## Thursday, February 21, 2019

### "Boltzmann vs foes": precursor to "QM vs anti-quantum zealots"

Yesterday, Ludwig Eduard Boltzmann would have had a chance to celebrate his 175th birthday if he hadn't killed that chance by hanging himself at age of 62, while vacationing with his wife Henriette and daughter Elsa (in Tybein) near Trieste, Northern Italy, Austria-Hungary.

The wife and daughter probably had a reason for some anxiety when they found him (the daughter found him first). But Boltzmann's reason powering the suicide were intellectually driven frustrations. And while it's sometimes said that the timing of his suicide was lousy because his ideas were going to win soon afterwards, I actually disagree.

If he were resurrected and if he were around, he would probably ask me whether there's a reasonable chance that the people will get more reasonable when it comes to the ideas required for his new statistical picture of thermodynamics and physics in general. I would probably answer "No" and he would hang himself again.

I could feel a bit guilty but my "No" answer would be a matter of scientific integrity because the anti-quantum zealots are nothing else than heirs to the high-profile idiots who opposed his ideas more than 100 years ago.

By the high-profile idiots, I mean primarily philosopher Auguste Comte, mathematician Henri Poincaré, chemist Wilhelm Ostwald, mathematician Georg Helm, engineer (and armchair physicist) William Rankine (who nevertheless gave the modern meaning to "energy"), and especially philosopher Ernst Mach.

To make things worse, I think that an important man who contributed to Boltzmann's suicide was also the chemist (Johann) Josef Loschmidt, an older colleague (and sometimes instructor) of Boltzmann's at the University of Vienna – despite the fact that the two men became friends. Loschmidt shared many of the misunderstandings with the list in the previous paragraph. Note that Loschmidt used to call himself "Austrian" but if he had lived a few decades later, such a German-speaking guy born in Carlsbad, Bohemia (who naturally went to work at a university in Prague) would be classified as a "Sudetenland German" and he would become a Czechoslovak citizen and probably a voter of Adolf Hitler's regional deputy Konrad Henlein. ;-)

OK, the reason why I believe that nothing much has changed about the stupidity of the philosophers and the bulk of armchair physicists since Boltzmann's years – truly technical fields in quantum physics are fortunately a different matter – and why I recommended Boltzmann to commit suicide again is that his ideas were a precursor to the new philosophy of physics brought by quantum mechanics which
• redirected the focus of physics from "pictures of reality" to "true propositions one can make about reality"
• changed the main Ansatz of the findings from "what is true" to "how likely is one evolution or another" (probabilistic character of physics)
• introduced the logical arrow of time to the laws of physics, despite the apparent mathematical time-reversal (or CPT-) symmetry of the microscopic laws of physics (this is a consequence of the previous two points)
You know, Boltzmann ended his life in 1906 but the changes in the "way of thinking needed to understand the world in terms of physics" were heavily overlapping with the changes that quantum mechanics required – and lots of people still have incredible and basically insurmountable psychological problems with these changes as of 2019.

OK, let's simplify his biography. Boltzmann was born in Vienna in 1844 – his paternal grandfather was a clock producer from Berlin and his father was a tax bureaucrat; his mother was born in Salzburg. His father died when Boltzmann was 15. Boltzmann did his PhD under Josef Stefan already in 1866 – at age of 22 – but although the Stefan-Boltzmann names could indicate otherwise, it was about the kinetic theory of gases, not about their radiation constant that Stefan formulated later in 1879 and Boltzmann partially derived in 1884.

Also in Vienna, Johann Strauss composed The Blue Danube in 1866, the same year when Boltzmann earned his PhD. It was first performed in 1867, the year of Austro-Hungarian Compromise.

A famous paper by Maxwell – who started to explain the thermal effects of gases by the atomic motion – was the first scientific paper that young Boltzmann read. But he got much further. He understood that entropy, a quantity that was only used in the continuous thermodynamic sense, had a statistical origin, as his tombstone reveals:$S = k\cdot \log W.$ Note that to get the quantity $$S$$ from phenomenological thermodynamics, whose units are "joules per kelvin" today (because it's a ratio of some energy and temperature), he needed to introduce a coefficient $$k$$ (with the same units), a universal constant of Nature that we call the Boltzmann constant for obvious reasons. Wonderfully enough, the new revised SI system of units says that $$k= 1.38064852\,{\rm J/K}$$ precisely.

(Note that in classical physics, $$W$$ was basically assumed to be a volume of a region in the phase space – which seems to be dimensionful. Boltzmann would have surely agreed it would be prettier if the argument of the logarithm were dimensionless. And indeed, that's what quantum mechanics does – $$W$$ is $$N$$, the number of microstates, and every "phase space cell" of volume $$(2\pi \hbar)^M$$ where $$M$$ is the number of coordinate-momentum pairs i.e. half of the number of coordinates on the phase space – that exponent is dictated by dimensional arguments – is basically assigned one microstate. That's another reason to believe that Boltzmann, who was able to discover his tombstone formula despite the need to exploit the ugly logarithm of a dimensionful expression, would love and quickly understand quantum mechanics if he were around in 1925. Maybe he would have discovered it earlier.)

On top of that, he derived the second law of thermodynamics (entropy isn't decreasing) from statistical physics. The proof is almost a mathematical one and is referred to as the H-theorem. Also, he derived a rather complex Boltzmann equation that allows you to explain the origin of apparently "time-reversal asymmetric" terms in equations of physics (such as those in viscosity and resistance) from statistical considerations.
The book on the left side offers you 282 pages of selected writings by Boltzmann – mostly texts for the public. Aside from essays on the obvious physical topics, you may enjoy e.g. a reply to Ostwald who wanted to interpret energy as mental happiness, an interpretation that wasn't quite equivalent to Rankine's physical energy, as Boltzmann argued. ;-)
An example. When solid objects move on some surface, there is friction. That is a word for a force $$F=ma$$ whose direction is proportional to the velocity (or at least has the same direction). That's surprising because the fundamental laws look like $$F=ma$$ – they include the position itself or its second derivatives. There are no terms with first derivatives in the fundamental equations – so how can the friction arise? But it can and Boltzmann not only understood why but also discovered (the classical edition) of the general mathematical framework that allows you to derive how these irreversible, time-reversal-asymmetric terms such as friction, diffusion, resistance, viscosity etc. arise.

Boltzmann also coined the adjective "ergodic", from Greek words "ergos" and "odos" (work, path). In equilibrium systems, the averages over time tend to be the same as averages over the phase space, and when it's so, we say that the behavior is "ergodic".

There are lots of equations and constants that most people who are close enough learned to "endorse" or parrot. But Boltzmann's Wikipedia page also quotes one seemingly "very philosophical" idea that Boltzmann is famous for, namely epistemological idealism. The first sentence of that Wikipedia page summarizes it crisply enough:
Epistemological idealism is a subjectivist position in epistemology that holds that what one knows about an object exists only in one's mind. It is opposed to epistemological realism.
The subsequent sentences explain that all our perceptions and knowledge are fundamentally of "mental nature". Lots of 19th century physicists, philosophers – and as I said, lots of would-be smart people in 2019 as well – have a fundamental problem with all these things. They exhibit knee-jerk negative reactions to anything "subjective", "mental" etc. They think that proper physics must be "materialist", "objective" etc. and they claim that any reduction of this basic axiom turns physics into a social science or a superstitious pseudoscience.

But it's not true at all. Epistemological idealism is absolutely needed to understand the statistical origin of the thermodynamic phenomena – and it's an unavoidable pillar of all of quantum mechanics, too. There's nothing "anthropomorphic" and nothing "religious" about this Boltzmannian-and-quantum kind of idealism. Instead, the new focus of physics is a method to distinguish "true and false propositions" or, more quantitatively, to determine the probability that a proposition about the objects around is true. So the idealism isn't "social"; instead, it is "mathematical" because the validity of propositions is what a mathematician is interested in.

With this basic change of perspective, we realize that the goal of physics isn't to create "pictures" of some "objective reality". Instead, physics means to assume that "some reality exists" but the existence is only objective in the very broad sense. Whenever we discuss "details about the reality", we need to discuss them in terms of "propositions about observables that we can make", and all these propositions must be treated just like propositions in an axiomatic system of mathematics that may be either proved or disproved (in physics, we may calculate their probabilities).

Indeed, a physicist becomes more analogous to a mathematician or a "spokesperson" who "speaks" about the reality, instead of a cook or an engineer who "directly holds it". But there's nothing wrong about a physicist's acting as a mathematician.

This basic paradigm shift has some technical consequences. And the irreversibility – and arrows of time – are the most easily "named" ones among these consequences. As I discussed many times, Boltzmann's picture means that when we calculate the probability that an incompletely specified state $$A$$ in the past (or "an ensemble" of states) evolves into the incompletely specified state $$B$$ in the future (or "another ensemble"), we need to sum the microstate-to-microstate probabilities over the states in $$B$$, but we need to average these microstate-to-microstate probabilities over the microstates in $$A$$ (either with uniform or non-uniform weights).

If we assume the equal weights, to simplify things a bit (it's often enough), the averaging adds the extra factor of $$1/N_{\rm initial}$$, the inverse number of initial microstates, from the definition of the arithmetic average. Because of this inverse proportionality, probabilities are large if the number of initial microstates is small – or smaller than $$N_{\rm final}$$, a seemingly analogous factor whose $$1/N_{\rm final}$$ is not included. And that's why the pure logic implies that $$N_{\rm initial} \lt N_{\rm final}$$ is favored if not mandatory, often with the sign $$\ll$$. And that's just a different way of saying that the entropy is increasing – we derived it from simple logic.

As you can see, the past and the future are treated completely differently and asymmetrically and there's nothing wrong about it. In fact, if you think about it, the summing-vs-averaging of the probabilities must follow these exact rules because they're a matter of the basic mathematical common sense. The probabilities over the final microstates are summed because we don't care what exact microstate from the ensemble appears as the final state; but they have to be averaged over the initial microstates because we don't know what is the right one.

The past is treated "intensively" because we know for sure that the total probability of all mutually exclusive initial microstates is 100%. This 100% "pie" has to be divided to different options, and that's why the probabilities have to be averaged over the options. On the other hand, while the sum of the probabilities of the mutually exclusive final microstates is also 100%, most of the possibilities are immediately excluded by the choice of the initial state and the "sum equals 100%" becomes pretty much vacuous. That's why the future or the final state is treated "extensively" – the probability of "B1 or B2" is simply the sum of the probabilities for "B1" and "B2" separately.

We can assume that the initial microstate is "generic" and not a special one that would lead to some special properties of the final state; but we can't assume the converse, that the final microstate is "generic", because by the defining properties of the past and future, the existence of the "special patterns" in the future state depends on the precise choice of the past state, but not vice versa.

You can psychologically deal with the asymmetric status of the past and the future in any way you want but what's important and independent of "philosophical or pedagogic preferences" is that this past-future asymmetry is
• completely compatible with the time-reversal-symmetric or CPT-symmetric microscopic laws of physics at the microstate level
• totally needed to explain thermodynamics as well as all macroscopic phenomena in terms of the fundamental laws, especially in quantum mechanics that can't work without the calculation of probabilities
Whether you want to imagine that Boltzmann has added "new axioms" (like some genericity of the initial state) or he has derived things like the H-theorem from "old axioms" isn't too clear and important. (What I recommend you is to assume that the physics toolkit needed to describe the reality is composed of the formula to calculate the probabilities of the $$A_i\to B_j$$ evolution among all pairs of the microstates; plus some probabilistic assumptions about the microstates themselves, and the fundamental ones are always those about the past $$A_i$$ because there is no teleology in science. And the second half was basically "discovered" by Boltzmann.) You may answer these questions (and assign the credit) according to your taste. But what's essential is that he has shown that this "idealist" way of thinking works to explain thermodynamics etc. And we really know that these derivations cannot work without this "idealist" thinking of Boltzmann's – or without the idealist thinking of quantum mechanics.

If you focus on some "objective pictures of the spacetime" that follow some time-reversal symmetric, reversible phenomena, and if this focus prevents you from considering any time-reversal-asymmetric or irreversible laws, then you are just wrong and incapable of understanding most of physics since the late 19th century and almost all physics since 1925.

Loschmidt had a serious problem to "believe" that irreversible phenomena for macroscopic objects could be compatible with the reversible laws for the microstates. And that's why we often talk about the "Loschmidt paradox" that is named after him. But all the high-profile names of Boltzmann's critics that I enumerated at the top had a problem with the "epistemological idealism" and its consequences. Some of them also had more non-philosophical problems with the whole atomic hypothesis. In particular, Mach – who lived up to 1916, accidentally the year when the general theory of relativity was published – was a life-long critic of the atomic hypothesis.

Mach suffered from cardiac arrest in 1898 and his life after that point was a combination of hospitals and Parliamentary politics. But even in 1898, the opposition to a basic paradigm such as the atomic hypothesis was pretty bad, I think. Physically wiser people – like Max Planck – obviously criticized Mach for those criticisms. We might say it's unsurprising that Mach couldn't get the whole depth of Boltzmann's ideas if he had a serious problem even with the very existence of the atoms.

But Boltzmann wasn't one big step ahead of the likes of Mach. He was at least two huge leaps ahead of them. He not only understood why the atomic hypothesis was needed but he also understood epistemological idealism and its essential character – and how to combine the atomic hypothesis and epistemological idealism to derive equations for macroscopic systems in physics!

As of 2019, the explicit opponents of the atomic hypothesis are rare, thank God. But you still hear lots of people – sometimes presenting themselves as intelligent people – who have a serious problem with epistemological idealism and all of its mathematically expressed consequences. I am talking about the statistical physics crackpots such as Sean Carroll as well as all the anti-quantum zealots. If someone considers the "epistemological idealism" as such a heresy that he couldn't think along its lines, it is very clear that such a person can properly understand nothing whatsoever about the relationship between the fundamental laws of physics on one side and the macroscopic phenomena on the other side!

It's sad that idiots like that are still so numerous in 2019 and they actually enjoy much of the limelight – and Boltzmann made a pretty intelligent decision to commit suicide in 1906.

And that's the memo.