The Path-Integral-Monte-Carlo Online-Interface



A demonstration of quantum effects in different one particle 1D Systems

This page shows a Java Applet where you can see a running PIMC-simulation while controlling all interesting parameters. The potential energy and the (relative) probability of x are calculated. Keep in mind that this is about a canonical enssemble, i.e. the energy is not constant, but the probability is given by Bolzmann's factor of the particular temperature.

The trotter number P gives the "degree of quantum-mechanicness" you allow for the simulation. P=1 means no quantum mechanics at all, i.e. the particle is represented by only one (classical) point. By increasing P you turn on the quantum mechanics, and will hopefully reach the "quantum limit" where further increases do not change your results.

Physics

We are talking about statistical mechanics. For calculating of any observable we need the partition function and related integrals like Int( E(x) * P(x) dx), where x means the configuration, E(x) the energy and P(x) the non normed probability of any given configuration. In these terms the partition function looks like Int( P(x) dx); and, in our case, P(x) should be the Bolzmann factor, because we want to control the temperature T.

With the help of Feynman's path-integrals and the primitive approximation (exp(eps(T+V)) appr.equals exp(epsT)exp(epsV)) we can map any quantum system onto a 'classical-like' system of "ring-polymers" - but be careful, this is only correct for a picture and NOT for the actual interaction between the single beads of the rings!

For the actual integration we use the same procedure thas is used for classical systems: the Monte-Carlo method. The name comes from the fact that random numbers are used to pick up one point after another in the configuration space. If we are able to visit each configuration with the apropriate probability, we only need to calculate the arithmetic average over the observables we are interested in and will end up having the correct statistical enssemble average.
This is possible by following Metropolis' algorithm. Here we need to calculate the energy-difference one would achieve by doing the "step" from a "present" configuration to the "next" configurtion (to be tested). If there is an energy loss the step will be accepted, and an energy gain is only accepted with the bolzmann-probability (i.e. take a random number between 0 and 1, and test if it is smaller than exp(-dE/kT)). If the tried step is rejected the old configuration is also the new one.
In our simulation there are two kinds of tested "steps":

  •  "one-Trotter move" (also known as "local move"): only one bead, called a Trotter-particle, is moved a random distance (dXoneMax at most)
  •  "all-Trotter move" (also known as "global move"): the whole particle, i.e. all beads, are moved the same random distance (dXallMax at most)

    • User-Interface

    In each (editable) textfield you can increase or decrease the number with cursor-up and cursor-down keys.
    If you enter a new value don't forget the return key to accept the changes.