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Stars form when cold cosmic nebulae spontaneously develop hot spots thatsteadily intensify until they reach fusion temperatures. Without this process,the universe would be dark and dead. Yet the spontaneous concentration of heatis exactly what the Second Law of Thermodynamics is in most cases supposed toforbid. The formation of protostars has been much discussed, for itsconsistency with the Second Law depends on a thermodynamical property that iscommon in systems whose strongest force is their own gravity, but otherwisevery rare: negative specific heat. Negative specific heat turns the worldupside down, thermodynamically; it implies that entropy increases when energyflows from lower to higher energy subsystems, opposite to the usual direction.Recent experiments have reported negative specific heat in melting atomicclusters and fragmenting nuclei, but these arguably represent transientphenomena outside the proper scope of thermodynamics. Here we show that thecounter-intuitive thermodynamics of spontaneous energy concentration can bestudied experimentally with trapped quantum gases, by using optical latticepotentials to realize weakly coupled arrays of simple dynamical subsystems thatshare the peculiar property of self-gravitating protostars, of having negativemicro-canonical specific heat. Numerical solution of real-time evolutionequations confirms the spontaneous concentration of energy in such arrays, withinitially dispersed energy condensing quickly into dense 'droplets'. Wetherefore propose laboratory studies of negative specific heat as an elusivebut fundamentally important aspect of thermodynamics, which may shed freshlight on the general problem of how thermodynamics emerges from mechanics.

An ultracold analogue to star formation: Spontaneous concentration of energy in trapped quantum gases