# Interdisciplinary Applied Mathematics

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dt

LC f + LD f,

where LC denotes the Liouville operator of the Hamiltonian system interacting with    conserved    forces    FC;    also,    LD    represents    the    dissipative    and

random terms.    If    the    last    term is    set    to zero,    we    obtain a    Hamiltonian

system, which admits the canonical Gibbs-Boltzmann distribution as a solution. That is, f e4(r.j, pi) = exp(— ^i p2/2mkBT — U/kBT) is a solution of

dfe q ~dt~

LC feq = 0.

However, in the presence of the extra two nonconservative forces the equilibrium distribution will be different from the above unless the condition

Ld feq = 0

is satisfied. This condition is satisfied if the amplitude of the random and dissipative forces and also the weight functions are related as expressed by equations (16.35a) and (16.36).

The time evolution of DPD particles is described by Newton’s law:

 dri = vidt, (16.37) dvi = — (Ffdt + FPdt + F?yfdt) . (16.38) m i i i

Here Ff = £= Ff is the total conservative force acting on particle i; FD and are defined similarly. The velocity increment due to the random force has a factor fdt instead of dt. This term represents Brownian motion, which is described readily by a standard Wiener process with a covariance kernel given by

RFF(ti,tj) =    L

We see therefore that (1/y) is the correlation length in time for this stochastic process.

###### 16-4-2 Numerical Integration

Unlike the MD equations, the DPD equations are stochastic, and this represents an extra degree of difficulty. In addition, the dissipative force depends on the    velocity,    which in    turn depends    on    the    force,    so    there is    nonlinear

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