The Experts below are selected from a list of 294 Experts worldwide ranked by ideXlab platform
Nándor Simányi - One of the best experts on this subject based on the ideXlab platform.
-
Further Developments of Sinai’s Ideas: The Boltzmann–Sinai Hypothesis
arXiv: Dynamical Systems, 2019Co-Authors: Nándor SimányiAbstract:In this chapter we present a brief survey of the rich and manifold developments of Sinai’s ideas, dating back to 1963, concerning his exact mathematical formulation of Boltzmann’s original Ergodic Hypothesis. These developments eventually lead to the 2013 proof of the so called “Boltzmann-Sinai Ergodic Hypothesis”.
-
Singularities and nonhyperbolic manifolds do not coincide
arXiv: Dynamical Systems, 2013Co-Authors: Nándor SimányiAbstract:We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the Ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.
-
the boltzmann sinai Ergodic Hypothesis in full generality
arXiv: Dynamical Systems, 2010Co-Authors: Nándor SimányiAbstract:In the Ergodic theory of semi-dispersing billiards the Local Ergodic Theorem, proved by Chernov and Sinai in 1987, plays a central role. So far, all existing proofs of this theorem had to use an annoying global Hypothesis, namely the almost sure hyperbolicity of singular orbits. (This is the so called Chernov--Sinai Ansatz.) Here we introduce some new geometric ideas to overcome this difficulty and liberate the proof from the tyranny of the Ansatz. The presented proof is a substantial generalization of my previous joint result with N. Chernov (which is a $2D$ result) to arbitrary dimensions. An important corollary of the presented ansatz-free proof of the Local Ergodic Theorem is finally completing the proof of the Boltzmann--Sinai Ergodic Hypothesis for hard ball systems in full generality.
-
Conditional proof of the Boltzmann-Sinai Ergodic Hypothesis
Inventiones mathematicae, 2009Co-Authors: Nándor SimányiAbstract:We consider the system of N (≥ 2) elastically colliding hard balls of masses m _1,…, m _ N and radius r on the flat unit torus $\mathbb{T}^{\nu}$ , ν ≥2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i.e. the full hyperbolicity and Ergodicity of such systems for every selection ( m _1,…, m _ N ; r ) of the external parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i.e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.
-
conditional proof of the boltzmann sinai Ergodic Hypothesis assuming the hyperbolicity of typical singular orbits
2006Co-Authors: Nándor SimányiAbstract:We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, . . . , mN and radius r on the flat unit torus T ν , ν ≥ 2. We prove the so called Boltzmann-Sinai Ergodic Hypoth- esis, i. e. the full hyperbolicity and Ergodicity of such systems for every selection (m1, . . . , mN; r) of the ex- ternal geometric parameters, under the assumption that almost every singular trajectory is geometrically hyperbolic (sufficient), i. e. the so called Chernov- Sinai Ansatz holds true for the model. The present proof does not use at all the formerly developed, rather involved algebraic techniques, instead it em- ploys exclusively dynamical methods and tools from geometric analysis.
V. B. Kokshenev - One of the best experts on this subject based on the ideXlab platform.
-
Ergodicity failure near structural glass transformations
Solid State Communications, 2001Co-Authors: V. B. KokshenevAbstract:Relaxation timescale for glass-forming materials is analyzed within a self-consistent description introduced within percolation theory treatment of the Adam-Gibbs model. The Ergodic-non-Ergodic phase diagram is proposed for the case of molecular supercooled liquids in terms of the Ergodic-phase instability temperature TE vs fragility. TE, being below and close to the glass-transformation temperature Tg, is established through violation of the Ergodic Hypothesis, i.e. by a crossover from the Gaussian ('Ergodic') to a non-Gaussian dynamics of evolution of clusters. The finite-size fractal cluster distribution is deduced from the known Stauffer scaling form. Crossover of the compact-structure ('Ergodic') clusters to the hole-like glassy clusters is attributed to their critical-size thermal fluctuations. © 2001 Published by Elsevier Science Ltd.
V I Pozdnyakova - One of the best experts on this subject based on the ideXlab platform.
-
mathematical modeling of random coupling between polarization modes in single mode optical fibers xiii conditions of applicability of the Ergodic Hypothesis for fiber ring interferometers
Optics and Spectroscopy, 2006Co-Authors: G B Malykin, V I PozdnyakovaAbstract:Some additional conditions of applicability of the Ergodic Hypothesis to fiber ring interferometers (FRIs) with a loop consisting of a single-mode optical fiber (SMOF) with random inhomogeneities are considered. It is shown by mathematical modeling that the change in the phase difference of counterpropagating waves at the FRI output with the SMOF temperature is not a stationary random process. However, in a fairly narrow temperature range, this dependence can be assumed to be locally stationary. The conditions determining this temperature range are formulated. It is shown for a fairly large ensemble of independent realizations of random inhomogeneities in an SMOF that, even when all conditions of Ergodicity are satisfied with a large margin, there will always be at least one realization violating strict Ergodicity. Thus, only conditional (approximate) Ergodicity occurs in this case. Nevertheless, in calculation of the FRI zero drift in this situation, averaging over an ensemble of independent realizations of random inhomogeneities in the SMOF of an FRI loop can be performed with sufficient accuracy. As a result, calculations are simplified significantly. In the general case, when at least one of the conditions of Ergodicity is not satisfied, averaging over temperature for each realization with subsequent averaging over the entire ensemble should be performed. It is shown also that, within this problem, we can speak only about quasi-Ergodicity or emulation of Ergodicity, since a change in the temperature of the SMOF of an FRI loop and successive enumeration of independent realizations of random inhomogeneities in the SMOF loop are radically different random processes. The parameters characterizing quasiperiodic temperature changes in the phase difference of counterpropagating waves at the FRI output are refined.
-
mathematical modeling of random coupling between polarization modes in single mode optical fibers x verification of the Ergodic Hypothesis for fiber ring interferometers
Optics and Spectroscopy, 2004Co-Authors: G B Malykin, V I PozdnyakovaAbstract:The problem of the validity of the Ergodic Hypothesis as applied to a fiber ring interferometer (FRI) is considered on the basis of a comparison between magnitudes of the zero drift of an FRI calculated upon changing temperature of the fiber and upon random realizations of inhomogeneities in a single-mode optical fiber (SMF). The physical nature and statistical characteristics of random inhomogeneities in an SMF, types of polarization nonreciprocity in an FRI, and thermo-optical parameters of an SMF are analyzed. An algorithm for calculation of the zero drift of an FRI on changing temperature is proposed. The conditions under which the Ergodic Hypothesis is satisfied in an FRI are formulated. In particular, it is shown that many random inhomogeneities have to be placed on the depolarization length of polychromatic radiation in the SMF loop of an FRI; otherwise, the zero drift of the FRI calculated by the method of averaging over an ensemble of independent realizations may significantly exceed its actual value. Numerical estimations are made. It is shown that thermostabilization of an FRI with a polychromatic radiation source may significantly reduce its zero drift.
Edmanuel Torres - One of the best experts on this subject based on the ideXlab platform.
-
Fractional Sampling Theorem for $\alpha$-Bandlimited Random Signals and Its Relation to the von Neumann Ergodic Theorem
IEEE Transactions on Signal Processing, 2014Co-Authors: Rafael Torres, Zandra Lizarazo, Edmanuel TorresAbstract:Considering that fractional correlation function and the fractional power spectral density, for α-stationary random signals, form a fractional Fourier transform pair. We present an interpolation formula to estimate a random signal from a temporal random series, based on the fractional sampling theorem for α-bandlimited random signals. Furthermore, by establishing the relationship between the sampling theorem and the von Neumann Ergodic theorem, the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach. Thus, the validity of the sampling theorem for random signals is closely linked to an Ergodic Hypothesis in the mean sense.
Takaharu Okajima - One of the best experts on this subject based on the ideXlab platform.
-
temporal variation in single cell power law rheology spans the ensemble variation of cell population
Biophysical Journal, 2017Co-Authors: Ryosuke Takahashi, Kaori Kuribayashishigetomi, Agus Subagyo, Kazuhisa Sueoka, John M Maloney, Krystyn J Van Vliet, Takaharu OkajimaAbstract:Changes in the cytoskeletal organization within cells can be characterized by large spatial and temporal variations in rheological properties of the cell (e.g., the complex shear modulus G∗). Although the ensemble variation in G∗ of single cells has been elucidated, the detailed temporal variation of G∗ remains unknown. In this study, we investigated how the rheological properties of individual fibroblast cells change under a spatially confined environment in which the cell translational motion is highly restricted and the whole cell shape remains unchanged. The temporal evolution of single-cell rheology was probed at the same measurement location within the cell, using atomic force microscopy-based oscillatory deformation. The measurements reveal that the temporal variation in the power-law rheology of cells is quantitatively consistent with the ensemble variation, indicating that the cell system satisfies an Ergodic Hypothesis in which the temporal statistics are identical to the ensemble statistics. The autocorrelation of G∗ implies that the cell mechanical state evolves in the ensemble of possible states with a characteristic timescale.