In an equilibrium system, the correlation between the ensemble average values of a dynamical variable

such as the velocity fluctuation ov = v – (v) at two times , and to should depend only on the time

difference, t = t2 – 1, and not on the absolute values of to and t2. This means if we write down the

time correlation function

C(t) = (ov(ti )ov(tz)gt;

and let $1 = 0,

C(t) = (ov(0)ov(t)) = (ov(-t)ov(0))

and switch the order of the two quantities in the angular brackets,

C(t) = (60(0)ov(-t)) = C(-t)

we can see there is symmetry in the time-correlation function C(t) = C(-t). 0

(i) What does this result suggest for the functional form of C(t) at short times? Can it be a simple

exponential (i.e. exp(-at)) decay?

(ii) If not, what other simple functional form would satisify C(t) = C(-t)?

(iii) Can you give a physical reason behind the functional form you suggest?Science