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## Mechanical interaction between macrosystems

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