It is a riddle and almost a scandal: If you let a particle travel fast through a landscape of randomly moving round troughs – like a frictionless ball sent through a set of circling, softly rounded “teacups” inserted into the floor (to be seated in for a ride at a country fair) – you will find that it loses speed on average.
This is perplexing because if you invert time before throwing in the ball, the same thing is bound to happen again – since we did not specify the direction of time beforehand in our frictionless fairy’s universe. So the effect depends only on the “hypothesis of molecular chaos” being fulfilled – lack of initial correlations – in Boltzmann’s 19th century parlance. Boltzmann was the first to wonder about this amazing fact – although he looked only at the opposite case of upwards-inverted cups, that is, repulsive particles.
The simplest example does away with fully 2-dimensional interaction. All you need is a light horizontal particle travelling back and forth in a frictionless 1-dimensional closed transparent tube, plus a single attractive, much heavier particle moving slowly up and down in a frictionless transversal 1-dimensional closed transparent tube of its own – towards and away from the middle of the horizontal tube while exerting a Newtonian attractive force on the light fast particle across the common plane. Then the energy-poor fast particle still gets statistically deprived of energy by the energy-rich heavy slow particle in a sort of “energetic capitalism.”
If now the mass of the heavy particle is allowed to go to infinity while its speed and the force exerted by it remain unchanged, we arrive at a periodically forced single-degree-of-freedom Hamiltonian oscillator in the horizontal tube. What could be simpler? But you again get “antidissipation” – a statistical taking-away of kinetic energy from the light fast particle by the heavy slow one.
A first successful numerical simulation was obtained by Klaus Sonnleitner in 2010 – still with a finite mass-ratio and hence with explicit energy conservation. Ramis Movassagh obtained a similar result independently and proved it analytically. Both publications did not yet look at the simpler – purely periodically forced – limiting case just described: A single-degree-of-freedom, periodically forced conservative system. The simplest and oldest paradigm in Poincaréan chaos theory as the source of big news?
If we invert the potential (Newtonian-repulsive rather than Newtonian-attractive), the light particle now gains energy statistically from the heavy guy – in this simplest example of statistical thermodynamics (which the system now turns out to be). Thus, chaos theory becomes the fundament of many-particle physics: both on earth with its almost everywhere repulsive potentials (thermodynamics) and in the cosmos with its almost everywhere attractive potentials (cryodynamics). The essence of two fundamental disciplines – statistical thermodynamics and statistical cryodynamics – is implicit in our periodically forced single-tube horizontal particle. That tube represents the simplest nontrivial example in Hamiltonian dynamics including celestial mechanics, anyhow. But it now reveals two miraculous new properties: “deterministic entropy” generation under repulsive conditions, and “deterministic ectropy” generation under attractive conditions.
I would love to elicit the enthusiasm of young and old chaos aficionados across the planet because this new two-tiered fundamental discipline in physics based on chaos theory is bound to generate many novel implications – from revolutionizing cosmology to taming the fire of the sun down here on earth. There perhaps never existed a more economically and theoretically promising unified discipline. Simple computers suffice for deriving its most important features, almost all still un-harvested.
Another exciting fact: The present proposal will be taken lightly by most everyone in academic physics because Lifeboat is not an anonymously refereed outlet. But many young people on the planet do own computers and will appreciate the liberating truth that “non-anonymous peer review” carries the day – with them at the helm. So, please, join in. I for one was so far unable to extract the really simplest underlying principle: Why is it possible to have a time-directed behavior in a non-time-directed reversible dynamics if that time-directedness does not come from statistics, as everyone believes for the better part of two centuries? What is the real secret? And why does the latter come in two mutually at odds ways? We only have scratched at the surface of chaos so far. Boltzmann used that term in a clairvoyant fashion, did he not? (For J.O.R.)