Fig. 2.

Emergence of macroscopic broken detailed balance in a complex system. (A) Two-spin Ising model with interactions J_αβ representing the strength of the influence of spin β on spin α (Left). If the interactions are asymmetric (such that $J_{\alpha \beta }\ne J_{\beta \alpha }$), then the system exhibits a loop of flux between spin states (Right). (B) Asymmetric SK model, wherein directed interactions are drawn independently from a zero-mean Gaussian with variance $1/N$, where N is the size of the system. (C) For an asymmetric SK model with N = 100 spins, we plot the probability distribution (color) and fluxes between states (arrows) for simulated time series at temperatures $T=0.1$ (Left), $T=1$ (Center), and $T=10$ (Right). To visualize the dynamics, the time series are projected onto the first two principal components of the combined data across all three temperatures. The scale is indicated in flux per timestep, and the disks represent 2-SD confidence intervals that arise due to finite data (Materials and Methods).