N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Brownian motors: noisy transport far from equilibrium

The large deviation approach to statistical mechanics

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

Stochastic resonance is said to be observed when increases in levels of unpredictable fluctuations— e.g., random noise—cause an increase in a metric of the quality of signal transmission or detection performance, rather than a decrease. This counterintuitive effect relies on system nonlinearities and on some parameter ranges being ''suboptimal''. Stochastic resonance has been observed, quantified, and described in a plethora of physical and biological systems, including neurons. Being a topic of widespread multidisciplinary interest, the definition of stochastic resonance has evolved significant- ly over the last decade or so, leading to a number of debates, misunderstandings, and controversies. Perhaps the most important debate is whether the brain has evolved to utilize random noise in vivo, as part of the ''neural code''. Surprisingly, this debate has been for the most part ignored by neuroscientists, despite much indirect evidence of a positive role for noise in the brain. We explore some of the reasons for this and argue why it would be more surprising if the brain did not exploit randomness provided by noise—via stochastic resonance or otherwise—than if it did. We also challenge neurosci- entists and biologists, both computational and experi- mental, to embrace a very broad definition of stochastic resonance in terms of signal-processing ''noise benefits'', and to devise experiments aimed at verifying that random variability can play a functional role in the brain, nervous system, or other areas of biology.

/pdf/what-is-stochastic-resonance-definitions-misconceptions-2wpsnjdehd.pdf

What Is Stochastic Resonance? Definitions, Misconceptions, Debates, and Its Relevance to Biology

The influence of weak external noise on the Lorenz model has been studied in parameter ranges where it is characterized by pairs of nonsymmetrical periodic attractors. The investigations, based on electronic analog and digital techniques, have shown that the noise can give rise to pronounced zero-frequency spectral peaks of complicated shape and to chaos.

/pdf/influence-of-noise-on-periodic-attractors-in-the-lorenz-racsjivdui.pdf

Influence of noise on periodic attractors in the Lorenz model: Zero-frequency spectral peaks and chaos.

A new form of heterodyning is reported in which a heterodyne signal at the difference frequency between an input signal and a reference signal can be enhanced by adding noise. The underlying mechanism is closely related to that of stochastic resonance. The dependences of the heterodyne signal and the signal-to-noise ratio on the amplitudes and frequencies of the input and reference signals have been investigated. Noise-induced enhancement of the heterodyning has been demonstrated both for white noise and for high-frequency noise (with the power spectrum centered at the frequency of the reference signal) added at the input.

/pdf/noise-enhanced-heterodyning-in-bistable-systems-31n3i7pk0u.pdf

Noise-enhanced heterodyning in bistable systems

Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondifferentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed.

Large Fluctuations in a Periodically Driven Dynamical System

The effect of fluctuations on the nonlinear response of an underdamped oscillator to an external periodic field at a subharmonic frequency has been investigated theoretically, numerically, and with an analog electronic circuit model. The system studied has often been analyzed in nonlinear optics in the context of two-photon absorption and second-harmonic generation. We consider its nonlinear spectroscopy. Its resonant nonlinear response is described over a broad range of the fluctuation intensities. It is shown that the fluctuation intensity can be used to "tune" the oscillator so as to maximize the nonlinear response. The dependence of the absorption cross section on the fluctuation intensity displays a clearly resolved maximum. If the eigenfrequency of the oscillator is a nonmonotonic function of its energy, the signal at the second harmonic displays a resonant peak at one of two different frequencies, depending on the noise intensity.

/pdf/resonant-subharmonic-absorption-and-second-harmonic-1omj8gwvyt.pdf

Resonant subharmonic absorption and second-harmonic generation by a fluctuating nonlinear oscillator

Activated escape is investigated for systems that are driven by noise whose power spectrum peaks at a finite frequency. Analytic theory and analog and digital experiments show that the system dynamics during escape exhibit a symmetry-breaking transition as the width of the fluctuational spectral peak is varied. For double-well potentials, even a small asymmetry may result in a parametrically large difference of the activation energies for escape from different wells.

/pdf/symmetry-breaking-of-fluctuation-dynamics-by-noise-color-awaxhzg915.pdf

N. D. Stein

Papers

Influence of noise on periodic attractors in the Lorenz model: Zero-frequency spectral peaks and chaos.

Noise-enhanced heterodyning in bistable systems

Large Fluctuations in a Periodically Driven Dynamical System

Resonant subharmonic absorption and second-harmonic generation by a fluctuating nonlinear oscillator

Symmetry breaking of fluctuation dynamics by noise color