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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
righthand 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
20060202