Thus H for a gas with the equilibrium Maxwellian distribution function does not change with time. Thus if we begin with a uniform gas having a non-equilibrium distribution function, H decreases until the gas relaxes to the equilibrium distribution when H attains a minimum value. Just as the entropy of an isolated system increased until it attains a maximum value, H decreases to reach a minimum. Although we began by assuming the reversibility of molecular interactions, we have ended up with the quantity H the evolution of which is not symmetric in time. It should be emphasized that the H -theorem is not the the same thing as the principle of increase of entropy, which is one of the statements of the second law of thermodynamics.
H is defined only for a uniform gas, whereas the entropy can be defined for complex systems.
On the other hand, the usual definition of entropy is only for system in thermodynamic equilibrium. Here we are considering the H function for gases not in thermodynamic equilibrium, i.e., gases not described by a Maxwellian distribution function. Only in the special case of a dilute gas in thermodynamic equilibrium can H and entropy both be defined, is there a simple relation between the two. Under these circumstances plus a constant.