  Let (M,J) be a dynamical model of macroscopic systems and (N,K) a less
microscopic model (i.e. a model involving less details) of the same macroscopic
systems; M and N are manifolds, J are vector fields on M, and K are vector
fields on N. Let P be the phase portrait corresponding to (M,J) (i.e. P is the
set of all trajectories in M generated by a family of vector fields in J), and
R the phase portrait corresponding to (N,K). Thermodynamics in its general
sense is a pattern recognition process in which R is recognized as a pattern in
P. In particular, the classical (both equilibrium and nonequilibrium)
thermodynamics arises in the investigation of relations between models (M,J)
and models without time evolution, i.e. models with K= 0. In such case R is a
submanifold of M composed of fixed points. Let Su mapping M to R be a
potential, called an upper entropy, generating the vector field J. The
equilibrium thermodynamic relation in N is the lower entropy Sd(y) defined by
Sd(y)=Su(x=y), where x is in M,y in N, and y is given by a final destination
(i.e. when the time goes to infinity) of x in the time evolution generated by
the vector field J. In this paper we show that if K is not zero (e.g. in
externally forced or, in other words, open systems), then the reduction also
provides thermodynamics (we call it flux-thermodynamics). If certain conditions
are satisfied, then the lower entropy Ss, that arises in the investigation of
the approach of J to K, is the time derivative of the lower entropy Sd arising
in the investigation of the approach of M to N as t goes to infinity.
