How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems
Abstract
The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there has been an ongoing controversy over whether the notion of the maximum entropy principle can be extended in a meaningful way to nonextensive, nonergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann–Gibbs–Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon–Khinchin axioms, the Graphic-entropies. We demonstrate that the MEP is a perfectly consistent concept for nonergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is to our knowledge the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.
Additional Information
© 2014 National Academy of Sciences. Freely available online through the PNAS open access option. Contributed by Murray Gell-Mann, April 4, 2014 (sent for review January 30, 2014) R.H. and S.T. thank the Santa Fe Institute for hospitality. M.G.-M. acknowledges the generous support of Insight Venture Partners and the Bryan J. and June B. Zwan Foundation. Author contributions: R.H., S.T., and M.G.-M. designed research, performed research, contributed new reagents/analytic tools, and wrote the paper. The authors declare no conflict of interest.Attached Files
Published - 6905.full.pdf
Files
Name | Size | Download all |
---|---|---|
md5:7d00be04c8bcc663e9a488decc51076b
|
702.3 kB | Preview Download |
Additional details
- PMCID
- PMC4024900
- Eprint ID
- 59836
- Resolver ID
- CaltechAUTHORS:20150824-094800526
- Insight Venture Partners
- Bryan J. and June B. Zwan Foundation
- Created
-
2015-08-24Created from EPrint's datestamp field
- Updated
-
2021-11-10Created from EPrint's last_modified field