Thermodynamic Depth of Causal States: When Paddling around in Occam's Pool Shallowness Is a Virtue
Abstract
Thermodynamic depth is an appealing but flawed structural complexity measure. It depends on a set of macroscopic states for a system, but neither its original introduction by Lloyd and Pagels nor any followup work has considered how to select these states. Depth, therefore, is at root arbitrary. Computational mechanics, an alternative approach to structural complexity, provides a definition for a system's minimal, necessary causal states and a procedure for finding them. We show that the rate of increase in thermodynamic depth, or {\it dive}, is the system's reversetime Shannon entropy rate, and so depth only measures degrees of macroscopic randomness, not structure. To fix this we redefine the depth in terms of the causal state representation$\epsilon$machinesand show that this representation gives the minimum dive consistent with accurate prediction. Thus, $\epsilon$machines are optimally shallow.
 Publication:

arXiv eprints
 Pub Date:
 August 1998
 DOI:
 10.48550/arXiv.condmat/9808147
 arXiv:
 arXiv:condmat/9808147
 Bibcode:
 1998cond.mat..8147C
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Adaptation;
 Noise;
 and SelfOrganizing Systems;
 Chaotic Dynamics;
 Nonlinear Sciences  Adaptation and SelfOrganizing Systems;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 11 pages, 9 figures, RevTeX