# Interdisciplinary Applied Mathematics

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For all particles within a sphere defined by rc we assign a probability (0 < rAt < 1) to predict an updated relative velocity from a Maxwelian distribution. So for a specific pair (ij) we first generate the relative velocity v?• • е.ц from a distribution Ёц ^/2к^Т/то, where Ё^„ is a Gaussian random variable with zero mean and with variance

ij

as before.

This procedure effectively sets up an Andersen thermostat; see Section 16.1.3. In particular, for rAt = 0 the system is not coupled to the thermostat, whereas for rAt = 1 the particle velocities are thermalized at every time step.

Two particularly attractive features of this method are:

energy conservation even at large time steps, and

the tracer diffusion properties of the fluid can match those of a real liquid, in conrast to other DPD versions.

• At time tn+1 we have

h = vn

2 m

C

ri = rn + Vn At,

Vi=Vi + ~Ff'(rj)At.

For all pairs rij < rc, select probability rAt. Obtain V0 ■ rij:

Aij —    ■ (v^ Vij) • rij,

vn+1 = vn + A •

Vi    Vi    ij,

vj

n+1 _ „,n+1

j

Aij ■

Compute physical quantities.

Conservation is one of the key conditions for accurate simulations of the canonical ensemble. To this end, we study how a DPD fluid reaches equilibrium using various discretization schemes. In particular, we set to zero all conservative forces, and we simulate in time the effective thermostat expressed by the balance of dissipative and random forces. The simulation is conducted in a 3D periodic box of size 10 x 10 x 10, where the length scale is defined by rc = 1, and a particle number density is set to p = 4; thus N = 4,000 particles. The random force strength is chosen as a = 3 in units of kBT and the dissipative force amplitude is 7 = 4.5.

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