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The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a
thermodynamic system A thermodynamic system is a body of matter and/or radiation separate from its surroundings that can be studied using the laws of thermodynamics. Thermodynamic systems can be passive and active according to internal processes. According to inter ...
not in a state of equilibrium; it was devised by
Ludwig Boltzmann Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...
in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. The classic example of such a system is a
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
with
temperature gradient A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature spatial gradient is a vector quantity with Dimensional analysis, ...
s in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
s making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
that the particle occupies a given very small region of space (mathematically the volume element d^3 \mathbf) centered at the position \mathbf, and has momentum nearly equal to a given momentum vector \mathbf (thus occupying a very small region of momentum space d^3 \mathbf), at an instant of time. The Boltzmann equation can be used to determine how physical quantities change, such as
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
energy and
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
, when a fluid is in transport. One may also derive other properties characteristic to fluids such as
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
,
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...
, and
electrical conductivity Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity in ...
(by treating the charge carriers in a material as a gas). See also convection–diffusion equation. The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.


Overview


The phase space and density function

The set of all possible positions r and momenta p is called the
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of the system; in other words a set of three
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
for each position coordinate ''x, y, z'', and three more for each momentum component , , . The entire space is 6-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al: a point in this space is , and each coordinate is parameterized by time ''t''. A relevant differential element is written d^3\mathbf \, d^3\mathbf = dx \, dy \, dz \, dp_x \, dp_y \, dp_z. Since the probability of molecules, which ''all'' have and within d^3\mathbf \, d^3\mathbf, is in question, at the heart of the equation is a quantity which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time . This is a
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: , defined so that, dN = f (\mathbf,\mathbf,t) \, d^3\mathbf \, d^3\mathbf is the number of molecules which ''all'' have positions lying within a volume element d^3\mathbf about and momenta lying within a momentum space element d^3\mathbf about , at time . Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region: \begin N & = \int\limits_\mathrm d^3\mathbf \int\limits_\mathrm d^3\mathbf\,f (\mathbf,\mathbf,t) \\ pt& = \iiint\limits_\mathrm \quad \iiint\limits_\mathrm f(x,y,z, p_x,p_y,p_z, t) \, dx \, dy \, dz \, dp_x \, dp_y \, dp_z \end which is a 6-fold integral. While is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one and is in question. It is not part of the analysis to use , for particle 1, , for particle 2, etc. up to , for particle ''N''. It is assumed the particles in the system are identical (so each has an identical
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
). For a mixture of more than one chemical species, one distribution is needed for each, see below.


Principal statement

The general equation can then be written asMcGraw Hill Encyclopaedia of Physics (2nd Edition), S. P. Parker, 1993, . \frac = \left(\frac\right)_\text + \left(\frac\right)_\text + \left(\frac\right)_\text, where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below. Note that some authors use the particle velocity instead of momentum ; they are related in the definition of momentum by .


The force and diffusion terms

Consider particles described by , each experiencing an ''external'' force not due to other particles (see the collision term for the latter treatment). Suppose at time some number of particles all have position within element d^3\mathbf and momentum within d^3\mathbf. If a force instantly acts on each particle, then at time their position will be \mathbf + \Delta \mathbf = \mathbf +\frac \, \Delta t and momentum . Then, in the absence of collisions, must satisfy f \left (\mathbf+\frac \, \Delta t,\mathbf+\mathbf \, \Delta t, t+\Delta t \right )\,d^3\mathbf\,d^3\mathbf = f(\mathbf, \mathbf,t) \, d^3\mathbf \, d^3\mathbf Note that we have used the fact that the phase space volume element d^3\mathbf \, d^3\mathbf is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume d^3\mathbf \, d^3\mathbf changes, so where is the ''total'' change in . Dividing () by d^3\mathbf \, d^3\mathbf \, \Delta t and taking the limits and , we have The total differential of is: where is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator, is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
, \frac = \mathbf_x\frac + \mathbf_y\frac + \mathbf_z \frac= \nabla_\mathbff is a shorthand for the momentum analogue of , and , , are Cartesian
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s.


Final statement

Dividing () by and substituting into () gives: \frac + \frac\cdot\nabla f + \mathbf \cdot \frac = \left(\frac \right)_\mathrm In this context, is the force field acting on the particles in the fluid, and is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since cannot be solved unless the collision term in is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions.


The collision term (Stosszahlansatz) and molecular chaos


Two-body collision term

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "" and is also known as the " molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions: \left(\frac\right)_\text = \iint g I(g, \Omega) (\mathbf,\mathbf_A, t) f(\mathbf,\mathbf_B,t) - f(\mathbf,\mathbf_A,t) f(\mathbf,\mathbf_B,t)\,d\Omega \,d^3\mathbf_B, where and are the momenta of any two particles (labeled as ''A'' and ''B'' for convenience) before a collision, and are the momenta after the collision, g = , \mathbf_B - \mathbf_A, = , \mathbf_B - \mathbf_A, is the magnitude of the relative momenta (see relative velocity for more on this concept), and is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle into the element of the solid angle , due to the collision.


Simplifications to the collision term

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook. The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form: \frac + \frac\cdot\nabla f + \mathbf \cdot \frac = \nu (f_0 - f), where \nu is the molecular collision frequency, and f_0 is the local Maxwellian distribution function given the gas temperature at this point in space. This is also called "relaxation time approximation".


General equation (for a mixture)

For a mixture of chemical species labelled by indices the equation for species is \frac + \frac \cdot \nabla f_i + \mathbf \cdot \frac = \left(\frac \right)_\text, where , and the collision term is \left(\frac \right)_ = \sum_^n \iint g_ I_(g_, \Omega) '_i f'_j - f_i f_j\,d\Omega\,d^3\mathbf, where , the magnitude of the relative momenta is g_ = , \mathbf_i - \mathbf_j, = , \mathbf'_i - \mathbf'_j, , and is the differential cross-section, as before, between particles ''i'' and ''j''. The integration is over the momentum components in the integrand (which are labelled ''i'' and ''j''). The sum of integrals describes the entry and exit of particles of species ''i'' in or out of the phase-space element.


Applications and extensions


Conservation equations

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density is given by n = \int f \,d^3\mathbf. The average value of any function is \langle A \rangle = \frac 1 n \int A f \,d^3\mathbf. Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus \mathbf \mapsto x_i and \mathbf \mapsto p_i = m v_i, where v_i is the particle velocity vector. Define A(p_i) as some function of momentum p_i only, whose total value is conserved in a collision. Assume also that the force F_i is a function of position only, and that ''f'' is zero for p_i \to \pm\infty. Multiplying the Boltzmann equation by ''A'' and integrating over momentum yields four terms, which, using integration by parts, can be expressed as \int A \frac \,d^3\mathbf = \frac (n \langle A \rangle), \int \frac\frac \,d^3\mathbf = \frac\frac(n\langle A p_j \rangle), \int A F_j \frac \,d^3\mathbf = -n F_j\left\langle \frac\right\rangle, \int A \left(\frac\right)_\text \,d^3\mathbf = \frac_\text (n \langle A \rangle) = 0, where the last term is zero, since is conserved in a collision. The values of correspond to moments of velocity v_i (and momentum p_i, as they are linearly dependent).


Zeroth moment

Letting A = m(v_i)^0 = m, the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: \frac\rho + \frac(\rho V_j) = 0, where \rho = mn is the mass density, and V_i = \langle v_i\rangle is the average fluid velocity.


First moment

Letting A = m(v_i)^1 = p_i, the
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation: \frac(\rho V_i) + \frac(\rho V_i V_j+P_) - n F_i = 0, where P_ = \rho \langle (v_i-V_i) (v_j-V_j) \rangle is the pressure tensor (the viscous stress tensor plus the hydrostatic
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
).


Second moment

Letting A = \frac = \frac, the
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of the particle, the integrated Boltzmann equation becomes the conservation of energy equation: \frac \left(u + \tfrac \rho V_i V_i\right) + \frac \left(u V_j + \tfrac \rho V_i V_i V_j + J_ + P_ V_i\right) - n F_i V_i = 0, where u = \tfrac \rho \langle (v_i-V_i) (v_i-V_i) \rangle is the kinetic thermal energy density, and J_ = \tfrac \rho \langle(v_i - V_i)(v_k - V_k)(v_k - V_k)\rangle is the heat flux vector.


Hamiltonian mechanics

In
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
, the Boltzmann equation is often written more generally as \hat \mathbf where is the Liouville operator (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and is the collision operator. The non-relativistic form of is \hat_\mathrm = \frac + \frac \cdot \nabla + \mathbf\cdot\frac\,.


Quantum theory and violation of particle number conservation

It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
, including the formation of the light elements in
Big Bang nucleosynthesis In physical cosmology, Big Bang nucleosynthesis (also known as primordial nucleosynthesis, and abbreviated as BBN) is a model for the production of light nuclei, deuterium, 3He, 4He, 7Li, between 0.01s and 200s in the lifetime of the universe ...
, the production of
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density ''f''. However, for a wide class of applications a well-defined generalization of ''f'' exists which is the solution of an effective Boltzmann equation that can be derived from first principles of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.


General relativity and astronomy

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by ''f''; in galaxies, physical collisions between the stars are very rare, and the effect of ''gravitational collisions'' can be neglected for times far longer than the age of the universe. Its generalization in
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
is \hat_\mathrm = p^\alpha\frac - \Gamma^\alpha_ p^\beta p^\gamma \frac = C where is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant phase space as opposed to fully contravariant phase space. In
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. In the very dense medium formed by the primordial plasma after the
Big Bang The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution ''f'' that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. This includes the formation of the light elements in
Big Bang nucleosynthesis In physical cosmology, Big Bang nucleosynthesis (also known as primordial nucleosynthesis, and abbreviated as BBN) is a model for the production of light nuclei, deuterium, 3He, 4He, 7Li, between 0.01s and 200s in the lifetime of the universe ...
, the production of
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
and baryogenesis.


Solving the equation

Exact solutions to the Boltzmann equations have been proven to exist in some cases; this analytical approach provides insight, but is not generally usable in practical problems. Instead,
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
(including finite elements and
lattice Boltzmann methods The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy- Pomeau-Pazzis and Frisch- Hasslacher- Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of s ...
) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from hypersonic aerodynamics in rarefied gas flows to plasma flows. An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result. Close to local equilibrium, solution of the Boltzmann equation can be represented by an
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
in powers of Knudsen number (the Chapman–Enskog expansion). The first two terms of this expansion give the
Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
and the
Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.


Limitations and further uses of the Boltzmann equation

The Boltzmann equation is valid only under several assumptions. For instance, the particles are assumed to be pointlike, i.e. without having a finite size. There exists a generalization of the Boltzmann equation that is called the Enskog equation. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s having a fixed radius. No further degrees of freedom besides translational motion are assumed for the particles. If there are internal degrees of freedom, the Boltzmann equation has to be generalized and might possess inelastic collisions. Many real fluids like
liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
s or dense gases have besides the features mentioned above more complex forms of collisions, there will be not only binary, but also ternary and higher order collisions. These must be derived by using the BBGKY hierarchy. Boltzmann-like equations are also used for the movement of cells. Since cells are
composite particle This is a list of known and hypothesized microscopic particles in particle physics, condensed matter physics and cosmology. Standard Model elementary particles Elementary particles are particles with no measurable internal structure; that is, ...
s that carry internal degrees of freedom, the corresponding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of
cancer Cancer is a group of diseases involving Cell growth#Disorders, abnormal cell growth with the potential to Invasion (cancer), invade or Metastasis, spread to other parts of the body. These contrast with benign tumors, which do not spread. Po ...
cells in tissue, morphogenesis, and
chemotaxis Chemotaxis (from ''chemical substance, chemo-'' + ''taxis'') is the movement of an organism or entity in response to a chemical stimulus. Somatic cells, bacteria, and other single-cell organism, single-cell or multicellular organisms direct thei ...
-related effects.


See also

* Vlasov equation * The Vlasov–Poisson equation * Landau kinetic equation * Fokker–Planck equation * Williams–Boltzmann equation * Derivation of Navier–Stokes equation from LBE * Derivation of Jeans equation from BE * Jeans's theorem * H-theorem


Notes


References

* . Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or
Landau Landau (), officially Landau in der Pfalz (, ), is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990), a long ...
equations. * * *


External links


The Boltzmann Transport Equation by Franz Vesely

Boltzmann gaseous behaviors solved
{{Statistical mechanics topics Eponymous equations of physics Partial differential equations Statistical mechanics Transport phenomena
Equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
1872 in science 1872 in Germany Thermodynamic equations