Nonlinear Hydrodynamics of Lattice-Gas Automata with Semi-Detailed Balance

1997 ◽  
Vol 08 (04) ◽  
pp. 653-674
Author(s):  
Alberto Suárez ◽  
Jean Pierre Boon
Keyword(s):  
Lattice Boltzmann ◽  
Lattice Gas ◽  
Equilibrium Distribution ◽  
Local Equilibrium ◽  
Transport Coefficients ◽  
Detailed Balance ◽  
Hydrodynamic Equations ◽  
Nonlinear Hydrodynamics ◽  
Far From Equilibrium ◽  
Dynamical Description

Equations governing the evolution of the hydrodynamic variables in a lattice-gas automaton, arbitrarily far from equilibrium, are derived from the micro-dynamical description of the automaton, under the condition that the local collision rules satisfy semi-detailed balance. This condition guarantees that a factorized local equilibrium distribution (for each node) of the Fermi–Dirac form is invariant under the collision step but not under propagation. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice-Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium. Linearization of the hydrodynamic equations derived here leads to Green–Kubo formulae for the transport coefficients.

scholarly journals Molecular dynamics lattice gas equilibrium distribution function for Lennard–Jones particles

2021 ◽  
Vol 379 (2208) ◽  
pp. 20200404 ◽  
Author(s):  
Aleksandra Pachalieva ◽  
Alexander J. Wagner
Keyword(s):  
Molecular Dynamics ◽  
Distribution Function ◽  
Lattice Boltzmann ◽  
Md Simulation ◽  
Lattice Gas ◽  
Equilibrium Distribution ◽  
Dynamics Simulation ◽  
Equilibrium Distribution Function ◽  
Weighted Sum ◽  
Time Step

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

scholarly journals Lattice Boltzmann Method to Analyse Fluid Flow Around a Circular Cylinder

2019 ◽  
Author(s):  
Dhruv Suri ◽  
Jayakrishnan Radhakrishnan ◽  
Raahil Nayak
Keyword(s):  
Numerical Simulation ◽  
Fluid Flow ◽  
Circular Cylinder ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Lattice Gas ◽  
Initial Conditions ◽  
Local Equilibrium ◽  
Collision Operator ◽  
Boltzmann Method

An overview of the Lattice Boltzmann Method has been presented with an in house algorithm for the numerical simulation of fluid flow around a circular cylinder. The linearization of the collision operator has been discussed for distributions not close to the local equilibrium state, and numerical simulation has been carried out for stable initial conditions up to a Reynolds number of 80. An overview of the lattice gas automata with regard to Boolean variables describing the particle occupation has also been defined. A comparison between the data obtained from the two dimensional fluid flow around the cylinder and previous experimentation has also been made.

scholarly journals Emergent colloidal currents across ordered and disordered landscapes

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Dominik Lips ◽  
Ralph L. Stoop ◽  
Philipp Maass ◽  
Pietro Tierno
Keyword(s):  
Magnetic Particles ◽  
Lattice Gas ◽  
Solvable Model ◽  
Model Systems ◽  
Suitable Model ◽  
Many Body ◽  
Emergent Phenomena ◽  
Density Relationship ◽  
Far From Equilibrium ◽  
Experimental Findings

AbstractMany-particle effects in driven systems far from equilibrium lead to a rich variety of emergent phenomena. Their classification and understanding often require suitable model systems. Here we show that microscopic magnetic particles driven along ordered and defective lattices by a traveling wave potential display a nonlinear current-density relationship, which arises from the interplay of two effects. The first one originates from particle sizes nearly commensurate with the substrate in combination with attractive pair interactions. It governs the colloidal current at small densities and leads to a superlinear increase. We explain such effect by an exactly solvable model of constrained cluster dynamics. The second effect is interpreted to result from a defect-induced breakup of coherent cluster motion, leading to jamming at higher densities. Finally, we demonstrate that a lattice gas model with parallel update is able to capture the experimental findings for this complex many-body system.

A GENERAL MULTIPLE-RELAXATION-TIME BOLTZMANN COLLISION MODEL

2007 ◽  
Vol 18 (04) ◽  
pp. 635-643 ◽  
Author(s):  
XIAOWEN SHAN ◽  
HUDONG CHEN
Keyword(s):  
Relaxation Time ◽  
Lattice Boltzmann ◽  
Lattice Structure ◽  
Transport Coefficients ◽  
Hermite Polynomials ◽  
Simple Extension ◽  
Relaxation Rates ◽  
Collision Model ◽  
Multiple Relaxation Time ◽  
Lattice Boltzmann Models

We formulate a simple extension to the Bhatnagar-Gross-Krook collision model by expanding the distribution function in Hermite polynomials and assigning a relaxation time to each hydrodynamic moment. By discretizing the velocity space, multiple-relaxation-time lattice Boltzmann models can be constructed. The transport coefficients are analytically calculated and numerically verified. At the lowest order, allowing different relaxation rates for the second and third Hermite components results in a variable Prandtl number. Comparing with the previously proposed multiple-relaxation-time lattice Boltzmann models, the present formulation is general in the sense that it is independent of the underlying lattice structure and does not require a procedure for transformation of base vectors.

Local Equilibrium Evaluation of Thermal Transport Coefficients in Simple Liquids

1968 ◽  
Vol 48 (2) ◽  
pp. 951-953 ◽  
Author(s):  
J. Misguich ◽  
G. Nicolis ◽  
J. A. Palyvos ◽  
H. Ted Davis
Keyword(s):  
Thermal Transport ◽  
Local Equilibrium ◽  
Transport Coefficients ◽  
Simple Liquids

Diffusion in Materials with Ionic and Electronic Disorder

1990 ◽  
Vol 210 ◽  
Author(s):  
Joachim Maier
Keyword(s):  
Ionic Conduction ◽  
Diffusion Theory ◽  
Local Equilibrium ◽  
Transport Coefficients ◽  
Electrochemical Techniques ◽  
Chemical Diffusion ◽  
Tracer Diffusion ◽  
Internal Defect ◽  
Concentration Cell ◽  
The Individual

AbstractBesides some necessary reviewing of conventional diffusion theory, this paper deals with the modification of the mass and charge transport equations originating from the occurence of internal defect-chemical reactions (especially valence changes of the defects) which can be considered to be in local equilibrium. It is shown how the phenomenological transport coefficients for chemical diffusion, tracer diffusion and ionic conduction depend on the individual defect diffusivities under such general conditions. Moreover, the evaluation formulae of well-known electrochemical techniques such as Wagner-Hebb polarization and concentration cell experiments have to be modified. Application is made to the influence of trapping effects in doped SrTi03, to the valence changes in YBa2Cu3O6+x as well as to the mixed conduction in orthorhombic PbO.

scholarly journals Phase Transitions in Lattice-Gas Models Far from Equilibrium

1984 ◽  
Vol 53 (8) ◽  
pp. 806-809 ◽  
Author(s):  
Henk van Beijeren ◽  
Lawrence S. Schulman
Keyword(s):  
Phase Transitions ◽  
Lattice Gas ◽  
Lattice Gas Models ◽  
Far From Equilibrium

scholarly journals Validity of the molecular-dynamics-lattice-gas global equilibrium distribution function

2019 ◽  
Vol 30 (10) ◽  
pp. 1941007 ◽  
Author(s):  
M. Reza Parsa ◽  
Aleksandra Pachalieva ◽  
Alexander J. Wagner
Keyword(s):  
Molecular Dynamics ◽  
Distribution Function ◽  
Lattice Gas ◽  
Equilibrium Distribution ◽  
Coarse Graining ◽  
High Volume ◽  
Equilibrium Distribution Function ◽  
Theory Approach ◽  
Lattice Boltzmann Methods ◽  
Definition Of

The molecular-dynamics-lattice-gas (MDLG) method establishes a direct link between a lattice-gas method and the coarse-graining of a molecular dynamics (MD) approach. Due to its connection to MD, the MDLG rigorously recovers the hydrodynamics and allows to validate the behavior of the lattice-gas or lattice-Boltzmann methods directly without using the standard kinetic theory approach. In this paper, we show that the analytical definition of the equilibrium distribution function remains valid even for very high volume fractions.

Application of Steepest-Entropy-Ascent Quantum Thermodynamics to Predicting Heat and Mass Diffusion From the Atomistic Up to the Macroscopic Level

2015 ◽  
Author(s):  
Guanchen Li ◽  
Michael R. von Spakovsky
Keyword(s):  
Spatial Scales ◽  
Local Equilibrium ◽  
Phenomenological Approach ◽  
Heat Diffusion ◽  
Quantum Thermodynamics ◽  
Macroscopic Property ◽  
State Evolution ◽  
Equilibrium Process ◽  
Far From Equilibrium ◽  
Non Equilibrium

Conventional first principle approaches for studying non-equilibrium or far-from-equilibrium processes all depend on the mechanics of individual particles or quantum states and as a result, require too many details of the mechanical features of the system to easily or even practically arrive at the value of a macroscopic property. In contrast, thermodynamics, which has been extremely successful in the stable equilibrium realm, provides an approach for determining a macroscopic property without going into the mechanical details. Nonetheless, such a phenomenological approach is not generally applicable to a non-equilibrium process except in the near-equilibrium realm and under the limiting local equilibrium and continuum assumptions, both of which prevent its application across all scales. To address these drawbacks, steepest-entropy-ascent quantum thermodynamics (SEAQT) can be used. It provides an ensemble-based, thermodynamics, first principles approach applicable to the entire non-equilibrium realm even that far-from-equilibrium and does so with a single kinematics and dynamics able to cross all temporal and spatial scales. Based on prior developments by the authors, this paper applies SEAQT to the study of mass and heat diffusion. Specifically, the study focuses on the thermodynamic features of far-from-equilibrium state evolution. Two kinds of size effects on the evolution trajectory, i.e., concentration and volume effects, are discussed.

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