There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

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.

Fluctuations in a periodically driven oscillator whose frequency of eigenoscillation depends nonmonotonically on its energy are investigated. Their character, and their evolution with the amplitude or frequency of the driving force, can be very different from those of oscillators with a monotonic dependence: the distinction reflects a transition between conventional, and a novel type of, nonlinear resonance.

Fluctuations in zero-dispersion nonlinear resonance.

Unlike the quasi-stationary escape flux, the flux on the time scales preceding the quasi-stationary stage may significantly depend on the initial state of the system. We analyze three characteristic initial stages: (i) the stable state of the noise-free system, i.e. the bottom of the potential well; (ii) the non-bottom state with given coordinate and velocity; (iii) a thermalized state. We prove rigorously and and demonstrate in simulations that, on the time scale of a period of eigenoscillation, the flux grows stepwise for cases (i) and (iii), and in an oscillatory manner for case (ii). Different steps/oscillations correspond to different topologies of the most probable escape path.

Characteristic types of evolution of noise-induced escape flux at short time scales

http://eprints.adm.unipi.it/493/01/marseille-journal_revised.pdf

Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport

An archetypal zer :-dispersionsystem, the tilted Duffing oscillator (TDO), is studied for weak dissipation under the action of a periodic force of intermediate amplitude, small enough for resonances to be possible but large enough for chaos to be possible too. It is found that, as in Hamiltonian systems, an overlap of resonances of different orders is important for the onset of chaos. The onset of chaos is also facilitated by certain global bifurcations of the averaged system.

Chaos in periodically driven dissipative zero-dispersion systems.

SUMMARY Numerical algorithms for the integration of stochastic differential equations
in the presence of white noise are introduced and compared. Algorithms for the
integration of stochastic correlated forces are also briefly reviewed. Finally,
a specialised algorithm for two dimensional systems is derived, having in mind
the integration of particles in the liquid state.

Riccardo Mannella

Papers

Fluctuations in zero-dispersion nonlinear resonance.

Characteristic types of evolution of noise-induced escape flux at short time scales

Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport

Chaos in periodically driven dissipative zero-dispersion systems.

Numerical integration of stochastic differential equations