Random forces in Langevin temperature control -- three questions

From: Blake Charlebois (bdc_at_mie.utoronto.ca)
Date: Fri Jul 15 2005 - 00:05:03 CDT

Hello everyone,

I would like to better understand the random force R applied in Langevin
temperature control.

BACKGROUND:

According to Tobias, Martyna, and Klein (J. Phys. Chem. 97(49):12959, 1993),
the governing equation (I will change variable names and use [...] for
subscripts) in, I assume, one dimension is

m[i]*a[i] = F[i] - m[i]*gamma[i]*v[i] + R[i](t)

and

"...R(t) is a Gaussian stochastic variable with zero mean and variance..."

<R[i](t)*R[i](t')> = 2*m[i]*gamma[i]*kB*T*delta(t-t')

Schneider and Stoll (Phys. Rev. B 17(3):1302, 1978) also give a description.

Here are my questions:

QUESTION ONE:
Variables t, t', and i appear on both sides of <R[i](t)*R[i](t')> =
2*m[i]*gamma[i]*kB*T*delta(t-t'). What type of averaging is <...> supposed
to represent here? Schneider and Stoll use <R[i](t)*R[k](t+tau)> =
2*m[i]*gamma[i]*kB*T*Kronecker_delta[i,k]*Dirac_delta(tau). Does this mean
that <...> denotes averaging over t, and that Tobias et al. have used
slightly confusing notation, or am I missing something?

QUESTION TWO:
It seems to me that the dimensions of the left- and right-hand sides of
<R(t)*R(t')> = 2*m*gamma*kB*T*delta(t-t') do not agree. What blunder am I
making here?

The dimensions of gamma are (time)^(-1).

The dimensions of 2*m*gamma*kB*T*delta(t-t') are
(mass)*(time)^(-1)*(force)*(distance) or (force)^2*(time).

The dimensions of <R(t)*R(t')> are (force)^2.

QUESTION THREE:
If delta(t-t') is the Dirac delta function, then the mean squared force is
infinity when t=t'. Schneider and Stoll, through mathematics I do not fully
understand at the moment, write R (they call it eta) in terms of random
variables and delta functions, allowing the elimination of the delta
functions when the equations of motion are integrated (I think). I have also
looked briefly at the NAMD 2.5 source code (file Sequencer.C, lines 278-286,
451-514; file Molecule.C, lines 3969-4160), but I do not fully understand it
either. In practice, what is the mean squared random force in a given
direction?

Any help with one or more of these questions would be much appreciated.

Blake Charlebois

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