Extropy

Date: Sat, 20 Jul 1996 14:37:16 -0700 (PDT)
From: Joel Henkel <jhenkel@juno.com>
To: quantum-d@teleport.com
Subject: Extropy

Here is a new notion I've come up with. It is a mirror image of entropy,
a mechanical concept, called extropy, a meta-mechanical concept. It
measures the complexity of an open system...

I. Definition of Extropy

From the Greek ex- out of and -tropy turn: The degree of diversity of
an ensemble of systems; the number of different kinds of systems. The
number of different kinds of degree of freedom in an ensemble of systems.

The new notion can be contrasted to entropy, from the Greek en- into
and -tropy turn:  In the statistical distribution within a single system,
entropy is the smoothness or evenness of the probability of possible
states of the system.

Entropy is defined mechanically, for a collection of the possible states
within a closed mechanical system. For example, the distribution of
molecular kinetic energy for a thermodynamic gas system in the kinetic
theory of gases. Extropy is defined in a larger context, the ensemble of
many different kinds of mechanical system. In this sense, it is a
meta-mechanical concept.

II. Hilbert Space Transformations and Tropy

In the description of a mechanical system in Hilbert or state space, all
system states are accessible to each other through unitary transformations.
The system is said to span its Hilbert space. In the description of a
meta-mechanical ensemble of mechanical systems, system states within
one mechanical system are inaccessible to states of another system. The
ensemble of systems divides Hilbert space into separate partitions, one
for each mechanical system. The partitions are connected only by non-
unitary transformations.

Entropy is defined only within a single partition of Hilbert space,
since it describes the distribution of probability within the possible
states of a single mechanical system.

Extropy is defined over the whole partitioned Hilbert space, since it
is a measure of the number of partitions in the space.

III. Extropy and the Evolution of Living, Open Systems

Mechanical systems are described by the time evolution of their states
through an equation of motion. This equation of motion is constrained
to operate on an isolated, or closed  mechanical system, since any
change of boundary conditions requires interaction of a system with
its environment, making it an open system. Change of boundary condition
is equivalent to considering a new mechanical system, creating an ensemble
of different mechanical systems. Hence, the notion of entropy is
constrained to closed systems. Extropy is able to describe the evolution
of open systems and their possible increase in complexity. Living
organisms are open systems, continuously interacting with their
environment.  In this sense, they are meta-mechanical and cannot be
described completely by equations of motion. Evolution for a species,
and growth for an individual, describes the change of a species or the
development of an organism's structure with irreversible (non-mechanical,
Bergsonian) time.

Joel