Partition function (statistical mechanics)
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium.
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy).
The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.
The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous.
The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach.
In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable.
In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms.
In this case we must describe the partition function using an integral rather than a sum.
As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h3N (where h is usually taken to be the Planck constant).
where gj is the degeneracy factor, or number of quantum states s that have the same energy level defined by Ej = Es.
The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above.
In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):
The classical form of Z is recovered when the trace is expressed in terms of coherent states[1] and when quantum-mechanical uncertainties in the position and momentum of a particle are regarded as negligible.
Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:
Equivalently, pi will be proportional to the number of microstates of the heat bath B with energy E − Ei:
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy.
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression.
[citation needed] In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system.
These results can be derived using the method of the previous section and the various thermodynamic relations.
Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting.
If the sub-systems have the same physical properties, then their partition functions are equal, ζ1 = ζ2 = ... = ζ, in which case
If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N!
The partition function can be related to thermodynamic properties because it has a very important statistical meaning.
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does not depend on s), ensuring that the probabilities sum up to one:
Other partition functions for different ensembles divide up the probabilities based on other macrostate variables.
As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature.
Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state
The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.
The grand partition function is sometimes written (equivalently) in terms of alternate variables as[2] where
