# Interdisciplinary Applied Mathematics

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law that uses molecular positions and molecular velocities. This is the basis of the molecular dynamics (MD) (see Section 16.1) but also for the lattice gas method. This is followed by the many-body kinetic model of Liouville, which employs distribution functions fN(x1, v1,…, xN, vN, t) that satisfy the Liouville equation

d_

at

N

E1

= i

+ >Vi-d^+ai

a

dvi

fN

0,

(15.47)

where    are the molecular accelerations. This is a 6N-dimensional conti

nuity equation; it can be simplified by coarse-graining, i.e., averaging over single-particle coordinates to obtain a distribution function of reduced order

fM = fl2…M<N =    fl2…N dZM +1 …dZN,

where dzk = dxk dvk, к = M +1,. ..,N. This averaging procedure results in the BBGKY hierarchy expressed by the equation

d

at

M

д    d

+ EQx + a* Qv

i=1

fM

Cm ,

(15.48)

where the right-hand-side term CM contains effects of intermolecular collisions represented by fM +1,…,fM+B, where B denotes the number of bodies involved in the interaction. The Boltzmann equation corresponds to B = 1; also, to obtain the Navier-Stokes equations we keep only the two lower levels, i.e., M = 1, 2 in the BBGKY hierarchy.

Next,    we    present    the    main results    of    all    three    aforementioned    lattice

approaches.

In the    last    half    of the    1980s, a    new    class    of    numerical    approaches was

developed for solving the Navier-Stokes equations indirectly. These new algorithms were based on discrete lattice models of interacting “particles,” whose continuum description could govern the continuum fluid flow equations. The most interesting of these methods was the cellular automaton model of Frisch, Hasslacher, and Pomeau (hereafter called FHP) (Frisch et al., 1986). The basic idea of lattice gas (or cellular automata (CA)) methods is to represent the fluid as an ensemble of interacting low-order-bit computers situated at regularly spaced lattice sites. In the FHP model of two-dimensional hydrodynamics, the underlying lattice is a close-packed equilateral triangular lattice with sites at triangle vertices. Each site has a seven-bit state    with    the    first    six    bits    specifying the presence    or    absence    of

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