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Equation of statistical mechanics
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium;
Boltzmann_equation
Equation used for physiological interfaces, polymer science, and semiconductors
The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the presence of one or more charged surfaces. This distribution
Poisson–Boltzmann_equation
Austrian mathematician and theoretical physicist (1844–1906)
Ludwig Eduard Boltzmann (/ˈbɔːltsˌmɑːn/ BAWLTS-mahn or /ˈboʊltsmən/ BOHLTS-muhn; German: [ˈluːtvɪç ˈeːduaʁt ˈbɔltsman]; 20 February 1844 – 5 September
Ludwig_Boltzmann
Equation in statistical mechanics
mechanics, Boltzmann's entropy formula (also known as the Boltzmann–Planck equation, not to be confused with the more general Boltzmann equation, which is
Boltzmann's_entropy_formula
Physical law on the emissive power of black body
The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature
Stefan–Boltzmann_law
The quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation, is the quantum mechanical modification of the Boltzmann equation, which gives
Quantum_Boltzmann_equation
Class of computational fluid dynamics methods
different interpretation of the lattice Boltzmann equation is that of a discrete-velocity Boltzmann equation. The numerical methods of solution of the
Lattice_Boltzmann_methods
Thermodynamic theorem
natural consequence of the kinetic equation derived by Boltzmann that has come to be known as Boltzmann's equation. The H-theorem has led to considerable
H-theorem
Specific probability distribution function, important in physics
Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
Maxwell–Boltzmann distribution
Maxwell–Boltzmann_distribution
Physical constant relating particle kinetic energy with temperature
The Boltzmann constant (kB or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic
Boltzmann_constant
Statistical mechanics framework
theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise
Chapman–Enskog_theory
Elliptic partial differential equation
charge density follows a Boltzmann distribution, then the Poisson–Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development
Poisson's_equation
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Model describing the departures from ideality in solutions of electrolytes and plasmas
superposition principle. Nevertheless, the two equations can be combined to produce the Poisson–Boltzmann equation. ∇ 2 ψ j ( r ) = − 1 ε 0 ε r ∑ i [ n i (
Debye–Hückel_theory
Partial differential equation
equations Boltzmann equation Convection–diffusion equation Klein–Kramers equation Kolmogorov backward equation Kolmogorov equation Langevin equation Master
Fokker–Planck_equation
an alternative to the Boltzmann equation in the case of Coulomb interaction. When used with the Vlasov equation, the equation yields the time evolution
Landau_kinetic_equation
Type of differential equation
parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations. One of the most important partial
Partial_differential_equation
Equations of motion for viscous fluids
with high gradients. For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement. Failing that, one may have to resort
Navier–Stokes_equations
System of differential equations
Richard Arnold. The collisionless Boltzmann equation, also called the Vlasov Equation is a special form of Liouville' equation and is given by: ∂ f ∂ t + v
Jeans_equations
Class of computational solid dynamics methods
The Lattice Boltzmann methods for solids (LBMS) are a set of methods for solving partial differential equations (PDE) in solid mechanics. The methods
Lattice Boltzmann methods for solids
Lattice_Boltzmann_methods_for_solids
Low-Density Gases
To describe non-equilibrium phenomena in rarefied gases, the Boltzmann transport equation must be used, which is the appropriate mathematical tool for
Rarefied_gas_dynamics
brain Boltzmann constant Boltzmann distribution Boltzmann equation Quantum Boltzmann equation Boltzmann factor Boltzmann machine Deep Boltzmann machine
List of things named after Ludwig Boltzmann
List_of_things_named_after_Ludwig_Boltzmann
Statistical distribution used in many-particle mechanics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal
Maxwell–Boltzmann_statistics
Formula for temperature dependence of rates of chemical reactions
described by the Arrhenius equation. The calculations for reaction rate constants involve an energy averaging over a Maxwell–Boltzmann distribution with E
Arrhenius_equation
French mathematician
study of the connections between interacting particle systems, the Boltzmann equation, and fluid mechanics. In 2008 she was awarded the European Mathematical
Laure_Saint-Raymond
Capacity of a material to conduct heat
expressions for thermal conductivity starting from the Boltzmann equation. The Boltzmann equation, in turn, provides a statistical description of a dilute
Thermal conductivity and resistivity
Thermal_conductivity_and_resistivity
American mathematician
in New York University. Lanford proved in 1975 the validity of the Boltzmann equation in a gas of particles under the laws of classical mechanics on short
Oscar_Lanford
is the Boltzmann constant. A simple derivation of the Boltzmann relation for the electrons can be obtained using the momentum fluid equation of the two-fluid
Boltzmann_relation
to the Poisson–Boltzmann equation (PBE), often referred to as FDPB-based methods (FDPB stands for "finite difference Poisson–Boltzmann"). The PBE is a
Protein_pKa_calculations
Resistance of a fluid to shear deformation
which derives expressions for the viscosity of a dilute gas from the Boltzmann equation. Consider a dilute gas moving parallel to the x {\displaystyle x}
Viscosity
Transport of dissolved species from the highest to the lowest concentration region
Ludwig Boltzmann, in the development of the atomistic backgrounds of the macroscopic transport processes, introduced the Boltzmann equation, which has
Diffusion
System where changes of output are not proportional to changes of input
systems. Algebraic Riccati equation Ball and beam system Bellman equation for optimal policy Boltzmann equation Colebrook equation General relativity Ginzburg–Landau
Nonlinear_system
Description of the time-evolution of plasma
Vlasov first argued that the standard kinetic approach, based on the Boltzmann equation, encounters fundamental limitations when applied to plasmas with long-range
Vlasov_equation
French mathematician (born 1956)
Medal. He was cited for his contributions to viscosity solutions, the Boltzmann equation, and the calculus of variations. He has also received the French Academy
Pierre-Louis_Lions
Relations between flows and forces, or gradients, in thermodynamic systems
difference) coefficients are equal. For many kinetic systems, like the Boltzmann equation or chemical kinetics, the Onsager relations are closely connected
Onsager_reciprocal_relations
In combustion, the Williams spray equation, also known as the Williams–Boltzmann equation, describes the statistical evolution of sprays contained in
Williams_spray_equation
Collision operator used in a computational fluid dynamics technique
operator) term refers to a collision operator used in the Boltzmann equation and in the lattice Boltzmann method, a computational fluid dynamics technique. It
Bhatnagar–Gross–Krook operator
Bhatnagar–Gross–Krook_operator
Physical constant equivalent to the Boltzmann constant, but in different units
constant) is denoted by the symbol R or R. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount
Gas_constant
Set of equations describing the dynamics of a system of many interacting particles
process of obtaining Boltzmann equation from Liouville equation is known as Boltzmann–Grad limit. Schematically, the Liouville equation gives us the time
BBGKY_hierarchy
Equation of the state of a hypothetical ideal gas
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior
Ideal_gas_law
Method in computational chemistry
condensed media or the difference of two solvation energies. The Poisson-Boltzmann equation (PB) describes the electrostatic environment of a solute in a solvent
Implicit_solvation
Principle in kinetic systems
space inversion and T is the time reversal. Detailed balance for Boltzmann's equation requires PT-invariance of collisions' dynamics, not just T-invariance
Detailed_balance
Stochastic differential equation
equilibrium, then the long-time solution to the Langevin equation must reduce to the Boltzmann distribution, which is the probability distribution function
Langevin_equation
Gas equation of state which accounts for non-ideal gas behavior
properties. Nevertheless, as Boltzmann observed, the van der Waals equation provides an essentially correct description. The vdW equation produces the critical
Van_der_Waals_equation
The equations in this article are classified by subject. S = k B ln Ω {\displaystyle S=k_{\mathrm {B} }\ln \Omega } , where kB is the Boltzmann constant
Table of thermodynamic equations
Table_of_thermodynamic_equations
Mathematical model which approximates the behavior of real gases
the degeneracy of states. The Sackur-Tetrode equation expresses the entropy of an ideal quantum Boltzmann gas for interparticle distances well above the
Ideal_gas
Expression of monatomic ideal gas entropy
independently as a solution of Boltzmann's gas statistics and entropy equations, at about the same time in 1912. The Sackur–Tetrode equation expresses the entropy
Sackur–Tetrode_equation
Equation used to calculate the electromigration of ions in a fluid
The Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It extends
Nernst–Planck_equation
Physical law in electrochemistry
In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction
Nernst_equation
Speed of sound wave through elastic medium
R is the molar gas constant, 8.31446261815324 J⋅mol−1⋅K−1; k is the Boltzmann constant, 1.380649×10−23 J⋅K−1; T is the absolute temperature; M is the
Speed_of_sound
theorist for the Boltzmann equation. It assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral
Cercignani_conjecture
Thermal engineering discipline concerning transfer of heat in physical systems
Stefan-Boltzmann equation: ϕ q = ϵ σ T 4 . {\displaystyle \phi _{q}=\epsilon \sigma T^{4}.} For radiative transfer between two objects, the equation is as
Heat_transfer
Topics referred to by the same term
lunar crater Boltzmann distribution Boltzmann equation Boltzmann's entropy formula Boltzmann relation Stefan–Boltzmann law Stefan–Boltzmann constant This
Boltzmann_(disambiguation)
French mathematician (born 1962)
partial differential equations. With Laure Saint-Raymond in 2004 he showed a connection of the weak solutions of the Boltzmann equation with the Leray solutions
François_Golse
ISSN 0002-9505. Chambré, P. L. (1952-11-01). "On the Solution of the Poisson-Boltzmann Equation with Application to the Theory of Thermal Explosions". The Journal
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Understanding of gas properties in terms of molecular motion
theory. Following the development of the Boltzmann equation, a framework for its use in developing transport equations was developed independently by David
Kinetic_theory_of_gases
Branch of astrophysics
collisionless Boltzmann equation, these moments are then related by various forms of continuity equations, of which most notable are the Jeans equations and Virial
Stellar_dynamics
Type of stochastic recurrent neural network
A Boltzmann machine (also called Sherrington–Kirkpatrick model with external field or stochastic Ising model), named after Ludwig Boltzmann, is a spin-glass
Boltzmann_machine
Concept
entropy perspective was introduced in 1870 by Austrian physicist Ludwig Boltzmann, who established a new field of physics that provided the descriptive
Entropy (statistical thermodynamics)
Entropy_(statistical_thermodynamics)
Property of a thermodynamic system
pp. 576–577. Jagannathan, Kannan (2019). "Anxiety and the Equation: Understanding Boltzmann's Entropy". American Journal of Physics. 87 (9): 765. Bibcode:2019AmJPh
Entropy
Equation in Brownian motion
the Boltzmann constant; T is the absolute temperature. This equation is an early example of a fluctuation-dissipation relation. Note that the equation above
Einstein relation (kinetic theory)
Einstein_relation_(kinetic_theory)
Equation relating the rate of an electrochemical reaction to the overpotential
Tafel Equation". "Applicability". "Derivation of the extended Butler–Volmer equation". "Connection between the Avogadro constant and the Boltzmann constant"
Tafel_equation
Measure of electrostatic effect and how far it persists
potential in the Poisson equation with their mean-field counterparts in the Boltzmann distribution yields the Poisson–Boltzmann equation: ε ∇ 2 Φ ( r ) = −
Debye_length
French mathematician and politician (born 1973)
differential equations involved in statistical mechanics, specifically the Boltzmann equation, where, with Laurent Desvillettes, he was the first to prove how quickly
Cédric_Villani
Force acting on charged particles in electric and magnetic fields
plasmas—more complex equations are required, such as the Boltzmann equation, the Fokker–Planck equation or the Navier–Stokes equations. These models go beyond
Lorentz_force
Taiwanese mathematician
of the Green’s functions for linearized Boltzmann equation, and invariant manifolds for stationary Boltzmann flows. Liu, Tai-Ping; Yu, Shih-Hsien (1999)
Shih-Hsien_Yu
Relation between two physical quantities which is specific to a substance
transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or
Constitutive_equation
(BGK) collision model used in the Boltzmann equation and in lattice Boltzmann methods and to the Gross–Pitaevskii equation which describes the ground state
Eugene_P._Gross
American mathematician (1923–1986)
the Boltzmann equation. He derived the Boltzmann equation from Liouville equation using BBGKY hierarchy under certain limits, known as Boltzmann–Grad
Harold_Grad
Formula for radiative heat transfer
when the medium changes with distance, Planck's Law and the Stefan-Boltzmann equation do not apply. This is often the case when dealing with atmospheres
Schwarzschild's equation for radiative transfer
Schwarzschild's_equation_for_radiative_transfer
Chemical kinetics equation
The Eyring equation (occasionally also known as Eyring–Polanyi equation) is an equation used in chemical kinetics to describe changes in the rate of a
Eyring_equation
Probabilistic problem-solving algorithm
In fluid dynamics, in particular rarefied gas dynamics, where the Boltzmann equation is solved for finite Knudsen number fluid flows using the direct simulation
Monte_Carlo_method
Probability distribution of energy states of a system
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure
Boltzmann_distribution
Models the mean time to failure of a semiconductor circuit due to electromigration
{\displaystyle Q} is the activation energy k {\displaystyle k} is the Boltzmann constant T {\displaystyle T} is the absolute temperature The model is
Black's_equation
Mathematics award
of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation." 2014 Seoul, South Korea Artur Avila University of Paris VII, France
Fields_Medal
Model in computational physics
by solving the Boltzmann equation or, when the correct description of long-range Coulomb interaction is necessary, by the Vlasov equation which contains
Plasma_modeling
American mathematician
differential equations, including work on geometric averaging operators, oscillatory integral operators, Fourier restriction, and the Boltzmann equation. He was
Philip_Gressman
Type of cellular automaton
precursor to the lattice Boltzmann methods. From lattice gas automata, it is possible to derive the macroscopic Navier–Stokes equations. Interest in lattice
Lattice_gas_automaton
Schwinger–Dyson equation Yang-Mills equations in gauge theory Boltzmann equation Continuity equation for conservation laws Diffusion equation Heat equation Kardar-Parisi-Zhang
List of named differential equations
List_of_named_differential_equations
Italian physicist and mathematician (1929–2010)
on the kinetic theory of gases. His contributions to the study of Boltzmann's equation include the proof of the H-theorem for polyatomic gases. The Cercignani
Carlo_Cercignani
Partial differential equation describing the evolution of temperature in a region
additional term may be introduced into the equation to account for radiative loss of heat. According to the Stefan–Boltzmann law, this term is μ ( u 4 − v 4 )
Heat_equation
Average uncertainty in variable's states
what is now known as information theory was first made by Boltzmann and expressed by his equation: S = k B ln W , {\displaystyle S=k_{\text{B}}\ln W,}
Entropy_(information_theory)
Molecular interface between a surface and a fluid
Electroosmotic pump Interface and colloid science Nanofluidics Poisson–Boltzmann equation Supercapacitor Dukhin, Andrei S.; Xu, Renliang (2025). Zeta Potential:
Double layer (surface science)
Double_layer_(surface_science)
French mathematician and academic (born 1978)
primarily in partial differential equations and mathematical physics (statistical mechanics, Boltzmann equation, Vlasov equation). Mouhot obtained his PhD in
Clément_Mouhot
Maxwell built a simple flywheel model of electromagnetism, and Ludwig Boltzmann built an elaborate mechanical model ("Bicykel") based on Maxwell's flywheel
History of Maxwell's equations
History_of_Maxwell's_equations
viscosity. The theoretical basis of the kinetic theory is given by the Boltzmann equation and Chapman–Enskog theory, which allow accurate statistical modeling
Temperature dependence of viscosity
Temperature_dependence_of_viscosity
Mathematical models for calculating viscosity
calculations are computer-intensive. Another approach utilises the Boltzmann equation, which describes the statistical behaviour of a thermodynamic system
Viscosity_models_for_mixtures
Study of motions and interactions of neutrons
experimental or industrial neutron beams. Neutron transport has roots in the Boltzmann equation, which was used in the 1800s to study the kinetic theory of gases
Neutron_transport
See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.
List of nonlinear partial differential equations
List_of_nonlinear_partial_differential_equations
Mathematic theory
Jeans, states that any steady-state solution of the collisionless Boltzmann equation depends on the phase space coordinates only through integrals of motion
Jeans's_theorem
Equation describing the transport of some quantity
Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes
Continuity_equation
Maxwell–Boltzmann distribution mentioned above, he also associated the kinetic energy of particles with their degrees of freedom. The Boltzmann equation for
History_of_thermodynamics
Female Innovator in China
the generalized Boltzmann equation for polyatomic gases, which resulted in it being called the Wang-Chang-Uhlenbeck ("WCU") equation. Upon returning to
Wang_Chengshu
Swedish mathematician
Boltzmann equation. Arkeryd earned his doctorate from Lund University in 1966, under the supervision of Jaak Peetre. Arkeryd, Leif: On the Boltzmann equation
Leif_Arkeryd
Transition rate formula
expressions that complement the golden rule in the semiclassical Boltzmann equation approach to electron transport. While the golden rule is commonly
Fermi's_golden_rule
Describes how neurons transmit electric signals
activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of V m {\displaystyle V_{m}} . In the original paper
Hodgkin–Huxley_model
Mathematical algorithm
partial differential equation with constant coefficients. More efficient ways of solving the linearized Poisson–Boltzmann equation have also been developed
Walk-on-spheres_method
Klein produces the Erlangen program on geometries. Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions
1872_in_Germany
Scientific application
effects of ionic strength mediated screening by evaluating the Poisson-Boltzmann equation at a finite number of points within a three-dimensional grid box.
DelPhi
Equation in machine learning
differential equations are a class of models in machine learning that combine neural networks with the mathematical framework of differential equations. These
Neural_differential_equation
BOLTZMANN EQUATION
BOLTZMANN EQUATION
BOLTZMANN EQUATION
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Happy; Carefree; Blissful
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Anglicized form of Greek AmÅs, AMOS means "strong." In the New Testament bible, this is the name of an ancestor of Christ.
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Of the Cardamom Creeper
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From the cross meadow.
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BOLTZMANN EQUATION
BOLTZMANN EQUATION
BOLTZMANN EQUATION
BOLTZMANN EQUATION
BOLTZMANN EQUATION
a.
Recurring once a month; monthly; gone through in a month; as, the menstrual revolution of the moon; pertaining to monthly changes; as, the menstrual equation of the sun's place.
n.
The system of equations required for the complete expression of the relations which exist between a set of quantities.
n.
The change, as of an equation or quantity, into another form without altering the value.
n.
Rank; degree; thus, the order of a curve or surface is the same as the degree of its equation.
n.
A surface whose equation in three variables is of the second degree. Spheres, spheroids, ellipsoids, paraboloids, hyperboloids, also cones and cylinders with circular bases, are quadrics.
n.
The bringing of any term of an equation from one side over to the other without destroying the equation.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
An identical equation.
n.
A spiral whose polar equation is r2/ = a; that is, a curve the square of whose radius vector varies inversely as the angle which the radius vector makes with a given line.
n.
Either of the two parts of an algebraic equation, connected by the sign of equality.
n.
That branch of algebra which treats of quadratic equations.
a.
Pertaining to terms of the second degree; as, a quadratic equation, in which the highest power of the unknown quantity is a square.
n.
A curve or surface whose equation is of the fourth degree in the variables.
n.
A curve of the fourth degree, invented by Pascal. Its polar equation is r = a cos / + b.
n.
The act of solving, or the state of being solved; the disentanglement of any intricate problem or difficult question; explanation; clearing up; -- used especially in mathematics, either of the process of solving an equation or problem, or the result of the process.
n.
The curve whose ordinates are proportional to the sines of the abscissas, the equation of the curve being y = a sin x. It is also called the curve of sines.
n.
Belonging to number; denoting number; consisting in numbers; expressed by numbers, and not letters; as, numerical characters; a numerical equation; a numerical statement.
v. t.
To bring, as any term of an equation, from one side over to the other, without destroying the equation; thus, if a + b = c, and we make a = c - b, then b is said to be transposed.
n.
The division of the terms of an equation by a known quantity that is involved in the first term.
n.
A quantity which may increase or decrease; a quantity which admits of an infinite number of values in the same expression; a variable quantity; as, in the equation x2 - y2 = R2, x and y are variables.