  The maximum particle kinetic energy that can be extracted from an initial
six-dimensional phase space distribution motivates the concept of free or
available energy. The free energy depends on the allowed operations that can be
performed. A key concept underlying the theoretical treatment of plasmas is the
Gardner free energy, where the exchange of the contents of equal phase volumes
is allowed. A second free energy concept is the diffusive free energy, in which
the contents of volumes are instead averaged. For any finite discretization of
phase space, the diffusive free energy is known to be less than the Gardner
free energy. However, despite the apparent fundamental differences between
these free energies, it is demonstrated here that the Gardner free energy may
be recovered from the continuous limit of the diffusive free energy, leading to
the surprise that macroscopic phase-space conservation can be achieved by
ostensibly entropy-producing microscopic operations.
