For some time, I have been wondering whether the Boltzmann brain paradox is a genuine one. Here’s how Wikipedia describes the paradox (emphases are mine):
In physics thought experiments, a Boltzmann brain is a self-aware entity that arises due to extremely rare random fluctuations out of a state of thermodynamic equilibrium. For example, in a homogeneous Newtonian soup, theoretically by sheer chance all the atoms could bounce off and stick to one another in such a way as to assemble a functioning human brain (though this would, on average, take vastly longer than the current lifetime of the universe).
The idea is indirectly named after the Austrian physicist Ludwig Boltzmann (1844–1906), who in 1896 published a theory that the Universe is observed to be in a highly improbable non-equilibrium state because only when such states randomly occur can brains exist to be aware of the Universe. One criticism of Boltzmann’s “Boltzmann universe” hypothesis is that the most common thermal fluctuations are as close to equilibrium overall as possible; thus, by any reasonable criterion, human brains in a Boltzmann universe with myriad neighboring stars would be vastly outnumbered by “Boltzmann brains” existing alone in an empty universe.
Boltzmann brains gained new relevance around 2002, when some cosmologists started to become concerned that, in many existing theories about the Universe, human brains in the current Universe appear to be vastly outnumbered by Boltzmann brains in the future Universe who, by chance, have exactly the same perceptions that we do; this leads to the absurd conclusion that statistically we ourselves are likely to be Boltzmann brains. Such a reductio ad absurdum argument is sometimes used to argue against certain theories of the Universe.
The two assumptions I’d like to question in this post are: (i) the assumption that a Boltzmann brain could self-assemble in the first place, given enough time; and (ii) the assumption that the spontaneous self-assembly of such a brain is more likely than the spontaneous formation of human observers.
Concerning (i), I would like to point out that the brain is a multi-layered organ containing 86 billion neurons, each of which is supported by glial cells and astrocytes. Within each neuron is a nucleus, surrounded by a cell body which receives signals via dendrites and sends signals along an axon, which in turn is insulated with a myelin sheath. The idea that individual atoms could arrange themselves into a brain containing tens of billions of these multi-layered structures is simply absurd, because such a “brain” would fall apart long before it formed. Without stable intermediate stages, a brain cannot spontaneously self-assemble out of atoms, even when given an infinite amount of time.
As for (ii): it seems to me that the main argument in its favor is that thermodynamically speaking, it’s easier for an infinite box of gas in equilibrium to assemble into X than it is for the gas to assemble into X + Y. Hence the spontaneous formation of a brain alone is far more likely than the self-assembly of a brain attached to a human body, which in turn is more likely to form spontaneously than an entire planet full of organisms (including humans), which is more likely to form spontaneously than a universe full of stars and planets (including our Earth). But this kind of reasoning ignores the possibility of “springboards,” or transitional stages which generate multiple paths, some of which lead to the desired target.
Thus the likelihood of a brain forming from a box of gas is far greater if the gas is first allowed to assemble into a primordial cell, as such a cell is capable of evolving into various kinds of organisms (including humans with brains). But evolution can only proceed in a physical environment; hence we need a planet for the cell and its descendants to live on. But a planet needs a source of energy to power the chemical reactions leading to life, which in turn requires energy to power its own internal processes; hence we need a star as well. And the formation of stars is rendered more likely by the postulation of an initial Big Bang, starting from a hot, dense state. In short: while the spontaneous formation of a universe, holus bolus, is many orders of magnitude less likely than the spontaneous formation of a brain, the formation of a human being possessing a brain within the framework of a Big Bang universe which is capable of generating stars and planets, some of which are capable of generating life, which subsequently evolves into the dazzling array of creatures we see on Earth, is far more likely than the spontaneous self-assembly of a Boltzmann brain – an event whose probability I have already argued is zero.
I’d like to finish with a closing thought. Is there something wrong with the way we define entropy? Currently, physicists define it in terms of the number of possible micro-states of a system which are consistent with its macro-description (e.g. “a human brain”). This, in essence, is what Boltzmann’s entropy formula expresses. However, such a definition ignores a system’s history, and makes no attempt to calculate the number of pathways by which the atoms in the system can reach their macro-level target. Boltzmann’s definition of entropy also leads many well-meaning but misinformed laypeople to mount fallacious arguments against biological evolution. Isn’t it about time we came up with a better definition, which doesn’t generate silly paradoxes? What do readers think?