Ludwig Boltzmann - The Boltzmann Equation

The Boltzmann Equation

The Boltzmann equation was developed to describe the dynamics of an ideal gas.

where ƒ represents the distribution function of single-particle position and momentum at a given time (see the Maxwell–Boltzmann distribution), F is a force, m is the mass of a particle, t is the time and v is an average velocity of particles.

This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. (See Hamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.

In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate boundary conditions. This first-order differential equation has a deceptively simple appearance, since ƒ can represent an arbitrary single-particle distribution function. Also, the force acting on the particles depends directly on the velocity distribution function ƒ. The Boltzmann equation is notoriously difficult to integrate. David Hilbert spent years trying to solve it without any real success.

The form of the collision term assumed by Boltzmann was approximate. However for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under shock wave conditions.

Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation — his famous H-theorem. However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with Loschmidt and others over Loschmidt's paradox ultimately ended in his failure.

Finally, in the 1970s E.G.D. Cohen and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently nonequilibrium statistical mechanics for dense gases and liquids focuses on the Green–Kubo relations, the fluctuation theorem, and other approaches instead.