Boltzmann at first (1872) claimed to have shown that his "minimum function" H (he actually used the German letter "E" which in resembles the English "H") could be shown to decrease to a minimum as a consequence of the dynamic evolution of a gas of colliding particles. Boltzmann counted the number of particles that would leave a small volume of phase space as a result of collisions and compared it to the number of particles that would enter the same volume. This was called the Stosszahlansatz (collision number estimate).

Already in the 1870's William Thomson (Lord Kelvin) and Josef Loschmidt had criticized this dynamical derivation on the grounds that if all the velocities of the particle were reversed the H function would increase (entropy would decrease). And Boltzmann had agreed with them, saying in 1877 that the only proper derivation of the H-Theorem is statistical. Systems would evolve toward macrostates with the greatest probability (the greatest number of microstates). Boltzmann's definition of entropy is proportional to the logarithm if the number of microstates W.

The proportionality factor k was introduced by Max Planck in 1900. He called it Boltzmann's constant.

The British scientists argued that there must be something causing "molecular disorder" or chaos that is introducing the irreversibility. Culverwell suggested it might be caused by the ether that was the presumed medium for electromagnetic waves. In 1890 he wrote:

We know that by means of the aether, bodies at a distance and wholly prevented from acting on each other molecularly, come to exactly the same temperature-equilibrium without any assistance from their collisions. Hence there is every reason to suppose that it is by the molecules interacting through the aether that the temperature-equilibrium is determined.

(Philosophical Magazine, 1890, Series 5, vol. 30, p. 95.)

Then in 1894 he argued that something must be preventing the reversibility, since a dynamical analysis leads to perfect reversibility.

The remarkable differences of opinion as to what the H-theorem is, and how it can be proved, show how necessary is the discussion elicited by my letter...

I say that if that proof does not somewhere or other introduce some assumption about averages, probability, or irreversibility, it cannot be valid.

(Nature, 1894, Vol. 51, p. 246.)

Culverwell's colleague S. H. Burbury used the terms "haphazard" and "chaos" to describe what is needed.

The objection that I understand to be made is that if you reverse all the velocities after collisions, the system will retrace its course with H increasing - which is supposed to be contrary to the thing proved...

I think the answer to this would be that any actual material system receives disturbances from without, the effect of which, coming at haphazard, is to produce the very distribution of coordinates which is required to make H diminish.

(Nature, 1894, Vol. 51, p. 78.)

In the first place the distribution of energy which is given by Boltzmann's Theorem is the only distribution which is permanent under the conditions postulated by this theorem. And in the second place, this law of distribution may break down entirely as soon as we admit an interaction, no matter how small, between the molecules and the surrounding ether. That such an interaction must exist is shown by the fact that a gas is capable of radiating energy. In fact, Boltzmann's Theorem rests on the assumption that the molecules of a gas form a conservative dynamical system, and it will appear that the introduction of a small dissipation function may entirely invalidate the conclusions of the theorem.* Thus we may regard the Boltzmann distribution as unstable, in the sense that a slight deviation from perfect conservation of energy may result in a complete redistribution of the total energy.

(Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 1901, Vol. 196, pp. 397-430)

If the mean free path in a gas is large compared to the mean distance of two neighboring molecules, then in a short time, completely different molecules than before will be nearest neighbors to each other. A molecular-ordered but molar-disordered distribution will most probably be transformed into a molecular-disordered one in a short time. Each molecule flies from one collision to another one so far away that one can consider the occurrence of another molecule, at the place where it collides the second time, with a definite state of motion, as being an event completely independent (for statistical calculations) of the place from which the first molecule came (and similarly for the state of motion of the first molecule). However, if we choose the initial configuration on the basis of a previous calculation of the path of each molecule, so as to violate intentionally the laws of probability, then of course we can construct a persistent regularity or an almost molecular-disordered distribution which will become an molecular-ordered at a particular time. Kirchhoff also makes the assumption that the state is molecular-disordered in his definition of the probability concept.

(Lectures on Gas Theory, 1896-8, tr. (S.G.Brush) p. 41)

Boltzmann was confident that probability played the major role and he prophetically described a future physics of "average values," eerily anticipating the "expectation values" of probabilistic indeterministic quantum physics.

Since today it is popular to look forward to the time when our view of nature will have been completely changed, I will mention the possibility that the fundamental equations for the motion of individual molecules will turn out to be only approximate formulas which give average values, resulting according to the probability calculus from the interactions of many independent moving entities forming the surrounding medium - as for example in meteorology the laws are valid only for average values obtained by long series of observations using the probability calculus. These entities must of course be so numerous and must act so rapidly that the correct average values are attained in millionths of a second.

(Lectures on Gas Theory, Section 91, p. 449, cited by Brush, Stephen G. 1973, Development of the Kinetic Theory of Gases, Chapter VIII, "Reversibility and Irreversibility.")