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

We outline the historical development of stochastic resonance (SR), a phenomenon in which the signal and/or the signal-to-noise ratio in a nonlinear system increase with increasing intensity of noise. We discuss basic theoretical ideas explaining and describing SR, and we review some revealing experimental data that place SR within the wider context of statistical physics. We emphasize the close relationship of SR to some effects that are well known in condensed-matter physics.

/pdf/stochastic-resonance-in-perspective-1krluatzeu.pdf

Stochastic resonance in perspective

A prehistory problem is formulated for large occasional fluctuations in noise-driven systems. It has been studied theoretically and experimentally, thereby illuminating the concept of optimal paths and making it possible to visualize and investigate them. The prehistory probability distribution measured for a white-noise-driven system, taken as an example, is shown to be in agreement with the theory.

/pdf/optimal-paths-and-the-prehistory-problem-for-large-3k8rc3vqe0.pdf

Optimal paths and the prehistory problem for large fluctuations in noise-driven systems.

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal-characteristics signatures of stochastic resonance (SR)-are reported for a monostable system. The results are shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

/pdf/stochastic-resonance-in-monostable-systems-46r4uhlhte.pdf

Stochastic resonance in monostable systems

A different form of stochastic resonance, in which the weak periodic force is applied multiplicatively (rather than additively) in the noise, has been investigated for a Brownian particle moving in a double-well potential. A regular periodic signal whose amplitude increases sharply with increasing noise intensity is shown to arise when the potential is asymmetric. The experimental measurements are in good agreement with a theoretical analysis.

/pdf/stochastic-resonance-for-periodically-modulated-noise-2i1fjsn0cu.pdf

Stochastic resonance for periodically modulated noise intensity.

The kinetics of a periodically driven nonlinear oscillator, bistable in a nearly resonant field, has been investigated theoretically and through analog experiments. An activation dependence of the probabilities of fluctuational transitions between the coexisting attractors has been observed, and the activation energies of the transitions have been calculated and measured for a wide range of parameters. The position of the kinetic phase transition (KPT), at which the populations of the attractors are equal, has been established. A range of critical phenomena is shown to arise in the vicinity of the KPT including, in particular, the appearance of a supernarrow peak in the spectral density of the fluctuations, and the occurrence of a high-frequency stochastic resonance (HFSR). The experimental measurements of the transition probabilities, the KPT line, the multipeaked spectral densities, the strength of the supernarrow spectral peak, and the HFSR effect are shown to be in good agreement with the theoretical predictions.

/pdf/supernarrow-spectral-peaks-and-high-frequency-stochastic-25q5pmm9jf.pdf

N. D. Stein

Papers

Stochastic resonance in perspective

Optimal paths and the prehistory problem for large fluctuations in noise-driven systems.

Stochastic resonance in monostable systems

Stochastic resonance for periodically modulated noise intensity.

Supernarrow Spectral Peaks and High Frequency Stochastic Resonance in Systems with Coexisting Periodic Attractors