Re: MD or MC

[mike #*at*# mycenae.CChem.Berkeley.EDU (Mike L. Greenfield)]

 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:
  #*at*# 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
 mike #*at*# mycenae.cchem.berkeley.edu