Statistical Mechanics of Dissipative Particle Dynamics

The stochastic differential equations corresponding to the updating algorithm of Dissipative Particle Dynamics (DPD), and the corresponding Fokker-Planck equation are derived. It is shown that a slight modification to the algorithm is required before the Gibbs distribution is recovered as the stationary solution to the Fokker-Planck equation. The temperature of the system is then directly related to the noise amplitude by means of a fluctuation-dissipation theorem. However, the correspondingly modified, discrete DPD algorithm is only found to obey these predictions if the length of the time step is sufficiently reduced. This indicates the importance of time discretisation in DPD. Recently, Hoogerbrugge and Koelman have introduced a new method for simulating hydrodynamic behaviour which has been coined Dissipative Particle Dynamics (DPD) [ 1,2]. This technique was conceived as an improvement over conventional molecular dynamics (MD) in order to describe complex hydrodynamic behaviour with computational efficiency. Although there have been attempts to study hydrodynamic phenomena with MD [3-61, the number of particles needed to obtain collective behaviour is extremely large. The reason is that hydrodynamic collective behaviour only appears for typical distances L much larger than the interparticle distance and for typical times T much larger than a collision time 7. Other techniques have also been introduced to deal with the computation burden of hydrodynamic flow. Lattice-Gas (LG) dynamics, which constitute a discrete caricature of MD [7] and the more efficient Lattice-Boltzmann dynamics (LB), which is essentially a discretisation of Boltzmann equation [8], both capture hydrodynamic behaviour. The main problem of LG/LB is that the lattice may induce spurious dynamics (due to the absence of perfect isotropy and, in LG, Galilean invariance). Although some of the problems can be eliminated with convenient lattices and rescaling of velocities, the problems show up in more severe forms when dealing with complex flows in complex boundaries as in immiscible mixtures or colloidal