On the relationship between Hamiltonian chaos and classical gravity

It is known that Hamiltonian equations of motion for low-dimensional chaotic systems are typically formulated using fractional derivatives. The evolution of such systems is governed by the fractional diffusion equation, which describes self-similar and non-Gaussian processes with strong intermittencies. We confirm, in this context, that the dynamics of a Brownian particle driven by space-time dependent fluctuations evolves towards Hamiltonian chaos and fractional diffusion. The corresponding motion of the particle has a time-dependent and nowhere vanishing acceleration. Invoking the equivalence principle of general relativity leads to the conclusion that fractional diffusion is locally equivalent to a transient gravitational field. It is shown that gravity becomes renormalizable as Newton’s constant converges towards a dimensionless quantity.