Editing Multi-particle collision dynamics
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[[Category:Computer simulation techniques]] | |||
Multi-particle collision dynamics (MPC), also known as stochastic rotation dynamics (SRD)<ref>[http://dx.doi.org/10.1007/978-3-540-87706-6_1 G. Gompper, T. Ihle, K. Kroll and R. G. Winkler "Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids", Advanced Computer Simulation Approaches for Soft Matter Sciences III, Advances in Polymer Science '''221''' p. 1 (2009)]</ref>, is a particle-based mesoscale simulation technique for complex fluids <ref>[http://dx.doi.org/10.1063/1.478857 A. Malevanets and R. Kapral "Mesoscopic model for solvent dynamics", Journal of Chemical Physics '''110''' pp. 8605-8613 (1999)]</ref>. Coupling of embedded particles to the coarse-grained solvent is achieved through [[molecular dynamics]] <ref>[http://dx.doi.org/10.1063/1.481289 A. Malevanets and R. Kapral "Solute molecular dynamics in a mesoscale solvent", Journal of Chemical Physics '''112''' pp. 7260-7269 (2000)]</ref>. | |||
==Method of simulation== | ==Method of simulation== | ||
The solvent is modelled as a set of <math>N</math> point particles of mass <math>m</math> with continuous coordinates <math>\vec{r}_{i}</math> and velocities <math>\vec{v}_{i}</math>. The simulation consists of streaming and collision steps. | The solvent is modelled as a set of <math>N</math> point particles of mass <math>m</math> with continuous coordinates <math>\vec{r}_{i}</math> and velocities <math>\vec{v}_{i}</math>. The simulation consists of streaming and collision steps. | ||
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During the streaming step, the coordinates of the particles are updated according to | During the streaming step, the coordinates of the particles are updated according to | ||
<math>\vec{r}_{i}(t+\delta t_{\mathrm{MPC}}) = \vec{r}_{i}(t) + \vec{v}_{i}(t) \delta t_{\mathrm{MPC}}</math> | |||
where <math>\delta t_{\mathrm{MPC}}</math> is a chosen simulation | where <math>\delta t_{\mathrm{MPC}}</math> is a chosen simulation time step which is typically much larger than a molecular dynamics [[time step]]. | ||
After the streaming step, interactions between the solvent particles are modelled in the collision step. The particles are sorted into collision cells with a lateral size <math>a</math>. Particle velocities within each cell are updated according to the collision rule | After the streaming step, interactions between the solvent particles are modelled in the collision step. The particles are sorted into collision cells with a lateral size <math>a</math>. Particle velocities within each cell are updated according to the collision rule | ||
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where <math>\vec{v}_{\mathrm{CMS}}</math> is the centre of mass velocity of the particles in the collision cell and <math>\hat{\mathbf{R}}</math> is a rotation matrix. In two dimensions, <math>\hat{\mathbf{R}}</math> performs a rotation by an angle <math>+\alpha</math> or <math>-\alpha</math> with probability <math>1/2</math>. In three dimensions, the rotation is performed by an angle <math>\alpha</math> around a random rotation axis. The same rotation is applied for all particles within a given collision cell, but the direction (axis) of rotation is statistically independent both between all cells and for a given cell in time. | where <math>\vec{v}_{\mathrm{CMS}}</math> is the centre of mass velocity of the particles in the collision cell and <math>\hat{\mathbf{R}}</math> is a rotation matrix. In two dimensions, <math>\hat{\mathbf{R}}</math> performs a rotation by an angle <math>+\alpha</math> or <math>-\alpha</math> with probability <math>1/2</math>. In three dimensions, the rotation is performed by an angle <math>\alpha</math> around a random rotation axis. The same rotation is applied for all particles within a given collision cell, but the direction (axis) of rotation is statistically independent both between all cells and for a given cell in time. | ||
If the structure of the collision grid defined by the positions of the collision cells is fixed, [[Galilean invariance]] is violated. | If the structure of the collision grid defined by the positions of the collision cells is fixed, [[Galilean invariance]] is violated. If is restored with the introduction of a random shift of the collision grid <ref>[http://dx.doi.org/10.1103/PhysRevE.67.066705 T. Ihle and D. Kroll "Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green-Kubo relations", Physical Review E '''67''' 066705 (2003)]</ref>. | ||
==References== | ==References== | ||
<references/> | <references/> | ||