  Stars form when cold cosmic nebulae spontaneously develop hot spots that
steadily intensify until they reach fusion temperatures. Without this process,
the universe would be dark and dead. Yet the spontaneous concentration of heat
is exactly what the Second Law of Thermodynamics is in most cases supposed to
forbid. The formation of protostars has been much discussed, for its
consistency with the Second Law depends on a thermodynamical property that is
common in systems whose strongest force is their own gravity, but otherwise
very rare: negative specific heat. Negative specific heat turns the world
upside down, thermodynamically; it implies that entropy increases when energy
flows from lower to higher energy subsystems, opposite to the usual direction.
Recent experiments have reported negative specific heat in melting atomic
clusters and fragmenting nuclei, but these arguably represent transient
phenomena outside the proper scope of thermodynamics. Here we show that the
counter-intuitive thermodynamics of spontaneous energy concentration can be
studied experimentally with trapped quantum gases, by using optical lattice
potentials to realize weakly coupled arrays of simple dynamical subsystems that
share the peculiar property of self-gravitating protostars, of having negative
micro-canonical specific heat. Numerical solution of real-time evolution
equations confirms the spontaneous concentration of energy in such arrays, with
initially dispersed energy condensing quickly into dense 'droplets'. We
therefore propose laboratory studies of negative specific heat as an elusive
but fundamentally important aspect of thermodynamics, which may shed fresh
light on the general problem of how thermodynamics emerges from mechanics.
