Complex Time and the Principle of Least Entropic Purpose

The Principle of Least Entropic Purpose is precisely isomorphic to the Principle of Least Action and was established by Parker et al. (Entropy 27, 2025, 131).  “Entropic Purpose” is defined (like Action) as a line integral on the appropriate Lagrangian: in this case the “purposive Lagrangian”, which is defined in complex time, is purely temporal (with no spatial component), and is formally proved valid by demonstrating that it satisfies the appropriate Euler-Lagrange equations. The variational description in entropic (hyperbolic) time is based on the Quantitative Geometrical Thermodynamics formalism, describing the trajectory across the complex temporal plane that a system traverses as it exhibits its Entropic Purpose (measured by how much Shannon Information is created in the process). Trajectories in complex time may not cross (and thereby create never-ending repetitive loops in contravention of the Second Law of Thermodynamics); yet may approach each other to enable quantum tunnelling across trajectories. This work rests on the result that a system’s (complex) Hamiltonian is equivalent to its (complex) entropy production (a result which assumes that time is also complexified). In the trivial reversible case the entropy production is zero for a non-zero Hamiltonian and is associated with double-helical geometries in space-time, whereas dissipative processes are instead marked by logarithmic spirals. Unitary entities (which have no parts at the characteristic length scale: see Entropy 27, 2025, 1119) may be formed by the addition of holomorphic structures, as components of stable (yet often dissipative) systems. Understanding how this applies to simple entities (such as photons, electrons, alpha particles, blackholes), and also more complex systems exhibiting entropic purpose, is a topic of ongoing research. In any case, time must be represented by a complex number in this formulation.