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

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

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

Monthly Notices is one of the three largest general primary astronomical research publications. It is an international journal, published by the Royal Astronomical Society. This article 1 describes its publication policy and practice.

Monthly Notices of the Royal Astronomical Society

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Numerical simulations of particulate suspensions via a discretized boltzmann equation: part 1. theoretical foundation

The Diffusion Tensor throughout the Heliosphere

Plasma turbulence is studied via direct numerical simulations in a two-dimensional spatial geometry. Using a hybrid Vlasov-Maxwell model, we investigate the possibility of a velocity-space cascade. A novel theory of space plasma turbulence has been recently proposed by Servidio {\it et al.} [PRL, {\bf 119}, 205101 (2017)], supported by a three-dimensional Hermite decomposition applied to spacecraft measurements, showing that velocity space fluctuations of the ion velocity distribution follow a broad-band, power-law Hermite spectrum $P(m)$, where $m$ is the Hermite index. We numerically explore these mechanisms in a more magnetized regime. We find that (1) the plasma reveals spectral anisotropy in velocity space, due to the presence of an external magnetic field (analogous to spatial anisotropy of fluid and plasma turbulence); (2) the distribution of energy follows the prediction $P(m)\sim m^{-2}$, proposed in the above theoretical-observational work; and (3) the velocity-space activity is intermittent in space, being enhanced close to coherent structures such as the reconnecting current sheets produced by turbulence. These results may be relevant to the nonlinear dynamics weakly-collisional plasma in a wide variety of circumstances.

Velocity-Space Cascade in Magnetized Plasmas: Numerical Simulations

The diffusive motion of charged particles in synthetic magnetic turbulence with different properties is investigated by using numerical simulations with unprecedented dynamical range, which allow us to ensure that both the inertial range and the long wavelength part of the turbulent spectrum are properly described. This is of particular importance in evaluating previous suggestions that parallel and perpendicular diffusion coefficients differ in their energy dependence, an assertion at odds with the many claims of universality of the ${D}_{\ensuremath{\perp}}$ and ${D}_{\ensuremath{\parallel}}$ as functions of particle energy. Cases with and without an ordered magnetic field are discussed. Results of the numerical simulations are compared with available theoretical models, for slab, slab/2D and isotropic turbulence. We find widespread evidence that universality is broken, and that the ratio ${D}_{\ensuremath{\perp}}/{D}_{\ensuremath{\parallel}}$ is not independent of energy. The implications of this finding for the physics of cosmic ray transport are discussed in depth.

Novel aspects of cosmic ray diffusion in synthetic magnetic turbulence

Recent efforts to improve and complete modulation theory are described, emphasizing factors that arise in attempts to understand the diffusion tensor based on turbulence theory and the theory of charged particle scattering. Direct numerical solutions of the transport equations are used to illustrate several sensitive factors affecting modulation that are difficult to understand either observationally or physically. First, the nature of perpendicular diffusion is still poorly understood. We introduce a new perpendicular diffusion formalism that can smoothly vary between the field line random walk limit and a recent treatment based on the Taylor-Green-Kubo equation. Second, this new perpendicular diffusion coefficient is shown to be sensitive to an outer scale of turbulence, the "ultrascale," which we know almost nothing about. Third, the radial variation of the ordinary correlation length is still poorly characterized, in part owing to the uncertain impact of pickup ion-driven turbulence in the outer heliosphere. These three factors encompass a very diverse set of possible solutions of the modulation equation, only a small part of which even remotely resembles available observations.

Toward an Ab Initio Theory of the Solar Modulation of Cosmic Rays

. This paper discusses the possibility that heating of the solar corona in open field-line regions emanating from coronal holes is due to a nonlinear cascade, driven by low-frequency or quasi-static magnetohydrodynamic fluctuations. Reflection from coronal inhomogeneities plays an important role in sustaining the cascade. Physical and observational constraints are discussed. Kinetic processes that convert cascaded energy into heat must occur in regions of turbulent small-scale reconnection, and may be similar in some respects to ion heating due to intense electron beams observed in the aurora.

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William H. Matthaeus

Papers

The Diffusion Tensor throughout the Heliosphere

Velocity-Space Cascade in Magnetized Plasmas: Numerical Simulations

Novel aspects of cosmic ray diffusion in synthetic magnetic turbulence

Toward an Ab Initio Theory of the Solar Modulation of Cosmic Rays

MHD turbulence and heating of the open field-line solar corona