I emailed the above letter on 10/27/09 to Normand Laurendeau, Professor of Combustion at Purdue University, but received no reply. I followed up with another letter dated 2/12/10, also with no reply. Since I thought his error was confined to his textbook, I moved on. I was subsequently informed of Andreas Albrecht’s web site http://albrecht.ucdavis.edu/, where I found his paper entitled “Cosmic Inflation and the Arrow of Time”. It included a concise description of Boltzmann’s rationale, summarized as follows:
The fact that such a large portion of all possible states for the system are associated with equilibrium, means that the special out-of-equilibrium initial states required for the arrow of time are very rare indeed. If one watched a random box of gas it would be in equilibrium almost all the time, a state with no arrow of time.
In the context of his paper, Albrecht’s state is synonymous with the macrostate in common usage in textbooks, so he is making the same mistake as Laurendeau. This aroused my curiosity and I subsequently discovered similar claims in other sources, such as Erwin Schrödinger’s Statistical Thermodynamics. In the fall of 2012, I decided to give Dr. Laurendeau a call, but found that, sadly, he had died a few months before.
Checking the references in the “Source Acknowledgements” section of his textbook, he identified Larry Caretto’s “Course Notes on Statistical Thermodynamics” (1968) as the source for “the evolution of the most probable macrostate”. I was then able to contact Larry Caretto at Cal State Northridge and a stimulating exchange ensued (see Caretto), which traced this idea as far back as Ralph Fowler’s Statistical Mechanics (1929). On page 127 Fowler states:
We may note here that S is closely connected to C, the total number of weighted complexions which represent all possible states of the assembly. It is not difficult to show if required that this total number of weighted complexions does not differ significantly from the number of weighted complexions which represent the approximately average state of the assembly.
On page 139 Fowler is mathematically explicit about the dominance of the most probable macrostate, stating that “Wmax/C or W(average state)/C is always effectively unity, expressing the fact that the possession of the average or most probable state is a normal property of the assembly” (C is equivalent to Laurendeau’s W). He gives Planck, Boltzmann and Gibbs as sources, but at the time I was unable to see where any of them explicitly addressed this point.
This left me at a loss until I came across a recently translated paper by Boltzmann (1877), where he states that “It is clear that every single uniform [micro]state distribution which establishes itself after a certain time given a defined initial [micro]state is equally as probable as every single nonuniform [micro]state distribution, comparable to the situation in the game of Lotto where every single quintet is as improbable as the quintet 12345. The higher probability that the [micro]state distribution becomes uniform with time arises only because there are far more uniform than nonuniform [micro]state distributions”. [my italics] While the term uniform refers to an even distribution of numbers in Lotto, it could also be taken to apply to a uniform distribution of molecules in space. This corresponds to Laurendeau’s statement that “we can actually demonstrate that a large majority of all possible microstates is affiliated with the most probable macrostate.”
He goes on to say that “The initial [macro]state in most cases is bound to be highly improbable and from it the system will always rapidly approach a more probable [macro]state until it finally reaches the most probable [macro]state, i.e., that of the heat equilibrium.” So it is apparent that Laurendeau’s belief that “a large majority of all possible microstates is affiliated with the most probable macrostate” originated with Boltzmann and has been passed down through generations of physicists to the present day.
Revised 6/4/23