504 Classical time-dependent statistical mechanics
and will resemble each other as they traverse the phase space. Inorder to see what the
existence of a correlation time implies for a correlation function, consider the special
casea(x) =b(x). The time correlation function
CAA(t) =〈a(0)a(t)〉=∫
dxf(x)a(x)a(xt(x)) (13.2.33)is known as anautocorrelation function. For very short times,a(xt(x)) anda(x) are
not very different, hence they are highly correlated. As long astis small compared
to the correlation time, the trajectory xt(x) appears to be particular to the initial
condition x, anda(xt(x)) remains correlated witha(x). However, for times longer
than the correlation time, the trajectory loses memory of its initialcondition and
a(xt(x)) anda(x) becomeuncorrelated. (Recall Fig. 3.7 which shows how rapidly two
very similar initial conditions diverge in time for a Lennard-Jones liquid.)Clearly, the
length of a correlation time depends on the nature of the system and the property
under consideration. However, the notion of two properties becoming uncorrelated
after a sufficiently long time is the basis of theOnsager regression hypothesis. The
latter states that in complex systems in which memory of initial condition is not
retained, the long-time behavior of a time correlation function is given by
lim
t→∞
CAB(t) =〈a〉〈b〉. (13.2.34)Clearly, this hypothesis does not apply to all systems and cannot beproved in general.
In fact, a harmonic oscillator is an example of a pathological system that has an
infinitely long memory of its initial condition and therefore violates theregression
hypothesis. However, for sufficiently chaotic systems with finite correlation times, the
regression hypothesis generally holds. Note that the initial value ofan autocorrelation
functionCAA(0) =〈a^2 〉is the equilibrium average ofa^2 (x). If we define a phase space
function
δa(x) =a(x)−〈a〉, (13.2.35)
then the corresponding macroscopic observableδA=〈δa〉= 0. However, the autocor-
relation function ofδAis
CδAδA(t) =〈δa(0)δa(t)〉=〈(a(x 0 )−〈a〉) (a(xt)−〈a〉)〉, (13.2.36)whose initial value is
CδAδA(0) =〈(a(x 0 )−〈a〉) (a(x 0 )−〈a〉)〉=〈a^2 〉−〈a〉^2 , (13.2.37)which is just the equilibrium fluctuation inA. In the remainder of this and in Chap-
ters 14 and 15, we will see that time correlation functions play a central role in the
theory of transport coefficients and vibrational spectra.
13.3 Applying linear response theory: Green–Kubo relations for
transport coefficients
13.3.1 Shear viscosity
The coefficient of shear viscosity (denotedη) is an example of a transport property
that characterizes the resistance of a system to flow under the action of a shearing