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

http://ltl.tkk.fi/ice/Kevo/noisereview.pdf

Shot noise in mesoscopic conductors

Self-propelled particles include both self-phoretic synthetic colloids and various microorganisms. By continually consuming energy, they bypass the laws of equilibrium thermodynamics. These laws enforce the Boltzmann distribution in thermal equilibrium: The steady state is then independent of kinetic parameters. In contrast, self-propelled particles tend to accumulate where they move more slowly. They may also slow down at high density for either biochemical or steric reasons. This creates positive feedback, which can lead to motility-induced phase separation (MIPS) between dense and dilute fluid phases. At leading order in gradients, a mapping relates variable-speed, self-propelled particles to passive particles with attractions. This deep link to equilibrium phase separation is confirmed by simulations but generally breaks down at higher order in gradients: New effects, with no equilibrium counterpart, then emerge. We give a selective overview of the fast-developing field of MIPS, focusing on theory and...

Motility-Induced Phase Separation

Stochastic models in biology.

A survey is given of the facts and fancies concerning the nonlinear Langevin or Ito equation. Actually, it is merely a pre-equation, which becomes an equation when an interpretation rule is added. The rules of Ito and Stratonovich differ, but both are mathematically consistent and therefore equally admissible conventions. The reason why they seem to lead to physical differences is that the Langevin approach used to arrive at the equation involves a tacit assumption. For systems with external noise this assumption can be justified, and it is then clear that the Stratonovich rule applies. Systems with internal noise, however, can only be properly described by a master equation and the Ito-Stratonovich controversy never enters. Afterward one is free to model the resulting fluctuations either with an Ito or a Stratonovich scheme, but that does not lead to any new information.

Itô versus Stratonovich

In order to derive the equations for dissipation and noise in a quantum mechanical system it is necessary to include the equations of motion of a suitably chosen bath interacting with the system. In this way the standard treatment arrives at an approximate master equation for the density matrix of the system, at the expense of a number ofad hoc assumptions. These assumptions are here scrutinized on the basis of an exactly soluble model. The conclusion is: the bath must obey certain specifications; the interaction must be weak; and the temperature must be so high that the interaction energy is within the classical domain rather than occurring in quanta. Some additional comments concerning dissipation in quantum mechanics are relegated to an appendix.

A soluble model for quantum mechanical dissipation

Diffusion in inhomogeneous media

Stochastic processes in physics and chemistry. Revised and enlarged ed.

The Hamilton equations for a Brownian particle involve its mass as a large parameter. As a consequence its motion is relatively slow and the fast motion of the surrounding fluid molecules can be eliminated by the standard method of eliminating fast variables. The result is equivalent to the known Langevin equation, plus additional higher orders. All coefficients are expressed in terms of correlation functions of the microscopic force on the particle.

N. G. van Kampen

Papers

Itô versus Stratonovich

A soluble model for quantum mechanical dissipation

Diffusion in inhomogeneous media

Stochastic processes in physics and chemistry. Revised and enlarged ed.

Brownian motion as a problem of eliminating fast variables