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Partition Function (other)
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Partition function may refer to: * Rotational partition function * Vibrational partition function *Partition function (number theory) *Partition function (mathematics), which generalizes its use in statistical mechanics and quantum field theory: **Partition function (statistical mechanics) **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 functio ... [...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 ''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 3''n'' − 5 for linear molecules and 3''n'' − 6 for non-linear ones.G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of ends in the digit 4 or 9, the number of partitions of will be divisible by 5. Definition and examples For a positive integer , is the ... [...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 underlies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (statistical Mechanics)
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate 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. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular 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 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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
