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Let us now examine a purely mechanical interaction between macrostates, where one
or more of the external parameters is modified, but there is no exchange of
heat energy. Consider, for the sake of simplicity, a situation where only
one external parameter
of the system is free to vary.
In general, the number of microstates
accessible to the system when the overall energy lies between
and
depends on the particular value of
, so we can write
.
When
is changed by the amount
, the energy
of a given
microstate
changes by
. The number of states
whose energy is changed from a value less than
to a value
greater than
when the parameter changes from
to
is given by
the number of microstates per unit energy range multiplied by the average
shift in energy of the microstates. Hence,
 |
(190) |
where the mean value of
is taken over all accessible
microstates (i.e., all states where the energy lies between
and
and
the external parameter takes the value
). The above equation can also be written
 |
(191) |
where
 |
(192) |
is the mean generalized force conjugate to the external parameter
(see Sect. 4.4).
Consider the total number of microstates between
and
. When the
external parameter changes from
to
, the number of states in this energy
range changes by
. This change is
due to the difference between the number of states which enter the
range because their energy is changed from a value less than
to one greater than
and the number which
leave
because their energy is changed from a value less than
to one
greater than
.
In symbols,
 |
(193) |
which yields
 |
(194) |
where use has been made of Eq. (191). Dividing both sides by
gives
 |
(195) |
However, according to the usual estimate
, the first term on the
right-hand side is of order
, whereas the second term is only
of order
.
Clearly, for a macroscopic system with many degrees of freedom,
the second term is utterly negligible, so we have
 |
(196) |
When there are several external parameters
, so that
, the above derivation is valid for
each parameter
taken in isolation. Thus,
 |
(197) |
where
is the mean generalized force conjugate to the parameter
.
Next: General interaction between macrosystems
Up: Statistical thermodynamics
Previous: Temperature
Richard Fitzpatrick
2006-02-02