CiteSeerX — Citation Query Reichl, A Modern Course (2023)

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Perfect spike-to-spike synchrony is studied in all-to-all coupled networks of identical excitatory, current-based, integrate-and-fire neurons with delta-impulse coupling currents and Poisson spiketrain external drive. This synchrony is induced by repeated cascading "total firing events," during which all neurons fire at once. In this regime, the network exhibits nearly periodic dynamics, switching between an effectively uncoupled state and a cascade-coupled total firing state. The probability of cascading total firing events occurring in the network is computed through a combinatorial analysis conditioned upon the random time when the first neuron fires and using the probability distribution of the subthreshold membrane potentials for the remaining neurons in the network. The probability distribution of the former is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analytically via an eigenfunction expansion. The latter is found using a central limit argument via a calculation of the cumulants of a single neuronal voltage. The influence of additional physiological effects that hinder or eliminate cascade-induced synchrony are also investigated. Conditions for the validity of the approximations made in the analytical derivations are discussed and verified via direct numerical simulations.

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The entropy time rate of systems described by nonlinear Fokker-Planck equations-which are directly related to generalized entropic forms-is analyzed. Both entropy production, associated with irreversible processes, and entropy flux from the system to its surroundings are studied. Some examples of known generalized entropic forms are considered, and particularly, the flux and production of the Boltzmann-Gibbs entropy, obtained from the linear Fokker-Planck equation, are recovered as particular cases. Since nonlinear Fokker-Planck equations are appropriate for the dynamical behavior of several physical phenomena in nature, like many within the realm of complex systems, the present analysis should be applicable to irreversible processes in a large class of nonlinear systems, such as those described by Tsallis and Kaniadakis entropies.

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The near-optimal performance of the turbo decoder has been a source of intrigue among communications engineers and information theorists, given its ad hoc origins that were seemingly disconnected from optimization theory. Naturally one would inquire whether the favorable performance might be explained by characterizing the turbo decoder via some optimization criterion or performance index. Recently, two such characterizations have surfaced. One draws from statistical mechanics and aims to minimize the Bethe approximation to a free energy measure. The other characterization involves constrained likelihood estimation, a setting perhaps more familiar to communications engineers. The intent of this paper is to assemble a tutorial overview of these recent developments, and more importantly to identify the formal mathematical duality between the two viewpoints. The paper includes tutorial background material on the information geometry tools used in analyzing the turbo decoder, and the analysis accommodates both the parallel concatenation and serial concatenation schemes in a common framework.

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The H-theorem [(df/dt) 0] for a free-energy functional, f = u − θs (with u and s representing, respectively, the internal energy and a generalized entropy of a given physical system), has been proven previously by making use of a nonlinear Fokker-Planck equation. Herein we focus on a nonlinear Fokker-Planck equation derived by means of a coarse-graining procedure on the equations of motion of a system of interacting vortices, under overdamped motion, in the absence of thermal noise (T = 0). In this case, we show that the parameter θ is directly related to the density as well as to the interactions among vortices. Generalized quantities such as entropy, internal energy, free energy, and heat capacity are analyzed for varying θ: important relations and physical behavior analogous to those of standard thermodynamics are found, showing that θ plays the role of an effective temperature. Estimates of θ in typical physical situations of different type-II superconductors are presented; in addition to this, possible experimental procedures for varying θ are proposed.

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A thermodynamic formalism is developed for a system of interacting particles under overdamped motion, which has been recently analyzed within the framework of nonextensive statistical mechanics. It amounts to expressing the interaction energy of the system in terms of a temperature θ , conjugated to a generalized entropy s q , with q = 2. Since θ assumes much higher values than those of typical room temperatures T θ, the thermal noise can be neglected for this system (T /θ 0). This framework is now extended by the introduction of a work term δW which, together with the formerly defined heat contribution (δQ = θds q ), allows for the statement of a proper energy conservation law that is analogous to the first law of thermodynamics. These definitions lead to the derivation of an equation of state and to the characterization of s q adiabatic and θ isothermic transformations. On this basis, a Carnot cycle is constructed, whose efficiency is shown to be η = 1 − (θ 2 /θ 1 ), where θ 1 and θ 2 are the effective temperatures of the two isothermic transformations, with θ 1 > θ 2 . The results for a generalized thermodynamic description of this system open the possibility for further physical consequences, like the realization of a thermal engine based on energy exchanges gauged by the temperature θ .

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A random multiplicative process with additive noise is described by a Langevin equation. We show that the fluctuation-dissipation relation is satisfied in the Langevin model, if the noise strength is not so strong.

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A recently proposed entropic criterion [P.V. Giaquinta and G. Guinta, Physica A 18'7, 145 (1992)] for the determination of phase transitions in simple Huids is applied to two-Huid models, a purely repulsive point Yukawa Quid, and a 6-12 Lennard-Jones system. Both the gas-liquid and the freezing transitions are investigated by means of integral equation theory, and assessed with simulation data available in the literature. Our results indicate that the entropic criterion is a reasonable tool for predicting the freezing transition at low temperatures, in particular for purely repulsive potentials. Comparison with other melting rules is less favorable when there is an important attractive component in the interaction. On the other hand, the determination of the gas-liquid critical point and the liquid side of the gas-liquid coexistence curve is merely qualitative. Our results, however, show the existence of a correlation between the gas-liquid transition and the location of one of the inQection points of the density-dependent excess residual entropy.

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The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.

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A model of superconducting vortices under overdamped motion is currently used for describing type-II superconductors. Recently, this model has been identified to a nonlinear Fokker-Planck equation and associated to an entropic form characteristic of nonextensive statistical mechanics, S 2 (t) ≡ S q=2 (t). In the present work, we consider a system of superconducting vortices under overdamped motion, following an irreversible process, so that by using the corresponding nonlinear Fokker-Planck equation, the entropy time rate [dS 2 (t)/dt] is investigated. Both entropy production and entropy flux from the system to its surroundings are analyzed. Molecular dynamics simulations are carried for this process, showing a good agreement between the numerical and analytical results. It is shown that the second law holds within the present framework, and we exhibit the increase of S 2 (t) with time, up to its stationary-state value.

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This article presents a phenomenological dynamic phase transition theorymodeling and analysis-for liquid helium-4. First we derive a time-dependent Ginzburg-Landau model for helium-4 by ͑1͒ separating the superfluid and the normal fluid densities with the superfluid density given in terms of a wave function and ͑2͒ using a unified dynamical Ginzburg-Landau model. One the one hand, the analysis of this model leads to phase diagrams consistent with the classical ones for liquid helium-4. On the other hand, it leads to predictions of ͑1͒ the existence of a metastable region H, where both solid and liquid He II states are metastable and appear randomly depending on fluctuations and ͑2͒ the existence of a switch point M on the curve, where the transitions changes types. It is hoped that these predictions will be useful for designing better physical experiments and lead to better understanding of the physical mechanism of superfluidity.