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Suppose we have a thermodynamic variable *x* with . What
happens if at some moment ?

**Thermodynamics Language:**- We have a
*restricted ensemble:*a set of systems with the given*x*. **Example:**- I prepared a set of beakers at the temperature
*T*and put it in the thermostat with the temperature_{0}*T*. **Kinetics and Equilibrium:**- A beaker at the temperature
*T*is*not*in the equilibrium with the environment--technically speaking, we*cannot*use thermodynamics! **A Trick:**- we say that it is
*partially equilibrated*--everything is equilibrated*for this temperature*. **General situation:**- we have a slowly changing variable
*x*, and say, that every other degree of freedom is fast.

(1) |

At we have *(why?).* For
*small* deviations is small. We can expand it in
series in , and

(2) |

(3) |

This formula is valid *only* for *t*>0: we can prepare an ensemble
with given *x _{0}* and observe it, but cannot go backward! This is
another manifestation of

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:

and therefore(4) |

Double averaging method:

- 1.
- Divide the ensemble into subensembles with given
*x*(0)=*x*_{i}. Each subensemble has many*microscopic*states with given*macroscopic*value*x*(0). - 2.
- Average
*x*(0)*x*(*t*)*in*a subensemble: - 3.
- Average over
*all*subensembles:

Since in a *subensemble* *x*(0) is given,

We obtained for the correlation function:

From (4) we obtain for negative(5) |

This function has different derivatives at : time irreversibility!

We have *several* variables *x*_{i}. Kinetic equations:

- 1.
- Translation of the time axis:
We see:
(6) - 2.
- Equations of motion are symmetric with respect to time inversion
:
(7) - 3.
- From (6) and (7)
(8)

© 1997

Tue Oct 28 22:13:33 EST 1997