Gibbs and afterwards Boltzmann and Professor Planck have put forward some very interesting ideas on this subject.
By following the route they have traced, we come to consider the principle as pointing out to us that a given system tends towards the configuration presented by the maximum probability, and, numerically, the entropy would even be the logarithm of this probability.

Thus two different gaseous masses, enclosed in two separate receptacles which have just been placed in communication, diffuse themselves one through the other, and it is highly improbable that, in their mutual shocks, both kinds of molecules should take a distribution of velocities which reduce them by a spontaneous phenomenon to the initial state.
We should have to wait a very long time for so extraordinary a concourse of circumstances, but, in strictness, it would not be impossible.

The principle would only be a law of probability.

Yet this probability is all the greater the more considerable is the number of molecules itself.

In the phenomena habitually dealt with, this number is such that, practically, the variation of entropy in a constant sense takes, so to speak, the character of absolute certainty.
But there may be exceptional cases where the complexity of the system becomes insufficient for the application of the principle of Carnot;-- as in the case of the curious movements of small particles suspended in a liquid which are known by the name of Brownian movements and can be observed under the microscope.