512 Classical time-dependent statistical mechanics
molecular dynamics simulations, one form might yield numerically more stable results
than the other.
13.3.3 Example: The harmonic oscillator
To illustrate the concept of a time correlation function more explicitly, we calculate
the position and velocity autocorrelation functions for a simple harmonic oscillator of
massmand frequencyωin the canonical ensemble. The Hamiltonian for the harmonic
oscillator is
H(x,p) =p^2
2 m+
1
2
mω^2 x^2. (13.3.36)From Section 1.3, we know that the solution (xt(x,p),pt(x,p)) of Hamilton’s equations
starting from an initial conditionxandpfor the position and momentum is
xt(x,p) =xcosωt+p
mωsinωtpt(x,p) =pcosωt−mωxsinωt. (13.3.37)Note that we have dropped the “0” subscript on the initial conditions, since we will
need to consider each point (x,p) in phase space as initial condition in order to calculate
the time corelation function. In Section 4.5, we showed that the classical canonical
partition function isQ(β) = 2π/βhω(see eqn. (4.5.19)). The position autocorrelation
function requires that eqn. (13.3.37) be integrated over all initial conditions weighted
by the canonical distribution exp[−βH(x,p)] according to the definition
Cxx(t) =1
hQ(β)∫∞
−∞dx∫∞
−∞dp[xxt(x,p)]e−βH(x,p). (13.3.38)Substitution of eqns. (13.3.37) into eqn. (13.3.38) gives
Cxx(t) = (βω)∫∞
−∞dx∫∞
−∞dp x(
xcosωt+
p
mωsinωt)
exp[
−β(
p^2
2 m+
1
2
mω^2 x^2)]
=
βω
2 πcosωt∫∞
−∞dx∫∞
−∞dp x^2 exp[
−β(
p^2
2 m+
1
2
mω^2 x^2)]
=
kT
mω^2
cosωt. (13.3.39)Because this correlation function never decays, the correlation time is infinite. In other
words, a harmonic oscillator never loses memory of its initial conditions and, therefore,
does not obey the Onsager regression hypothesis.
13.4 Calculating time correlation functions from molecular dynamics
The use of the Green–Kubo relations for transport coefficients requires the calcula-
tion of classical time correlation functions, which, as we will show in this section,