Reversing the irreversible: From limit cycles to emergent time symmetry

In 1979 Penrose hypothesized that the arrows of time are explained by the hypothesis that the fundamental laws are time irreversible [R. Penrose, in General Relativity: An Einstein Centenary Survey (1979)]. That is, our reversible laws, such as the standard model and general relativity are effective, and emerge from an underlying fundamental theory which is time irreversible. In [M. Cortês and L. Smolin, Phys. Rev. D 90, 084007 (2014), 10.1103/PhysRevD.90.084007; 90, 044035 (2014), 10.1103/PhysRevD.90.044035; 93, 084039 (2016), 10.1103/PhysRevD.93.084039] we put forward a research program aiming at realizing just this. The aim is to find a fundamental description of physics above the Planck scale, based on irreversible laws, from which will emerge the apparently reversible dynamics we observe on intermediate scales. Here we continue that program and note that a class of discrete dynamical systems are known to exhibit this very property: they have an underlying discrete irreversible evolution, but in the long term exhibit the properties of a time reversible system, in the form of limit cycles. We connect this to our original model proposal in [M. Cortês and L. Smolin, Phys. Rev. D 90, 084007 (2014), 10.1103/PhysRevD.90.084007], and show that the behaviors obtained there can be explained in terms of the same phenomenon: the attraction of the system to a basin of limit cycles, where the dynamics appears to be time reversible. Further than that, we show that our original models exhibit the very same feature: the emergence of quasiparticle excitations obtained in the earlier work in the space-time description is an expression of the system's convergence to limit cycles when seen in the causal set description.