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

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.

https://iopscience.iop.org/article/10.1088/0305-4470/9/8/029/pdf

Topology of cosmic domains and strings

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems

The Inverse scattering transform fourier analysis for nonlinear problems

Topology of Cosmic Domains and Strings

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

A two-stage scheme is presented to study structural perturbations of the sine-Gordon equation. Although the method is based upon the inverse scattering transform, detailed knowledge of this technical apparatus is not necessary in order to effect the calculations. In the first stage, slow modulations of speeds and positions for the soliton components are computed. The radiation resulting from the acceleration of the solitons is then calculated as a first-order correction through an easily constructed radiative Green's function. The method is exemplified by using it to study several outstanding problems that arise in applications of the Josephson transmission line. In particular we consider: (i) the pinning of flux quanta by microshorts, (ii) the quantum flux shuttle, (iii) annihilation conditions for fluxon-antifluxon collisions, (iv) breather decay, and (v) radiation from a moving fluxon.

Perturbation analysis of fluxon dynamics

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

A new set of canonically conjugate variables 1s introduced for the penoclic Korteweg-de Vries equation and the periodic Toda lattice. These vanables are used for reducing both equations to a nonlinear system which can be integrated in terms of theta functions. It becomes clear that the discrete and the continuous problems are, in a sense, Jsomorph1c. Action variables are defined by loop integrals, and the basic oscillation frequencies are computed. In the infinite-period limit, these action variables tend to the ones used in the canonical description of the inverse-scattering solution method.

Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)

In this paper, we offer an explanation for how selectivity for orientation could be produced by a model with circuitry that is based on the anatomy of V1 cortex. It is a network model of layer 4Cα in macaque primary visual cortex (area V1). The model consists of a large number of integrate-and-fire conductance-based point neurons, both excitatory and inhibitory, which represent dynamics in a small patch of 4Cα—1 mm2 in lateral area—which contains four orientation hypercolumns. The physiological properties and coupling architectures of the model are derived from experimental data for layer 4Cα of macaque. Convergent feed-forward input from many neurons of the lateral geniculate nucleus sets up an orientation preference, in a pinwheel pattern with an orientation preference singularity in the center of the pattern. Recurrent cortical connections cause the network to sharpen its selectivity. The pattern of local lateral connections is taken as isotropic, with the spatial range of monosynaptic excitation exceeding that of inhibition. The model (i) obtains sharpening, diversity in selectivity, and dynamics of orientation selectivity, each in qualitative agreement with experiment; and (ii) predicts more sharpening near orientation preference singularities.

/pdf/a-neuronal-network-model-of-macaque-primary-visual-cortex-v1-148fiq73nq.pdf

David W. McLaughlin

Papers

The soliton: A new concept in applied science

Perturbation analysis of fluxon dynamics

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)

A neuronal network model of macaque primary visual cortex (V1): Orientation selectivity and dynamics in the input layer 4Cα