Editing H-theorem (section)

== Definition and meaning of Boltzmann's ''H'' ==

The ''H'' value is determined from the function ''f''(''E'', ''t'') ''dE'', which is the energy distribution function of molecules at time ''t''. The value ''f''(''E'', ''t'') ''dE'' is the number of molecules that have kinetic energy between ''E'' and ''E'' + ''dE''. ''H'' itself is defined as

:<math> H(t) = \int_0^\infty f(E,t) \left( \ln\frac{f(E,t)}{\sqrt{E}} - 1 \right) \, dE. </math>

For an isolated ideal gas (with fixed total energy and fixed total number of particles), the function ''H'' is at a minimum when the particles have a [[Maxwell–Boltzmann distribution]]; if the molecules of the ideal gas are distributed in some other way (say, all having the same kinetic energy), then the value of ''H'' will be higher. Boltzmann's ''H''-theorem, described in the next section, shows that when collisions between molecules are allowed, such distributions are unstable and tend to irreversibly seek towards the minimum value of ''H'' (towards the Maxwell–Boltzmann distribution).

(Note on notation: Boltzmann originally used the letter ''E'' for quantity ''H''; most of the literature after Boltzmann uses the letter ''H'' as here. Boltzmann also used the symbol ''x'' to refer to the kinetic energy of a particle.)