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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; see partition function (mathematics) for
In other words, each single-particle level is a separate, tiny grand canonical ensemble. By the Pauli exclusion principle, there are only two possible microstates for the single-particle level: no particle (energy E = 0), or one particle (energy E = ε). The resulting partition function for that single-particle level therefore has just two terms:
For now let us refer to these single-particle stationary states as orbitals (to avoid confusing these "states" with the total many-body state), with the provision that each possible internal particle property (spin or polarization) counts as a separate orbital. Each orbital may be occupied by a particle (or particles), or may be empty.
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
The denominator in equation 1 is a normalizing factor so that the ratios : add up to unity — in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system).
In statistical mechanics, the translational partition function, is that part of the partition function resulting from the movement (translation) of the center of mass.For a single atom or molecule in a low pressure gas, neglecting the interactions of molecules, the canonical ensemble can be approximated by: [1]
The total canonical partition function of a system of identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions : [1] =! with: = /, where is the degeneracy of the jth quantum level of an individual particle, is the Boltzmann constant, and is the absolute temperature of system.
If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers N 1 (first kind of particle), N 2 (second kind of ...