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MOL:  Programs for molecular simulation with Lennard-Jones potential.

The 'mol' programs implement simulation of systems of molecules
interacting according to the artificial Lennard-Jones potential
function.  The full range of techniques supported by this software are
implemented for this module, including Metropolis Monte Carlo,
molecular dynamics, simulated tempering, annealed importance sampling,
and Hamiltonian importance sampling.

The molecules are simulated in a space of 1, 2, or 3 dimensions, with
each dimension wrapping around after some distance (ie, "periodic
boundary conditions" are used).  The molecules interact according the
the Lennard-Jones potential, which for each pair of molecules is given
by

min { 4 scale [ (width/dist)^12 - (width/dist)^6 ],
max (0, maxpe - 4 scale (dist/width)^2) }

Here, "dist" is the distance between the two molecules (defined as the
distance of the shortest line connecting them in the wrapped-around
space).  The scale, width, and maxpe parameters are specified when
setting up a simulation.  The maxpe parameter can be infinite, in
which case there is no upper limit on the potential.  No "cut off" in
the potential is used - the potential for all pairs is calculated,
even when the contribution of a pair to the overall potential is tiny.

The total potential energy for the system is the sum of the the pair
potentials for each distinct pair of molecules in the system.  The
basic task of the molecular simulation programs is to sample from the
canonical distribution with this potential energy, at a temperature of
one (using units in which Boltzmann's constant is one).  Sampling from
both the NVT ensemble (constant number of molecules, volume, and
temperature) and the NPT ensemble (constant number of molecules,
pressure, and temperature) are supported.

For the NPT ensemble, there is an additional term in the potential
energy relating to the current volume.  If the state were recorded in
terms of volume, this term would be the volume times the pressure, P,
minus the entropy for N particles in volume V, -N log(V).  However,
what is recorded in the state is actually the log of the length of a
dimension, L, which gives volume equal to exp(LD)), where D is the
dimensionality.  The appropriately transformed term for the potential
energy is then

P exp(LD) - (N+1) LD

Although the simulation always proceeds as if the desired distribution
is that at a temperature of one, the equivalent of the canonical
distribution at a temperature other than one can be obtained by
changing the "scale" parameter (and "maxpe" if it's not infinite) in
the pair potential.  For the NPT ensemble, the pressure also needs to
be changed.  One then needs to also rescale things when interpreting
the results - in particular, one should look at the "U" quantity
rather than "E", and "p" rather than "P" (see mol-quantities.doc).

Some of the Monte Carlo methods involve momentum variables, with
associated kinetic energy, for which each molecule is assumed to have
unit mass.