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Function that encodes the dependence of a coupling parameter on the energy scale
In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, β(g), encodes the dependence of a coupling parameter
Beta_function_(physics)
Topics referred to by the same term
to: Beta function (physics), details the running of the coupling strengths Dirichlet beta function, closely related to the Riemann zeta function Gödel's
Beta function (disambiguation)
Beta_function_(disambiguation)
The beta function in accelerator physics is a function related to the transverse size of the particle beam at the location s along the nominal beam trajectory
Beta function (accelerator physics)
Beta_function_(accelerator_physics)
Second letter of the Greek alphabet
predictor X. In statistics, beta may represent type II error, or regression slope. Dirichlet beta function Some uses of beta in physics and engineering include:
Beta
Function in thermodynamics and statistical physics
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium.[citation needed] Partition functions are
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Topics referred to by the same term
the free dictionary. Renormalization group equation may refer to: Beta function (physics) Callan–Symanzik equation Exact renormalization group equation This
Renormalization group equation
Renormalization_group_equation
Topics referred to by the same term
response to a pressure change (symbol "β") Beta function (physics), also β(g)—a function in quantum field theory Beta particle, a name used to refer to high-energy
Beta_(disambiguation)
Type of radioactive decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron)
Beta_decay
Mathematical function common in physics
The stretched exponential function f β ( t ) = e − t β {\displaystyle f_{\beta }(t)=e^{-t^{\beta }}} is obtained by inserting a fractional power law into
Stretched exponential function
Stretched_exponential_function
Result of repeatedly applying a mathematical function
combined functions, such as g(f(x)). Given the iteration velocity, or beta function (physics), v(x)=∂fn(x)∂n|n=0 for the nth iterate of the function f, we
Iterated_function
Symbols for constants, special functions
neurotransmitter noradrenaline in neuroscience B represents the beta function β represents: the thermodynamic beta, equal to (kBT)−1, where kB is the Boltzmann constant
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Phenomenon in quantum chromodynamics
Lund string model Gluon field strength tensor Asymptotic freedom Beta function (physics) Yang–Mills existence and mass gap Cornell potential § Calculation
Color_confinement
Characteristic quantity of plasmas
The beta of a plasma, symbolized by β, is the ratio of the plasma pressure (p = nkBT) to the magnetic pressure (pmag = B2/2μ0). The term is commonly used
Plasma_beta
Generating function in integrable systems
{\displaystyle \tau } -function of hypergeometric type. In particular, choosing r j = r j β := e j β {\displaystyle r_{j}=r_{j}^{\beta }:=e^{j\beta }} for some small
Tau function (integrable systems)
Tau_function_(integrable_systems)
Measure of the coldness of a system
has units of energy. The thermodynamic beta was originally introduced in 1971 (as Kältefunktion "coldness function") by Ingo Müller [de], one of the proponents
Thermodynamic_beta
of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which
List of mathematical functions
List_of_mathematical_functions
Method of solution to differential equations
Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. The Green's function as used in physics is usually
Green's_function
Type of radioactive decay
In nuclear physics, double beta decay (ββ decay) is a type of radioactive decay in which two neutrons are simultaneously transformed into two protons,
Double_beta_decay
Correlators of field operators
{\textstyle \beta ={\frac {1}{k_{\text{B}}T}}} .) Note regarding signs and normalization used in these definitions: The signs of the Green functions have been
Green's function (many-body theory)
Green's_function_(many-body_theory)
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
} , β {\displaystyle \beta } and γ {\displaystyle \gamma } are all complex numbers. The one-parameter Mittag-Leffler function is defined as E α ( z )
Prabhakar_function
Parameter describing the strength of a force
In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous. If a beta function is positive, the corresponding
Coupling_constant
Generalization of the concept from statistical mechanics
{\displaystyle Z(\beta )=\sum _{x_{i}}\exp \left(-\beta H(x_{1},x_{2},\dots )\right)} The function H is understood to be a real-valued function on the space
Partition function (mathematics)
Partition_function_(mathematics)
Thermodynamic potential
and the name Helmholtz energy. In physics, the symbol F is also used in reference to free energy or Helmholtz function. The Helmholtz free energy is defined
Helmholtz_free_energy
Theorized type of radioactive decay
this two-neutrino double beta decay. This conventional double beta decay is allowed in the Standard Model of particle physics. It has thus both a theoretical
Neutrinoless double beta decay
Neutrinoless_double_beta_decay
Mathematical model of ferromagnetism in statistical mechanics
_{1}=e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\cosh \beta h)^{2}-2\sinh 2\beta J}}=e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\sinh \beta h)^{2}+e^{-2\beta
Ising_model
unrestricted Hartree–Fock wave function and its use in second-order Møller–Plesset perturbation theory". Journal of Chemical Physics. 91 (3): 1789–1795. Bibcode:1989JChPh
Spin_contamination
Statistical mechanics model for phase transitions
about the system is encoded in the partition function, Z = ∑ i e − β E i , {\displaystyle Z=\sum _{i}e^{-\beta E_{i}},} where the sum runs over all possible
Lee–Yang_theory
Possible outcome of renormalization in physics
positive β {\displaystyle \beta } -function with the strong coupling behavior β ( g ) = β ∞ g {\displaystyle \beta (g)=\beta _{\infty }g} . It signifies
Quantum_triviality
Mathematical technique in thermal field theory
}={\frac {1}{\beta }}\sum _{i\omega }g(i\omega )={\frac {1}{2\pi i\beta }}\oint g(z)h_{\eta }(z)\,dz,} As in Fig. 1, the weighting function generates poles
Matsubara_summation
Inverse functions of sin, cos, tan, etc.
used in engineering, navigation, physics, and geometry. There are several notations for the inverse trigonometric functions. The most common uses the prefix
Inverse trigonometric functions
Inverse_trigonometric_functions
Physical model defined on a lattice
can define the partition function Z = ∑ σ ∈ C exp ( − β E ( σ ) ) {\displaystyle Z=\sum _{\sigma \in {\mathcal {C}}}\exp(-\beta E(\sigma ))} and there
Lattice_model_(physics)
Temperature at which the partition function of a statistical-mechanical system diverges
5 . {\displaystyle \alpha ={\frac {aV}{(2\pi \beta )^{3/5}}}.} Then the asymptotic partition function is given by Z q ( V o , T ) = ( 1 β − β o ) α −
Hagedorn_temperature
Physical systems hotter than any other
beta }+2e^{-1\beta }+e^{-2\beta }\\[3pt]&=1+2e^{-\beta }+e^{-2\beta }\\[6pt]E(T)&={\frac {0e^{-0\beta }+2\times 1e^{-1\beta }+2e^{-2\beta }}{Z}}\\[3pt]&={\frac
Negative_temperature
Statistical description for the behavior of fermions
the function: f ( n i ) = ln W + α ( N − ∑ n i ) + β ( E − ∑ n i ε i ) . {\displaystyle f(n_{i})=\ln W+\alpha \left(N-\sum n_{i}\right)+\beta \left(E-\sum
Fermi–Dirac_statistics
Generalization of the Meijer G-function and the Fox–Wright function
generalization of the Fox H-function was given by Ram Kishore Saxena. A further generalization of this function, useful in physics and statistics, was provided
Fox_H-function
Scientific field of study
behaviour. Particle physics & Nuclear physics: a Feynman diagram representing beta decay. These are just some of the many branches of physics. Others include
Physics
Random matrix with gaussian entries
For all β = 1 , 2 , 4 {\displaystyle \beta =1,2,4} cases, the GβE(N) ensemble is defined with density function ρ ( W N ) = 1 Z e − β 4 ∑ i = 1 N W N
Gaussian_ensemble
Property of gauge theories in particle physics
quark flavors. Asymptotic freedom can be derived by calculating the beta function describing the variation of the theory's coupling constant under the
Asymptotic_freedom
Intrinsic quantum property of particles
Group Theory in Physics: An Introduction to Symmetry Principles, Group Representations, and Special Functions In Classical And Quantum Physics. London, England
Spin_(physics)
Concept in theoretical physics
1982. The RG in particle physics was reformulated in more practical terms by Callan and Symanzik in 1970. The above beta function, which describes the "running
Renormalization_group
Mathematical entity to describe the probability of each possible measurement on a system
In quantum physics, a quantum state is a mathematical entity that represents a physical system. Quantum mechanics specifies the construction, evolution
Quantum_state
Disordered magnetic state
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, despite cooperative behavior in freezing of spins at a temperature
Spin_glass
Evolutionary equation under renormalization group flow
In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the
Callan–Symanzik_equation
Function defined by a hypergeometric series
j-invariant, a modular function, is a rational function in λ ( τ ) {\displaystyle \lambda (\tau )} . Incomplete beta functions Bx(p, q) are related by
Hypergeometric_function
Parameter used to calculate the volume change of a fluid or solid in response to pressure
J. (1973). "Compressibility of water as a function of temperature and pressure". Journal of Chemical Physics. 59 (10): 5529–5536. Bibcode:1973JChPh..59
Compressibility
Locally defined function in general relativity
\sigma (x,x')={\frac {1}{2}}\eta _{\alpha \beta }(x-x')^{\alpha }(x-x')^{\beta }.} Obviously Synge's function can be defined also in Riemannian manifolds
Synge's_world_function
American biophysicist (born 1960)
of theoretical problems at the interface of physics and biology, centers around whether various functions of living beings are optimal, and (if so) whether
William_Bialek
Generating function for quantum correlation functions
statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Function space of all functions whose derivatives are rapidly decreasing
mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives of all orders are rapidly decreasing. This
Schwartz_space
"Detailed comparison of the Williams–Watts and Cole–Davidson functions". Journal of Chemical Physics. 73 (7): 3348–3357. Bibcode:1980JChPh..73.3348L. doi:10
Cole–Davidson_equation
Probability distribution
{\displaystyle F_{\beta }} denote the cumulative distribution function of the Tracy–Widom distribution with given β {\displaystyle \beta } . It can be defined
Tracy–Widom_distribution
American physicist (1947–2001)
in quantum gauge theory, most notably, his 1972 calculation of the beta function to two-loop accuracy. His pioneering work in the days of FORTRAN and
William_E._Caswell
Idealization of a large number of atomic-sized systems
{A}}={\frac {\sum _{i}A_{i}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}.} The generalized version of the partition function provides the complete framework
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Berzelium Beta-M Beta-decay stable isobars Beta (plasma physics) Beta (velocity) Beta barium borate Beta decay Beta function (disambiguation) Beta function (physics)
Index_of_physics_articles_(B)
Topological invariants concerning BPS states
}~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GW}}(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M
Gopakumar–Vafa_invariant
Limit of sequence of smooth functions
other function. Then it follows that ∮ ∂ D g ( β ) d β = − ∫ R d ∇ x 1 x ∈ D ⋅ n x g ( x ) d x . {\displaystyle \oint _{\partial D}\,g(\beta )\;d\beta =-\int
Laplacian_of_the_indicator
ds^{N}\exp(-\beta U_{1})\delta (U_{1}-U_{0}-\Delta U)}{Q_{1}}}} where in p 1 {\displaystyle p_{1}} is a configurational part of a partition function p 1 ( Δ
Overlapping distribution method
Overlapping_distribution_method
Ensemble of states at constant pressure
+ p V i ) {\displaystyle Z^{-1}e^{-\beta (E_{i}+pV_{i})}} , where Z {\displaystyle Z} is the partition function, E i {\displaystyle E_{i}} is the internal
Isothermal–isobaric_ensemble
Characterization of nuclide stability
In nuclear physics, the valley of stability (also called the belt of stability, nuclear valley, energy valley, or beta stability valley) is a characterization
Valley_of_stability
Model in electromagnetism
the stretched exponential function can be a viable alternative that has one parameter less. For β = 1 {\displaystyle \beta =1} the Havriliak–Negami equation
Havriliak–Negami_relaxation
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Extension of the factorial function
integral of the second kind. (Euler's integral of the first kind is the beta function.) The relationship to the factorial can be shown by induction. The value
Gamma_function
Wave shaped like the sine function
are needed. In physics, a sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the
Sine_wave
Lattice model of statistical mechanics
modified Bessel functions evaluated at β J {\displaystyle \beta J} , namely ω n = 2 π I n ( β J ) {\displaystyle \omega _{n}=2\pi I_{n}(\beta J)} . For any
Classical_XY_model
Expectation or average of the falling factorial of a random variable
]}={\binom {n}{r}}{\frac {B(\alpha +r,\beta )r!}{B(\alpha ,\beta )}}=(n)_{r}{\frac {B(\alpha +r,\beta )}{B(\alpha ,\beta )}}} The rth raw moment of a random
Factorial_moment
Continuous probability distribution
X-\log(1-X)} is the logit function. If X ∼ G u m b e l ( μ X , β ) {\displaystyle X\sim \mathrm {Gumbel} (\mu _{X},\beta )} and Y ∼ G u m b e l ( μ Y
Logistic_distribution
Two-dimensional conformal field theory
{\left[\beta ^{2(\beta +\beta ^{-1})}\lambda ^{-i}\right]^{{\frac {\beta ^{-1}-\beta }{2}}+P_{1}+P_{2}+P_{3}}\prod _{\pm ,\pm }\Upsilon _{\beta }({\frac
Liouville_field_theory
Millennium Prize Problem
existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
+\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos
List of trigonometric identities
List_of_trigonometric_identities
Describes energy between atoms
ϕ α β {\displaystyle \phi _{\alpha \beta }} is a pair-wise potential function, ρ β {\displaystyle \rho _{\beta }} is the contribution to the electron
Embedded_atom_model
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 is used
Softmax_function
Probability distribution in physics
probability density function is expressed in terms of hypergeometric functions. The Holtsmark distribution has applications in plasma physics and astrophysics
Holtsmark_distribution
1968 physics-related discovery
theoretical physics, the Veneziano amplitude refers to the discovery made in 1968 by Italian theoretical physicist Gabriele Veneziano that the Euler beta function
Veneziano_amplitude
Locality condition in quantum field theory
~ β α {\displaystyle {\tilde {S}}_{\beta \alpha }} must be proportional to a momentum conserving delta function δ ( Σ p ) {\displaystyle \delta (\Sigma
Cluster_decomposition
Theory in supersymmetric gauge theory
Seiberg, Nathan (May 1988). "Supersymmetry and non-perturbative beta functions". Physics Letters B. 206 (1): 75–80. Bibcode:1988PhLB..206...75S. doi:10
Seiberg–Witten_theory
Numerical technique for solving quantum Hamiltonians
Exact diagonalization (ED) is a numerical technique used in physics to determine the eigenstates and energy eigenvalues of a quantum Hamiltonian. In this
Exact_diagonalization
Random temperature-influenced deviations of particles from their average state
β ) {\displaystyle {\mathcal {Z}}(\beta )} , which is referred to as the partition function, or generating function. The latter name is due to the fact
Thermal_fluctuations
Polynomial sequence
gamma function. In the special case that the four quantities n {\displaystyle n} , n + α {\displaystyle n+\alpha } , n + β {\displaystyle n+\beta } , n
Jacobi_polynomials
Concept in information theory
(1998). "Minimum uncertainty for antisymmetric wave functions". Letters in Mathematical Physics. 43 (3): 233–248. arXiv:quant-ph/9706015. Bibcode:1997quant
Entropic_uncertainty
Generalization of Gaussian distribution
the probability density function f ( x ) = β C q e q ( − β x 2 ) {\displaystyle f(x)={{\sqrt {\beta }} \over C_{q}}e_{q}(-\beta x^{2})} where e q ( x )
Q-Gaussian_distribution
State of matter of many bosons
{1}{(\beta E_{c})^{\alpha }}}={\frac {f}{(\hbar \omega \beta )^{3}}}} where V(r) = mω2r2/2 is the harmonic potential. It is seen that Ec is a function of
Bose_gas
Description of a quantum-mechanical system
"Schrödinger's original struggles with a complex wave function". American Journal of Physics. 88 (6): 433–438. Bibcode:2020AmJPh..88..433K. doi:10.1119/10
Schrödinger_equation
Quantity in relativistic physics
Bessel functions: ∑ m = 1 ∞ ( J m − 1 2 ( m β ) + J m + 1 2 ( m β ) ) = 1 1 − β 2 . {\displaystyle \sum _{m=1}^{\infty }\left(J_{m-1}^{2}(m\beta )+J_{m+1}^{2}(m\beta
Lorentz_factor
Integral transform useful in probability theory, physics, and engineering
{\displaystyle g(E)} defines the partition function. That is, the canonical partition function Z ( β ) {\displaystyle Z(\beta )} is given by Z ( β ) = ∫ 0 ∞ e −
Laplace_transform
Exponential function Beta function Gamma function Riemann zeta function Riemann hypothesis Generalized Riemann hypothesis Elliptic function Half-period
List of complex analysis topics
List_of_complex_analysis_topics
Least squares approximation of linear functions to data
}}=(\beta _{1},\beta _{2},\dots ,\beta _{n}),} it is desired to find the parameters β j {\displaystyle \beta _{j}} such that the model function "best"
Linear_least_squares
Mechanism of beta decay proposed in 1933
particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed
Fermi's_interaction
Protein structure
that more than 600 proteins with various function such as oxidase, dismutase, and amylase contain the beta barrel structure. In many cases, the strands
Beta_barrel
Tensor index notation for tensor-based calculations
derivative of a scalar function, a contravariant vector and a covariant vector are: f ; β = f , β {\displaystyle f_{;\beta }=f_{,\beta }} A α ; β = A α ,
Ricci_calculus
Statistical mechanics of quantum-mechanical systems
as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively
Quantum_statistical_mechanics
Symmetry in statistical physics
{\displaystyle \beta (K):=\xi {\frac {dK}{d\xi }}={\frac {2s(K)(1-s(K))}{(1+s(K))[ds(K)/dK]}}.} This function is the beta function of renormalization
Kramers–Wannier_duality
Relationship between programs and proofs
_{\alpha ,\beta ,1}:\alpha \times \beta \to \alpha } and π α , β , 2 : α × β → β {\displaystyle \pi _{\alpha ,\beta ,2}:\alpha \times \beta \to \beta } evaluation:
Curry–Howard_correspondence
Model of magnetic hysteresis
{\displaystyle \alpha =\beta } . We now present the Preisach density function, denoted μ ( α , β ) {\displaystyle \mu (\alpha ,\beta )} . This function describes the
Preisach_model_of_hysteresis
Distribution of variables which satisfies a stability property under linear combinations
} and β {\displaystyle \beta } , but possibly different values of μ and c. Not every function is the characteristic function of a legitimate probability
Stable_distribution
2025 video game
Teyon faced allegations during Rennsport's closed beta stage regarding the game's internal physics code, which was found to be identical to the isiMotor
Rennsport
Coupling constant divergence at high energies
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength)
Landau_pole
Special functions of several complex variables
theta functions have useful applications in topics such as number theory: "in how many ways can a number be written as a sum of squares?" physics: "how
Theta_function
Third letter of the Greek alphabet
of in graph theory Gamma radiation in nuclear physics Shear strain in physics Surface tension in physics The photon, the elementary particle of light and
Gamma
Class of chemical compounds
Beta-glucans, β-glucans comprise a group of β-D-glucose polysaccharides (glucans) naturally occurring in the cell walls of plants (including cereals),
Beta-glucan
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