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.
The Boltzmann equation is an equation for the time t evolution of the distribution (properly a density) function 'f'(x, p, t) in one-particle phase space, where x and p are position and momentum, respectively. The distribution is defined such that
f(\mathbf{x},\mathbf{p},t)\,d\mathbf{x}\,d\mathbf{p}
is the number of molecules which, at time t, have positions lying within a volume element d3x about x and momenta lying within a momentum-space element d3p about p.[1].
Consider those particles described by f experiencing an external Force F. Then f must satisfy, in absence of collisions
f(\mathbf{x}+\frac{\mathbf{p}}{m}\,dt,\mathbf{p}+\mathbf{F}\,dt,t+dt)\,d\mathbf{x}\,d\mathbf{p} = f(\mathbf{x},\mathbf{p},t)\,d\mathbf{x}\,d\mathbf{p},
saying that if some particles are at time t in \mathbf{x} with momentum \mathbf{p}, at time t + dt, they will (all) be in \mathbf{x}+\frac{\mathbf{p}}{m}\mathrm{d}t, with momentum \mathbf{p} + \mathbf{F}\mathrm{d}t.
However, since collisions do occur, the particle density in the phase-space volume dx dp changes.
f(\mathbf{x}+\frac{\mathbf{p}}{m}dt,\mathbf{p} + \mathbf{F}dt,t+dt) \,d\mathbf{x}\,d\mathbf{p} - f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} = \left. \frac{\partial f(\mathbf{x},\mathbf{p},t)}{\partial t} \right|_{\mathrm{coll}} \, d\mathbf{x} \, d\mathbf{p} \, dt
Dividing the equation by dx dp dt and taking the limit, we can get the Boltzmann equation
\frac{\partial f}{\partial t} + \frac{\partial f}{\partial \mathbf{x}} \cdot \frac{\mathbf{p}}{m} + \frac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F} = \left. \frac{\partial f}{\partial t} \right|_{\mathrm{coll}}.
F(x, t) is the force field acting on the particles in the fluid, and m 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. The collisionless Boltzmann equation is often mistakenly called the Liouville equation (the Liouville Equation is an N-particle equation).
