The thermodynamic efficiency of the Lorenz system

Abstract

We study the thermodynamic efficiency of the Malkus-Lorenz waterwheel. For this purpose, we derive an exact analytical formula that describes the efficiency of this dissipative structure as a function of the phase space variables and the constant parameters of the dynamical system. We show that, generally, as the machine is progressively driven far from thermodynamic equilibrium by increasing its uptake of matter from the environment, it also tends to increase its efficiency. However, sudden drops in the efficiency are found at critical bifurcation points leading to chaotic dynamics. We relate these discontinuous crises in the efficiency to a reduction of the attractor's average value projected along the phase space dimensions that contribute to the rate of entropy generation in the system. In this manner, we provide a thermodynamic criterion that, presumably, governs the evolution of far-from-equilibrium dissipative systems towards their self-assembly and synchronization into increasingly complex networks and structures.

Orbit quantization in a retarded harmonic oscillator

“…1(b). In the following section we show that, as we push the system even further from thermodynamic equilibrium [30] by increasing the effects of the time-delay feedback, the oscillator undergoes further bifurcation phenomena producing a second quantized excited orbit, at a higher energy level.…”

“…Of course, this is only possible at the expense of an energy input in the system, which must come from external field sources [13,24]. Therefore, the present dynamical system must be regarded as as a non-equilibrium open physical system, whose nonlinear periodic motion can be interpreted as a cyclic thermodynamic engine [30]. Due to the existence of energy losses, these dynamical systems are frequently named dissipative structures.…”

The Electrodynamic Origin of the Wave-Particle Duality

“…In particular, it has been claimed that the quantum potential can produce the symmetry breaking of the Lorentz group. Importantly, we recall that symmetry breaking is an essential feature in the study of nonlinear dynamics [12].…”