Relation To The Partition Function
A system kept at constant volume and temperature is described by the canonical ensemble. The probability to find the system in some energy eigenstate r is given by:
where
Z is called the partition function of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.
The average internal energy of the system is the expectation value of the energy and can be expressed in terms of Z as follows:
If the system is in state r, then the generalized force corresponding to an external variable x is given by
The thermal average of this can be written as:
Suppose the system has one external variable x. Then changing the system's temperature parameter by and the external variable by dx will lead to a change in :
If we write as:
we get:
This means that the change in the internal energy is given by:
In the thermodynamic limit, the fundamental thermodynamic relation should hold:
This then implies that the entropy of the system is given by:
where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes where is the ground state degeneracy. The partition function in this limit is where is the ground state energy. Thus, we see that and that:
Read more about this topic: Helmholtz Free Energy
Famous quotes containing the words relation to the, relation to, relation and/or function:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
“Art should exhilarate, and throw down the walls of circumstance on every side, awakening in the beholder the same sense of universal relation and power which the work evinced in the artist, and its highest effect is to make new artists.”
—Ralph Waldo Emerson (1803–1882)
“My function in life is not to be a politician in Parliament: it is to get something done.”
—Bernadette Devlin (b. 1947)