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Boltzmann equation



As Wikipedia defines it:

The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in a fluid. It is one of the most important equations of non-equilibrium statistical mechanics, the area of statistical mechanics that deals with systems far from thermodynamic equilibrium; for instance, when there is an applied temperature gradient or electric field. The Boltzmann equation is used to study how a fluid transports physical quantities such as heat and charge, and thus to derive transport properties such as electrical conductivity, Hall conductivity, viscosity, and thermal conductivity.Solutions to the 140-year-old Boltzmann equation were not known until 2010.

ft+fxpm+fpF=ft| coll\frac{\partial f}{\partial t} + \frac{\partial f}{\partial \vec{x}} \frac {\vec{p}}{m} + \frac{\partial f}{\partial \vec{p}} F = \frac{\partial f}{\partial t}|_coll

F(x,t)F(\vec{x}, t) is the force field acting on the particles in the fluid, and mm 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.