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Class: Langevin_Dynamics

Introduction

This class handles systems that obey Langevin Dynamics. It is kept separated from other class for ease-of-use.

Force calculators

void ParSim::Langevin_Dynamics::Neighbours_search(ParticleSystem &ps)

void ParSim::Langevin_Dynamics::Neighbours_search_PBC(ParticleSystem &ps)
Force calculator with WCA interaction and PBC
void ParSim::Langevin_Dynamics::Force_NL_PBC_WCA(ParSim::ParticleSystem &ps)
  1. This Force Calculator uses Neighbour lists of each particle object to calculate force.
  2. Interaction: WCA
  3. Boundary: Periodic boundary conditions.

Integrators

This section contains information for the integrators.

  • We need a new integrator other than conventional ones that integrate the Langevin Dynamics eqautions.
  • Optimal one is by Leimkuhler and Matters 2013a.

BOAOB algorithm: LM Integrators

Warning

Equations for theta evolution involve omega dependent force. BOAOB method is best for forces that are derived from a position dependent potentianl only. We are still using the same integrator for theta.

LM integration is carried out by the following two methods in parts

First Part
void ParSim::Langevin_Dynamics::LM_Intergrator1(ParSim::ParticleSystem &parsym, int step)
  1. Updates \(v(t)\) to \(v(t+(dt/2))\)
  2. Updates \(x(t)\) to \(x(t+(dt/2))\) using \(v(t+(dt/2))\)
  3. Adds Fluctuation and Dissipation changing \(v(t+(dt/2))\) to \(v'(t+(dt/2))\)
  4. Updates \(x(t+(dt/2))\) to \(x(t + dt)\) using \(v'(t+(dt/2))\)

For updating \(v'(t+(dt/2))\) to \(v(t+dt)\) we need to calculate forces (that are position dependent) using \(x(t + dt)\);

Second part
void ParSim::Langevin_Dynamics::LM_Intergrator2(ParSim::ParticleSystem &parsym, int step)
  1. Updates \(v'(t+(dt/2))\) to \(v(t+dt)\) using new forces \(f(t+dt)\)

Evolvers

Task of an Evolver is to bring together appropriate force calculators and intergrators in the correct order of execution as required by the integration scheme you are using.

LM Evolver

LM evolver
void ParSim::Langevin_Dynamics::evolve_system_LM_NL(ParticleSystem &parsym, int step) {

  // Update Neighbour list after every NL_tau steps
  if (step == 0 || step % this->NL_tau == 0) {
    Neighbours_search(parsym);
  }
  // LM integration: (x(t),v(t)) -- > (x(t + dt),v(t + dt))

  // i) Calculate force f(t) using (x(t)) --
  Force_NL_PBC_WCA(parsym);

  // ii) Update (x(t), v(t))  to (x(t+dt) , v'(t+(dt/2))
  LM_Intergrator1(parsym, step);

  // iii) Again calculate force f(t+dt) from x((t+dt))
  Force_NL_PBC_WCA(parsym);

  //  iv) Update v'(t+(dt/2)) to v(t+dt) --------
  LM_Intergrator2(parsym, step);
}