If you thought the name of this blog comes from me thinking to be a possible reincarnation of Boltzmann, then you were almost but not quite right. Boltzmann’s brain is a serious scientific paradox at the intersection of statistical physics and cosmology with the (still serious) consequence that the Universe could be filled with zombies, including ones having the same brain as Boltzmann. How can this be? Is this paradoxical? And what are the alternative options?
Boltzmann (1844 – 1906) dedicated his life to deriving the laws of thermodynamics from the assumption that the world surrounding us is built out of tiny atoms that obey Newton’s laws of motion—at a time when the very idea of an atom was not even accepted! In particular, he was interested in deriving the second law of thermodynamics, but due to the inbuilt time-reversal symmetry of Newton’s laws of motion, Boltzmann had to eventually admit (not unlikely this was caused by debates with Loschmidt):
The Second Law can never be proved mathematically by means of the equations of dynamics alone.
L. Boltzmann in Nature 51, 413 (1895)
So, if the dynamical laws do not imply the second law, of what else is it a consequence then? This problem haunts physicists and philosophers until today.
Boltzmann’s suggestion
Boltzmann had more than one idea about it, but in the same paper from 1895 he suggests the following:
We assume that the whole universe is, and rests for ever, in thermal equilibrium. The probability that one (only one) part of the universe is in a certain state, is the smaller the further this state is from thermal equilibrium; but this probability is greater, the greater is the universe itself. If we assume the universe great enough, we can make the probability of one relatively small part being in any given state (however far from the state of thermal equilibrium), as great as we please. We can also make the probability great that, though the whole universe is in thermal equilibrium, our world is in its present state.
L. Boltzmann in Nature 51, 413 (1895)
In different words: Imagine the universe to be a giant box filled with particles that are in thermal equilibrium. Then, if the box is large enough (or if we wait long enough), we will at some point observe a fluctuation away from equilibrium that is sufficient to explain our current low entropy state. Oddly enough, Boltzmann attributes this idea to his “old assistant, Dr. Schuetz”. Perhaps because he is already suspecting some strong counterarguments against this idea?
Side note: If you are a historian of science, please let me know whether there is any evidence for the existence of this “Dr. Schuetz”!
The paradox
Here comes the problem. A box containing a single brain having the impression of living in a low entropy Universe (with its rich biology, societies, solar systems, galaxies, etc.) has a much higher entropy than a box actually containing this low entropy Universe (with all its biology, societies, solar systems, galaxies, etc.). Conversely, the probability that the box spontaneously fluctuates from thermal equilibrium into a state containing the actual low entropy state is much smaller than the probability that the box spontaneously creates just a single brain—a “Boltzmann brain”—with the impression of this Universe. And with “much smaller” I mean much much much smaller.[for estimates ask Roger Penrose] This is illustrated in the figure below.
Put differently, since humanity started to exist, we have looked deeper and deeper into the sky, and discovered more and more stars, planets, and eventually galaxies. All those stars, planets and galaxies are out of equilibrium and, therefore, they are in a very atypical low entropy state. Why are we continuing to discover low entropy objects out there? If the Universe just locally fluctuated away from its global thermal equilibrium state, we should expect to see only thermal noise beyond what we have seen already.
All this sounds quite discomforting, in particular because it implies that it is much more likely that you are a Boltzmann brain that fluctuates “back” into thermal equilibrium and ceases to exist within the next seco
OK, jokes aside. What are the alternatives?
Alternative (1): The standard Answer
The standard answer is to claim that the Universe started in an extremely low entropy state, and in some sense this initial low entropy state is synonymous with the big bang.
At first sight this looks also paradoxical because the state shortly after the big bang was an extremely dense, hot and homogeneous plasma (whose reminiscence is known as the cosmic microwave background), i.e., exactly something which qualifies as thermal equilibrium according to conventional statistical mechanics. But this is only true if we exclude gravity from the picture. If we include gravity in the calculation, then the state after the big bang is indeed very far from equilibrium, and it’s entropy is much smaller compared to the entropy it could have if all the energy would collapse into a black hole. Luckily, gravity is weak, and the gravitational degrees of freedom only slowly accumulate entropy such that structure, including us, could form in the galaxy (and, luckily, we are still far away from a hypothetical maximum entropy state of the Universe).
Does this sound better than Boltzmann’s idea? To be honest, I have doubts. After all, this approach simply postulates by fiat an extremely unlikely initial condition without explaining it. What was there before? While one can simply accept this as a fact (after all, differential equations require the specification of boundary conditions to be solved), there is no observable difference between postulating this peculiar ad hoc initial condition or assuming that it was created by a fluctuation. Indeed, the latter option has an attractive property: it is compatible with a time reversal symmetric and eternal Universe.
I admit, this conclusion was reached by picturing the Universe as a giant box filled with particles. This is inaccurate at best (and completely wrong at worst). So, cosmologist might know better reasons to accept alternative (1) and reject Boltzmann’s (Schuetz’?) proposal. But this actually directly leads to…
Alternative (2): Who cares about entropy?
This sounds provocative, but it is a fact that nobody has yet presented any viable definition of entropy for the entire Universe (not even verbally). Why? Because everything we know about thermodynamics relies on the ability to put a kind of box around the entire setup, to regularize it if you want, or more mathematically speaking to approximate it with a finite dimensional Hilbert space.
Indeed, imagine a bunch of particles in free space—without any box! These particles would never equilibrate or thermalize, but just fly around forever. In contrast, once you put a box around it, the particles can come to thermal equilibrium, and entropy has a well defined maximum that it can reach and maintain for a long time (provided there are enough particles).
But to the best of our knowledge, we can not put any box around the Universe. Many cosmologists believe that it is truly infinite. And even it were finite, we currently observe space to stretch at an accelerating rate, suggesting that we perhaps never reach the equilibration time because the expansion of the Universe happens faster than its content equilibrates. Even a more down to earth problem haunts physicists until today: how to microscopically define entropy for gravitating systems? There is this beautiful Bekenstein-Hawking formula for black holes, but what it microscopically means remains a subject of research since 50 years, not to mention the problem of defining gravitational entropy for systems different from black holes.
But are there any other options besides entropy? There are, but I cannot cover them here and now. Suffice it to mention that it is indeed possible to identify an arrow of time even within time reversal symmetric Newtonian mechanics, as demonstrated, e.g., in a beautiful paper by Barbour, Koslowski and Mercati [1].[References below]
Alternative (3): First person first
Perhaps the conundrum and absurdity of Boltzmann’s brain also teaches us that we have to fundamentally switch perspective. After all, we have no physical theory of what an observer really “feels” or “experiences” we can start from. Perhaps, then, the question why there is any world with observers in the first place (may they be “ordinary” or Boltzmann brains) requires an explanation not within this world.
Interestingly, this idealistic or solipsistic philosophical perspective (observer first, external world second) has been recently formulated as a rigorous mathematical theory by Markus Müller [2]. It is based on two main assumptions:
- observer states can be identified with binary strings x (such as x = 010001001110),
- the probability that the observer state transitions to the state yx (with y another binary string, e.g., y = 111) is determined by algorithmic probability.
Stay with me! Algorithmic probability sounds like a technical monster[it is], but its basic idea is that higher probabilities are given to observer states y that can be generated by simpler or shorter computer programs running on input x. For instance, consider Kepler having collected all these data about the planets, which we encode as a binary string and identify with Kepler’s observer state. Then, what is the probability that in the next years the planets will follow Kepler’s law? Algorithmically, this probability is very high because these data can be generated with a very short computer program, namely Newton’s laws. Thus, in some sense algorithmic probability (also known as universal probability or Solomonoff induction) mathematically formalizes the idea behind Occam’s razor.
So, what follows from these two assumptions above for “our world out there”? Well, surprisingly, the theory first predicts that observers will find themselves in worlds that follows simple “natural laws” based on simple initial conditions. No big bang or low entropy fluctuation is needed for that.
Second, since observer states are the fundamental ontological objects of this theory, it does not even make sense to ask “am I a Boltzmann brain in a simulation or am I really a part of a low entropy Universe?” Both “I”s are just the same! However, it makes sense for you to speculate about the nature of other observers out there. But then, surprisingly, one finds that the probability for a “third person” Boltzmann brain is very small. The reason is, simply speaking, that it is algorithmically very complicated to generate a Boltzmann brain by chance in contrast to generating an ordinary brain within your world (which, remember, follows simple natural laws).
To conclude, Markus Müller’s approach almost inverts the problem: instead of predicting extremely high probabilities for Boltzmann brains (as in the standard account given initially), it predicts very low probabilities. Poor Boltzmann brains…
Summary
Boltzmann’s brain is a serious scientific paradox at the intersection of statistical physics and cosmology that can not yet be considered as solved. This is perhaps unsurprising given its relation to the possible deepest mystery of the Universe: how can it be that there actually “is” something? what does it mean to be an observer?
Note that Boltzmann’s brain is a purely classical paradox. The quantum mechanical many worlds interpretation would merely add the possibility of multiple simultaneously existing Boltzmann brains that fluctuate into existence from nowhere (having perhaps the sensation of very different universes). In some sense, my colleagues and me have recently simulated exactly that using Schrödinger’s equation [3], but I spare you the details for another post.
Thus, Boltzmann’s brain continues to haunt physicists for good reasons. But don’t worry. According to this paradox it is also likely that all your enemies just exist in your imagination and decay to dust and ashes tomorrow. If that is not a reason to celebrate…
References
[1] J. Barbour, T. Koslowski, and F. Mercati, Phys. Rev. Lett. 113, 181101 (2014). ArXiv.
[2] M. Müller, Quantum 4, 301 (2020). ArXiv.
[3] P. Strasberg, T.E. Reinhard, and J. Schindler, arXiv 2304.10258 (2023). ArXiv.
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