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 p_x, p_y, p_z. The entire space is 6-dimensional: a point in this space is (r, p) = (x, y, z, p_x, p_y, p_z), and each coordinate is parameterized by time t. The small volume ("differential volume element") is written

$${\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} =dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}.}$$

Since the probability of N molecules, which all have r and p within ${\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} }$, is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function: f(r, p, t), defined so that,

$${\displaystyle dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} }$$

is the number of molecules which all have positions lying within a volume element ${\displaystyle d^{3}\mathbf {r} }$ about r and momenta lying within a momentum space element ${\displaystyle d^{3}\mathbf {p} }$ about p, at time t.^[5] Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

$${\displaystyle {\begin{aligned}N&=\int \limits _{\mathrm {momenta} }d^{3}\mathbf {p} \int \limits _{\mathrm {positions} }d^{3}\mathbf {r} \,f(\mathbf {r} ,\mathbf {p} ,t)\\[5pt]&=\iiint \limits _{\mathrm {momenta} }\quad \iiint \limits _{\mathrm {positions} }f(x,y,z,p_{x},p_{y},p_{z},t)\,dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}\end{aligned}}}$$

which is a 6-fold integral. While f 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 r and p is in question. It is not part of the analysis to use r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N for particle N.

It is assumed the particles in the system are identical (so each has an identical mass m). 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^[6]

$${\displaystyle {\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\text{force}}+\left({\frac {\partial f}{\partial t}}\right)_{\text{diff}}+\left({\frac {\partial f}{\partial t}}\right)_{\text{coll}},}$$

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 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.^[6]

Note that some authors use the particle velocity v instead of momentum p; they are related in the definition of momentum by p = mv.

Final statement

Dividing (3) by dt and substituting into (2) gives:

$${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }}$$

In this context, F(r, t) is the force field acting on the particles in the fluid, and m 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 f cannot be solved unless the collision term in f 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.

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:^[2]

$${\displaystyle \left({\frac {\partial f}{\partial t}}\right)_{\text{coll}}=\iint gI(g,\Omega )[f(\mathbf {r} ,\mathbf {p'} _{A},t)f(\mathbf {r} ,\mathbf {p'} _{B},t)-f(\mathbf {r} ,\mathbf {p} _{A},t)f(\mathbf {r} ,\mathbf {p} _{B},t)]\,d\Omega \,d^{3}\mathbf {p} _{B},}$$

where p_A and p_B are the momenta of any two particles (labeled as A and B for convenience) before a collision, p′_A and p′_B are the momenta after the collision,

$${\displaystyle g=|\mathbf {p} _{B}-\mathbf {p} _{A}|=|\mathbf {p'} _{B}-\mathbf {p'} _{A}|}$$

is the magnitude of the relative momenta (see relative velocity for more on this concept), and I(g, Ω) is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the solid angle dΩ, 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.^[7] 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:

$${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\nu (f_{0}-f),}$$

where ${\displaystyle \nu }$ is the molecular collision frequency, and ${\displaystyle f_{0}}$ is the local Maxwellian distribution function given the gas temperature at this point in space.

Conservation equations

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.^[8]: 163 For a fluid consisting of only one kind of particle, the number density n is given by

$${\displaystyle n=\int f\,d^{3}\mathbf {p} .}$$

The average value of any function A is

$${\displaystyle \langle A\rangle ={\frac {1}{n}}\int Af\,d^{3}\mathbf {p} .}$$

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 ${\displaystyle \mathbf {x} \mapsto x_{i}}$ and ${\displaystyle \mathbf {p} \mapsto p_{i}=mv_{i}}$, where ${\displaystyle v_{i}}$ is the particle velocity vector. Define ${\displaystyle A(p_{i})}$ as some function of momentum ${\displaystyle p_{i}}$ only, which is conserved in a collision. Assume also that the force ${\displaystyle F_{i}}$ is a function of position only, and that f is zero for ${\displaystyle p_{i}\to \pm \infty }$. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

$${\displaystyle \int A{\frac {\partial f}{\partial t}}\,d^{3}\mathbf {p} ={\frac {\partial }{\partial t}}(n\langle A\rangle ),}$$

$${\displaystyle \int {\frac {p_{j}A}{m}}{\frac {\partial f}{\partial x_{j}}}\,d^{3}\mathbf {p} ={\frac {1}{m}}{\frac {\partial }{\partial x_{j}}}(n\langle Ap_{j}\rangle ),}$$

$${\displaystyle \int AF_{j}{\frac {\partial f}{\partial p_{j}}}\,d^{3}\mathbf {p} =-nF_{j}\left\langle {\frac {\partial A}{\partial p_{j}}}\right\rangle ,}$$

$${\displaystyle \int A\left({\frac {\partial f}{\partial t}}\right)_{\text{coll}}\,d^{3}\mathbf {p} =0,}$$

where the last term is zero, since A is conserved in a collision. The values of A correspond to moments of velocity ${\displaystyle v_{i}}$ (and momentum ${\displaystyle p_{i}}$, as they are linearly dependent).

Zeroth moment

Letting ${\displaystyle A=m(v_{i})^{0}=m}$, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:^{[8]: 12, 168}

$${\displaystyle {\frac {\partial }{\partial t}}\rho +{\frac {\partial }{\partial x_{j}}}(\rho V_{j})=0,}$$

where ${\displaystyle \rho =mn}$ is the mass density, and ${\displaystyle V_{i}=\langle v_{i}\rangle }$ is the average fluid velocity.

First moment

Letting ${\displaystyle A=m(v_{i})^{1}=p_{i}}$, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:^{[8]: 15, 169}

$${\displaystyle {\frac {\partial }{\partial t}}(\rho V_{i})+{\frac {\partial }{\partial x_{j}}}(\rho V_{i}V_{j}+P_{ij})-nF_{i}=0,}$$

where ${\displaystyle P_{ij}=\rho \langle (v_{i}-V_{i})(v_{j}-V_{j})\rangle }$ is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).

Second moment

Letting ${\displaystyle A={\frac {m(v_{i})^{2}}{2}}={\frac {p_{i}p_{i}}{2m}}}$, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:^{[8]: 19, 169}

$${\displaystyle {\frac {\partial }{\partial t}}(u+{\tfrac {1}{2}}\rho V_{i}V_{i})+{\frac {\partial }{\partial x_{j}}}(uV_{j}+{\tfrac {1}{2}}\rho V_{i}V_{i}V_{j}+J_{qj}+P_{ij}V_{i})-nF_{i}V_{i}=0,}$$

where ${\textstyle u={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{i}-V_{i})\rangle }$ is the kinetic thermal energy density, and ${\textstyle J_{qi}={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{k}-V_{k})(v_{k}-V_{k})\rangle }$ is the heat flux vector.

Hamiltonian mechanics

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as

$${\displaystyle {\hat {\mathbf {L} }}[f]=\mathbf {C} [f],}$$

where L 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 C is the collision operator. The non-relativistic form of L is

$${\displaystyle {\hat {\mathbf {L} }}_{\mathrm {NR} }={\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} }}\,.}$$

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,^[9] 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.^[10]

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 is^[11]

$${\displaystyle {\hat {\mathbf {L} }}_{\mathrm {GR} }=p^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}-\Gamma ^{\alpha }{}_{\beta \gamma }p^{\beta }p^{\gamma }{\frac {\partial }{\partial p^{\alpha }}},}$$

where Γ^α_βγ 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 (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.^[12][13]

In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation.^[14] More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity.^[9] 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 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.^[10] This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.