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## Grand canonical ensemble
In statistical mechanics, the It is convenient to use the grand canonical ensemble when the number of particles of the system cannot be easily fixed. Especially in quantum systems, e.g., a collection of bosons or fermions, the number of particles is an intrinsic property (rather than an external parameter) of each quantum state. And fixing the number of particles will cause certain mathematical inconvenience.
## Additional recommended knowledge
## The partition functionClassically, the partition function of the grand canonical ensemble is given as a weighted sum of canonical parition functions with different number of particles, where denotes the partition function of the canonical ensemble at temperature , of volume , and with the number of particles fixed at . (In the last step, we have expanded the canonical partition function, and is the Boltzmann constant, the second sum is performed over all microscopic states, denoted by with energy . ) Quantum mechanically, the situation is even simpler (conceptually). For a system of bosons or fermions, it is often mathematically easier to treat the number of particles of the system as an intrinsic property of each quantum (eigen-)state, . Therefore the partition function can be written as The parameter is called fugacity (the easiness of adding a new particle into the system). The chemical potential is directly related to the fugacity through - .
And the chemical potential is the Gibbs free energy per particle. (We haved used fugacity instead of chemical potential in defining the partition function. This is because fugacity is an independent parameter of partition function to control the number of particles, as temperature to control the energy. On the other hand, the chemical potential itself contains temperature dependence, which may lead to some confusion. ) ## Thermodynamic quantitiesThe average number of particles of the ensemble is obtained as And the average internal energy is
## Statistics of bosons and fermionsFor a quantum mechanical system, the eigenvalues (energies) and the corresponding eigenvectors (eigenstates) of the Hamiltonian (the energy function) completely describe the system. For a macroscopic system, the number of eigenstates (microscopic states) is enormous. Statistical mechanics provides a way to average all microscopic states to obtain meaningful macroscopic quantities. The task of summing over states (calculating the partition function) appears to be simpler if we do not fix the total number of particles of the system. Because, for a noninteracting system, the partition function of grand canonical ensemble can be converted to a product of the partition functions of individual Each The is the energy of the mode. For fermions, can be 0 or 1 (no particle or one particle in the mode). For bosons, . The upper (lower) sign is for fermions (bosons) in the last step. The total partition function is then a product of the ones for individual modes. ## Quantum mechanical ensembleAn ensemble of quantum mechanical systems is described by a density matrix. In a suitable representation,
a density matrix where So the trace of It is also assumed that the ensemble in question is Suppose where i-th energy eigenstate. If a system i-th energy eigenstate has n number of particles, the corresponding observable, the _{i}number operator, is given by
From classical considerations, we know that the state has (unnormalized) probability Thus the grand canonical ensemble is the mixed state The grand partition, the normalizing constant for |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Grand_canonical_ensemble". A list of authors is available in Wikipedia. |