Statistical Physics
Maxwell and Boltzmann
I started reading Abraham Pais’ 1982 book, Subtle is the Lord: the science and the life of Albert Einstein last week. I never took a university level course in relativity or quantum physics, but I did study some undergraduate math and physics, and right off the bat, some of the concepts are not exactly intuitive, over 100 years later.
I also failed thermodynamics three times, however. When I was a student I thought this was due to my lack of discipline and focus (it was), but I once again have enough time on my hands to make a concerted effort to explore the weird philosophical implications of this topic, even if it defeats me again.
The chapters about Einstein’s life are enlightening and easy to read, and while the “science” chapters are also enlightening, they are challenging. These chapter titles aren’t italicized in the table of contents, and they constitute the majority of the book’s contents. I’ve been collecting some notes here as I’ve been reading. It‘s time-consuming, but I don’t think it makes sense to speed through this stuff;— slowing down seems to help.
There is a monument on Ludwig Boltzmann’s grave, in the Central Cemetery of Vienna, which contains the following formula:
\[\begin{equation} S = k \; log \, W \end{equation}\]Boltzmann never wrote the equation in this form, where \(S\) is the Entropy of a closed system at thermodynamic equilibrium, and \(W\) is the combination of distinct microscopic states available to the system (which he called complexions). In fact, this form of the equation, as well as the constant \(k\), which carries his name, was first used by Max Planck in his formula to describe the spectral density of Blackbody Radiation:
\[\begin{equation} \rho(\nu, T) = \frac{8 \pi h \nu^3}{c^3} \frac{1}{e^{h \nu / k T} - 1} \end{equation}\]The version of this equation that Boltzmann introduced in his 1872 paper also contains what Pais calls the \(H\) theorem. He defined a quantity called \(H\) in terms of the velocity distribution of particles in a system, with the property \(dH/dT \leq 0\), so that \(H\) can be identified with Entropy. In an important 1877 paper which describes the Second Law of Thermodynamics, Boltzmann proved that universal increase in Entropy (decrease in \(H\)) cannot be an absolute law, but a matter of overwhelming probability.
This principle was as sacred to Einstein as the law of Conservation of Energy.
One of Einstein’s insights was to reverse the direction of Boltzmann’s relation between \(S\) and \(W\). Instead of using the probabilities of the microscopic state of the system to determine an expression for Entropy, he suggested using the Entropy to deduce the probability. In other words, instead of “Boltzmann logic” where \(W \rightarrow S\), he introduced “Einstein logic”, where \(S \rightarrow W\).
He also observed that “neither Herr Boltzmann nor Herr Planck has given a definition of \(W\).” While Boltzmann (and everyone else) considered \(W\) as the originating concept: the number of different ways of organizing the particles in a system,— usually an ideal gas, Einstein’s genius allowed him to extend the idea of quantization from ideal gases to light,— by reversing the direction of causality.
Other notes
Boltzmann’s constant \(k\) is one of the seven ”defining constants“ in the SI system, and since 2011 is no longer experimentally determined, but instead defined with a value of exactly \(1.380649×10^{−23} \; J/K\).
Boltzmann provided two definitions of thermodynamic probability for a system of N particles. The first, which Einstein accepted, was based on the evolution in time of the particles circulating on a 6N-dimensional surface of constant energy, later called the \(\Gamma\)-space. The second is a counting definition — the combinations of all possible energy levels for these particles.
Pais wonders about Einstein’s reluctance to accept the status quo on the counting problem, and other concepts widely accepted in quantum physics. For example, he says that Einstein had subtle ways of expressing his dissatisfaction with the conventions of quantum physics, for example, in his references to the Wave Function \(\Psi\) — in German he always used the mathematical term die Psifunktion instead of the more conventional die Wellenfunktion.
Frank Lambert calls the ”disorder” metaphor to describe Entropy a “cracked crutch.”
Next: The Reality of Molecules