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

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of ``Brownian motors.'' In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications.

/pdf/artificial-brownian-motors-controlling-transport-on-the-1553poaezw.pdf

Artificial Brownian motors: Controlling transport on the nanoscale

Advanced Mathematical Methods for Scientists and Engineers

A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.

/pdf/calculating-free-energies-using-average-force-4jlluavu6y.pdf

Calculating free energies using average force

The hydrodynamic equations for a gas of hard spheres with dissipative dynamics are derived from the Boltzmann equation. The heat and momentum fluxes are calculated to Navier-Stokes order and the transport coefficients are determined as explicit functions of the coefficient of restitution. The dispersion relations for the corresponding linearized equations are obtained and the stability of this linear description is discussed. This requires consideration of the linear Burnett contributions to the energy balance equation from the energy sink term. Finally, it is shown how these results can be imbedded in simpler kinetic model equations with the potential for analysis of more complex states.

Hydrodynamics for granular flow at low density

The dynamics for a system of hard spheres with dissipative collisions is described at the levels of statistical mechanics, kinetic theory, and simulation. The Liouville operator(s) and associated binary scattering operators are defined as the generators for time evolution in phase space. The BBGKY hierarchy for reduced distribution functions is given, and an approximate kinetic equation is obtained that extends the revised Enskog theory to dissipative dynamics. A Monte Carlo simulation method to solve this equation is described, extending the Bird method to the dense, dissipative hard-sphere system. A practical kinetic model for theoretical analysis of this equation also is proposed. As an illustration of these results, the kinetic theory and the Monte Carlo simulations are applied to the homogeneous cooling state of rapid granular flow.

Dissipative dynamics for hard spheres

This research was partially supported by Grant No. PB96- 0534 from the Direccion General de Investigacion Cientifica y Tecnica (Spain)

/pdf/homogeneous-cooling-state-of-a-low-density-granular-flow-2w5gatdg1g.pdf

Homogeneous cooling state of a low-density granular flow

A kinetic model for a dilute multicomponent gas system is proposed. It is constructed by replacing the Boltzmann collision operator with a relaxation‐time term, in the same manner as in the Bhatnagar–Gross–Krook (BGK) model for a single gas. The model contains several parameters that are determined by keeping some of the main properties of the Boltzmann description. In contrast to previous works, the BGK equation is recovered when mechanically identical particles are considered. Thus the model can be expected to apply to systems in which masses are comparable. The transport properties to the Navier–Stokes level are studied and Onsager’s reciprocal relations are found to hold.

A kinetic model for a multicomponent gas

Steady simple shear flows of smooth inelastic spheres are studied by means of a model kinetic equation and also of a direct Monte Carlo simulation method. Both approaches are based on the Enskog equation and provide for each other a test of consistency. The dependence of the granular temperature and of the shear and normal stresses on both the solid fraction and the coefficient of restitution is analysed. Quite a good agreement is found between theory and simulations in all cases. Also, simplified expressions based on the analytical solution of the model for small dissipation are shown to describe fairly well the simulation results even for not small inelasticity. A critical comparison with previous theories is carried out.

J. Javier Brey

Papers

Hydrodynamics for granular flow at low density

Dissipative dynamics for hard spheres

Homogeneous cooling state of a low-density granular flow

A kinetic model for a multicomponent gas

Kinetic theory of simple granular shear flows of smooth hard spheres