In an equilibrium system, the correlation between the ensemble average values of a dynamical variablesuch as the velocity fluctuation ov = v – (v) at two times , and to should depend only on the timedifference, t = t2 – 1, and not on the absolute values of to and t2. This means if we write down thetime correlation functionC(t) = (ov(ti )ov(tz)>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 simpleexponential (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?
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I think I need to figure out what functional form has the symmetry f(t)=f(-t). I was thinking f(t) might be either
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