  We define the entropy S and uncertainty function of a squeezed system
interacting with a thermal bath, and study how they change in time by following
the evolution of the reduced density matrix in the influence functional
formalism. As examples, we calculate the entropy of two exactly solvable
squeezed systems: an inverted harmonic oscillator and a scalar field mode
evolving in an inflationary universe. For the inverted oscillator with weak
coupling to the bath, at both high and low temperatures, $S\to r $, where r is
the squeeze parameter. In the de Sitter case, at high temperatures, $S\to
(1-c)r$ where $c = \gamma_0/H$, $\gamma_0$ being the coupling to the bath and H
the Hubble constant. These three cases confirm previous results based on more
ad hoc prescriptions for calculating entropy. But at low temperatures, the de
Sitter entropy $S\to (1/2-c)r$ is noticeably different. This result, obtained
from a more rigorous approach, shows that factors usually ignored by the
conventional approaches, i.e., the nature of the environment and the coupling
strength betwen the system and the environment, are important.
