Title: Entropy? Honest!,
Abstract: Here we deconstruct, and then in a reasoned way reconstruct, the concept of
"entropy of a system," paying particular attention to where the randomness may
be coming from. We start with the core concept of entropy as a COUNT associated
with a DESCRIPTION; this count (traditionally expressed in logarithmic form for
a number of good reasons) is in essence the number of possibilities---specific
instances or "scenarios," that MATCH that description. Very natural (and
virtually inescapable) generalizations of the idea of description are the
probability distribution and of its quantum mechanical counterpart, the density
operator.
We track the process of dynamically updating entropy as a system evolves.
Three factors may cause entropy to change: (1) the system's INTERNAL DYNAMICS;
(2) unsolicited EXTERNAL INFLUENCES on it; and (3) the approximations one has
to make when one tries to predict the system's future state. The latter task is
usually hampered by hard-to-quantify aspects of the original description,
limited data storage and processing resource, and possibly algorithmic
inadequacy. Factors 2 and 3 introduce randomness into one's predictions and
accordingly degrade them. When forecasting, as long as the entropy bookkeping
is conducted in an HONEST fashion, this degradation will ALWAYS lead to an
entropy increase.
To clarify the above point we introduce the notion of HONEST ENTROPY, which
coalesces much of what is of course already done, often tacitly, in responsible
entropy-bookkeping practice. This notion, we believe, will help to fill an
expressivity gap in scientific discourse. With its help we shall prove that ANY
dynamical system---not just our physical universe---strictly obeys Clausius's
original formulation of the second law of thermodynamics IF AND ONLY IF it is
invertible. Thus this law is a TAUTOLOGICAL PROPERTY of invertible systems!