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3. Monte Carlo Simulation For Mac Free
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The MonteCarlo scheme is a common useful way to sample systems in various ensembles.

The main idea of this technique is ImportanceSampling -

Namely, starting the simulation at a microstate that is part of the wanted ensemble, and then moving on to sample a new microstate with probabilities
that match the ensemble's microstates distribution (the Boltzmann distribution).

Even though the dynamics of the simulations might not mimic the real dynamics of the system, the equilibrium properties sampled by this technique
should correspond to the real system's equilibrium properties.

For further reading on the Monte Carlo scheme, please refer to: UnderstandingMolecularSimulations / Frenkel & Smit, Chapter 3

Examples

• Hard Discs - Monte Carlo simulation of hard discs, in the NVT ensemble

• Hard Discs with PBC- Same simulation, with Periodic Boundary Conditions.
  This is done to remove the impact of the simulation walls - practically simulating an infinite size system.

• Lennard-Jones Particles in NVT - Monte Carlo simulation of Lennard Jones particles, in the NVT ensemble.
  Add sampling routines to follow any desired property!

Direct Simulation Monte Carlo

• Lennard-Jones Particles in NPT- Monte Carlo simulation of Lennard Jones particles, in the NPT ensemble.
  Using displacement moves as well as volume change moves.
  Add sampling routines to follow any desired property!

• Parallel Tempering - Using the Parallel tempering technique to sample a double well system.

• Grand Canonical MC - Simulation of an ideal lattice gas in the grand canonical ensemble.
  A monte carlo move of particle addition/removal is introduced, in addition to the regular displacement moves.

• Grand Canonical MC, Activity Based- Same as the previous simulation, with a different acceptance crtiria for particle addition/removal.

• Grand Canonical MC, an Ising model- A grand canonical ensemble simulation on a lattice, with nearest neighbor interactions between particles

Monte Carlo Simulation For Mac Free

• Grand Canonical MC, Parallel Tempering - Paraller tempering on the Ising model system, sampling in the grand canonical ensemble.

Monte Carlo Simulation For Tolerance Stackup

Exercises

Monte Carlo Simulation Software For Mac

1. RadialDistributions - Calculate the radial distribution function of hard discs at various densities.
   
2. MaxwellBoltzmanmDistribution - Use the Lennard-Jones scripts given above to show the Maxwell-Boltzmann distribution of the particles
   at various phases.
   Compare your results for a gas, liquid and solid. What has changed?
   To find out how to sample each of these phases, use the following phase diagram: Picture taken from: PhaseDiagramofColloidalSolutions; N. Asherie, A. Lomakin, G.B. Benedek, Physical Review Letters (1996) 77, 23, pp. 4832-4835
   Where:
   • is the critical density (in reduced units)
   • is the critical temperature (in reduced units)
   • is the triple point temperature (in reduced units)
     
3. HarmonicOscillators - Test the assumption that a subset of a system of constant energy will in practice look like it's being coupled to a heat bath.
   Write a simulation of a set of harmonic oscillators, each holding an integer 'amount' of energy.
   At each time step, two random oscillators should be able to transfer one energy quanta from one oscillator to the other, as long as the 'donating'
   oscillator has a non-zero energy.
   What would be a typical histogram of the energy of one such oscillator? Does it depend on system size?
   Does your answer change if you start with all the energy stored in just one oscillator?

   Energy distribution in coupled Harmonic Oscillators

   
   
4. Whyuseparalleltempering? - Write a script that will follow a particle in the same double well system described in Parallel Tempering.
   Compare the timings for both methods to sample sufficiently.
   Would you say using parallel tempering is useful in that case?
   For timings, use the tic & toc functions.
   
5. GrandCanonicalLennard-Jones - Modify the existing LJ scripts (Lennard-Jones Particles in NVT or Lennard-Jones Particles in NPT)
   to sample at the Grand Canonical ensemble.
   How can you tell what was the chemical potential at the initial version?

Monte Carlo Simulation For Mac Os