Computational Physics

8.5 Molecular dynamics methods for different ensembles 225

in velocity influences the equilibration time and the kinetic energy fluctuations.
If the rate is high, equilibration will proceed quickly, but as the velocity updates
are uncorrelated, they will destroy the long time tail of the velocity autocorrela-
tion function. Moreover, the system will then essentially perform a random walk
through phase space, which means that it moves relatively slowly. If on the other
hand the rate is low, the equilibration will be very slow. The rateRcollisionsfor which
wall collisions are best mimicked by Andersen’s procedure is of the order of

Rcollisions∼

κ

kBn^1 /^3 N^2 /^3

(8.73)

whereκis the thermal conductivity of the system, andn,Nthe particle density and
number respectively [32] (see Problem 8.9). Andersen’s method leads to a canonical
distribution for allN. The proof of this statement needs some theory concerning
Markov chains and is therefore postponed to Section 15.4.3, where we consider the
application of this method to lattice field theories.
For evaluating equilibrium expectation values for time- and momentum-
independent quantities, the full canonical distribution (8.72) is not required: a
canonical distribution in the positional coordinates

ρ(R)=e−U(R)/(kBT) (8.74)

is sufficient since the momentum part can be integrated out for momentum-
independent expectation values. For a sufficiently large system the total kinetic
energy of a canonical system will evolve towards its equilibrium value 3NkBT/ 2
and fluctuations around this value are very small. We might therefore force the
kinetic energy to have a value exactly equal to the one corresponding to the desired
temperature. This means that we replace the narrow distribution of the kinetic
energy by a delta-function

ρ(Ekin)→δ[Ekin− 3 (N− 1 )kBT/ 2 ]. (8.75)

The simplest way of achieving this is by applying a simple velocity rescaling pro-
cedure as described in the previous section(Eqs. (8.14)and(8.15))afterevery
integration step rather than occasionally:

pi→pi

√

3 / 2 (N− 1 )kBT

Ekin

. (8.76)

This method can also be derived by imposing a constant kinetic energy via a Lag-
range multiplier term added to the Lagrangian of the isolated system [33]. It turns
out [34] that this velocity rescaling procedure induces deviations from the canonical
distribution of order 1/

√

N, whereNis the number of particles.
Apart from the rescaling method, which is ratherad hoc, there have been attempts
to introduce the coupling via an extra force acting on the particles with the purpose