De Sitter Thermodynamics in the canonical ensemble

Abstract

The existing thermodynamics of the cosmological horizon in de Sitter spacetime is established in the micro-canonical ensemble, while the thermodynamics of black hole horizons is 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, which has negligible mass 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 so as 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 in the sense that it preserves the "thermodynamic consistency", which means that the state variables satisfy not only the four laws of thermodynamics but also the appropriate differential relations of state variables with thermodynamic functions; e.g. partial derivatives of the free energy yield the entropy, pressure, and so on. The special role of cosmological constant in de Sitter thermodynamics is also revealed.