Philosophy of Statistical Mechanics?

Collapsed pillars of modern building.

Let us continue with the post Three Elephants of Modern Physics taking a closer look at one of them. 

We learn from Stanford Encyclopedia of Philosophy the following about Statistical Mechanics SM: 

• Statistical Mechanics is the third pillar of modern physics, next to quantum theory and relativity theory. 
• Its aim is to account for the macroscopic behaviour of physical systems in terms of dynamical laws governing the microscopic constituents of these systems and probabilistic assumptions.
• Philosophical discussions in statistical mechanics face an immediate difficulty because unlike other theories, statistical mechanics has not yet found a generally accepted theoretical framework or a canonical formalism. 
• For this reason, a review of the philosophy of SM cannot simply start with a statement of the theory’s basic principles and then move on to different interpretations of the theory.

This is not a very good start, but we continue learning: 

• Three broad theoretical umbrellas: “Boltzmannian SM” (BSM), “Boltzmann Equation” (BE), and “Gibbsian SM” (GSM).
• BSM enjoys great popularity in foundational debates due to its clear and intuitive theoretical structure. Nevertheless, BSM faces a number of problems and limitations
• There is no way around recognising that BSM is mostly used in foundational debates, but it is GSM that is the practitioner’s workhorse.
• So what we’re facing is a schism whereby the day-to-day work of physicists is in one framework and foundational accounts and explanations are given in another framework.
• This would not be worrisome if the frameworks were equivalent, or at least inter-translatable in relatively clear way...this is not the case.
• The crucial conceptual questions (concerning BE) at this point are: what exactly did Boltzmann prove with the H-theorem?

This is the status today of the third pillar of modern physics formed by Boltzmann 1866-1906 and Gibbs 1902 as still being without a generally accepted theoretical framework, despite 120 years of deep thinking by the sharpest brains of modern physics. 

Is this something to worry about? If one the pillars apparently is shaky, what about the remaining two pillars? Who cares?

Recall that SM was introduced to rationalise the 2nd Law of Thermodynamics stating irreversibility of macroscopic systems based on deterministic reversible exact microscopics. This challenge was taken up by Boltzmann facing the question: If all components of a system are reversible, how can it be that the system is irreversible? From where does the irreversibility come? The only way forward Boltzmann could find was to replace exact determinism of microscopics by randomness/statistics as a form of inexactness. 

 

In the modern digital world the inexactness can take the form of finite precision computation performed with a certain number of digits (e g single or double precision). Here the microscopics is deterministic up the point of keeping only a finite number of digits, which can have more or less severe consequences on  macroscopic reversibility. This idea is explored in Computational Thermodynamics offering a 2nd Law expressed in the physical quantities of kinetic energy, internal energy, work and turbulent dissipation without need to introduce any concept of entropy.

Replacing SM by precisely defined finite precision computation gives a more solid third pillar. But this is new and not easily embraced by analytical/theoretical mathematicians/physicists not used to think in terms of computation, with Stephen Wolfram as notable exception.  

PS1 To meet criticism that the Stosszahlansatz underlying the H-theorem stating that particles before collision are uncorrelated, simply assumes what has to proved (irreversibility), Boltzmann argued:

• But since this consideration has, apart from its tediousness, not the slightest difficulty, nor any special interest, and because the result is so simple that one might almost say it is self-evident I will only state this result.

Convincing?

PS2 Connecting to the previous post, recall that the era of quantum mechanics was initiated in 1900 by Planck introducing statistics of "energy quanta" inspired by Boltzmann's statistical mechanics, to explain observed atomic radiation spectra, opening the door to Born's statistical interpretation in 1927 of the Schrödinger wave function as the "probability of finding an electron" at some specific location in space and time, which is the text book wisdom still today. Thus the pillar of quantum mechanics is also weakened by statistics. The third pillar of relativity is free of statistics, but also of physics, and so altogether the three pillars offer a shaky foundation of modern physics.  Convinced?