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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2010

Broken Ergodicity in Two-Dimensional Homogeneous Magnetohydrodynamic Turbulence

Two-dimensional (2-D) homogeneous magnetohydrodynamic (MHD) turbulence has many of the same qualitative features as three-dimensional (3-D) homogeneous MHD turbulence.The se features include several ideal invariants, along with the phenomenon of broken ergodicity. Broken ergodicity appears when certain modes act like random variables with mean values that are large compared to their standard deviations, indicating a coherent structure or dynamo.Recently, the origin of broken ergodicity in 3-D MHD turbulence that is manifest in the lowest wavenumbers was explained. Here, a detailed description of the origins of broken ergodicity in 2-D MHD turbulence is presented. It will be seen that broken ergodicity in ideal 2-D MHD turbulence can be manifest in the lowest wavenumbers of a finite numerical model for certain initial conditions or in the highest wavenumbers for another set of initial conditions.T he origins of broken ergodicity in ideal 2-D homogeneous MHD turbulence are found through an eigen analysis of the covariance matrices of the modal probability density functions.It will also be shown that when the lowest wavenumber magnetic field becomes quasi-stationary, the higher wavenumber modes can propagate as Alfven waves on these almost static large-scale magnetic structures

Shebalin, John V.↗
NASA NTRS2006

Granular Contact Forces: Proof of "Self-Ergodicity" by Generalizing Boltzmann's Stosszahlansatz and H Theorem

Ergodicity is proved for granular contact forces. To obtain this proof from first principles, this paper generalizes Boltzmann's stosszahlansatz (molecular chaos) so that it maintains the necessary correlations and symmetries of granular packing ensembles. Then it formally counts granular contact force states and thereby defines the proper analog of Boltzmann's H functional. This functional is used to prove that (essentially) all static granular packings must exist at maximum entropy with respect to their contact forces. Therefore, the propagation of granular contact forces through a packing is a truly ergodic process in the Boltzmannian sense, or better, it is self-ergodic. Self-ergodicity refers to the non-dynamic, internal relationships that exist between the layer-by-layer and column-by-column subspaces contained within the phase space locus of any particular granular packing microstate. The generalized H Theorem also produces a recursion equation that may be solved numerically to obtain the density of single particle states and hence the distribution of granular contact forces corresponding to the condition of self-ergodicity. The predictions of the theory are overwhelmingly validated by comparison to empirical data from discrete element modeling.

Metzger, Philip T.↗
NASA NTRS2003

Applications of Ergodic Theory to Coverage Analysis

The study of differential equations, or dynamical systems in general, has two fundamentally different approaches. We are most familiar with the construction of solutions to differential equations. Another approach is to study the statistical behavior of the solutions. Ergodic Theory is one of the most developed methods to study the statistical behavior of the solutions of differential equations. In the theory of satellite orbits, the statistical behavior of the orbits is used to produce 'Coverage Analysis' or how often a spacecraft is in view of a site on the ground. In this paper, we consider the use of Ergodic Theory for Coverage Analysis. This allows us to greatly simplify the computation of quantities such as the total time for which a ground station can see a satellite without ever integrating the trajectory, see Lo 1,2. More over, for any quantity which is an integrable function of the ground track, its average may be computed similarly without the integration of the trajectory. For example, the data rate for a simple telecom system is a function of the distance between the satellite and the ground station. We show that such a function may be averaged using the Ergodic Theorem.

Ergodic Theory↗
DOE OSTI2025

Dipolar Aleppo lattice: Ground state ordering and ergodic dynamics in the absence of vertex frustration

We introduce the Aleppo spin ice geometry, another variation of decimated square ice patterns, which in contrast to similar systems previously studied, does not exhibit vertex frustration. Using synchrotron-based photoemission electron microscopy, we directly visualize low-energy states achieved after thermal annealing, in addition to temperature-dependent moment fluctuations. The results reveal the observation of ground state patterns and the absence of ergodicity-breaking dynamics. Our observations further confirm vertex frustration to be an important criterion for the emergence of ergodicity transitions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Towards absolutely stable ergodicity breaking in two and three dimensions

We propose physically reasonable systems capable of avoiding ergodicity at infinite time in the thermodynamic limit, even with generic perturbations and when coupled to a heat bath. In two dimensions, the rainbow loop soup has (stretched) exponentially numerous absolutely stable nonergodic states with diverging energy but vanishing energy density. In three dimensions the rainbow membrane soup has (stretched) exponentially numerous nonergodic states with diverging energy barriers, leading to infinite-time robust ergodicity breaking that even survives coupling to a nonzero temperature heat bath. We describe our results in the language of exact emergent symmetries and demonstrate how the systems avoid common instabilities. Furthermore, our construction naturally connects to quantum dimer models, topologically ordered systems, the group word construction, and Hamiltonians whose low-energy eigenstates exhibit anomalous entanglement entropy.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity

We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools.

Eigenstate thermalization↗
NASA NTRS2010

Broken Ergodicity in MHD Turbulence

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series, where the inherent periodic box serves as a surrogate for a bounded astrophysical plasma. Independent Fourier coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. The eigenvalues at lowest wave number can be very small, resulting in a large-scale coherent structure: a turbulent dynamo. This is seen in computations and a theoretical explanation in terms of 'broken ergodicity' contains Taylor s theory of force-free states. An important problem for future work is the case of real, i.e., dissipative flows. In real flows, broken ergodicity and coherent structure are still expected to occur in MHD turbulence at the largest scale, as suggested by low resolution simulations. One challenge is to incorporate coherent structure at the largest scale into the theory of turbulent fluctuations at smaller scales.

Shebalin, John V.↗
NASA NTRS2010

Broken Ergodicity in Ideal, Homogeneous, Incompressible Turbulence

We discuss the statistical mechanics of numerical models of ideal homogeneous, incompressible turbulence and their relevance for dissipative fluids and magnetofluids. These numerical models are based on Fourier series and the relevant statistical theory predicts that Fourier coefficients of fluid velocity and magnetic fields (if present) are zero-mean random variables. However, numerical simulations clearly show that certain coefficients have a non-zero mean value that can be very large compared to the associated standard deviation. We explain this phenomena in terms of broken ergodicity', which is defined to occur when dynamical behavior does not match ensemble predictions on very long time-scales. We review the theoretical basis of broken ergodicity, apply it to 2-D and 3-D fluid and magnetohydrodynamic simulations of homogeneous turbulence, and show new results from simulations using GPU (graphical processing unit) computers.

Morin, Lee↗
NASA NTRS2011

Broken Ergodicity in MHD Turbulence in a Spherical Domain

Broken ergodicity (BE) occurs in Fourier method numerical simulations of ideal, homogeneous, incompressible magnetohydrodynamic (MHD) turbulence. Although naive statistical theory predicts that Fourier coefficients of fluid velocity and magnetic field are zero-mean random variables, numerical simulations clearly show that low-wave-number coefficients have non-zero mean values that can be very large compared to the associated standard deviation. In other words, large-scale coherent structure (i.e., broken ergodicity) in homogeneous MHD turbulence can spontaneously grow out of random initial conditions. Eigenanalysis of the modal covariance matrices in the probability density functions of ideal statistical theory leads to a theoretical explanation of observed BE in homogeneous MHD turbulence. Since dissipation is minimal at the largest scales, BE is also relevant for resistive magnetofluids, as evidenced in numerical simulations. Here, we move beyond model magnetofluids confined by periodic boxes to examine BE in rotating magnetofluids in spherical domains using spherical harmonic expansions along with suitable boundary conditions. We present theoretical results for 3-D and 2-D spherical models and also present computational results from dynamical simulations of 2-D MHD turbulence on a rotating spherical surface. MHD turbulence on a 2-D sphere is affected by Coriolus forces, while MHD turbulence on a 2-D plane is not, so that 2-D spherical models are a useful (and simpler) intermediate stage on the path to understanding the much more complex 3-D spherical case.

Shebalin, John V.↗
DOE OSTI2026

Microcanonical Kinetics of Water-Mediated Proton Transfer in 4ABAH + ·(H 2 O) n = 4–6 Clusters (ABA = Aminobenzoic Acid): A Model System for Size-Dependent Relaxation to Ergodic Behavior

Here, we leverage the unique properties of the 4ABAH + · (H 2 O) n clusters (ABA = 4-aminobenzoic acid, n = 4−6) to quantitatively address how a finite, isolated system evolves into an ergodic condition starting from localized arrangements in configuration space. This system adopts two distinct structural isomers in which water molecules cluster around the cationic centers of its two protomers with widely separated positive charge centers. These isomers arise from excess proton attachment to either the acid (O) or amino (N) group on opposite sides of the benzene ring. Both forms are captured and kinetically trapped using cryogenic ion methods and then selectively vibrationally excited through their mutually exclusive IR bands involving NH and OH stretching fundamentals. Because the IR excitation lies below the water binding energy, the system can evolve to explore slow, rare events that lead to the interconversion between the two isomers. The rates of these intracluster reactions are determined by using a pump−probe scheme involving ∼5 ns IR pump and UV probe lasers. The rates occur on the microsecond time scale, leading to steady state populations of the isomers, thus revealing the cluster size-dependent fractionation between the two species at microcanonical equilibrium. The steady state distributions are correlated with the expected trend in the cluster size-dependent reaction energetics, which are in turn consistent with changes in the relative densities of states of the two species. These results thus provide an unusually clear example in which complex, protic-solvent-mediated chemical transformations are captured within a finite system at a precisely determined internal energy.

Rana, Abhijit [Yale Univ., New Haven, CT (United S↗
DOE OSTI2025

Ergodic Lagrangian dynamics in a superhero universe

We present a fictional scenario that, while undeniably whimsical, provides the foundation for a unique exercise in extended problem solving, physics analysis, and quantitative model development. Starting with the foundational premise of the Wild Cards shared-world superhero universe, we demonstrate how a variety of concepts appropriate to the advanced undergraduate level—ergodicity, functional analysis, Lagrangian mechanics, and the ever-important simplifying approximation—can be combined into a rich, coherent mathematical model. The goal of this case study is to develop a useful pedagogical exercise in exploring an open-ended research question that presents, at first glance, no clear path forward. Being both eclectic and lengthy, this exercise offers a unique way for students to apply their core physics and mathematics education. It is perhaps best used within a senior honors seminar or within a brief (e.g., January term) elective class.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1974

The ergodic decomposition of stationary discrete random processes

The ergodic decomposition is discussed, and a version focusing on the structure of individual sample functions of stationary processes is proved for the special case of discrete-time random processes with discrete alphabets. The result is stronger in this case than the usual theorem, and the proof is both intuitive and simple. Estimation-theoretic and information-theoretic interpretations are developed and applied to prove existence theorems for universal source codes, both noiseless and with a fidelity criterion.

Gray, R. M.↗
NASA NTRS1985

Relations between ergodicity and observability

Aspects of global observability (GO) for some standard examples of chaotic systems are considered, the underlying philosophy being that such systems should be easy to observe and should be observed by large classes of scalar functions. Consideration is given to GO for piecewise monotone maps of the interval (a standard example of a chaotic system). Attention is then given to GO for ergodic translations on compact (metrizable) topological groups; the most familiar examples are the irrational translation flows on the n-torus.

Drager, L.↗