We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

Lattice boltzmann method for fluid flows

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

We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.

https://iopscience.iop.org/article/10.1209/0295-5075/17/6/001/pdf

Lattice BGK Models for Navier-Stokes Equation

We present a novel method for simulating hydrodynamic phenomena. This particle-based method combines features from molecular dynamics and lattice-gas automata. It is shown theoretically as well as in simulations that a quantitative description of isothermal Navier-Stokes flow is obtained with relatively few particles. Computationally, the method is much faster than molecular dynamics, and the at same time it is much more flexible than lattice-gas automata schemes.

https://iopscience.iop.org/article/10.1209/0295-5075/19/3/001/pdf

Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics

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

We show that a class of deterministic lattice gases with discrete Boolean elements simulates the Navier-Stokes equation, anc can be used to design simple, massively parallel computing machines.

Lattice-Gas Automata for the Navier-Stokes Equation

Hydrodynamical phenomena can be simulated by discrete lattice gas models obeing cellular automata rules (U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56, 1505, (1986); D. d'Humieres, P. Lallemand, and U. Frisch, Europhys. Lett. 2, 291, (1986)). It is here shown for a class of D-dimensional lattice gas models how the macrodynamical (large-scale) equations for the densities of microscopically conserved quantities can be systematically derived from the underlying exact ''microdynamical'' Boolean equations. With suitable restrictions on the crystallographic symmetries of the lattice and after proper limits are taken, various standard fluid dynamical equations are obtained, including the incompressible Navier-Stokes equations in two and three dimensions. The transport coefficients appearing in the macrodynamical equations are obtained using variants of fluctuation-dissipation and Boltzmann formalisms adapted to fully discrete situations.

Lattice gas hydrodynamics in two and three dimensions

We have used a semiclassical method developed earlier to compute the particle spectrum of a field theory in two-dimensional space-time defined by the (sine-Gordon) Lagrangian $\frac{1}{2}{({\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\varphi})}^{2}+(\frac{{m}^{4}}{\ensuremath{\lambda}}){cos[(\frac{\sqrt{\ensuremath{\lambda}}}{m})\ensuremath{\varphi}]\ensuremath{-}1}$. For weak coupling we find a heavy particle, the soliton, corresponding to a peculiar classical field configuration and an antisoliton. Below the soliton-antisoliton threshold there are a large number of further states. They can be viewed either as soliton-antisoliton bound states or as bound states of $n$ of the usual quanta of the theory. The "elementary particle" $\ensuremath{\varphi}$ is the lowest of these. As the coupling increases, the higher states successively unbind, decaying into soliton-antisoliton pairs. At $\frac{\ensuremath{\lambda}}{{m}^{2}}=4\ensuremath{\pi}$, the "elementary particle" unbinds leaving only solitons and antisolitons for $\frac{\ensuremath{\lambda}}{{m}^{2}}g4\ensuremath{\pi}$. Comparing our semiclassical results with recent exact results of Coleman and with perturbation theory, we find that the semiclassical calculations are exact. This field theory seems similar to the hydrogen atom for which the Bohr-Sommerfeld quantization rules give the energy levels exactly. We also treat a ${\ensuremath{\varphi}}^{4}$ theory in weak coupling and carry out a number of calculations which provide nontrivial illustrations of the semiclassical method.

Particle spectrum in model field theories from semiclassical functional integral techniques

We discuss the application of semiclassical quantization methods to two-dimensional model field theories for which exact nontrivial classical solutions are known analytically This yields results which cannot be reached by ordinary perturbation methods In particular, we obtain extended objects which can be considered as prototypes for hadrons We study their quantum corrections and renormalization We also develop a method for including fermions

/pdf/nonperturbative-methods-and-extended-hadron-models-in-field-59evclpnys.pdf

Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons

This is the first of a series of papers on the use of semiclassical approximations to find particle states in field theory. The meaning of the WKB approximation is examined from a functional-integral approach. Special emphasis is placed on the distinction between a true WKB or semiclassical approach and the weak-coupling approximation to it. Other topics include the center-of-mass motion of particle states and some problems special to field theory such as multiple-particle states, statistics, and infinite-volume systems. Ultraviolet divergences are touched on, but are dealt with more thoroughly in the following paper where specific models are examined. The central result of this series is that certain kinds of nonlinear field theories have extended particle solutions which survive quantization. The most interesting of these objects, which are reminiscent of hadrons, come from theories with spontaneous symmetry breaking.

/pdf/nonperturbative-methods-and-extended-hadron-models-in-field-kz7cblez6f.pdf

Brosl Hasslacher

Papers

Lattice-Gas Automata for the Navier-Stokes Equation

Lattice gas hydrodynamics in two and three dimensions

Particle spectrum in model field theories from semiclassical functional integral techniques

Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons

Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods