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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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