Off-topic: Yuri Milner, Facebook's Zuckerberg, and Google's Brin launched a Life Sciences counterpart of the Milner Prize, the same money. Because it's about life sciences, the chairman of the foundation is the chairman of Apple.
Ludwig Boltzmann was born on February 20th, 1844, in Vienna, the capital of the Austrian Empire. He hanged himself 62 years later, on September 5th, 1906, near Trieste, (then) also in the Austrian Empire, where he was on vacations with his wife Henriette von Aigentler and a daughter. They had 3 daughters and 2 sons; Boltzmann probably suffered from undiagnosed bipolar disorder.I consider Boltzmann to be not only the #1 person behind classical statistical physics but also the latest "forefather" of quantum mechanics. His name appears in something like 100 TRF blog entries.
Fine. His grandfather was a clock manufacturer while his father Ludwig Georg Boltzmann (who died when his son-physicist was 15) was an IRS official. He got some good teachers such as Loschmidt and Stefan (yes, they share the Stefan-Boltzmann law) and was exposed to the contributions by 19th century giants such as Maxwell rather early on. He collaborated with Kirchhoff and Helmholtz, among others. On the contrary, he was later the adviser to Lise Meitner, Paul Ehrenfest, and many others.
Already when he was 20+ years old, he wrote a dissertation on kinetic theory of gases, the main subject he revolutionized. Graz, proper Austria's second largest city (an important one for Slovenes), was his most successful workplace.
All his major contributions to physics are linked to statistical mechanics – the microscopic "explanation" of the laws of thermodynamics. They include the Boltzmann [transport] equation for a probability distribution on the phase space\[
\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{collisions}
\] but also the Stefan-Boltzmann law for the total radiated energy (scaling as the fourth power of the absolute temperature; \(4\) is indeed generalized to the dimension of the spacetime if you change it)\[
j^{\star} = \sigma T^{4}.
\] and the Boltzmann distribution saying that Nature exponentially suppresses the likelihood that things jump to higher energy levels – the colder the temperature is, the more speedy the suppression becomes:\[
{N_i \over N} = {g_i e^{-E_i/(k_BT)} \over Z(T)}.
\] Most famously, his tomb offers the visitors his statistical interpretation of the entropy:
In this equation, \(S=k\cdot \log W\), and in many other equations, we see Boltzmann's constant \[
k=k_B= 1.38\times 10^{-23}\,{\rm J/K}.
\] This is the ultimate constant to convert between kelvins (temperature) and joules (energy per a degree of freedom such as an atomic one etc.). In other words, it's the conversion factor between statistical physics and thermodynamics; mature physicists set it equal to one, \(k=1\), much like in the case of \(c=\hbar=1\). The numerical value is small because people had only been familiar with the "thermodynamic limit" in which the number of atoms is very large, effectively infinite. In this \(N\to\infty\) limit, the thermal energy per atom becomes comparable to \(kT\) which is numerically about as small as \(k\) itself. A large number of atoms (or photons) form a "continuum" and the energy is smooth.
Let me mention that \(W\) in Boltzmann's tomb trademark equation looks simple but it stands for a rather complicated word, Wahrscheinlichkeit. That simply means "probability" although a more correct explanation is "frequency of occurrence of a macrostate: how many microstates correspond to it" (the natural probability of one particular microstate is the inverse of this number, so its logarithm is the same thing as the logarithm of the number with a minus sign).
You may also wonder why the entropy is denoted \(S\). Well, I can tell you something. The symbol as well as the word "entropy" were introduced by Clausius in 1865. The word "entropy" was deliberately chosen to be similar to "energy". The word "ἐνέργεια" i.e. "energeia" means "activity" or "operation" in Greek; similarly, "τροπη" i.e. "trope" is a transformation. The symbol \(S\) wasn't explained but given the fact that the front page of a major related article by Clausius mentioned Sadi Carnot as the ultimate guru in the theory of heat (he was in the first wave of proponents of entropy), it's likely that \(S\) actually stands for "Sadi".
Ludwig Boltzmann mastered lots of the combinatorial exercises we often learn in the context of the classical statistical physics (factorials to calculate the number of arrangements, and so on). But he has spent many years with efforts to prove the second law of thermodynamics. A particular proof of this sort was the proof of the H-theorem in 1872.
Controversies about the provability of the second law
Even though Boltzmann's proof of the H-theorem is obviously right and is a template to prove the second law of thermodynamics in any microscopic theory or any formalism (including quantum mechanics), it has been attacked by irrational criticisms from the very beginning. Poor Boltzmann spent a lot of energy and created lots of entropy by the defense of his important insight and I am sure that the crackpots who criticized him contributed to his decision to commit suicide.
The so-called Loschmidt irreversibility paradox is named after Boltzmann's former teacher, Johann Joseph Loschmidt (born in Carlsbad, Czech lands), and it is a deep misunderstanding of the origin of the second law of thermodynamics (and the arrow of time). The basic logic behind this "paradox" is that the microscopic laws are time-reversal-symmetric (or at least CPT-symmetric, if you discuss a generic Lorentz-invariant quantum field theory; the impact of the CPT symmetry is almost identical). So it shouldn't be possible to derive time-reversal-asymmetric conclusions such as the second law of thermodynamics (entropy is increasing with time but not decreasing with time).
Well, if you state it in this way, it looks tautologically impossible to prove the second law. However, what this argument completely misses is the fact that the second law of thermodynamics is a statement about statistical or probabilistic quantities such as the entropy. And to derive such statements, we actually need more than just the "microscopic dynamical laws" of physics; we also need the probability calculus. The probability calculus applied to statements about events in time is intrinsically time-reversal-asymmetric. And this asymmetry, the "logical arrow of time", is imprinted to time-reversal asymmetries in other contexts.
For example, Boltzmann's H-theorem proves that the thermodynamic arrow of time (the direction of time in which the entropy increases) is inevitably and provably aligned with the logical arrow of time.
Technically, the critics were attacking the assumption of molecular chaos and were claiming that it was the key assumption that made Boltzmann's proof vacuous or invalid or whatever they would say. But this is a complete misunderstanding of this assumption. Molecular chaos is just a technical assumption about the velocity of the gas molecules etc. – they're uncorrelated etc. in the initial state – which allows one to calculate certain things analytically.
But even if we introduced arbitrary correlations or other features of the initial probability distribution, it would still be guaranteed – up to negligible, exponentially supertiny probabilities – that the entropy is actually going to increase with time! If you want the entropy to decrease with time, it is not enough to start with an initial state that contains correlations between velocities. You need to start with some totally unnatural correlations between the positions and velocities that are exponentially unlikely and that happen to evolve in a way that reduces entropy for some time. The only way to "calculate" what the correlations required to start a decline of the entropy are is to actually evolve a desired low-entropy final state backwards in time. But this can't occur naturally.
The very fact that the entropy increases in the real world has nothing whatsoever to do with any details of the molecular chaos assumption. The actual reason why the second law – a maximally time-reversal-asymmetric fact about Nature – holds is that it is a claim of probabilistic character. And probability calculus has an inevitable, intrinsic, logical arrow of time. When you calculate the probabilities for a transition between macrostates \(A\to B\), you need to sum the probabilities over the possible final microstates \(B_j\), but you need to average (not sum) over the possible initial microstates \(A_i\). Think about the origin of this summing and averaging. They are unavoidable consequences of "pure logic". For the final (mutually exclusive) microstates, the probabilities are simply summed; for the initial states, the fixed "total prior probability" must be divided among many microstates.
Averaging and summing are different things. As explained in dozens of TRF posts, this difference is reflected by an extra factor of \(1/N_A\sim \exp(-S_A)\) and guarantees that\[
\frac{P(A\to B)}{P(B^*\to A^*)} \sim \exp[(S_B-S_A)/k].
\] The asterisks represent the time-reversal- or CPT-transformed states. (The signs of velocities/momenta etc. are reverted.)
Because the entropy difference \(S_B-S_A\) is typically incomparably larger than \(k\), Boltzmann's constant, and because the probabilities can't exceed one, it's clear that the right hand side above is either zero or infinity in the thermodynamic limit and only one of the probabilities from the numerator or denominator (the probability of the process where the entropy increases) may be nonzero. This is the actual reason why the second law of thermodynamics holds.
Glimpses of the quantum, Bohrian thinking about the world
Why was it so hard for the people to understand these things? And why it's so hard to some physicists – such as Brian Greene or Sean Carroll – even today, more than 100 after these discoveries were made? Well, I think that the reason is the same as the reason why certain physicists (including the two I have mentioned) can't understand the foundations of quantum mechanics. In fact, they face almost the same problem. Let me explain why.
Niels Bohr is the author of a quote that's been mentioned on this blog many times:
There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature...I have mentioned that this is the key psychological obstacle for many people in the context of quantum mechanics. They're permanently looking for a classical model. There is an objective reality, they believe, described by the values of some mathematical objects (functions): the mathematical functions and the objective reality are isomorphic to each other and observations are only passive reflections of this underlying reality.
As quoted in "The philosophy of Niels Bohr" by Aage Petersen, in the Bulletin of the Atomic Scientists Vol. 19, No. 7 (September 1963); The Genius of Science: A Portrait Gallery (2000) by Abraham Pais, p. 24, and Niels Bohr: Reflections on Subject and Object (2001) by Paul McEvoy, p. 291
In quantum mechanics, things are different. The fundamental things are propositions that we can make about observables (properties of physical systems) at various moments. The laws of quantum mechanics relate the truth value (and, more generally, probability) of these propositions directly and there is no way to reduce them to a classical model or objective reality in between. Although all of us laughed when we were kids and we were told about philosophers who question the existence of objective reality, it's nevertheless true that at the fundamental level, objective reality doesn't exist. It is just an emergent, approximate concept.
But what I haven't emphasized sufficiently often is that Bohr's quote actually applies to classical statistical physics as well. Many people misunderstand the proposition-based character of physics in this context which leads them to invent lots of irrational criticisms against classical statistical physics, too. What's going on?
The critics are still imagining that Nature is found in a particular, deterministically evolving microstate, and they try to evaluate the second of law of thermodynamics from this viewpoint. In classical physics, it is tolerable to imagine that there is a particular, deterministically evolving microstate behind all the information about Nature; in quantum physics, it isn't tolerable, as Bohr's quote above pointed out.
However, even in classical statistical physics, the existence of such a particular, deterministically evolving microstate is completely irrelevant for the validity of the second law of thermodynamics. Why? Because the second law of thermodynamics isn't a statement about a particular microstate in the phase space (or Hilbert space) at all! It is an intrinsically statistical statement about a collection of such microstates or about a probability distribution and its evolution.
Let me give you an example. The second law says, for example, that...
...if you place a hot bowl of soup on a cold table and measure the soup-table temperature difference 20 minutes later, it's almost guaranteed that you will get a smaller number than you obtained at the beginning.Let's analyze it a little bit. The first thing to notice is that the sentence above is a proposition, not an object. It is a probabilistic proposition and quantitatively speaking, it is true because we may show that for the proposition to be wrong, the entropy would have to decrease but the probability of such a process is exponentially tiny, something like \(p\sim \exp(-10^{26})\). The smallness of this number is what we mean by "almost guaranteed".
Fine. How it is possible that such a time-reveral-asymmetric proposition follows from the time-reversal-symmetric microscopic dynamical laws in physics? To see the answer, we must realize that the proposition deals with macroscopic objects such as a hot bowl of soup and table. It's important to notice that these phrases don't represent any particular microstates – precise arrangements of atoms. It's very important to acknowledge that these phrases represent macrostates – statistical mixtures of microstates that look macroscopically (almost) indistinguishable. That's true for the soup in the initial state as well as the soup in the final state.
It is a technical detail whether these statistical mixtures are "uniform" (the same probability for all of the microstates) or not; that's just the difference between microcanonical and canonical or grand canonical ensembles. These mixtures don't have to be uniform. What matters is that there are many microstates that have comparably large probabilities to be realizations of the concepts such as a hot bowl of soup. That's why the proposition "soup will cool down" above is a proposition about a transition between an initial state and a final state. And it is a proposition that is appropriately "statistically averaged or summed" over the microstates.
As I have emphasized above (and in dozens of older TRF blog entries), the right way to statistically treat such combined propositions about many microstates or macrostates is to sum over the possible final microstates, but average over the possible initial microstates (with weights identified with some "prior probabilities" that depend on other subjective choices and knowledge, but you won't lose much if you assume that all initial microstates in a set are equally likely). When you sum-and-average these transition probabilities for microstates of the soup+table, you will get a result that is totally time-reversal-asymmetric. The entropy increases because the summing/averaging asymmetry favors a larger number of "fellow microstates" for the final state and a lower number of "fellow microstates" for the initial state.
It's the mathematical logic, pure probability calculus applied to propositions about things that occur at various moments of time, that is the source of the time-reversal asymmetry. One doesn't need any time-reversal asymmetry of the microscopic laws. Indeed, these laws are time-reversal-symmetric (or at least CPT-invariant in the case of quantum field theories but the CPT-invariance plays the same role).
There is nothing paradoxical about the validity of the claim "soup will cool down". Every sane person knows that it's true. And Nature doesn't need any time-reversal-asymmetric terms in the microscopic equations of motion for the elementary particles to cool down the damn soup! It just cools down because of basic statistical considerations.
Things would be different if we made a statement about a particular microstate of the hot bowl of soup. Would a particular microstate of soup (in contact with a cold table) evolve into a microstate that looks like hotter soup or colder soup? Now, this question isn't completely well-defined. You would have to tell me what microstate you are actually asking about. And by this comment, I really mean that you would have to tell me the \(10^{26}\) positions and velocities of elementary particles in the soup with amazing precision that I need to make the prediction.
No one ever does that in the real world and it isn't really needed because we know that with insanely unlikely exceptions, whatever the initial state of the soup is, the soup will just cool down within a second, after a minute, it will always cool down. A macroscopic decrease of the entropy is virtually impossible. It will never happen anywhere in the Universe during its lifetime (except for possibly super long timescales such as the Poincaré recurrence time).
What about the unlikely exceptions? Yes, a tiny fraction of the states, \(1/\exp[(S_B-S_A)/k]\) of them, will evolve in a way that decreases the entropy. But these exceptional states can't be isolated by any natural condition for their velocities that would only look what their values are "now" (in the initial state). The only way to define these special states is to say that these are the states that will just happen to evolve into low-entropy final states.
Indeed, if you define the initial state of the soup in this way (it is a microstate that evolves into a lower-entropy final state by the microscopic equations of motion), the right statement about its evolution will be that it will evolve into a lower-entropy final state. But this proposition will be an uninteresting tautology. What the actual second law of thermodynamics is concerned with is something entirely different: realizable initial states of hot soup and the phrase "soup" that is simply defined as a statistical mixture of these microstates which forces you to use statistical and probabilistic methods to evaluate the probabilities and truth values of propositions!
The critics are also entirely wrong that the same comments would apply to the final state. It is not true that the final microstate state of the soup is "generic". Instead, among the equally high-entropy states, it is an extremely special microstate because it has evolved from a lower-entropy initial state and there are just "few" of these low-entropy initial microstates. We are told that it has evolved; it is a part of the homework exercise we were supposed to solve!
When we discuss the evolution of a bowl of soup on the table, it would be totally incorrect to think that "bowl of soup" in the final state denotes an equal statistical mixture of all conceivable similar high-entropy microstates of the soup and the table. Indeed, the very formulation of the problem says that the bowl of soup was sitting on the table so the final soup did evolve from an initial state, and it must therefore be a special microstate.
The key difference between the initial state and the final state is that it is legitimate to assume that all the allowed microstates in the initial state are comparably likely; but it is not legitimate to assume that all the macroscopically similar final microstates are equally likely. The latter claim about the final state is illegitimate simply because we aren't allowed to choose the probabilities of the final microstates; they are – by the very definition of the adjective "final" or the noun "future" – determined from the probability distributions in the initial state and the properties of the initial state in general. The future evolves from the past, not vice versa!
On the other hand, the initial state evolved from some states at even earlier instants of time and the formulation of the problem says nothing about those. That's why it's allowed to organize our knowledge about the initial state as a statistical mixture of microstates of a certain kind – where all the microstates are comparably represented. That's right because the only thing that we know about the initial state is that it is a hot bowl of soup etc.; this state doesn't have any special "micro" properties. The final microstate does have some special "micro" properties (correlations) because – as the description of the very problem says – this final state evolved from an initial state that was also a soup. We can't revert this statement and say that the initial state evolved from the final state because – by definition of the words "initial" and "final", it just hasn't.
As you can see, the critics reach wrong conclusions because they're sloppy about what we know and what we do not know. But it's a part of their philosophy to be sloppy about "what we know" because they think that the knowledge is an irrelevant spiritual subjectivist solipsist stuff that has nothing to do with physics. So they behave as if they knew the exact microstate. But that's a complete fallacy. They do not know the exact microstate and if they assume that they do, and that the initial state is very special, they inevitably reach wrong conclusions that are easily falsified by observations. Statistical claims about the soup or any objects in thermodynamics are all about our knowledge and ignorance. It's very important to distinguish what we know and what we do not know and what we partially (probabilistically) know and what the probabilities are.
Classical statistical physics is about the careful derivations of true (or extremely likely) claims about systems with many degrees of freedom out of some other true (or extremely likely) assumptions that we were told to be valid. It is all about propositions. Quantum mechanics has upgraded this principle to a new level because it became impossible – even in principle – to assume that there is a particular "objective reality" (microstate at each moment) at all. Nevertheless, the feature – that physics is about making right propositions, not about mindlessly visualizing a "model of the precise thing" – was already present in classical statistical physics because even classical statistical physics tells us that interesting statements about Nature (or soup) are statements encoding partial and probabilistic knowledge of some features of Nature (or soup) and we should be very interested in how these statements are related to each other.
So the deluded folks who helped to drive Ludwig Boltzmann to suicide were direct predecessors of the contemporary anti-quantum zealots in the same sense in which Ludwig Boltzmann himself was a forefather of modern physics, a Niels Bohr prototype.
And that's the memo.
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