Multi-particle collision dynamics
Encyclopedia
Multi-particle collision dynamics (MPC), also known as stochastic rotation dynamics (SRD), is a particle-based mesoscale simulation technique for complex fluids which fully incorporates thermal fluctuations and hydrodynamic interactions. Coupling of embedded particles to the coarse-grained solvent is achieved through molecular dynamics
.
point particles of mass 
with continuous coordinates 
and velocities 
. The simulation consists of streaming and collision steps.
During the streaming step, the coordinates of the particles are updated according to
where
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
. Particle velocities within each cell are updated according to the collision rule
where
is the centre of mass velocity of the particles in the collision cell and 
is a rotation matrix. In two dimensions, 
performs a rotation by an angle 
or 
with probability 
. In three dimensions, the rotation is performed by an angle 
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. It is restored with the introduction of a random shift of the collision grid.
Explicit expressions for the diffusion coefficient and viscosity
derived based on Green-Kubo relations
are in excellent agreement with simulations.
The simulation parameters define the solvent properties, such as
where
is the dimensionality of the system.
A typical choice for normalisation is
. To reproduce fluid-like behaviour, the remaining parameters may be fixed as 
.
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
.
Method of simulation
The solvent is modelled as a set of point particles of mass with continuous coordinates and velocities . The simulation consists of streaming and collision steps.During the streaming step, the coordinates of the particles are updated according to
where
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
. Particle velocities within each cell are updated according to the collision rulewhere
is the centre of mass velocity of the particles in the collision cell and is a rotation matrix. In two dimensions, performs a rotation by an angle or with probability . In three dimensions, the rotation is performed by an angle 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
Galilean invariance
Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames...
is violated. It is restored with the introduction of a random shift of the collision grid.
Explicit expressions for the diffusion coefficient and viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
derived based on Green-Kubo relations
Green-Kubo relations
The Green–Kubo relations give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...
are in excellent agreement with simulations.
Simulation parameters
The set of parameters for the simulation of the solvent are:- solvent particle mass
- average number of solvent particles per collision box
- lateral collision box size
- stochastic rotation angle
- kT (energy)KT (energy)kT is the product of the Boltzmann constant, k, and the temperature, T. This product is used in physics as a scaling factor for energy values in molecular-scale systems , as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy...
- time step
The simulation parameters define the solvent properties, such as
- mean free pathMean free pathIn physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
- diffusion coefficient
- shear viscosity
- thermal diffusivityThermal diffusivityIn heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...
where
is the dimensionality of the system.A typical choice for normalisation is
. To reproduce fluid-like behaviour, the remaining parameters may be fixed as .Applications
MPC has become a notable tool in the simulations of many soft-matter systems, including- colloidColloidA colloid is a substance microscopically dispersed evenly throughout another substance.A colloidal system consists of two separate phases: a dispersed phase and a continuous phase . A colloidal system may be solid, liquid, or gaseous.Many familiar substances are colloids, as shown in the chart below...
dynamics - polymerPolymerA polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
dynamics - vesicles
- active systems