Chemistry - Boltzmann equation

The Boltzmann equation describes the statistical distribution of particles 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 current, and thus to derive transport-related properties such as electrical conductivity, Hall conductivity, viscosity, and thermal conductivity.

The Boltzmann equation is

$\frac{\partial f}{\partial t} + \frac{\partial f}{\partial \mathbf{r}} \cdot \frac{\mathbf{p}}{m} + \frac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F} = \left. \frac{\partial f}{\partial t} \right|_{\mathrm{coll}}$

where t denotes time, r position, and p momentum. F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term ∂f/∂t|_coll on the right hand side describes the effect of collisions between particles, and has to be independently supplied; it is zero if the particles do not collide with one another. Finally, the quantity f(r, p, t), which is known as the distribution function, is defined as follows:

$f(\mathbf{r},\mathbf{p},t) \frac{d^3r\,d^3p}{h^3}\equiv$ the mean number of particles with center of mass located within a small volume d³r near the point r, and momentum within a range d³p near p, at time t.

The quantity h is customarily inserted to make f a dimensionless quantity. For classical statistical mechanics, this is just a matter of convention, since it does not show up in the final results of calculations. For quantum statistical mechanics, one usually takes h to be Planck's constant, so that f stands for the occupancy of a cell in phase space delineated by the Heisenberg uncertainty principle.

The Boltzmann equation is also often used in the field of dynamics, especially galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f. When the collision term is null, the equation is also named non-collisional Boltzmann equation; in galaxies, physical collisions between the stars are very rare, and the effect gravitational collisions can be neglected for times far longer than the age of the universe.

Boltzmann equation

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