# Interdisciplinary Applied Mathematics

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These forces can be interpreted as follows: The conservative forces cause the fluid particles to be as evenly distributed in space as possible as a result of certain “pressures” among them. The frictional forces represent viscous resistances between different parts of the fluid. Finally, the stochastic forces represent degrees of freedom that have been eliminated during the coarse-graining process. The last two forces effectively implement a thermostat, so that thermal equilibrium is achieved. Correspondingly, the amplitude of these forces is dictated by the fluctuation-dissipation theorem (Espanol and Warren, 1995), which ensures that in thermodynamic equilibrium the system will have a canonical distribution. All three forces are modulated by a weight function that specifies the range of interaction between the particles and renders the interaction local.

The distinguishing feature of the DPD forces is that they conserve momentum, and therefore, the DPD model satisfies mass and momentum conservation, which are responsible for the hydrodynamic behavior of a fluid at large scales (Espanol, 1995). Also, by changing the conservative interactions between the fluid particles, one can easily construct “complex” fluids, such as polymers, colloids, amphiphiles, and mixtures.

In summary, the DPD method is characterized by the following conditions:

Position and velocity variables are continuous, as in MD, but the time step is updated in discrete steps, as in LBM.

The conservative forces between DPD particles are soft-repulsive, which makes it possible to extend the simulations to longer time scales compared to MD.

Hydrodynamic behavior is expected at much smaller particle numbers than in classical MD.

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