In physics, specifically non-equilibrium statistical mechanics, the Boltzmann equation or Boltzmann transport equation describes the statistical behaviour of a fluid not in thermodynamic equilibrium, i.e. when there are temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random (and biased) transport of particles. It was devised by Ludwig Boltzmann in 1872.
The equation arises not by statistical analysis of all the individual positions and momenta of each particle in the fluid; rather by considering the probability that a number of particles all occupy a very small region of space (mathematically written d3r, where d means "differential", a very small change) centered at the tip of the position vector r, and have very nearly equal small changes in momenta from a momentum vector p, 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, and other properties characteristic to fluids such as viscosity, thermal conductivity also electrical conductivity (by treating the charge carriers in a material as a gas) can be derived. See also convection-diffusion equation.
The equation is a linear stochastic partial differential equation, since the function to solve the equation for is a continuous random variable. In fact - the problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.
Read more about Boltzmann Equation: The Force and Diffusion Terms, The Collision Term (Stosszahlansatz) and Molecular Chaos, General Equation (for A Mixture)
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