  The physical foundations of a variety of emerging technologies --- ranging
from the applications of quantum entanglement in quantum information to the
applications of nonequilibrium bulk and interface phenomena in microfluidics,
biology, materials science, energy engineering, etc. --- require understanding
thermodynamic entropy beyond the equilibrium realm of its traditional
definition. This paper presents a rigorous logical scheme that provides a
generalized definition of entropy free of the usual unnecessary assumptions
which constrain the theory to the equilibrium domain. The scheme is based on
carefully worded operative definitions for all the fundamental concepts
employed, including those of system, property, state, isolated system,
environment, process, separable system, system uncorrelated from its
environment, and parameters of a system. The treatment considers also systems
with movable internal walls and/or semipermeable walls, with chemical reactions
and/or external force fields, and with small numbers of particles. The
definition of reversible process is revised by introducing the new concept of
scenario. The definition of entropy involves neither the concept of heat nor
that of quasistatic process; it applies to both equilibrium and nonequilibrium
states. The role of correlations on the domain of definition and on the
additivity of energy and entropy is discussed: it is proved that energy is
defined and additive for all separable systems, while entropy is defined and
additive only for separable systems uncorrelated from their environment;
decorrelation entropy is defined. The definitions of energy and entropy are
extended rigorously to open systems. Finally, to complete the discussion, the
existence of the fundamental relation for stable equilibrium states is proved,
in our context, for both closed and open systems.
