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); 90, 044035 (2014); 93, 084039 (2016)] 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)], 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.

Figure 1

The basin of attraction of one limit cycle in a discrete deterministic dynamical system. Image courtesy of Andy Wuensche from [2].

Figure 2

The complete state space of a discrete deterministic dynamical system, showing several limit cycles and their basins of attraction. Image courtesy of Andy Wuensche from [2].

Figure 3

Deterministic energetic causal set model, showing the quick capture by a limit cycle. Model with 20 lineages and $10^4$ total steps. In this and the following figures, the story is told in four panels. The upper left panel shows the history plotted in the emergent $1+1$ dimensional spacetime. The upper right panel plots the events in the order by which they take place in the simulation. The lower left panel plots the first 1% of events in the sequence and the lower right plots the last 1% of events. In these we connect the events in the order generated, which gives us a trajectory.

Figure 4

Plot of the values of the numerical value of the pasts in each lineage of events, for each step, and for a model with 8 lineages. Note that this is a plot of the value of the pasts of events, and not of their positions $x$, as in other plots. Just like it happens for the value of positions, the evolution of each the pasts also enters a limit cycle promptly. Together with Fig. 3 this shows that the pair $\{z_i^{\pm },\mathrm{\Sigma }(z_i^{\pm })\}$ in the thick present collectively exhibit limit cycles.

Figure 5

(Number of independent $\mathrm{families}=20$; Number of events $10^5$) Stochastic energetic causal set model, showing the quick capture by a limit cycle. Panels shown as in Fig. 3. In this stochastic model the stark transition between the first 1% and the last 1% of events, in this and later figures, depicts clearly the system’s evolution toward regularity and limit cycles.

Figure 6

(Number of independent $\mathrm{families}=20$; Number of events $10^5$) Here we show an example of another model to illustrate variation between simulation runs.

Figure 7

(Number of independent $\mathrm{families}=20$; Number of events $10^5$) Here we show an example of another model to illustrate variation between simulation runs.

Figure 8

(Number of independent $\text{families}=30$; Number of events $10^5$) If we increase the number of families in the simulation, as expected the time scale for entering the limit cycles is increased.