  Unlike typical condensed-matter systems, ultra-cold atoms loaded into optical
lattices allow separate control of both the particle number and system size. As
a consequence, there are two distinct "thermodynamic" limits that can be
defined for these systems: i) "infinite-volume limit" at constant finite
density, and ii) "empty-lattice limit" at constant particle number. To probe
the difference between these two limits and their crossover, we consider a
partially occupied lattice and study the transport of non-interacting fermions
and fermions interacting at the mean-field level into the unoccupied region. In
the infinite-volume limit, a finite steady-state current emerges. On the other
hand, in the empty-lattice limit there is no finite steady-state current. By
changing the initial filling, we find a smooth crossover between the two
limits. Our predictions may be verified using available experimental tools and
demonstrate a fundamental difference between isolated small systems such as
ultra-cold atoms and conventional condensed-matter systems.
