The H-Theorem Formulation

Formulating the H-theorem Boltzmann considered a simple gas consisting of spherical molecules, possessing only translational energy, and are subject to no external forces. If the gas state is uniform in space, so that the velocity distribution function / is independent of r, the Boltzmann equation (2.184) reduces to  

Moreover, Boltzmann introduced a novel function H t), defined in terms of a complete integral over all values of the molecular velocities  

The function H t) thus depends on the mode of distribution of the molecular velocities only [20], 

Multiplying the reduced form of Boltzmann equation (2.184) by (1 + In /) and integrating over the phase space, the results is  

Substitution of the expression on the LHS of (2.236) by use of (2.237), and with some further manipulation of the collision term, the reduced Boltzmann equation (2.236) becomes  

We first derive the //-theorem in an approximate manner, starting out by introducing the differential //-property function defined by  

To determine how H changes in time we consider a uniform state in which the distribution function / is independent of position r and no external forces 

H is thus given by the integral over all velocities. Then, iL is a number, independent of r, but a function of t, depending only on the mode of distribution of the molecular velocities [12]. 

Considering that (1 + In/) represents a summation invariant property, (2.219) can then be expressed as  

---