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

Kim and Lee (Phys. Rev. E 52, 473; 1995) report relativity-induced resonances in periodically driven oscillators. We comment that zero-dispersion nonlinear resonance (ZDNR) will occur in some of the systems considered, outline the physical origins of the ZDNR, and propose an explanation of a discrepancy noted by Kim and Lee between their theoretical and numerical values of the energy at the stationary stable points of Poincare sections.

/pdf/comment-on-nonlinear-resonance-and-chaos-in-the-relativistic-1k1ulbk6ya.pdf

Comment on "Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems".

Zero-dispersion peaks (ZDP's), which can arise in the fluctuation spectra of noise-driven underdamped oscillators for which the dependence of the eigenfrequency $\ensuremath{\omega}(E)$ on energy $E$ possesses a maximum or minimum, have been investigated by means of analog electronic experiments. Two different model systems were studied: a tilted Duffing oscillator and a model of a bistable superconducting quantum-interference device (SQUID). It is demonstrated experimentally that, for strong enough intensity $T$ of Gaussian pseudowhite noise (equivalent to temperature in a thermal system), the shape of a ZDP becomes universal, independent of the system under investigation. Its evolution with $T$ is also shown to exhibit universal features, being governed by a single parameter provided that $T$ exceeds a critical value ${T}_{c}$, below which the ZDP disappears abruptly. The hierarchy of universalities connected to particular types of extrema in $\ensuremath{\omega}(E)$ is discussed. The results are of relevance to underdamped SQUID's and, in particular, to the recently discovered phenomenon of zero-dispersion stochastic resonance.

/pdf/universality-of-zero-dispersion-peaks-in-the-fluctuation-15qp6h4iei.pdf

Universality of zero-dispersion peaks in the fluctuation spectra of underdamped nonlinear oscillators.

High frequency stochastic resonance (SR) phenomena, associated with fluctuational transitions between coexisting periodic attractors, have been investigated experimentally in an electronic model of a single-well Duffing oscillator bistable in a nearly resonant field of frequency $\omega_F$. It is shown that, with increasing noise intensity, the signal/noise ratio (SNR) for a signal due to a weak trial force of frequency $\Omega \sim \omega_F$ at first decreases, then {\it increases}, and finally decreases again at higher noise intensities: behaviour similar to that observed previously for conventional (low frequency) SR in systems with static bistable potentials. The stochastic enhancement of the SNR of an additional signal at the mirror-reflected frequency $\vert \Omega - 2 \omega_F \vert$ is also observed, in accordance with theoretical predictions. Relationships with phenomena in nonlinear optics are discussed.

High Frequency Stochastic Resonance in Periodically Driven Systems

The change of the character of the response of a nonlinear system to a low-frequency periodic field induced by external noise is analyzed by means of analog electronic simulation and theoretically. In general, noise of sufficient intensity linearizes the response. For certain parameter ranges, however, an increase in the noise intensity can sometimes have the opposite effect and is shown to delinearize the response. The physical origins of these contrary behaviors are discussed.

Noise-Induced Linearization and Delinearization

The currents generated by noise-induced activation processes in a periodic potential are investigated analytically, by digital simulation and by performing analog experiments. The noise is taken to be quasimonochromatic and the potential to be a smoothed sawtooth. Two analytic approaches are studied. The first involves a perturbative expansion in inverse powers of the frequency characterizing quasimonochromatic noise and the second is a direct numerical integration of the deterministic differential equations obtained in the limit of weak noise. These results, together with the digital and analog experiments, show that the system does indeed give rise, in general, to a net transport of particles. All techniques also show that a current reversal exists for a particular value of the noise parameters.

/pdf/ratchet-driven-by-quasimonochromatic-noise-2d9v4x658w.pdf

N. D. Stein

Papers

Comment on "Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems".

Universality of zero-dispersion peaks in the fluctuation spectra of underdamped nonlinear oscillators.

High Frequency Stochastic Resonance in Periodically Driven Systems

Noise-Induced Linearization and Delinearization

Ratchet driven by quasimonochromatic noise