PIMC live

Check out the Path Integral simulation applet on John Shumway's page. You can change the temperature and see the paths localize at high temperature and spread out at low temperature.

Pseudorandom number generators and the stock market

As correlations are discovered in psuedorandom number generators used in simulations, the correlations are removed by creating better generators. So as time marches on, the state of the art PRNG's should be getting more and more random.

As correlations are discovered in the stock market, they are removed as the resevoir of correlation is drained for its profit. So the stock market should be getting more and more random as time goes on (especially as faster computers and more sophisticated statistical techniques are used)

Experimentally switching between liquid and solid phases?

Looking at Grochola's free energy path, I thought it was a nice computational technique.

The March 2004 issue of Physics Today has an article on controlling atoms and ions in optical traps (it's actually about using these for quantum information processing). On page 41, it describes putting the atoms in a optical lattice. The depth of the wells and the hopping matrix elements can be controlled by varying the intensity of the lasers creating the lattice.

This seems like one could experimentally switch between solid and liquid states without going through a phase transition.

There are some differences - I don't know if they are important or not. First, the experimental setup is a quantum system, Grochola's paper describes a classical system. Second, the confining potential is different. In the experimental setup, atoms are confined to any lattice site, but are not tethered to a particular lattice site.

The Physics Today article mentions a Mott insulator transition when the confining potential becomes strong enough. This transition seems due to the superfluid (quantum) nature of the liquid state. If the experiment were performed on a non-superfluid system, this transition would not happen ??

Update (3/16/2004): I forgot the final stage in the path - turning off the confining potential to get a stable solid state. I'm guessing the lattice spacing of the optical lattice is at least the laser wavelength or greater, so it's much larger than the lattice spacing of a normal solid. But some atoms can have their scattering length tuned - could it be made large enough to be comparable to the optical lattice spacing? In that case the atoms would behave as hard spheres (most importantly, there would likely not be enough attraction to keep the solid together), so an external potential would be required. Most of these setups seem to generate a harmonic confining potential (a box-like potential seems much harder to generate??), so the question is - what is the behavior of hard spheres in a harmonic potential?

For this system, there are two parameters - the number of particles and the strength of the external potential. I suspect there would be three regimes - one all liquid, one with a solid center surrounded by a liquid exterior, and one would be all solid. To take the thermodynamic (large N) limit, it seems that one would need to increase the particle number and decrease the external potential strength simultaneously to keep the central density or pressure constant (otherwise the pressure/density at the center would increase with added particles). Or maybe such a limit doesn't exist?

The Way of Antisymmetry

The traditional QMC wavefunction is the product of a determinant of single particle orbitals and a Jastrow factor (which contains the electron-electron correlation). The determinant ensures the required spatial antisymmetry of the wavefunction, and is computable in O(N³) time (where N is the number of electrons). But it is not the most general form possible.

The most general form is obtained by explicitly antisymmetrizing: summing the values of a function with all possible permutations of the electron coordinates (and multiplying by the sign appropriate for the permutation). For example, with 2 particles and a function f(r₁,r₂), the antisymmetrized form is f(r₁,r₂) - f(r₂,r₁).

Unfortunately, the number of terms is N!. (The determinant form is obtained by applying this procedure when the function has the special form of the product of functions that each depend only on a single electron coordinate - these functions are the single particle orbitals. In the two particle example, f = f₁(r₁)f₂(r₂). The antisymmetrized form is f₁(r₁)f₂(r₂) - f₁(r₂)f₂(r₁), which is the determinant of a matrix.)

Dario Bressanini, Massimo Mella, Gabriele Morosi, and Luca Bertini have several papers using this form of the wavefunction for small molecules, and they get very good results. The papers can be found on Massimo Mella's publication list (edit 12/3/2005 - removed nonworking link). They are (this may not be a complete list):

• Many-electron correlated exponential wavefunctions. A Quantum Monte Carlo application to H2 and He2+
  
  Chemical Physics Letters 240, 566 (1995)
  
  Paper available at this Citeseer page
  
  
• Nonadiabatic wavefunctions as linear expansions of correlated exponentials. A Quantum Monte Carlo application to H2+ and Ps2
  
  Chem. Phys. Lett. 272, 370 (1997)
  
  Paper available at this Citeseer page
  
  
• Linear expansions of correlated functions: a variational Monte Carlo case study
  
  Int. J. Quantum. Chem. 74, 23 (1999)
  
  Paper available at this Citeseer page
  
  
• Explicitly correlated trial wave functions in quantum Monte Carlo calculations of excited state of Be and Be-
  
  J. Phys. B, (2001)
  

Not all N! terms will give a significant contribution to the result. Is it possible to use techniques similar to O(N) approaches to reduce the number of terms? Is it possible to sample terms from the sum? These might have trouble near the nodes, since the wavefunction vanishes due to the cancelation of all the terms.

DMC of porphyrin

An all electron calculation of a large molecule (large by QMC standards, anyway):
Quantum Monte Carlo for electronic excitations of free-base porphyrin by Alan Aspuru-Guzik, Ouafae El Akramine, Jeffrey Grossman, and William Lester. The porphyrin they used has the formula C₂₀ N₄ H₁₄, so 168 electrons.

Solid vs. Liquid, The Free Energy Battle

Determining the free energy difference between the solid and liquid states is essential for accurately mapping the phase diagram of a substance. Typically, one determines the free energies of the solid and liquid states through separate thermodynamic integrations, where each integration has one end point connected to a state with an analytically known free energy. In the solid state, the Einstein crystal (springs attaching particles to lattice sites) is often used as a starting point. In the liquid state, the low density gas can be used as an end point. Other more complicated scalings of the potential are also possible for the integration path.

A direct integration path between the solid and liquid states would be useful.
Gregory Grochola has published one such path in "Constrained fluid lambda-integration: Constructing a reversible thermodynamic path between the solid and liquid state"(JCP 120, 2122).

The basic path is to start with the liquid state and turn off the intermolecular potential. Then turn on a potential constraining particles to lattice sites (He used a Gaussian potential). The last step is to turn off the lattice potential while turning the intermolecular potential back on.

He mentions at the end that this method is not very computationally efficient. I would expect using Bennett's method for free energy differences would cut down on the number of intermediate states needed.

Questions

1. Compare and contrast this path with the phase switching Monte Carlo technique (there's a reference in the article)
   
2. Thermodynamic integration requires there be no phase transitions on the path. Is it immediately obvious that this path has no transitions? Under what conditions, if any, would a phase transition occur on the path?
   

Hard Spheres on the Move

Cluster algorithms increase efficiency for systems on a lattice by flipping large numbers of connected spins at once. In "Cluster Monte Carlo algorithms", Werner Krauth summarizes work on how such algorithms can be used in a continuum hard sphere simulation.

The basic idea is to rotate particles by &pi around a randomly chosen axis. (The key observation is that this transformation is its own inverse). A practical algorithm (the pocket algorithm) chooses a collection of particles that can be rotated without any hard spheres overlapping. It starts with a single particle and rotates it. If it overlaps with any other particles, they are added to the collection. Then the new particles are tested in the same way, and the collection is built up of all the particles that need to be rotated to prevent any overlaps.

At high densities, this algorithm will fail because the pocket cluster will contain all (or most) of the particles in the system, and simply rotate the entire configuration. Despite this limitation, there are systems where these moves are useful (see the paper for some examples).

On another subject, suppose you wanted to keep the trial box size fixed, and see how the dynamics varied with density (which would be useful if one were studying the actual MC dynamics) At higher densities, the efficiency would get quite bad. Hirosi Watanabe, et al tackle this problem in A rejection-free Monte Carlo method for the hard-disk system. They use geometry rather than random sampling to determine the allowable move area (and acceptance probability). To actually choose a point, they use a rejection method (it's not clear why this method is called "rejection-free", then. Maybe because the rejections are not essential for maintaining detailed balance, and rejected moves don't need to be included in the average.). At the step of choosing a new point, the area they choose from is greatly reduced from the original trial area.