250 Molecular dynamics simulations
The expectation value ofv^2 is determined in a similar way. Using(8.151)and
(8.144)we find
〈[v(t)]^2 〉=v^20 exp(− 2 γt/m)+
q
2 γm( 1 −e−^2 γt/m), (8.154)which for largetreduces to
〈[v(∞)]^2 〉=
q
2 γm. (8.155)
According to(8.152),vdepends linearly on the random forcesR(t)and as the
latter are distributed according to a Gaussian, the same will hold for the velocity.
The width is given by(8.155), so we have
P[v(t)]=(
γm
πq) 1 / 2
exp[−mv(t)^2 γ/q] (8.156)for larget. This is precisely the Maxwell distribution if we write
q= 2 kBTγ, (8.157)so this equation defines the value ofqnecessary to obtain a system with temperature
T. In Chapter 12 we shall discuss Langevin types of equations in a more formal
way, using the Fokker–Planck equation.
The velocity autocorrelation function can also be obtained from (8.152):
〈v( 0 )v(t)〉=〈v( 0 )^2 〉e−γt/m. (8.158)
The absence of a long time tail in this correlation function reflects the oversim-
plifications in the construction of the Langevin equation, in particular the absence
of correlations in the random force and the fact that the frictional force does not
depend on the ‘history’ of the system.
The results presented here are easily generalised to more than one dimension.
However, including a force acting between the heavy particles causes problems if
this force exhibits correlations with the random force, andEq. (8.157)is no longer
valid in that case. Such correlation effects are often neglected and the systematic
force is simply added to the friction and the Langevin term.
A further refinement is the inclusion of memory kernels in the forces, similar to
the approach inEq. (8.143). In that case, the random force is no longer uncorrelated –
it is constructed with correlations in accordance with the fluctuation-dissipation
theorem[64]:
〈R( 0 )R(t)〉=〈v^2 〉γ(t). (8.159)
However, this equation is again no longer valid if external forces are included. The
approach with memory kernels has led to a whole industry of so-called generalised
Langevin-dynamics simulations [ 64 – 67 ].