Long-time relaxation in pilot-wave theory

Abstract

We initiate the study of relaxation to quantum equilibrium over long timescales in pilot-wave theory. We simulate the time evolution of the coarse-grained H-functionH(t) for a two-dimensional harmonic oscillator. For a (periodic) wave function that is a superposition of the first 25 energy states we confirm an approximately exponential decay ofH over five periods. For a superposition of only the first four energy states we are able to calculateH(t) over 50 periods. We find that, depending on the set of phases in the initial wave function,H can decay to a large nonequilibrium residue exceeding 10% of its initial value or it can become indistinguishable from zero (the equilibrium value). We show that a large residue inH is caused by a tendency for the trajectories to be confined to sub-regions of configuration space for some wave functions, and that this is less likely to occur for larger numbers of energy states (if the initial phases are chosen randomly). Possible cosmological implications are briefly discussed.

Extreme quantum nonequilibrium, nodes, vorticity, drift and relaxation retarding states

“…This mechanism, called 'quantum relaxation', occurs spontaneously through a process that is directly analogous to the classical relaxation (from statistical nonequilibrium to statistical equilibrium) of a simple isolated mechanical system as described by the second thermodynamical law. Demonstration of the validity of quantum relaxation [11,15,5,14,12,13] has meant that the need to postulate agreement with quantum probabilities outright may be dispensed with. It has also prompted the consideration of 'quantum nonequilibrium'-that is, violations of standard quantum probabilities, which in principle could be observed experimentally †.…”

Geometrical hypotheses underlying wave functions and their emerging dynamics

“…The strategy is to impose mild distributions of unknown initial conditions for the physical system at hand, and show that equations (29,65) generally take us, at a coarse grained level, to the result that the degrees of freedom of the physical system get distributed according to the area function after some time evolution, reaching quantum equilibrium. This is work in progress [26][27][28]. However, what is important for us here is the verification that the association of the area function with a probability measure is a consequence of the postulation of equations (29) and (65): it is not postulated a priori, it comes as a second step in the formulation of the theory.…”

“…The coordinates of the source space have dimensions of length, [x a ] = L, and the metrics g ab and h αβ are dimensionless. If one wants to normalize the constant V in Eq (27). to unity, then [f α ] = L −m/2 .…”

Mechanism for nonlocal information flow from black holes

“…In pilot-wave theory, the initial Bornrule distribution (3) enjoys the property of 'equivariance' under the equations of motion (1) and (2): such a distribution evolves into a Born-rule distribution ρ(q, t) = |ψ(q, t)| 2 at later times (see Section 2). Furthermore, as one might expect from the analogy with thermal physics, in appropriate circumstances initial nonequilibrium distributions relax towards equilibrium, as has been shown through extensive numerical simulations [9][10][11][12][13][14]. Such relaxation may be quantified in terms of a subquantum H-function [7]…”

“…whose coarse-grained value is found to decay approximately exponentially with time [11,12,14]. While there is as yet no experimental evidence for violations of the Born rule, the theoretical study of quantum nonequilibrium opens up a large domain of potentially new physics which might one day be observable (or at least testable).…”