Collisionless Boltzmann equation

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, 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 (mathematically written ${\displaystyle d^{3}{\bf {r}}}$, where d means "differential") centered at the position ${\displaystyle {\bf {r}}}$, and has momentum nearly equal to a given momentum vector ${\displaystyle {\bf {p}}}$ (thus occupying a very small region of momentum space ${\displaystyle d^{3}{\bf {p}}}$), at an instant of time.

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