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!