Ergodic Hypothesis Research Articles

A large-N expansion for minimum bias

Despite being the overwhelming majority of events produced in hadron or heavy ion collisions, minimum bias events do not enjoy a robust first-principles theoretical description as their dynamics are dominated by low-energy quantum chromodynamics. In this paper, we present a novel expansion scheme of the cross section for minimum bias events that exploits an ergodic hypothesis for particles in the events and events in an ensemble of data. We identify power counting rules and symmetries of minimum bias from which the form of the squared matrix element can be expanded in symmetric polynomials of the phase space coordinates. This expansion is entirely defined in terms of observable quantities, in contrast to models of heavy ion collisions that rely on unmeasurable quantities like the number of nucleons participating in the collision, or tunes of parton shower parameters to describe the underlying event in proton collisions. The expansion parameter that we identify from our power counting is the number of detected particles N and as N → ∞ the variance of the squared matrix element about its mean, constant value on phase space vanishes. With this expansion, we show that the transverse momentum distribution of particles takes a universal form that only depends on a single parameter, has a fractional dispersion relation, and agrees with data in its realm of validity. We show that the constraint of positivity of the squared matrix element requires that all azimuthal correlations vanish in the N → ∞ limit at fixed center-of-mass energy, as observed in data. The approach we follow allows for a unified treatment of small and large system collective behavior, being equally applicable to describe, e.g., elliptic flow in PbPb collisions and the “ridge” in pp collisions. We also briefly comment on power counting and symmetries for minimum bias events in other collider environments and show that a possible ridge in e+e− collisions is highly suppressed as a consequence of its symmetries.

The averaging principle for non-autonomous slow-fast stochastic differential equations and an application to a local stochastic volatility model

In this work we study the averaging principle for non-autonomous slow-fast systems of stochastic differential equations. In particular in the first part we prove the averaging principle assuming the sublinearity, the Lipschitzianity and the Holder's continuity in time of the coefficients, an ergodic hypothesis and an L2-bound of the fast component. In this setting we prove the weak convergence of the slow component to the solution of the averaged equation. Moreover we provide a suitable dissipativity condition under which the ergodic hypothesis and the L2-bound of the fast component, which are implicit conditions, are satisfied.In the second part we propose a financial application: we consider a slow-fast local stochastic volatility model and prove the weak convergence of the model to a local volatility one. Then under a risk neutral measure we show that the prices of the derivatives, possibly path-dependent, converge to the ones calculated using the limit model.

Wealth Inequality and the Ergodic Hypothesis: Evidence from the United States

Many studies of wealth inequality make the ergodic hypothesis that rescaled wealth converges rapidly to a stationary distribution. Under this assumption, changes in distribution are expressed as changes in model parameters, reflecting shocks in economic conditions, with rapid equilibration thereafter. Here we test the ergodic hypothesis in an established model of wealth in a growing and reallocating economy. We fit model parameters to historical data from the United States. In recent decades, we find negative reallocation, from poorer to richer, for which no stationary distribution exists. When we find positive reallocation, convergence to the stationary distribution is slow. Our analysis does not support using the ergodic hypothesis in this model for these data. It suggests that inequality evolves because the distribution is inherently unstable on relevant timescales, regardless of shocks. Studies of other models and data, in which the ergodic hypothesis is made, would benefit from similar tests.

Ergodic decay laws in Newtonian and relativistic chaotic scattering

In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.

Ergodic theory for energetically open compressible fluid flows

The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier–Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff–Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any t∈R). In particular, the ergodic averages converge for any trajectory provided its ω-limit set in the trajectory space supports a unique (in law) stationary solution.

A New Noise Reduction Concept for the FFT EIS Method Using the Ergodic Hypothesis for Fuel Cell Measurement

I propose a new noise reduction concept for FFT (Fast Fourier transform) Electrochemical Impedance Spectroscopy (EIS) using the ergodic hypothesis. Instead of a conventional time-averaged impedance, an ensemble average of impedances with slightly different frequencies (quasi-ensemble-median value) is defined as an experimentally obtained appropriate impedance. This approximation is acceptable in some conditions and will contribute to faster measurement. By applying this method to a hydrogen fuel cell, a high-accuracy measurement of fuel cell impedance between approx. 0.38 Hz and 10 kHz was performed within three seconds. The results from FRA and the proposed FFT method in the steady state were almost the same. The averaged percentage error throughout the whole of the measured frequencies was around ±1%, and the maximum difference was around −6% at approx. 1 Hz.

Chiroptical spectroscopy of a freely diffusing single nanoparticle

Chiral plasmonic nanoparticles can exhibit strong chiroptical signals compared to the corresponding molecular response. Observations are, however, generally restricted to measurements on stationary single particles with a fixed orientation, which complicates the spectral analysis. Here, we report the spectroscopic observation of a freely diffusing single chiral nanoparticle in solution. By acquiring time-resolved circular differential scattering signals we show that the spectral interpretation is significantly simplified. We experimentally demonstrate the equivalence between time-averaged chiral spectra observed for an individual nanostructure and the corresponding ensemble spectra, and thereby demonstrate the ergodic principle for chiroptical spectroscopy. We also show how it is possible for an achiral particle to yield an instantaneous chiroptical response, whereas the time-averaged signals are an unequivocal measure of chirality. Time-resolved chiroptical spectroscopy on a freely moving chiral nanoparticle advances the field of single-particle spectroscopy, and is a means to obtain the true signature of the nanoparticle’s chirality.

General and consistent statistics for cosmological observations

This paper focuses on two aspects of the statistics of cosmological observables that are important for the next stages of precision cosmology. First, we note that the theory of reduced angular $N$-point spectra has only been developed in detail up to the trispectrum case and in a fashion that makes it difficult to go beyond. To fill this gap, here we present a constructive approach that provides a systematic description of reduced angular $N$-point spectra and their covariance matrices, for arbitrary $N$. Second, we focus on the common practice in the literature on cosmological observables, which consists in simply discarding a part of the expression, namely, the terms containing fields evaluated at the observer position. We point out that this is not justified beyond linear order in perturbation theory, as these terms contribute to all the multipoles of the corresponding spectra and with a magnitude that is of the same order as the rest of the non-linear corrections. We consider the possibility that the reason for neglecting these terms is a conceptual discomfort when using ensemble averages, which originates in an apparent tension between the ergodic hypothesis and the privileged position of the observer on the light-cone. We clarify this subtle issue by performing a careful derivation of the relation between the theoretical statistical predictions and the observational estimators for all $N$. We conclude that there is no inconsistency whatsoever in ensemble-averaging fields at and near the observer position, thus clearing the way for consistent and robust high-precision calculations.

A statistical solution to the chaotic, non-hierarchical three-body problem.

The three-body problem is arguably the oldest open question in astrophysics and has resisted a general analytic solution for centuries. Various implementations of perturbation theory provide solutions in portions of parameter space, but only where hierarchies of masses or separations exist. Numerical integrations1 show that bound, non-hierarchical triple systems of Newtonian point particles will almost2 always disintegrate into a single escaping star and a stable bound binary3,4, but the chaotic nature of the three-body problem5 prevents the derivation of tractable6 analytic formulae that deterministically map initial conditions to final outcomes. Chaos, however, also motivates the assumption of ergodicity7-9, implying that the distribution of outcomes is uniform across the accessible phase volume. Here we report a statistical solution to the non-hierarchical three-body problem that is derived using the ergodic hypothesis and that provides closed-form distributions of outcomes (for example, binary orbital elements) when given the conserved integrals of motion. We compare our outcome distributions to large ensembles of numerical three-body integrations and find good agreement, so long as we restrict ourselves to 'resonant' encounters10 (the roughly 50 per cent of scatterings that undergo chaotic evolution). In analysing our scattering experiments, we identify 'scrambles' (periods of time in which no pairwise binaries exist) as the key dynamical state that ergodicizes a non-hierarchical triple system. The generally super-thermal distributions of survivor binary eccentricity that we predict have notable applications to many astrophysical scenarios. For example, non-hierarchical triple systems produced dynamically in globular clusters are a primary formation channel for black-hole mergers11-13, but the rates and properties14,15 of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.

The ergodicity problem in economics

The ergodic hypothesis is a key analytical device of equilibrium statistical mechanics. It underlies the assumption that the time average and the expectation value of an observable are the same. Where it is valid, dynamical descriptions can often be replaced with much simpler probabilistic ones — time is essentially eliminated from the models. The conditions for validity are restrictive, even more so for non-equilibrium systems. Economics typically deals with systems far from equilibrium — specifically with models of growth. It may therefore come as a surprise to learn that the prevailing formulations of economic theory — expected utility theory and its descendants — make an indiscriminate assumption of ergodicity. This is largely because foundational concepts to do with risk and randomness originated in seventeenth-century economics, predating by some 200 years the concept of ergodicity, which arose in nineteenth-century physics. In this Perspective, I argue that by carefully addressing the question of ergodicity, many puzzles besetting the current economic formalism are resolved in a natural and empirically testable way. This Perspective argues that ergodicity — a foundational concept in equilibrium statistical physics — is wrongly assumed in much of the quantitative economics literature. By asking the extent to which dynamical problems can be replaced by probabilistic ones, many economics puzzles are resolved in a natural and empirically testable fashion.

INFLUENCE OF SOLAR ACTIVITY ON THE HUMAN’S CARDIAC RHYTHM UNDER STRESS

The aim: To determine the effect of solar activity on the cardiac rhythm indices of apparently healthy men and women, smokers and non-smokers, depending on the stress load (physical and psycho-emotional). Methods. The registration of myocardium bioelectric activity parameters was carried out in the participants of the experiment (26 people) for 2 years (spring, autumn, 95 days), using ECG sensor «Fazagraf», the cardiosignals were handled on-line (www.geliomed.kiev.ua). Long time-series data were mathematically processed using Microsoft Office Excel and STATISTICA programs. The change of geomagnetic storminess was estimated using Кр-index. Results. Biophysical monitoring data ofcardiac rhythm was analyzed. In accordance with the ergodic hypothesis, a statistical analysis of balance factor values of the T-wave (T) of ECG was performed using time series for 4 groups with common traces - gender and tobacco smoking. Using the Pearson's chi-squared test, it was proved that the samples obeyed the normal distribution law. Significant differences in the cardiac rhythm were revealed after physical and emotional loads in smokers in comparison with non-smokers, regardless of the geomagnetic situation. The mean values of the T-wave balance factor changes were calculated after the stress load relative to the rest state (ΔT) for the indicated groups of subjects, for periods with a quiet geomagnetic environment (Kp < 16) and with high geomagnetic activity (Kp > 16). Significant differences (р < 0.05) in the parameters of cardiac rhythm changes under the load in the magnetic disturbed days in comparison with the unperturbed period were found only in smokers. The greatest differences were recorded in women-smokers under psycho-emotional stress and in men smokers after physical activity. Conclusion. Physical load and psycho-emotional stress are risk factors for the cardiovascular system of smokers, respectively, in days with high sunspot activity.

Statistical theory of the polarization on the Poincaré sphere.

The random nature of light emitters requires that the theory to study polarization must adopt a statistical approach. Although the Stokes parameters are proposed in this sense, these are implemented as temporal averages, appealing to an ergodic hypothesis, without discussing which probability distributions are treated. Therefore, here the polarization dynamics on the Poincaré sphere of random light sources, whose phases and amplitudes of the electric fields are statistically independent, is described through the von Mises-Fisher distribution. This allows us to relate the degree of polarization and the normalized Stokes parameters with the concentration parameter and the mean direction of the von Mises-Fisher distribution. Also, it is found that their marginal distributions have the same statistical behavior as the Eliyahu-Brosseau distribution.

Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap

Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

Macroscopic energy barrier and rate-independent hysteresis in martensitic transformations

A thermodynamic theory for the lower-bound hysteresis in martensitic transformations is developed. It is shown that the elastic energy generated by the transformation-induced crystal lattice misfit makes the system free energy a nonlinear function of the volume fractions of the phases, lifts the additivity principle and ergodicity hypothesis of the conventional Gibbsian thermodynamics, and produces a macroscopic energy barrier between the parent and product phases. Since the energy barrier is proportional to the macroscopic volume of the system and thus cannot be surmounted by the thermally assisted nucleation of the new phase, a rate-independent hysteresis of thermodynamic nature is produced, which sets the lower bound for the total transformation hysteresis. This lower-bound hysteresis is characterized by two critical temperatures of the martensitic and reverse transformations, one during cooling and another during heating, with their difference defining the corresponding transformation hysteresis. The necessary undercooling and overheating are intrinsic material properties determined by the transformation elastic energy. The theory is tested against experimental observations, explains the ubiquitous temperature hysteresis in the martensitic transformations, and correlates the hysteresis with the volumetric misfit strain of the transformations. The proposed theory is generic and is applicable to any displacive (diffusionless and martensitic) phase transformations.

Estimation of the parameters of stochastic differential equations for sea clutter

When the sea surface is illuminated by a radar sufficiently far away and in the absence of a coherent target, the scattered electromagnetic signal, called ‘sea clutter’ is unpredictable and can be represented by a stochastic process. A model based on stochastic differential equations and consistent with previous statistical models (K distribution) has been proposed. It depends on three parameters: ${\cal A}$ A , ${\cal B}$ B , and $\alpha $ α . To estimate them, maximum likelihood estimators for ${\cal A}$ A and ${\cal B}$ B , and an estimator based on the hypothesis of ergodicity for $\alpha $ α are proposed in this study. The authors compare three expressions for the transition probabilities: the exact one, Euler's approximation, and Nowman's approximation. By regenerating the trajectories from the same Brownian increments, they can quantify the typical error made on the sea clutter from the typical error made on the estimated parameters. Though the exact transition probabilities minimise the error on the sea clutter, they show that an approximation such as Euler's is sufficient.

Quantization of fields by averaging classical evolution equations

This paper extends the formalism for quantizing field theories via a microcanonical quantum field theory and Hamilton's principle to classical evolution equations. These are based on the well-known correspondence under a Wick rotation between quantum field theories and 4-D statistical mechanical theories. By placing quantum field theories on a 4+1-D under Wick rotation to 5-D, expectations of observables are calculated for a microcanonical field theory averaging Hamiltonian flow over a fifth spacelike dimension, a technique common in lattice gauge simulations but not in perturbation theory. In a novel demonstration, averaging pairs of external lines in the classical Feynman diagrams over the fifth dimension generates diagrams with loops and vacuum fluctuations identical to Standard Model diagrams. Because it is microcanonical, this approach, while equivalent for standard quantum fields theories in the Standard Model, is able to quantize theories that have no canonical quantization. It is also unique in representing expectations as averages over solutions to an ordinary, classical PDE rather than a path integral or operator based approaches. Hence, this approach draws a clear connection between quantum field theory and classical field theory in higher dimensions which has implications towards how quantum effects are interpreted. In particular, it raises questions about how violations of the ergodic hypothesis could influence quantum measurements even in standard, non-statistical quantum field theory.

Ergodic Hypothesis of Consciousness

A hypothesis that explains the action principle of consciousness on the base of extremely simple mathematical model of a plane wave is suggested. The doctrine of reincarnation of consciousness in the physical world is indicated as a consequence of the mathematical property of the cyclicity of a complex number. Based on the consideration of the notion of a Vibration the zonal structure of the world nearest to human perception is shown. An interpretation of the attraction and repulsion forces origin, effects of dark matter, dark energy, levitation, teleportation, the law of karma, as well as Heisenberg uncertainty principle in terms of interacting vibrations are given. The work will be interesting to everyone who is looking for the answer to the question: What is the essence of consciousness?.

From Plateau to Plain\u2014Using Space-for-Time Substitution in Geoheritage Interpretation, Elbsandsteingebirge, Germany

The southern part of Elbsandsteingebirge in east Germany represents tableland morphology, with more than a dozen of residual tabular hills—plateaus, mesas and buttes. Their morphological diversity and close proximity to one another create an opportunity to use the ergodic principle (i.e. space-for-time substitution) as a framework for geoheritage interpretation and the development of a thematic geotourist trail. The virtual trail proposed here links eight localities, from the largest and least dissected plateau remnant (Grosser Zschirnstein) to the scatter of allochthonous boulders (Zeisighübel), claiming that they may illustrate the sequential development of erosional landforms in sandstone tablelands. It is argued that ergodic principle is a powerful explanatory tool designed for the general public and more advanced visitors, even though such an approach may not be an accurate reflection of the actual landscape evolution and does not take into account the rates of landform change. The concept is applicable to other localities, and there is usually an option to add further elements (sites) to the existing routes.

Time average estimation in the fraction-of-time probability framework

Abstract The problem of time average estimation is addressed in the fraction-of-time probability framework. In this approach, the observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. Under mild regularity assumptions on temporal cumulants of the signal, a central limit theorem (CLT) is proved for the normalized error of the time average estimate, where the normalizing factor is the square root of the observation interval length. This rate of convergence is the same as that obtained in the classical stochastic approach, but is derived here without resorting to mixing assumptions. That is, no ergodicity hypothesis is needed. Examples of functions of interest in communications are provided whose time average estimate satisfies the CLT. For the class of the almost-periodic functions, a non normal limit distribution for the normalized error of the time average estimate can be obtained when the normalizing factor equals the observation interval length. The limit distribution, nevertheless, depends on the sequence adopted to perform the limit, a result that cannot be obtained in the classical stochastic approach. Numerical examples illustrate the theoretical results and an application to test the presence of a nonzero-mean cyclostationary signal is presented.

Onset of Floquet thermalization

In presence of interactions, a closed, homogeneous (disorder-free) many-body system is believed to generically heat up to an `infinite temperature' ensemble when subjected to a periodic drive: in the spirit of the ergodicity hypothesis underpinning statistical mechanics, this happens as no energy or other conservation law prevents this. Here we present an interacting Ising chain driven by a field of time-dependent strength, where such heating onsets only below a threshold value of the drive amplitude, above which the system exhibits non-ergodic behaviour. The onset appears at {\it strong, but not fast} driving. This in particular puts it beyond the scope of high-frequency expansions. The onset location shifts, but it is robustly present, across wide variations of the model Hamiltonian such as driving frequency and protocol, as well as the initial state. The portion of nonergodic states in the Floquet spectrum, while thermodynamically subdominant, has a finite entropy. We find that the magnetisation as an {\it emergent} conserved quantity underpinning the freezing; indeed the freezing effect is readily observed, as initially magnetised states remain partially frozen {\it up to infinite time}. This result, which bears a family resemblance to the Kolmogorov-Arnold-Moser theorem for classical dynamical systems, could be a valuable ingredient for extending Floquet engineering to the interacting realm.