Partition Function

In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. 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. Other types of partition functions can be defined for different circumstances. A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. Equivalently, the members of the ensemble can be considered loosely-coupled to each other so that they can share the total energy. The distribution of the total energy amongst the possible dynamical states (i.e. the members of the ensemble) is given by the partition function. A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy. By contrast, in the microcanonical ensemble, the energy of each individual system is fixed. A full development of the concept of the canonical ensemble is given in the article on the partition function. An example of the mathematical formulation of the canonical ensemble as a probability measure expressed in the language of measure theory is given in the article on the Potts model. In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. The grand canonical ensemble frequently provides the most convenient avenue for calculations. [edit]

Partition function The partition function of the grand canonical ensemble with a matrix/operator formalism is:

Here μ is the chemical potential, β the inverse temperature, sometimes also adorned with the inverse of the Boltzmann constant.

is the

Hamiltonian of the system class considered, the operator that counts the total number of particles in one system. [edit]

Discrete summation formalism The partition function of the grand canonical ensemble is given by

The sum of the index i is over all the energy states of the system. The sum over the index j is over all the number of partitions, where Nj gives the number of particles in partition j. [edit]

Characteristic state function The characteristic state function for the grand canonical ensemble is the quantity PV. This is because the ensemble satisfies the property Retrieved from "http://en.wikipedia.org/wiki/Grand_canonical_ensemble"