1999/01/01 by James P. Crutchfield, Cosma Rohilla Shalizi · 2 citations
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Protein Structure and Dynamics #Statistical Mechanics and Entropy #Advanced Thermodynamics and Statistical Mechanics
paper · doi:10.1103/physreve.59.275
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
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 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---\ensuremathε-machines---and show that this representation gives the minimum dive consistent with accurate prediction. Thus, \ensuremathε-machines are optimally shallow.