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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, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
not in a state of equilibrium, devised by
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
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 with temperature gradients 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 corpuscule 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 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
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
s 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 the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
that the particle occupies a given very small region of space (mathematically the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
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 In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
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 defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
,
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, and electrical conductivity (by treating the charge carriers in a material as a gas). See also
convection–diffusion equation The convection–diffusion equation is a combination of the diffusion equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferr ...
. The equation is a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
integro-differential equation In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivati ...
, 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 of the system; in other words a set of three coordinates 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 coor ...
al: a point in this space is , and each coordinate is parameterized by time ''t''. The small volume ("differential
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
") 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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
: , 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 In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
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 The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
). For a mixture of more than one
chemical species A chemical species is a chemical substance or ensemble composed of chemically identical molecular entities that can explore the same set of molecular energy levels on a characteristic or delineated time scale. These energy levels determine the wa ...
, 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 chemica ...
of particles, and "coll" is the
collision In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great fo ...
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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
(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 of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator, is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, \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 (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
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 interaction Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
s, is often called the
Vlasov equation The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma ...
. 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 Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
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 In the kinetic theory of gases in physics, the molecular chaos hypothesis (also called ''Stosszahlansatz'' in the writings of Paul Ehrenfest) is the assumption that the velocities of colliding particles are uncorrelated, and independent of positi ...
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 The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A. Classical mechanics In one dimension (non-relativistic) We begin with relative motion in the classi ...
for more on this concept), and is the
differential cross section In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle o ...
of the collision, in which the relative momenta of the colliding particles turns through an angle into the element of the
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
, 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.


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 w_i, where w_i is the particle velocity vector. Define A(p_i) as some function of momentum p_i only, which 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 = 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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
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 w_i\rangle is the average fluid velocity.


First moment

Letting A = m(v_i)^1 = p_i, the momentum 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 (w_i-V_i) (w_j-V_j) \rangle is the pressure tensor (the
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
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 e ...
).


Second moment

Letting A = \frac = \frac, the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
of the particle, the integrated Boltzmann equation becomes the conservation of energy equation: \frac(u + \tfrac\rho V_i V_i) + \frac (uV_j + \tfrac\rho V_i V_i V_j + J_ + P_V_i) - nF_iV_i = 0, where u = \tfrac \rho \langle (w_i-V_i) (w_i-V_i) \rangle is the kinetic thermal energy density, and J_ = \tfrac \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle is the heat flux vector.


Hamiltonian mechanics

In
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, the Boltzmann equation is often written more generally as \hat \mathbf where is the
Liouville operator In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajecto ...
(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 equation The quantum Boltzmann equation, also known as the Uehling-Uhlenbeck equation, is the quantum mechanical modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typ ...
s 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 f ...
, including the formation of the light elements in
Big Bang nucleosynthesis In physical cosmology, Big Bang nucleosynthesis (abbreviated BBN, also known as primordial nucleosynthesis) is the production of nuclei other than those of the lightest isotope of hydrogen ( hydrogen-1, 1H, having a single proton as a nucleu ...
, the production of
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
and
baryogenesis In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, i.e. the imbalance of matter (baryons) and antimatter (a ...
. 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.


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 In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...
. Its generalization in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. is \hat_\mathrm = p^\alpha\frac - \Gamma^\alpha_ p^\beta p^\gamma \frac, where is the
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
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 f ...
the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the
early universe The chronology of the universe describes the history and future of the universe according to Big Bang cosmology. Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, wit ...
often attempt to take into account the effects of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment
quantum coherence In physics, two wave sources are coherent if their frequency and waveform are identical. Coherence is an ideal property of waves that enables stationary (i.e., temporally or spatially constant) interference. It contains several distinct concepts ...
and the spatial extension of the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
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. This includes the formation of the light elements in
Big Bang nucleosynthesis In physical cosmology, Big Bang nucleosynthesis (abbreviated BBN, also known as primordial nucleosynthesis) is the production of nuclei other than those of the lightest isotope of hydrogen ( hydrogen-1, 1H, having a single proton as a nucleu ...
, the production of
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
and
baryogenesis In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, i.e. the imbalance of matter (baryons) and antimatter (a ...
.


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 th ...
(including
finite elements The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
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 sol ...
) 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 to ...
in powers of
Knudsen number The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is name ...
(the Chapman–Enskog expansion). The first two terms of this expansion give the
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are 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 ...
and the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. 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 exist a generalization of the Boltzmann equation that is called the Enskog equation. The collision term is modified in Enskog equations such that finite size of particles, for example particles can be a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
having a fixed radius. No further degree of freedoms besides translational motion are assumed for the particles. If there are internal degree of freedom, the Boltzmann equation has to be generalized and might possess inelastic collisions. Many real fluids like liquids 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
BBGKY hierarchy In statistical physics, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equ ...
. Boltzmann-like equations are also used for the movement of cells. Since cells are
composite particle This is a list of known and hypothesized particles. Elementary particles Elementary particles are particles with no measurable internal structure; that is, it is unknown whether they are composed of other particles. They are the fundamental ob ...
s that carry internal degree of freedom, the corresonding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of
cancer Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...
cells in tissue,
morphogenesis Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of deve ...
, and chemotaxis-related effects.


See also

*
Vlasov equation The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma ...
* The Vlasov–Poisson equation *
Fokker–Planck equation In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag force ...
* Williams–Boltzmann equation * Derivation of Navier–Stokes equation from LBE * Derivation of Jeans equation from BE * Jeans's theorem *
H-theorem In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules. L. Boltzmann,Weitere Studien über das Wä ...


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 equation Landau ( pfl, Landach), 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 ...
s. * * * *


External links


The Boltzmann Transport Equation by Franz Vesely

Boltzmann gaseous behaviors solved
{{Statistical mechanics topics Partial differential equations Statistical mechanics Transport phenomena Equations of physics Equation 1872 in science 1872 in Germany Thermodynamic equations