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{p}. The average value of any function is \langle A \rangle = \frac 1 n \int A f \,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 \mathbf{x} \mapsto x_i and \mathbf{p} \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{\partial f}{\partial t} \,d^3\mathbf{p} = \frac{\partial }{\partial t} (n \langle A \rangle), \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 A p_j \rangle), \int A F_j \frac{\partial f}{\partial p_j} \,d^3\mathbf{p} = -n F_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle, \int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3\mathbf{p} = \frac{\partial }{\partial t}_\text{coll} (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 of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: 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.
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 \hat{\mathbf{L}}_\mathrm{GR}[f] = p^\alpha\frac{\partial f}{\partial x^\alpha} - \Gamma_{\beta\gamma}^\alpha p^\beta p^\gamma \frac{\partial f}{\partial p^\alpha} = C[f], 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 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 and
general relativity. 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. This includes the formation of the light elements in
Big Bang nucleosynthesis, the production of
dark matter and
baryogenesis. == Solving the equation ==