h.jensen@imperial.ac.uk, rpazuki@gmail.com, g.pruessner@imperial.ac.uk, p.tempesta@fis.ucm.es
The volume of phase space may grow super-exponentially (“explosively”) with the number of degrees of freedom for certain types of complex systems such as encountered in biology or neuroscience, where components interact and create new emergent states. Standard ensemble theory can break down for such cases as we show in an example a simple model reminiscent of complex systems where new collective states emerge. We present a rigorously defined entropy which is extensive in the micro canonical, equal probability, ensemble for super-exponentially growing phase spaces. This entropy may be useful in determining probability measures in analogy with how statistical mechanics establishes statistical ensembles by maximising entropy.
Especially evolutionary biologists have for some time discussed the explosive growth of phase space. Stuart Kauffman [1] points out: Because these evolutionary processes typically cannot be pre-stated, the very phase space of biological, economic, cultural, legal evolution keeps changing in unpre-statable ways. In physics, we can always pre-state the phase space, hence can write laws of motion, hence can integrate them to obtain the entailed becoming of the physical system. Similar difficulties are pointed out by Nors Nielsen and Ulanowicz [2] in their discussion of ontic openness, or combinatorial explosion, encountered in developmental biological processes. This suggests that the size of the configuration spaces for complex systems, such as those encountered in biology, grow super-exponentially.
As a stylistic model of such systems we introduce the following “Pairing Model”, which consists of N coins in various configurations. Each coin can either show head or tail, so the available phase space grows like W (N ) = 2N. However, for N > 1 we allow two possibilities. The first is that a coin behaves as when it is isolated and assumes one of the two possible single particle states, namely head or tail. The second possibility is that a coin enters into a paired state with another coin as if they were sticking to each other. The available number of configurations for N coins flows the iterative equation
W(N+1)=2W(N)+NW(N-1)
We explore how problems arise when the standard Gibbs-Boltzmann micro-canonical and canonical ensembles is applied to the Pairing Coins. We then present an axiomatically based entropy which remains extensive on the micro-canonical ensemble and we suggest that through the maximum entropy approach this Group Theoretic entropy may be useful in deriving probabilistic weights for ontic open systems.
[1] Kauffman, S. Children Of Newton And Modernity, vol. https://edge.org/response-detail/23801 (Edge, 2013).
[2] Nielsen, S. N. & Ulanowicz, R. Ontic openness: An absolute necessity for all developmental processes. Ecol. Model. 222, 2908 (2011).