The Boltzmann brain is a reductio ad absurdum response to Ludwig Boltzmann's early explanation for the low-entropy state of our universe.
In this physics thought experiment, a Boltzmann brain is a fully formed brain, complete with memories of a full human life in our universe, that arises due to extremely rare random fluctuations out of a state of thermodynamic equilibrium. Theoretically over a period of time on the order of hundreds of billions of years, by sheer chance atoms in a void could spontaneously come together in such a way as to assemble a functioning human brain. As with any brain in such circumstances, it would almost immediately stop functioning and begin to deteriorate.
The idea is ironically named after the Austrian physicist Ludwig Boltzmann (1844-1906), who in 1896 published a theory that tried to account for the fact that we find ourselves in a universe that is not as chaotic as the budding field of thermodynamics seemed to predict. He offered several explanations, one of them being that the universe, even one that is fully random (or at thermal equilibrium), would spontaneously fluctuate to a more ordered (or low-entropy) state. One criticism of this "Boltzmann universe" hypothesis is that the most common thermal fluctuations are as close to equilibrium overall as possible; thus, by any reasonable criterion, actual humans in the actual universe would be vastly less likely than "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 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. When applied to more recent theories about the multiverse, Boltzmann brain arguments are part of the unsolved measure problem of cosmology.