the light elements in Big Bang nucleosynthesis, the production of dark matter 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 (including finite elements and lattice Boltzmann methods) 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 powers of Knudsen number (the Chapman–Enskog expansion). The first two terms of this expansion give the Euler equations and the Navier–Stokes equations. 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 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. Boltzmann-like equations are also used for the movement of cells. Since cells are composite particles that carry internal degree of freedom, the corresponding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of cancer cells in tissue, morphogenesis, and chemotaxis-related effects. See also Vlasov equation The Vlasov–Poisson 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