Subject: Re: MD or MC

From: [email-masked] (Mike L. Greenfield)
Date: Wed, 9 Nov 94 23:20:34 PST

In terms of calculating equilibrium properties, another difference
between MD and MC arises when simulating polymer systems subject to
constraints.  Imposing bond length (and bond angle) constraints
changes the equilibrium sampling distribution in configuration space,
since different techniques naturally use a different model of the
polymer intramolecular interactions.  If no Fixman potential is used
in the MD force calculations, a typical MD simulation will sample a
model polymer with "rigid" constraints.  An MC simulation in which
these constraints are imposed during each attempted move will sample a
model polymer with "flexible constraints in the limit of infinite
stiffness".  Though these two models sound the same, they sample
configuration space differently, and equilibrium results obtained by
the two methods will differ.

In a rigid model, the bond length (and angle) constraint enters
directly into the equations of motion via the Lagrangian.

In a flexible model, bond lengths (and angles) are constrained to
their desired value by a harmonic potential.  In the infinite
stiffness limit, the coefficients of the harmonic potential approach
infinity, and thus constrain the system.
(exp -V/kT = exp -(l-l_0)^2/kT = delta function centered at l=l_0)

The differences between these models arise from their different
treatment of the kinetic energy.  Go and Scheraga recommend using the 
flexible model in the limit of infinite stiffness.

For discussions of "rigid" vs "infinitely stiff" polymer models, see
the classic paper of Go and Scheraga:

@Article{Go760,
  author = 	 "Nobuhiro G\={o} and Harold A. Scheraga",
  title = 	 "On the Use of Classical Statistical Mechanics in the
		  Treatment of Polymer Chain Conformation",
  journal = 	 "Macromolecules",
  year = 	 1976,
  volume = 	 9,
  pages = 	 "535--542"
}

Other references include

M. Fixman,  Proc Nat Acad Sci USA 71:3050 (1974) (Fixman potential)
M. Fixman,  J Chem Phys 69:1527 (1978) (Fixman potential)
E. Helfand, J Chem Phys 71:5000 (1979)
P. Pechukas "   "    "  72:6320 (1980) (comment on Helfand's paper)
M.R. Pear and J.H. Weiner J Chem Phys, 71:212(1979), 72:3939 (1980)

To obtain consistent results with the two techniques, it is necessary
to use a Fixman potential (see Fixman, Go & Scheraga above) s.t.
molecular dynamics samples according to the flexible model.  I do not
know what effect this potential has on the dynamics aspects of MD.

Mike Greenfield

[email-masked]