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Suppose we have a thermodynamic variable x with . What happens if at some moment ?
At we have (why?). For small deviations is small. We can expand it in series in , and
This formula is valid only for t>0: we can prepare an ensemble with given x0 and observe it, but cannot go backward! This is another manifestation of time irreversibility.
For space-dependent problems we defined the correlation function . For time-dependent case we define time correlation function . We want to calculate . This is the averaging over the ensemble of systems with arbitrary x(0). Note this difference with the previous case: there we had x(0) fixed, and used restricted ensembles. Now we return to the general case and consider equilibrium ensemble.
Translation along the time axis:
Double averaging method:
Since in a subensemble x(0) is given,
with generalized susceptibility .
We obtained for the correlation function:
From (4) we obtain for negative t that , or
We have several variables xi. Kinetic equations:
Now we have multi-variate correlation functions:
© 1997 Boris Veytsman and Michael Kotelyanskii