### Non-Commutative Theory of Nonequilibrium Reveals Cantor Triadic Set in a Rich Ensemble of Coalescing Distributions

**Authors:** Jérôme Chauvet

**Comments:** 24 pages. Keywords: nonequilibrium, non-commutativity, chronon, Planck's time,
Cantor set, Poisson process, coalescence, nuclear magnetic resonance

Mathematics of non-commutative spaces is a rapidly growing research field, which has to
date found convincing proof of its legitimacy in the nature, precisely, in quantum systems. In
this paper, I evaluate the extension of fundamental non-commutativity to the theory of
chemical equilibrium in reactions, of which little is known about its phenomenological
implication. To do so, I assume time to be fundamentally discrete, with time values taken at
integer multiples of a time quantum, or chronon. By integrating chemical ordinary differential
equations (ODE) over the latter, two non-commutative maps are derived. The first map allows
excluding some hypothetical link between chemical Poisson process and uncertainty due to
non-commutativity, while the second map shows that, in first-order reversible schemes, orbits
generate a rich collection of non-equilibrium statistics, some of which have their support close
to the Cantor triadic set, a feature never reported for the Poisson process alone. This study
points out the need for upgrading the current chemical reaction theory with
noncommutativity-dependent properties.

**Category:** Biochemistry