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Oxford Theoretical PhysicsStatistical Mechanics and Fluid SimulationJulia Yeomans |
is given by:
is the strength of the random force,
is a weighting function of the
particle separation 
.
The relative velocity between the particles is 
and 
is the unit vector between them.
is given by:
is the strength of the random force,
is a second weighting function
and 
are random elements whose mean and standard deviations
obey the following correlations:
Working from a Kinetic Thoery point of view, we developed a Fokker-Planck-Boltzmann equation [ 9 ] for the evolution of the one particle distribution function. The difference between this and the traditional Boltzmann equation being in the presence of second derivative operators, resulting from the presence of the random force. Following the Chapman Enskog method [ 11 ] , it is then possible to derive expressions for the viscosities and self-diffusion coefficients of the DPD system. [ 10 ].
Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics.
The original DPD paper. The particle-based method is introduced and its similarities to Molecular Dynamics and Lattice Cellular Automata are indicated. Theory and simulations demonstrate that a quantative description of isothermal Navier-Stokes Flow is obtained with few particles. The advantages of speed over Molecular Dynamics and flexibility over Lattice Gases are emphasised.
Dynamic Simulation of Hard-Sphere suspensions under steady shear.
DPD is used to study the flow of a suspension of hard spheres in a steady shear system. The authors investigate the combination of hydrodynamic interaction and sphere configurations in systems that are far from equilibrium. For large volume fractions (up to 35%), the authors obtain excellent agreement with experimental results from the literature.
Computer simulation of rheological phenomena in dense colloidal suspensions with dissipative particle dynamics.
Colloid suspensions of hard spheres and rods are studied using the DPD method. The viscosity is measured as a function of shear rate and volume fraction of suspension. Shear thinning is observed and the results are in good agreement with experimental results. The viscosity of a dilute rod suspension shows excellent agreement with theoretical predictions. A discussion is given of the use of DPD in such simulations.
Computer simulation of Dilute Polymer solutions with the dissipative particle dynamics method.
Computer simulations of domain growth and phase separation in 2-dimensional binary immiscible fluids using dissipative particle dynamics.
The DPD system is used to study a binary fluid system in 2 dimensions. Following a symmetric quench, the domain size is observed numerically to grow with power laws t^(1/2) and t^(2/3). After an asymmetric quench, only the t^(1/2) law is observed. Lack of momentum conservation results in the observation of t^(1/3) growth at short times. The surface tension is measured using bubble simulations and the results compared with Laplace's law. The DPD method is compared with lattice gases, lattice boltzmann and langevin dynamics.
Statistical Mechanics of Dissipative Particle Dynamics
In the limit of infinitesimal time step, the DPD algorithm is examined as a set of coupled stochastic differential equations. A modifiaction to the algorithm is made to ensure that the resulting Fokker-Planck equation has a Gibbs distribution and the temperature of this distribution is derived from a fluctuation-dissipation theorem. The deviations from these predictions for finite time step are emphasised.
Simulation of a confined polymer in solution using the dissipative particle dynamics method.
A bead and spring polymer model is used with the DPD method. The polymers are confined between two walls, resulting in different relaxation times and polymer statistics parallel and perpendicular to these walls. These effects are put forwards as an explanation of a width-dependent rheology for nanoscale systems.
Dissipative Particle Dynamics: the equilibrium for finite time-step
An analysis is made of the DPD model in the case of finite time step. The temperature for this full algorithm is derived under the assumption that the equilbrium distribution remains approximately Gibbsian in form. The theoretical predictions are observed to give excellent agreement with numerical simulations.
Fokker-Planck-Boltzmann equation for dissipative particle dynamics
The algorithm for Dissipative Particle Dynamics (DPD), as modified by Espagnol and Warren, is used as a starting point for proving an H-theorem for the free energy and deriving hydrodynamic equations. Equilibrium and transport properties of the DPD fluid are explicitly calculated in terms of the system parameters for the continuous time version of the model.
Static and Dynamic Properties of Dissipative Particle Dynamics
ABSTRACT: The algorithm for the DPD fluid, the dynamics of which is conceptually a combination of molecular dynamics, Brownian dynamics and lattice gas automata, is designed for simulating rheological properties of complex fluids on hydrodynamic time scales. This paper calculates the equilibrium and transport properties (viscosity, self-diffusion) of the thermostated DPD fluid explicitly in terms of the system parameters. It is demonstrated that temperature gradients cannot exist, and that there is therefore no heat conductivity. Starting from the N-particle Fokker-Planck, or Kramers' equation, we prove an H-theorem for the free energy, obtain hydrodynamic equations, and derive a nonlinear kinetic equation (the Fokker-Planck-Boltzmann equation) for the single particle distribution function. This kinetic equation is solved by the Chapman-Enskog method. The analytic results are compared with numerical simulations.