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Topics referred to by the same term
Partition function may refer to: Partition function (statistical mechanics), a function used to derive thermodynamic properties Rotational partition function
Partition_function
Function in thermodynamics and statistical physics
partition function describes the statistical properties of a system in thermodynamic equilibrium.[citation needed] Partition functions are functions of
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Number of partitions of an integer
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because
Partition function (number theory)
Partition_function_(number_theory)
Decomposition of an integer as a sum of positive integers
same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n)
Integer_partition
Generalization of the concept from statistical mechanics
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition
Partition function (mathematics)
Partition_function_(mathematics)
Generating function for quantum correlation functions
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Divide and conquer sorting algorithm
gets swapped with other elements in the partition function. Therefore, the index returned in the partition function isn't necessarily where the actual pivot
Quicksort
Some remarkable congruences for the partition function
mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: p(5k+4)≡0(mod5),p(7k+5)≡0(mod7)
Ramanujan's_congruences
Family of probability distributions related to the normal distribution
justifies calling A the log-normalizer or log-partition function. Now, the moment-generating function of T(x) is M T ( u ) ≡ E [ exp ( u T T ( x
Exponential_family
The vibrational partition function traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of
Vibrational partition function
Vibrational_partition_function
Statistical mechanics model for phase transitions
finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of
Lee–Yang_theory
In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system Δ {\displaystyle
Kostant_partition_function
or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the
Polymer_field_theory
Topological invariants concerning BPS states
invariants can be viewed as a partition function in topological quantum field theory. They are proposed to be the partition function in Gopakumar–Vafa form:
Gopakumar–Vafa_invariant
Conjecture in algebraic geometry
curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's
Witten_conjecture
Smooth approximation of one-hot arg max
partition function, often denoted by Z; and the factor β is called the coldness (or thermodynamic beta, or inverse temperature). The softmax function
Softmax_function
Function in Chemistry
rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. The total canonical partition function Z {\displaystyle
Rotational_partition_function
Temperature at which the partition function of a statistical-mechanical system diverges
to the high energy collisions was first proposed by E. Fermi. The partition function of the fireballs can be written in two forms, one in terms of its
Hagedorn_temperature
Array of nonnegative integers in combinatorics
MacMahon also mentions the generating functions of plane partitions. The formula for the generating function can be written in an alternative way, which
Plane_partition
Even integers as sums of two primes
are believed to be of roughly comparable difficulty. The Goldbach partition function associates to each even integer the number of ways it can be decomposed
Goldbach's_conjecture
Parallel programming model
Each Map function output is allocated to a particular reducer by the application's partition function for sharding purposes. The partition function is given
MapReduce
Model in statistical mechanics generalizing the Ising model
interaction energy. This partition function is written as a function of the interaction V to emphasize that it is only a function of the interaction, and
Potts_model
High-temperature expansion in statistical mechanics
expansion or hopping expansion) is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting
Cluster_expansion
Formal power series
expansions of many special functions and enumerate partition functions. In particular, we recall that the partition function p(n) is generated by the reciprocal
Generating_function
Model in statistical mechanics
β=1kT and absolute temperature T and the Boltzmann constant k, the partition function ZN(K≡βJ,L≡βJ∗)=∑{σ}exp(K∑⟨ij⟩Hσiσj+L∑⟨ij⟩Vσiσj). The critical temperature
Square_lattice_Ising_model
Physical function in thermodynamics
statistical mechanics, the translational partition function, q T {\displaystyle q_{T}} is that part of the partition function resulting from the movement (translation)
Translational partition function
Translational_partition_function
Statistical mechanics model for crystal lattices
Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows. For two-dimensional
Ice-type_model
Unsolved problem in mathematics
arbitrary m, r, are there infinitely many values of n such that the partition function at n is congruent to r mod m? More unsolved problems in mathematics
Newman's_conjecture
configurations. For a triangular lattice with N sites, the grand partition function is Z ( z ) = ∑ n z n g ( n , N ) = 1 + N z + 1 2 N ( N − 7 ) z 2 +
Hard_hexagon_model
{Q}}}{n}}\right)\right]} The quantum concentration can be derived from the canonical partition function of an ideal gas particle with the free particle Hamiltonian. For a
Quantum_concentration
Formulation of quantum mechanics
is extremely similar to the partition function in statistical mechanics. Indeed, it is sometimes called a partition function, and the two are essentially
Path-integral_formulation
Mathematical model of ferromagnetism in statistical mechanics
{\displaystyle Z_{\beta }=\sum _{\sigma }e^{-\beta H(\sigma )}} is the partition function. For a function f {\displaystyle f} of the spins ("observable"), one denotes
Ising_model
Function used to generate other functions
system's dynamics. Common examples are the partition function of statistical mechanics, the Hamiltonian, and the function which acts as a bridge between two sets
Generating_function_(physics)
Physics of many interacting particles
ensemble that reflects the knowledge about that system. Once the partition function for an ensemble has been determined for a given system, that system
Statistical_mechanics
Generating function in integrable systems
Tau functions also appear as matrix model partition functions in the spectral theory of random matrices, and may also serve as generating functions, in
Tau function (integrable systems)
Tau_function_(integrable_systems)
Term in number theory and combinatorics
in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan.
Rank_of_a_partition
Function over multiple rows in SQL
(10 rows) The PARTITION BY clause groups rows into partitions, and the function is applied to each partition separately. If the PARTITION BY clause is
Window_function_(SQL)
Quantum chromodynamics on a lattice
Gene supercomputer. After Wick rotation, the path integral for the partition function of QCD takes the form Z = ∫ D U e − S [ U ] = ∫ ∏ x , μ d U μ ( x
Lattice_QCD
Mathematical technique
transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and
Transfer-matrix method (statistical mechanics)
Transfer-matrix_method_(statistical_mechanics)
Model describing the adsorption of a mono-layer of gas molecules on an ideal flat surface
partition function of the N A {\displaystyle N_{A}} adsorbed molecules by taking a product of the individual partition functions (refer to Partition function
Langmuir_adsorption_model
published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences. p(5n + 4) ≡ 0 (mod 5)
Crank_of_a_partition
Software package used to predict RNA interactions
Partition function for Interacting RNAs (piRNA) is a parallel C++ package to compute joint and individual partition functions for two RNA sequences. From
Partition function for Interacting RNAs
Partition_function_for_Interacting_RNAs
Thermodynamic potential
the partition functions are constants with respect to taking averages and that the free energy is proportional to minus the logarithm of the partition function
Helmholtz_free_energy
spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each integer partition of
Spt_function
Modified partition function
inverse temperature β is defined as a modification of the standard partition function: Tr [ ( − 1 ) F e − β H ] {\displaystyle {\textrm {Tr}}[(-1)^{F}e^{-\beta
Witten_index
Theorem in statistical mechanics
states that if partition functions of certain models in statistical field theory with ferromagnetic interactions are considered as functions of an external
Lee–Yang_theorem
Breakdown of parity at the quantum level
there an ambiguity in the partition function of a factor of -1. We want to know when the choice of sign of the partition function is ill-defined. The possibility
Parity_anomaly
State of matter
count all microstates though use of a partition function. The use of statistical mechanics and the partition function is an important tool throughout all
Gas
Summability method in physics
the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function can be computed explicitly. Consider a scalar field
Zeta_function_regularization
Set of functions from a topological space to [0,1] which sum to 1 for any input
mathematics, a partition of unity on a topological space X {\displaystyle X} is a set R {\displaystyle R} of continuous functions from X {\displaystyle
Partition_of_unity
Pictorial representation of the behavior of subatomic particles
The field's partition function is the normalization factor on the bottom, which coincides with the statistical mechanical partition function at zero temperature
Feynman_diagram
Theorem in classical statistical mechanics
the same results can be obtained by an alternative method using the partition function. A diatomic gas can be modelled as two masses, m1 and m2, joined by
Equipartition_theorem
Function whose domain is the positive integers
congruences for the functions. See Ramanujan tau function for some examples. Extend the domain of the partition function by setting p(0) = 1. p(n)=1n∑1≤k≤nσ(k)p(n−k)
Arithmetic_function
estimating chemical potential. For two N particle systems 0 and 1 with partition function Q 0 {\displaystyle Q_{0}} and Q 1 {\displaystyle Q_{1}} , from F (
Overlapping distribution method
Overlapping_distribution_method
Mathematical statistics distance measure
statistics, θ is canonical natural parameters, and A(θ) is the log-partition function. The KL divergence between two distributions p(x|θ1) and p(x|θ2) is
Kullback–Leibler_divergence
Mathematical limit applied in statistical physics
where Z {\displaystyle Z} is most commonly the partition function, or a similar thermodynamic function. It is typically used to simplify the calculation
Replica_trick
Method of calculating chiral anomalies
It uses the correspondence between functional determinants and the partition function, effectively making use of the Atiyah–Singer index theorem. Suppose
Fujikawa_method
Expectation value of time-ordered quantum operators
can be treated separately. Effective action Green's function (many-body theory) Partition function (mathematics) Source field The − i {\displaystyle -i}
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Ordered partition of a set Partition calculus Partition function (quantum field theory) Partition function (statistical mechanics) Partition of an interval
List_of_partition_topics
Ensemble of states at constant pressure
Z^{-1}e^{-\beta (E_{i}+pV_{i})}} , where Z {\displaystyle Z} is the partition function, E i {\displaystyle E_{i}} is the internal energy of the system in
Isothermal–isobaric_ensemble
Ensemble of states at a constant temperature
{\displaystyle \textstyle P={\frac {1}{Z}}e^{-E/(kT)},} using the canonical partition function Z = e − F / ( k T ) {\displaystyle \textstyle Z=e^{-F/(kT)}} rather
Canonical_ensemble
Approximation of physical behavior
combinatorial problems arise that make things like computing the partition function of a system difficult. MFT is an approximation method that often makes
Mean-field_theory
Monster and modular connection
AdS/CFT dual to a holomorphic CFT with central charge c=24, and the partition function of the CFT is precisely j-744, i.e., the graded character of the moonshine
Monstrous_moonshine
Concept in combinatorics
on the partition function and for representation theory of the symmetric group, especially modular representation theory. A partition λ = ( λ 1 , … , λ
Core_partition
Probability distribution of energy states of a system
The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values
Boltzmann_distribution
Particular relationship between the partition function of an ensemble
characteristic state function or Massieu's potential in statistical mechanics refers to a particular relationship between the partition function of an ensemble
Characteristic_state_function
Set of quantities in probability theory
published in 1929, Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs
Cumulant
26-dimensional string theory
Physical quantities are then constructed from the (Euclidean) partition function and N-point function: Z = ∑ h = 0 ∞ ∫ D g m n D X μ N exp ( − I [ g , X ]
Bosonic_string_theory
Quantum field theory
to obtain all the n-point functions with perturbation theory. Using LSZ reduction formula we get from the n-point functions the corresponding process
Yang–Mills_theory
Field theory involving topological effects in physics
See (Schwarz 2000). In Schwarz-type TQFTs, the correlation functions or partition functions of the system are computed by the path integral of metric-independent
Topological quantum field theory
Topological_quantum_field_theory
Topics referred to by the same term
Look up partition in Wiktionary, the free dictionary. Partition may refer to: Partition (1987 film), directed by Ken McMullen Partition (2007 film), directed
Partition
molecular conformations in the partition function. In this sense, the symmetry number depends upon how the partition function is formulated. For example,
Symmetry_number
Function related to statistics and probability theory
likelihood function. In general, for a likelihood function depending on the parameter vector θ {\textstyle \mathbf {\theta } } that can be partitioned into
Likelihood_function
Physical model defined on a lattice
{\displaystyle {\mathcal {C}}} is also finite. We can define the partition function Z = ∑ σ ∈ C exp ( − β E ( σ ) ) {\displaystyle Z=\sum _{\sigma \in
Lattice_model_(physics)
Statistical distribution used in many-particle mechanics
= ∑ i N i {\displaystyle \textstyle N=\sum _{i}N_{i}} , Z is the partition function: Z = ∑ i g i e − ε i / k B T {\displaystyle \textstyle Z=\sum
Maxwell–Boltzmann_statistics
Expansion coefficients in statistical mechanics
virial coefficients is a cluster expansion of the grand canonical partition function Ξ = ∑ n λ n Q n = e ( p V ) / ( k B T ) {\displaystyle \Xi =\sum _{n}{\lambda
Virial_coefficient
Concept
_{\text{mic}}} is the microcanonical partition function Z can {\displaystyle Z_{\text{can}}} is the canonical partition function Z gr {\displaystyle {\mathcal
Entropy (statistical thermodynamics)
Entropy_(statistical_thermodynamics)
First sector of partitioned PC computer disk
are divided into partitions, each partition notionally containing a file system. The MBR also contains executable code to function as a loader for the
Master_boot_record
Basic integral in elementary calculus
suitable functions, including every continuous function on a closed bounded interval, these Riemann sums approach a single limiting value as the partitions of
Riemann_integral
Set of random variables
i}(x_{\{k\}})} is simply a dot product over field configurations, and Z is the partition function: Z = ∑ x ∈ X exp ( ∑ k w k ⊤ f k ( x { k } ) ) . {\displaystyle
Markov_random_field
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Relation of magnetization to applied magnetic field and temperature
magnetic moments are aligned with the field can be calculated from the partition function. For a single particle, this is Z 1 = ∑ n = 0 , 1 e − E n β = e μ
Curie's_law
Two dimensional conformal field theory
is considered to be the simplest minimal model with a non-diagonal partition function in Virasoro characters, as well as the simplest non-trivial CFT with
Critical three-state Potts model
Critical_three-state_Potts_model
than) a given one. Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways
List of mathematical functions
List_of_mathematical_functions
Topological quantum field theory
multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer
Chern–Simons_theory
Specific probability distribution function, important in physics
— in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system). Because velocity
Maxwell–Boltzmann distribution
Maxwell–Boltzmann_distribution
Description of the behaviour of bosons
expression for a grand partition function and replacing E {\displaystyle E} with N ε {\displaystyle N\varepsilon } , the grand partition function takes the form
Bose–Einstein_statistics
German mathematician
Ono, he developed a finite algebraic formula for the values of the partition function. He was named to the 2023 class of Fellows of the American Mathematical
Jan_Hendrik_Bruinier
Number associated with self-avoiding walks
to b {\displaystyle b} . The aim of the proof is to show that the partition function Z ( x ) = ∑ γ : a → H x ℓ ( γ ) = ∑ n = 0 ∞ c n x n {\displaystyle
Connective_constant
Mathematical constant used in combinatorics
occur in spin ice, and the same calculation of the partition function applies in that case. The function f {\displaystyle f} also counts the number of 3-colorings
Lieb's_square_ice_constant
Description of the effect of adding a quadrant to a lattice
k and l are arranged as follows: For a lattice with N sites, the partition function is Z N = ∑ a l l s p i n s ∏ a l l f a c e s w ( σ i , σ j , σ k
Corner_transfer_matrix
Function defined by multiple sub-functions
piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into
Piecewise_function
Special functions of several complex variables
007. Eric W. Weisstein (2022-03-11). "Partition Function P". Eric W. Weisstein (2022-03-11). "Partition Function Q". Abramowitz, Milton; Stegun, Irene
Theta_function
Graph representation in quantum mechanics
dimension. Tensor networks can represent many-body wave functions and density operators, partition functions, Hamiltonians and other linear maps, probability
Tensor_network
Integral of the Gaussian function, equal to sqrt(π)
statistical mechanics, to find its partition function. Although no elementary function exists for the error function, as can be proven by the Risch algorithm
Gaussian_integral
tool that calculates a maximum expected accuracy alignment using partition function posterior probabilities. Base pair probabilities are estimated using
Probalign
Method in computational chemistry
function of the Boltzmann-weighted integral over phase space (i.e. a partition function), the free energy difference between two macroscopic states cannot
Thermodynamic_integration
Model of a crystalline solid
capacities. Heat capacity is obtained through the use of the canonical partition function of a simple quantum harmonic oscillator. Z = ∑ n = 0 ∞ e − E n / k
Einstein_solid
Indian-born Canadian mathematician (1921–2006)
the 1960s Subbarao began to study the congruence properties of the partition function, p(n), which became one of his favourite problems. For example, he
Mathukumalli_V._Subbarao
In Bayesian probability theory
that the posterior is a proper probability. It is related to the partition function in statistical mechanics. Given a set of independent identically distributed
Marginal_likelihood
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