ergodicity in SD for all series lengths. Meanwhile, explicitly encoding the cascade dynamics that can generate temporally correlated non-Gaussian noise offers a way to restore ergodicity to our causal models. Specifically, fractal and multifractal properties encode both scale-invariant power-law correlations and their variety, respectively—features that index the underlying cascade parameters. Fractal and multifractal descriptors of long-range correlated non-Gaussian processes show no ergodicity breaking and hence, provide a more stable explanation for the long-range correlated non-Gaussian form of biological and psychological processes. Fractal and multifractal descriptors offer a path to restoring ergodicity to causal modeling in these fields.

Ergodic descriptors of non-ergodic stochastic processes.

The stochastic processes underlying the growth and stability of biological and psychological systems reveal themselves when far-from-equilibrium. Far-from-equilibrium, non-ergodicity reigns. Non-ergodicity implies that the average outcome for a group/ensemble (i.e. of representative organisms/minds) is not necessarily a reliable estimate of the average outcome for an individual over time. However, the scientific interest in causal inference suggests that we somehow aim at stable estimates of the cause that will generalize to new individuals in the long run. Therefore, the valid analysis must extract an ergodic stationary measure from fluctuating physiological data. So the challenge is to extract statistical estimates that may describe or quantify some of this non-ergodicity (i.e. of the raw measured data) without themselves (i.e. the estimates) being non-ergodic. We show that traditional linear statistics such as the standard deviation, coefficient of variation and root mean square can break ergodicity. Time series of statistics addressing sequential structure and its potential nonlinearity: fractality and multi-fractality, change in a time-independent way and fulfil the ergodic assumption. Complementing traditional linear indices with fractal and multi-fractal indices would empower the study of stochastic far-from-equilibrium biological and psychological dynamics.

Magnetic Helicity and the Geodynamo

We present theoretical and computational results in magnetohydrodynamic turbulence that we feel are essential to understanding the geodynamo. These results are based on a mathematical model that focuses on magnetohydrodynamic (MHD) turbulence, but ignores compressibility and thermal effects, as well as imposing model-dependent boundary conditions. A principal finding is that when a turbulent magnetofluid is in quasi-equilibrium, the magnetic energy in the internal dipole component is equal to the magnetic helicity multiplied by the dipole wavenumber. In the case of the Earth, measurement of the exterior magnetic field gives us, through boundary conditions, the internal poloidal magnetic field. The connection between magnetic helicity and dipole field in the liquid core then gives us the toroidal part of the internal dipole field and a model value of 3 mT for the average core dipole magnetic field. Here, we present the theoretical analysis and numerical simulations that lead to these conclusions. We also test an earlier assertion that differential oblateness may be related to dipole alignment, and while there is an effect, rotation appears to be far more important. In addition, the relationship between dipole quasi-stationarity, broken ergodicity and broken symmetry is clarified. Lastly, we discuss how inertial waves in a rotating magnetofluid can affect dipole alignment.

Nonergodicity in silo unclogging: Broken and unbroken arches.

We report an experiment on the unclogging dynamics in a two-dimensional silo submitted to a sustained gentle vibration. We find that arches present a jerking motion where rearrangements in the positions of their beads are interspersed with quiescent periods. This behavior occurs for both arches that break down and those that withstand the external perturbation: Arches evolve until they either collapse or get trapped in a stable configuration. This evolution is described in terms of a scalar variable characterizing the arch shape that can be modeled as a continuous-time random walk. By studying the diffusivity of this variable, we show that the unclogging is a weakly nonergodic process. Remarkably, arches that do not collapse explore different configurations before settling in one of them and break ergodicity much in the same way than arches that break down.

Coexisting Ordered States, Local Equilibrium-like Domains, and Broken Ergodicity in a Non-turbulent Rayleigh-B\xe9nard Convection at Steady-state

A challenge in fundamental physics and especially in thermodynamics is to understand emergent order in far-from-equilibrium systems. While at equilibrium, temperature plays the role of a key thermodynamic variable whose uniformity in space and time defines the equilibrium state the system is in, this is not the case in a far-from-equilibrium driven system. When energy flows through a finite system at steady-state, temperature takes on a time-independent but spatially varying character. In this study, the convection patterns of a Rayleigh-Bénard fluid cell at steady-state is used as a prototype system where the temperature profile and fluctuations are measured spatio-temporally. The thermal data is obtained by performing high-resolution real-time infrared calorimetry on the convection system as it is first driven out-of-equilibrium when the power is applied, achieves steady-state, and then as it gradually relaxes back to room temperature equilibrium when the power is removed. Our study provides new experimental data on the non-trivial nature of thermal fluctuations when stable complex convective structures emerge. The thermal analysis of these convective cells at steady-state further yield local equilibrium-like statistics. In conclusion, these results correlate the spatial ordering of the convective cells with the evolution of the system’s temperature manifold.

Magnetohydrodynamic turbulence and the geodynamo

Ideal magnetohydrodynamic (MHD) turbulence contains an intrinsic statistical mechanism called ‘broken ergodicity’ for producing an energetic, large-scale, quasi-stationary coherent structure. Since MHD turbulence occurs in the Earth’s liquid outer core, the implication is that the Earth’s dipole magnetic field may be due to MHD turbulence, per se. The purpose of this work is to examine how ideal MHD results apply to the real (i.e., dissipative) MHD turbulence found in the outer core. Prior theoretical and computational results will be briefly reviewed, after which new results will be presented. The significant effects caused by variations in the time-dependence of the forcing will be discussed. It will be also shown that dissipative MHD turbulence driven with isotropic forcing produces large-scale, coherent structures similar to those appearing in ideal MHD turbulence. A connection between spherical harmonic degree and turbulence wave number will be made, allowing for a qualitative comparison of numerical magnetic energy spectra with the International Geomagnetic Reference Field. In addition, a mechanism for explaining the dominance of the Earth’s axial dipole component over its azimuthal ones will also be described. These results indicate that MHD turbulence, per se, may underlie the geodynamo.

Ergodicity Breaking and Localization of the Nicolai Supersymmetric Fermion Lattice Model

We investigate dynamics of the supersymmetric fermion lattice model defined by Hermann Nicolai. We provide its local fermionic constants of motion that exist infinitely many. These generate hidden local supersymmetries that the Nicolai model possesses in addition to its defining dynamical supersymmetry. The existence of such local constants directly implies the breaking ergodicity of the model in the sense of Mazur. At zero temperature, there are infinitely many degenerated classical ground states. We discuss these MBL-like properties. First, we show the delocalization scenario proposed by De Roeck-Huveneers can not naively apply to the Nicolai model at zero temperature despite its disorder-free translation-invariant quantum interaction. Second, we discuss the quantum integrability of the Nicolai model based on the proposal by Caux-Mossel.

Ergodicity breaking dynamics of arch collapse

Flows in hoppers and silos are susceptible to clogging due to the formation of arches at the exit. The failure of these arches is the key to reinitiation of flow, yet the physical mechanism of failure is not well understood. Experiments on vibrated hoppers exhibit a broad distribution of the duration of clogs. Using numerical simulations of a hopper in two dimensions, we show that arches become trapped in locally stable shapes that are explored dynamically under vibrations. The shape dynamics, preceding failure, break ergodicity and can be modeled as a continuous-time random walk with a broad distribution of waiting, or trapping, times. We argue that arch failure occurs as a result of this random walk crossing a stability boundary, which is a first-passage process that naturally gives rise to a broad distribution of unclogging times.

Termination of Biological Function at Low Temperatures: Glass or Structural Transition?

Energy of life is produced by electron transfer in energy chains of respiration or photosynthesis. A small input of free energy available to biology puts significant restrictions on how much free energy can be lost in each electron-transfer reaction. We advocate the view that breaking ergodicity, leading to violation of the fluctuation-dissipation theorem (FDT), is how proteins achieve high reaction rates without sacrificing the reaction free energy. Here we show that a significant level of nonergodicity, represented by a large extent of the configurational temperature over the kinetic temperature, is maintained in the entire physiological range for the cytochrome c electron transfer protein. The protein returns to the state consistent with the FDT below the crossover temperature close to the temperature of the protein glass transition. This crossover leads to a sharp increase in the activation barrier of electron transfer and is displayed by a kink in the Arrhenius plot for the reaction rate constant.

On the Possibility of Calculating Entropy, Free Energy, and Enthalpy of Vitreous Substances.

A critical analysis for the arguments in support of, and against, the traditional approach to thermodynamics of vitreous state is provided. In this approach one presumes that there is a continuous variation of the entropy in the glass-liquid transition temperature range, or a “continuous entropy approach” towards 0 K which produces a positive value of the entropy at T → 0 K. I find that arguments given against this traditional approach use a different understanding of the thermodynamics of glass transition on cooling a liquid, because it suggests a discontinuity or “entropy loss approach” in the variation of entropy in the glass-liquid transition range. That is based on: (1) an unjustifiable use of the classical Boltzmann statistics for interpreting the value of entropy at absolute zero; (2) the rejection of thermodynamic analysis of systems with broken ergodicity, even though the possibility of analogous analysis was proposed already by Gibbs; (3) the possibility of a finite change in entropy of a system without absorption or release of heat; and, (4) describing the thermodynamic properties of glasses in the framework of functions, instead of functionals. The last one is necessary because for glasses the entropy and enthalpy are not functions of the state, but functionals, as defined by Gibbs’ in his classification.

Anomalous behaviour of mutual information in finite flocks

The existing consensus is that flocks are poised at criticality, entailing long correlation lengths and a maximal value of Shannon mutual information in the large-system limit. We show, by contrast, that for finite flocks which do not truly break ergodicity in the long-observation-time limit, mutual information may not only fail to peak at criticality —as observed for other critical systems— but also diverge as noise tends to zero. This result carries implications for other finite-size, out-of-equilibrium systems, where observation times may vary widely compared to time scales of internal system dynamics; thus it may not be assumed that mutual information locates the phase transition.