The probability of a Maxwellian distribution changing to a different distribution, say will all the molecules moving to the left, is extremely improbable. While deriving the Boltzmann equation we have explicitly assume the reversibility of the microscopic processes. Still the evolution of the distribution function according to the Boltzmann equation turns out to be irreversible. Hence, the Boltzmann equation must hold the key to understanding how irreversibility at the macroscopic level arises out of reversible microscopic processes. We limit out discussion to a proof of the H -theorem, which give a hint at the origins of macroscopic irreversibility.
H is a quantity per unit volume.
Conversely if then and and the produce is less than zero. 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.