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• Partition_function_(mathematics) abstract "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 appearance of the partition function in maximum entropy methods and the algorithms derived therefrom.The partition function ties together many different concepts, and thus offers a general framework in which many different kinds of quantities may be calculated. In particular, it shows how to calculate expectation values and Green's functions, forming a bridge to Fredholm theory. It also provides a natural setting for the information geometry approach to information theory, where the Fisher information metric can be understood to be a correlation function derived from the partition function; it happens to define a Riemannian manifold.When the setting for random variables is on complex projective space or projective Hilbert space, geometrized with the Fubini–Study metric, the theory of quantum mechanics and more generally quantum field theory results. In these theories, the partition function is heavily exploited in the path integral formulation, with great success, leading to many formulas nearly identical to those reviewed here. However, because the underlying measure space is complex-valued, as opposed to the real-valued simplex of probability theory, an extra factor of i appears in many formulas. Tracking this factor is troublesome, and is not done here. This article focuses primarily on classical probability theory, where the sum of probabilities total to one.".
• Partition_function_(mathematics) wikiPageID "16846849".
• Partition_function_(mathematics) wikiPageLength "19239".
• Partition_function_(mathematics) wikiPageOutDegree "142".
• Partition_function_(mathematics) wikiPageRevisionID "701178966".
• Partition_function_(mathematics) wikiPageWikiLink Artificial_intelligence.
• Partition_function_(mathematics) wikiPageWikiLink Boltzmann_distribution.
• Partition_function_(mathematics) wikiPageWikiLink Bures_metric.
• Partition_function_(mathematics) wikiPageWikiLink C*-algebra.
• Partition_function_(mathematics) wikiPageWikiLink Canonical_ensemble.
• Partition_function_(mathematics) wikiPageWikiLink Category:Entropy_and_information.
• Partition_function_(mathematics) wikiPageWikiLink Category:Partition_functions.
• Partition_function_(mathematics) wikiPageWikiLink Clique_(graph_theory).
• Partition_function_(mathematics) wikiPageWikiLink Complex_projective_space.
• Partition_function_(mathematics) wikiPageWikiLink Configuration_space.
• Partition_function_(mathematics) wikiPageWikiLink Conjugate_variables_(thermodynamics).
• Partition_function_(mathematics) wikiPageWikiLink Conservation_of_energy.
• Partition_function_(mathematics) wikiPageWikiLink Corpus_linguistics.
• Partition_function_(mathematics) wikiPageWikiLink Correlation_function.
• Partition_function_(mathematics) wikiPageWikiLink Countable_set.
• Partition_function_(mathematics) wikiPageWikiLink Covariance_matrix.
• Partition_function_(mathematics) wikiPageWikiLink Critical_exponent.
• Partition_function_(mathematics) wikiPageWikiLink Differential_operator.
• Partition_function_(mathematics) wikiPageWikiLink Dynamical_system.
• Partition_function_(mathematics) wikiPageWikiLink Effective_action.
• Partition_function_(mathematics) wikiPageWikiLink Energy.
• Partition_function_(mathematics) wikiPageWikiLink Entropy.
• Partition_function_(mathematics) wikiPageWikiLink Equivalence_class.
• Partition_function_(mathematics) wikiPageWikiLink Ergodicity.
• Partition_function_(mathematics) wikiPageWikiLink Euclidean_distance.
• Partition_function_(mathematics) wikiPageWikiLink Expected_value.
• Partition_function_(mathematics) wikiPageWikiLink Exponential_family.
• Partition_function_(mathematics) wikiPageWikiLink Fisher_information.
• Partition_function_(mathematics) wikiPageWikiLink Fisher_information_metric.
• Partition_function_(mathematics) wikiPageWikiLink Fredholm_theory.
• Partition_function_(mathematics) wikiPageWikiLink Fubini–Study_metric.
• Partition_function_(mathematics) wikiPageWikiLink Fugacity.
• Partition_function_(mathematics) wikiPageWikiLink Functional_integration.
• Partition_function_(mathematics) wikiPageWikiLink Generalized_coordinates.
• Partition_function_(mathematics) wikiPageWikiLink Generalized_forces.
• Partition_function_(mathematics) wikiPageWikiLink Generating_function.
• Partition_function_(mathematics) wikiPageWikiLink Genomics.
• Partition_function_(mathematics) wikiPageWikiLink Gibbs_measure.
• Partition_function_(mathematics) wikiPageWikiLink Grand_canonical_ensemble.
• Partition_function_(mathematics) wikiPageWikiLink Greens_function.
• Partition_function_(mathematics) wikiPageWikiLink Ground_state.
• Partition_function_(mathematics) wikiPageWikiLink Group_(mathematics).
• Partition_function_(mathematics) wikiPageWikiLink Group_action.
• Partition_function_(mathematics) wikiPageWikiLink Hamiltonian_mechanics.
• Partition_function_(mathematics) wikiPageWikiLink Hilbert_space.
• Partition_function_(mathematics) wikiPageWikiLink Hopfield_network.
• Partition_function_(mathematics) wikiPageWikiLink Information_geometry.
• Partition_function_(mathematics) wikiPageWikiLink Information_theory.
• Partition_function_(mathematics) wikiPageWikiLink Integral.
• Partition_function_(mathematics) wikiPageWikiLink Invariant_measure.
• Partition_function_(mathematics) wikiPageWikiLink Ising_model.
• Partition_function_(mathematics) wikiPageWikiLink Jacobian_matrix_and_determinant.
• Partition_function_(mathematics) wikiPageWikiLink Karush–Kuhn–Tucker_conditions.
• Partition_function_(mathematics) wikiPageWikiLink Lagrange_multiplier.
• Partition_function_(mathematics) wikiPageWikiLink Lagrangian_mechanics.
• Partition_function_(mathematics) wikiPageWikiLink Lattice_model_(physics).
• Partition_function_(mathematics) wikiPageWikiLink Manifold.
• Partition_function_(mathematics) wikiPageWikiLink Markov_logic_network.
• Partition_function_(mathematics) wikiPageWikiLink Markov_property.
• Partition_function_(mathematics) wikiPageWikiLink Markov_random_field.
• Partition_function_(mathematics) wikiPageWikiLink Matrix_(mathematics).
• Partition_function_(mathematics) wikiPageWikiLink Measure_(mathematics).
• Partition_function_(mathematics) wikiPageWikiLink Metric_tensor.
• Partition_function_(mathematics) wikiPageWikiLink Normalizing_constant.
• Partition_function_(mathematics) wikiPageWikiLink Observable.
• Partition_function_(mathematics) wikiPageWikiLink Operator_(mathematics).
• Partition_function_(mathematics) wikiPageWikiLink Partition_function_(quantum_field_theory).
• Partition_function_(mathematics) wikiPageWikiLink Partition_function_(statistical_mechanics).
• Partition_function_(mathematics) wikiPageWikiLink Path_integral_formulation.
• Partition_function_(mathematics) wikiPageWikiLink Power_set.
• Partition_function_(mathematics) wikiPageWikiLink Principle_of_maximum_entropy.
• Partition_function_(mathematics) wikiPageWikiLink Probability_amplitude.
• Partition_function_(mathematics) wikiPageWikiLink Probability_density_function.
• Partition_function_(mathematics) wikiPageWikiLink Probability_distribution.
• Partition_function_(mathematics) wikiPageWikiLink Probability_measure.
• Partition_function_(mathematics) wikiPageWikiLink Probability_space.
• Partition_function_(mathematics) wikiPageWikiLink Probability_theory.
• Partition_function_(mathematics) wikiPageWikiLink Projective_Hilbert_space.
• Partition_function_(mathematics) wikiPageWikiLink Propagator.
• Partition_function_(mathematics) wikiPageWikiLink Quadratic_form.
• Partition_function_(mathematics) wikiPageWikiLink Quantum_field_theory.
• Partition_function_(mathematics) wikiPageWikiLink Quantum_mechanics.
• Partition_function_(mathematics) wikiPageWikiLink Quantum_state.
• Partition_function_(mathematics) wikiPageWikiLink Random_variable.
• Partition_function_(mathematics) wikiPageWikiLink Renormalization_group.
• Partition_function_(mathematics) wikiPageWikiLink Riemannian_manifold.
• Partition_function_(mathematics) wikiPageWikiLink Scalar_potential.
• Partition_function_(mathematics) wikiPageWikiLink Sigma-algebra.
• Partition_function_(mathematics) wikiPageWikiLink Simplex.
• Partition_function_(mathematics) wikiPageWikiLink Source_field.
• Partition_function_(mathematics) wikiPageWikiLink Space_(mathematics).
• Partition_function_(mathematics) wikiPageWikiLink Spontaneous_symmetry_breaking.
• Partition_function_(mathematics) wikiPageWikiLink Statistical_mechanics.
• Partition_function_(mathematics) wikiPageWikiLink Thermodynamic_beta.
• Partition_function_(mathematics) wikiPageWikiLink Thermodynamic_free_energy.

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