de Sitter thermodynamics in the canonical ensemble
  The existing thermodynamics of the cosmological horizon in de-Sitter
spacetime is established in the micro-canonical ensemble, while thermodynamics
of black hole horizons are established in the canonical ensemble. Generally in
the ordinary thermodynamics and statistical mechanics, both of the
micro-canonical and canonical ensembles yield the same equation of state for
any thermodynamic system. This implies the existence of a formulation of
de-Sitter thermodynamics based on the canonical ensemble. This paper reproduces
the de-Sitter thermodynamics in the canonical ensemble. The procedure is as
follows: We put a spherical wall at the center of de-Sitter spacetime, whose
mass is negligible and perfectly reflects the Hawking radiation coming from the
cosmological horizon. Then the region enclosed by the wall and horizon settles
down to a thermal equilibrium state, for which the Euclidean action is
evaluated and the partition function is obtained. The integration constant
(subtraction term) of Euclidean action is determined to reproduce the equation
of state (e.g. entropy-area law) verified already in the micro-canonical
ensemble. Our de-Sitter canonical ensemble is well-defined to preserve the
"thermodynamic consistency", which means that the state variables satisfy not
only the four laws of thermodynamics but also the appropriate differential
relations with thermodynamic functions; e.g. partial derivatives of the free
energy give the entropy, pressure, and so on. The special role of cosmological
constant in de-Sitter thermodynamics is also revealed.
