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,

(5.34) |

where the mean value of is taken over all accessible microstates (i.e., all states for which the energy lies between and , and the external parameter takes the value ). The previous equation can also be written

where

(5.36) |

is the mean generalized force conjugate to the external parameter . (See Section 4.4.)

Consider the total number of microstates whose energies lies 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 that enter the range because their energy is changed from a value less than to one greater than , and the number that leave because their energy is changed from a value less than to one greater than . In symbols,

(5.37) |

which yields

(5.38) |

where use has been made of Equation (5.35). Dividing both sides by gives

(5.39) |

However, according to the usual estimate (see Section 3.8), 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

(5.40) |

where use has been made of Equation (5.30).

When there are several external parameters, , so that , the previous derivation is valid for each parameter taken in isolation. Thus,

where is the mean generalized force conjugate to the parameter . (See Section B.2.)