# Interdisciplinary Applied Mathematics

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P = РквТ + 7^7/    _ rJ’) ‘ F?j

j>i    ‘

where V is the volume. Also, the density is a free parameter, and for p > 3 the fluid behaves as liquid. However, since the number of interactions between particles scales linearly with density, the required computational time per unit volume increases with the square of the density. Therefore, in the simulations it suffices to work with the lowest value of p = 3 for computational efficiency (Groot and Warren, 1997).

The requirement of canonical distribution sets two conditions on the weight functions and the amplitudes of the dissipative and random forces.

Specifically, we have that

uD (r ij )= [uR(rij )]2    (16.35a)

and

a2 = 2jksT,    (16.36)

where T is the system temperature and kB the Boltzmann constant. The weight function takes the form

uR(nj)

1rij/rc for rij < rc, 0    for rij > rc,

where rc is the cutoff radius. This is simply a convenient model to localize the interactions.

In the initial formulation of DPD by (Hoogerburgge and Koelman, 1992), the above conditions were not satisfied and energy was not conserved. This was corrected by (Espanol and Warren, 1995), who employed solutions of the Fokker-Planck equation. Let us consider the distribution function f (ri, pi, t), which describes the probability of finding the system in a state with particles located at ri having momenta pi at time t. The time evolution of this distribution is expressed by the Fokker-Planck equation