From automata to fluid flow: Comparisons of simulation and theory

The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered. Although the equation is a derivative of lattice gas automata, it may be interpreted as a Lagrangian finite-difference method for the numerical simulation of the discrete-velocity Boltzmann equation that makes use of a BGK collision operator. As a result, it is not surprising that numerical instability of lattice Boltzmann methods have been frequently encountered by researchers. We present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constantdensity uniform mean flow. The linear stability depends on the following parameters: the distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wavenumber of the perturbations. This parameter space is too large to compute the complete stability characteristics. We report some stability results for a subset of the parameter space for a 7-velocity hexagonal lattice, a 9-velocity square lattice and a 15-velocity cubic lattice. Results common to all three lattices are 1) the BGK relaxation time τ must be greater than 1 2 corresponding to positive shear viscosity, 2) there exists a maximum stable mean velocity for fixed values of the other parameters and 3) as τ is increased from 1 2 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger τ .

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“…al. [26]). However, the Burnett level terms are expected to become negligible as the global Knudsen number becomes small.…”

Section: Review Of Chapman-enskog Methodsmentioning

confidence: 99%

“…The lattice Boltzmann equation, equation (26), is an explicit scheme for the computation of the particle population associated with each discrete velocity. It is a nonlinear scheme due to the use of the equilibrium distribution function in the collision term.…”

Section: Theory Developmentmentioning

confidence: 99%

Stability Analysis of Lattice Boltzmann Methods

Sterling

Chen

1996

Journal of Computational Physics

383

232

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“…The equations of motion are replaced by rules that govern the discrete evolution of the lattice configuration, and in many cases, this method has provided an excellent description of complex phenomena. Famous examples include lattice gas automata (17)(18)(19), which were used for the description of complex phenomena in flow dynamics, and models for diffusion (20)(21)(22) and percolation (23) processes. AB modeling can be considered a generalization of CA where the model system is (in general) not required to be on a lattice and the rules can take any form including adaptive elements and goals-directed behavior.…”

Section: Agent-based Model Of Molecular Self-assemblymentioning

confidence: 99%

An agent-based approach for modeling molecular self-organization

Troisi

Wong

Ratner

2004

Proc. Natl. Acad. Sci. U.S.A.

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Agent-based modeling is a technique currently used to simulate complex systems in computer science and social science. Here, we propose its application to the problem of molecular self-assembly. A system is allowed to evolve from a separated to an aggregated state following a combination of stochastic, deterministic, and adaptive rules. We consider the problem of packing rigid shapes on a lattice to verify that this algorithm produces more nearly optimal aggregates with less computational effort than comparable Monte Carlo simulations.agent-based simulation ͉ molecular self-assembly ͉ nanostructure prediction S elf-organization is one of the most fascinating phenomena in nature. It appears in such apparently disparate arenas as crystal growth, the regulation of metabolism, and dynamics of animal and human behavior (1-3). One of the great challenges in the field of complexity is the definition of the common patterns that make possible the emergence of order from apparently disordered systems. Although it is not proven that selforganization in apparently unrelated systems can be studied within a unique framework, many researchers are trying to identify new methods for interpretation of this aspect of complexity that allow a unified view. One example is given by the scale invariant networks that appear to offer a good perspective for many complex systems (4). Another possibility is the study of emergent phenomena through agent-based (AB) modeling. This developed, almost in parallel, in computer science (5-7) and social science (8-10) and has proved to be an excellent tool for the study of self-organizing computer programs, robots, and individuals. In this paper, after defining briefly the principles of AB modeling, we explore the possibility that such a modeling paradigm could be useful for the study of self-organizing chemical systems, complementing the currently used stochastic (Monte Carlo) or deterministic (molecular dynamics) methods. Agent-Based Model of Molecular Self-AssemblyBackground. An agent is a computer system that decides for itself (11,12). After sensing the environment it takes decisions based on some rules. A typical example is a thermostat that switches on and off the heating after sensing the temperature. An intelligent agent is capable of ''flexible'' autonomous actions: (i) It interacts with other agents and its environment; (ii) its actions (rules) might change in time as a result of this interaction; and (iii) the agent shows goal-directed behavior (i.e., it takes the initiative to satisfy a goal). An AB simulation is a simulation with many intelligent agents interacting among themselves and with the environment. In a typical AB simulation of social behavior, the agents are the individuals that take rational decision based on their neighbors' decisions. Very interesting social phenomena have been recently investigated for example by Axelrod (13) (cooperation), Epstein (14) (social instability), and Helbing (15) (crowd modeling).The great advantage of this modeling technique is that the emerg...

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“…LGCA can model a wide range of phenomena including the diffusion of ideal gases and fluids [70], reaction-diffusion processes [18] and population dynamics [111]. Dormann provides a wonderful introduction to CA [34].…”

mentioning

confidence: 99%

On Cellular Automaton Approaches to Modeling Biological Cells

Alber

Kiskowski

Glazier

et al. 2003

Mathematical Systems Theory in Biology, Communications, Computation, and Finance

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Abstract. We discuss two different types of Cellular Automata (CA): lattice-gasbased cellular automata (LGCA) and the cellular Potts model (CPM), and describe their applications in biological modeling.LGCA were originally developed for modeling ideal gases and fluids. We describe several extensions of the classical LGCA model to self-driven biological cells. In particular, we review recent models for rippling in myxobacteria, cell aggregation, swarming, and limb bud formation. These LGCA-based models show the versatility of CA in modeling and their utility in addressing basic biological questions.The CPM is a more sophisticated CA, which describes individual cells as extended objects of variable shape. We review various extensions to the original Potts model and describe their application to morphogenesis; the development of a complex spatial structure by a collection of cells. We focus on three phenomena: cell sorting in aggregates of embryonic chicken cells, morphological development of the slime mold Dictyostelium discoideum and avascular tumor growth. These models include intercellular and extracellular interactions, as well as cell growth and death.

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From automata to fluid flow: Comparisons of simulation and theory

Cited by 104 publications

References 18 publications

Stability Analysis of Lattice Boltzmann Methods

Stability Analysis of Lattice Boltzmann Methods

An agent-based approach for modeling molecular self-organization

On Cellular Automaton Approaches to Modeling Biological Cells

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