Has the definition of "statistical mechanics" changed from its original meaning?

Question

I guess this never registered with me when I read the Feynman Lectures on Physics in the past. But I have wondered, from time to time, what distinguishes statistical mechanics from, say, kinetic theory.

Like most people, I assumed the distinction was that statistical mechanics is a probabilistic treatment of thermal physics. Thermodynamics deals with heat at a macroscopic level using concepts such as "heat flow" and refinements of the ideal gas law, etc. So that's different from kinetic theory which deals with concepts of actual molecular motion. To my mind, I always thought of statistical mechanics filling in the the middle part between kinetic theory and thermodynamics.

But, notice what Feynman says statistical mechanics is:

We have discussed some of the properties of large numbers of intercolliding atoms. The subject is called kinetic theory, a description of matter from the point of view of collisions between the atoms. Fundamentally, we assert that the gross properties of matter should be explainable in terms of the motion of its parts.

We limit ourselves for the present to conditions of thermal equilibrium, that is, to a subclass of all the phenomena of nature. The laws of mechanics which apply just to thermal equilibrium are called statistical mechanics, and in this section we want to become acquainted with some of the central theorems of this subject.

http://www.feynmanlectures.caltech.edu/I_40.html

Considering Feynman held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology, I tend to take his characterization of statistical mechanics seriously. That is, considering Tolman wrote "the book" on statistical mechanics: The Principles of Statistical Mechanics, By: Richard C. Tolman

So what is the proper definition of "statistical mechanics"?

Answer 1

I would say that the definition of statistical mechanics is "the application of probability theory to physical systems with a large number of degrees of freedom".

Now, historically we can say that the first time statistical mechanics appeared was in 1859, when Maxwell published his paper on the distribution of molecular velocities (even tough there were some precursors, like Daniel Bernoulli's Hydrodynamica). Successively, Boltzmann and Gibbs, following slightly different approaches, laid the theoretical foundations of the field.

Note that in this period the notion that matter was actually made of small entities called "atoms" or "molecules" was not accepted by the entire scientific community, and many eminent physicists, like Ernst Mach and Wilhelm Ostwald, openly opposed atomic theory. It was not until the early 1900s, most notably after the works of Einstein on Brownian motion and the subsequent experiments of Jean Baptiste Perrin, that people started to accept the existence of atoms/molecules as a fact. For an interesting account of the early experimental test on atomic theory, I would suggest the book Atoms, by Jean Perrin, which is freely available online here.

In the early days, statistical mechanics was not known as "statistical mechanics". Since the first works on what we would today call statistical mechanics were on the physics of gases, it was more widely known as "kinetic theory of gases" or simply "gas theory" (as in Boltzmann's Lectures on Gas Theory, Vorlesungen über Gastheorie in German). Therefore, from an historical point of view the kinetic theory of gases is the precursor, or the "old name" if you want, of statistical mechanics.

If we have to believe what is reported on Wikipedia, the term "statistical mechanics" was used for the first time by Gibbs in its 1906 paper "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics". Taking a look at the google n-gram for kinetic theory, statistical mechanics and statistical physics you can see that the "kinetic theory" starts to be used around 1870 (remember, the paper by Maxwell on the distribution of molecular velocity was published in 1859) while the term "statistical mechanics" starts to be used around 1900 (as also "statistical physics", which for some reason is much less common).

From a modern point of view, we would rather say that the kinetic theory of gases is a subset of statistical mechanics. Feynman is probably referring to the fact that usually in statistical mechanics we deal with equilibrium phenomena, while kinetic theory sometimes deals with non-equilibrium phenomena (see the H theorem for example). However, we also have the field of non-equilibrium statistical mechanics, which includes the non-equilibrium part of kinetic theory.

To sum up: from an historical point of view, kinetic theory is the precursor, or the "old name" of statistical mechanics. From a modern point of view, it is a subset of statistical mechanic that mainly deals with gases, or fluids in general.

Answer 2

Kinetic Theory was developed by Boltzmann in an attempt to obtain thermodynamics by direct application of the equations of motion. Boltzmann's goal was to describe the evolution of a microstate in time and hopefully show that this evolution leads inescapably to the equilibrium state.

The term Statistical Mechanics was coined by Gibbs with his book Elementary Principles in Statistical Mechanics. It is distinguished from Kinetic Theory in that it does not attempt to follow the evolution of a system in time. Instead, it considers all microstates that are possible under the macroscopic specifications of the system, assign a probability to each of them and calculates thermodynamic properties as an average over all microstates based on the assigned probabilities. This approach turns out to work very well.

To connect Boltzmann's approach to Gibbs's one must invoke ergodicity, a term first coined by Boltzmann and refers to the notion that a microsystem by virtue of the laws of motion alone will eventually visit every possible microstate where it would spend a fraction of time equal to the probability assigned to that microstate by Gibbs. To date the connection between Boltzmann and Gibbs remains elusive.

So what is the definition of statistical mechanics? It is as Feynmann describes it. It is a theory for the equilibrium state of a system of particles that interact via the laws of mechanics, based on the statistical postulate that all possible microscopic configurations contribute to the measurable properties of the macroscopic system according to well defined probabilities.