514 Classical time-dependent statistical mechanics
...λ=1λ=2λ=Kttt(^02) ∆t 4 ∆t
t
0 2 ∆t 4 ∆t
(a) (b)
Fig. 13.5 Pictorial representations of (a) the calculation of the time correlation function
CAB(n∆t) at times ∆t,2∆t,3∆t, and 4∆t via eqn. (13.4.1), and (b) the calculation of
CAB(n∆t) at points ∆tand 2∆tvia eqn. (13.4.3). In each panel, terms connected by a
square bracket are multiplied. All similar brackets are then averaged.
approximately applies, so that the microcanonical distribution is approximately equiv-
alent to the canonical distribution; 2) solutions to Hamilton’s equations are ergodic
enough to generate an adequate sampling off 0 (H(x)). The first assumption implies
that microcanonical temperature fluctuations are negligibly small (recall that such
fluctuations decrease as 1/
√
N, whereNis the number of particles), while the second
ensures that the trajectory can serve as a means both of sampling the equilibrium
distribution and generating the dynamics of the system. If both conditions hold, then
the calculation of time correlation functions can be simplified considerably.
Consider rewriting eqn. (13.2.30) in a form that exploits the equivalence of the
time and phase space averages of an ergodic system (see Section 3.7):
CAB(τ) =∫
dxf 0 (H(x))a(x)b(xτ(x)) = lim
T→∞1
T
∫T
0dt a(xt)b(xt+τ). (13.4.2)Eqn. (13.4.2) seems to embody a contradiction. First, it implies that each configuration
xtgenerated by solving Hamilton’s equations is an independent sampling off 0 (H(x)).
At the same time, however, it exploits the unique dependence of a trajectory on its
initial conditions, which implies that any point xtof the trajectory can be uniquely de-
termined from any other point in the trajectory. Thus, the point xt+τevolves uniquely
from xtforτ >0. These two conditions might appear incompatible, for how can the
point xtbe both an independent sampling fromf 0 (H(x)) and a source for any other
point xt+τof the trajectory? Once again, the existence of a finite correlation time re-
solves the paradox. If the time correlation function eventually decays to zero on a time
scale specified by the correlation time and the total timeTof a finite-time trajectory is
much larger than the correlation time, then we can imagine breaking atrajectory into
segments of length similar to the correlation time. Each segment canthen be regarded