1998/08/13 by James P. Crutchfield, Cosma Rohilla Shalizi, Crutchfield, James P. +1
Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #adap-org #chao-dyn #cond-mat.stat-mech #nlin.AO #nlin.CD
paper · pdf · doi:10.48550/arxiv.cond-mat/9808147
11 pages, 9 figures, RevTeX
arxiv created 1998/08/13 · arxiv updated 2009/11/30
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 follow-up 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 reverse-time 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---ε-machines---and show that this representation gives the minimum dive consistent with accurate prediction. Thus, ε-machines are optimally shallow.