Path Integral Monte Carlo

a presentation by Keola Wierschem

Metropolis Monte Carlo

• based on random sampling

• an exact simulation technique

• error from approximations in atomic interactions,

finite-size effects, and use of pseudo-random numbers

Sampling Technique

• start with any configuration of particles

• systematically propose and accept/reject new particle positions

• must satisfy detailed balance

• must be able to reach all possible configurations in a finite number of iterations

• an iteration occurs when a new position has been proposed

and accepted/rejected for each particle in the system

Accept or Reject?

• large based on Hamiltonian of system

• large ie kinetic and potential

• large changes with temperature and forces

Path Integrals

• developed by R. P. Feynman mid-20th century

• describe quantum particles as classical ”ring-polymers”

• use concept of imaginary time (inverse of temperature)

• allow one to understand the quantum effects at low temperature in terms of the better understood classical behavior at higher temperature

Path Integral Monte Carlo

• uses path integrals to reduce the temperature of interactions in standard Monte Carlo

• more paths = lower temperature

• has been used to study superfluidity, a quantum effect

occuring at low temperatures

Superfluidity

• A superfluid has zero viscosity, or resistance to flow

• Helium is the observed superfluid to date

• Bulk helium becomes superfluid at about 2.17 K

• PIMC can be used to search for new superfluids

• Using computer simulation, one can study any number of systems with very little cost Figure 1: Initial configuration of humans

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Figure 2: Relaxed configuration of humans

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Figure 3: Relaxed configuration of humans adjusted by KEG

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Summary

• Monte Carlo is a powerful technique based on random sampling

• Path integrals, when applied to MC, allow one to simulate the quantum effects of particles that occur at low temperatures

Figure 4: from Ceperly and Gordillo: superfluid hydrogen on silver doped with alkali metal