Partition Function (statistical Mechanics), Partition Function
Partition function may refer to: * Partition function (statistical mechanics), a function used to derive thermodynamic properties ** Rotational partition function, partition function for the rotational modes of a molecule ** Vibrational partition function, partition function for the vibrational modes of a molecule ** Partition function (quantum field theory), partition function for quantum path integrals * Partition function (mathematics), generalization of the statistical mechanics concept * Partition function (number theory) In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function i ...

Partition Function (statistical Mechanics)
In physics, a partition function describes the statistics, statistical properties of a system in thermodynamic equilibrium. Partition functions are function (mathematics), functions of the thermodynamic state function, state variables, such as the temperature and volume. Most of the aggregate thermodynamics, thermodynamic variables of the system, such as the energy, total energy, Thermodynamic free energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular Thermodynamic free energy, free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the Environment (systems), environment at fixed temperature, volume, an ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Rotational Partition Function
In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. Definition The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta: Z = \frac with: \zeta = \sum_j g_j e^ , where g_j is the degeneracy of the ''j''th quantum level of an individual particle, k_\text is the Boltzmann constant, and T is the absolute temperature of system. For molecules, under the assumption that total energy levels E_j can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom) E_j = \sum_i E_j^i = E_j^\text + E_j^\text + E_j^\text + E_j^\text + E_j^\text and the number of degenerate states are given as products of the single contributions g_j = \prod_i g_j^i = g_j^\text g_j^\text g_j^\text g_j^\text g_j^\text, where "trans", "ns", "rot", "vib ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Vibrational Partition Function
The vibrational partition functionDonald A. McQuarrie, ''Statistical Mechanics'', Harper & Row, 1973 traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. Definition For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text(T) = \prod_j where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_ is the energy of the ''j''th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . For an isolated molecule of ''N'' atoms, the number of vibrational modes (i.e. values of ''j'') is for linear molecules and for non-linear ones.G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945 In crystals, the vib ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Partition Function (quantum Field Theory)
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams. Generating functional Scalar theories In a d-dimensional field theory with a real scalar field \phi and action S phi/math>, the partition function is defined in the path integral formalism as the functional : Z = \int \mathcal D\phi \ e^ where J(x) is a fictitious source current. It acts as a generating functional for arbitrary n-point correlation functions : G_n(x_1, \dots, x_n) = (-1)^n \frac \frac\bigg, _. The derivatives used here ar ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Partition Function (mathematics)
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 partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the Hopfield network), and applications such as genomics, corpus linguistics and artificial intelligence, which employ Markov networks, and Markov logic networks. The Gibbs measure is also the unique measure that has the property of maximizing the entropy for a fixed expectation value of the energy; this under ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]