We’ve reached a problematic nexus.
I think I’ll just quote Darrigol (page 220 of his
Exegetical book on Boltzmann) and then go from there:
The seductive simplicity of the combinatorial entropy tricked Boltzmann into regarding it as mostly independent of the dynamics of the system and as qualitatively different from his earlier statistico-mechanical definitions of the entropy. Although he later realized that this was not the case, many of his readers and followers from Planck – who first wrote S = k log W – to modern textbook writers, have placed the combinatorial entropy at the forefront of statistical mechanics.
In this bit of exegetical work, Darrigol is covering some of the same problematic area that he first went over in his 1992 From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory, which is really a very sympathetic treatment of Bohr, but starts with puzzling over Planck’s problems with Boltzmann and ends with Dirac’s problems with Dirac.
And what is the problem here? Or there?
As we will see, there is a problem for Planck and Zermelo since they don’t find the discrete constructions of statistico-mechanical definitions of the entropy particularly plausible – they wanted an answer in terms of continuous functions and aethereal energies if possible. Historically, as Darrigol notes, Planck’s eventual restatement of Boltzmann’s combinatorial entropy has been enshrined even on Boltzmann’s grave. The problems with combinatorial entropy don’t seem to be the kind of thing that people like Planck or Zermelo wanted to deal with, they just want it accepted as it was and without going into its limitations. After all, combinatorial entropy comes along with an irreversible arrow of time – which combinatorial entropy gives you but statistico-mechanical definitions of the entropy do not quite exactly since nothing guarantees statistico-mechanical irreversibility and its little friend the Arrow of Time except pure chance.
Essentially, combinatorial entropy closes off events that are outside the boxes constructed by probabilities within probabilities in its journey to maximum entropy and complete equilibrium. While in a sense it is probably true that there is a vast, even infinite, range of things that probably don’t happen very often if at all ever, that doesn’t mean the probabilistic construction that manipulates the boundaries of what can happen is necessarily always a good way to address a problem or a simulation or discrete objects in a construction. The statistico-mechanical definitions of the entropy make the edges of the parameterized boxes involved in defining entropy much clearer. You can see, for example, that the idea of any overall entropy for very large systems needs some arbitrary specifications and/or boundaries and all the more so for large regions interpreted in terms of General Relativity. I doubt that my science blog will ever go into such things as simulating causal sets in terms of General Relativity with discretized quantum gravity, but when you stick to the statistico-mechanical definitions of the entropy, such things are possible at least in simulations. So we will join Boltzmann and Darrigol and avoid Planck and Zermelo for the moment and go into the details as Boltzmann covered them in dealing with Planck and Zermelo.
Another problem with the statistico-mechanical definitions of the entropy seems to be about what to do with the “arrow of time.” Does time really need an arrow at all? As far as I can tell time only becomes worthy of an arrow if it isn’t a coordinate built into spacetime, for example as in Special and General Relativity. Time seems to work fine without an arrow in SR and GR ( you can foliate a consistent time measure outward from any event – of course, your time in passing probably won’t match that foliation, but you can compare different foliations if you want to do some relative arrows of time) so that would seem to suggest that things like the “Arrow of Time” and the “Heat Death of the Universe” are problems with the fidgeting ghosts of the late nineteenth-century aethers more than anything else.
But, as we will see, such fidgeting was inherent in the aether/continuum-based approaches of Planck and Zermelo and Boltzmann barely managed to get them roughly on course, though they happily re-interpreted his versions of entropy once he was dead.