Boltzmann equation

The Boltzmann equation or Boltzmann transport equation describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.
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 particles 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 that the particle occupies a given very small region of space centered at the position, and has momentum nearly equal to a given momentum vector , at an instant of time.
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity. 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 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-dimensional: a point in this space is, and each coordinate is parameterized by time t. A relevant differential element is written
Since the probability of molecules, which all have and within, 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:, defined so that,
is the number of molecules which all have positions lying within a volume element about and momenta lying within a momentum space element 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:
which is a 6-fold integral. While is associated with a number of particles, the phase space is for one-particle, 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. 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 as
where the "force" term corresponds to the forces exerted on the particles by an external influence, the "diff" term represents the diffusion 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.
Suppose at time some number of particles all have position within element and momentum within. If a force instantly acts on each particle, then at time their position will be and momentum. Then, in the absence of collisions, must satisfy
Note that we have used the fact that the phase space volume element is constant, which can be shown using Hamilton's equations. However, since collisions do occur, the particle density in the phase-space volume changes, so
where is the total change in. Dividing by and taking the limits and, we have
The total differential of is:
where is the gradient operator, is the dot product,
is a shorthand for the momentum analogue of, and,, are Cartesian unit vectors.

Final statement

Dividing by and substituting into gives:
In this context, is the force field acting on the particles in the fluid, and is the mass 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 "Stosszahlansatz" 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:
where and are the momenta of any two particles before a collision, and are the momenta after the collision,
is the magnitude of the relative momenta, 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:
where is the molecular collision frequency, and 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
where, and the collision term is
where, the magnitude of the relative momenta is
and is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand. 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
The average value of any function is
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 and, where is the particle velocity vector. Define as some function of momentum only, whose total value is conserved in a collision. Assume also that the force is a function of position only, and that f is zero for. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
where the last term is zero, since is conserved in a collision. The values of correspond to moments of velocity .

Zeroth moment

Letting, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:
where is the mass density, and is the average fluid velocity.

First moment

Letting, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:
where is the pressure tensor.

Second moment

Letting, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:
where is the kinetic thermal energy density, and is the heat flux vector.

Hamiltonian mechanics

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as
where is the Liouville operator describing the evolution of a phase space volume and is the collision operator. The non-relativistic form of is

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, including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter 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.