Scale-free dynamics and reservoir computing in a non-stationary Freedericksz model

Scale-free behavior is commonly associated with stationary dynamics and thresholded events, assumptions that are violated in many physical systems exhibiting slow drifts over time. Here, we investigate weakly and strongly driven regimes of Freedericksz dynamics in a model of nematic liquid crystals. To characterize the weakly driven, non-stationary regime without thresholding or stationarity assumptions, we use a squared temporal-increment measure of the director field. Numerical simulations show that this squared-increment activity follows a robust power-law distribution over several decades. Using mean-field avalanche theory as a scaling framework, we derive the associated exponent and show that it is largely independent of microscopic model parameters. We further demonstrate finite-size scaling, with distributions obtained for different system sizes collapsing onto a common scaling function after rescaling by a characteristic cutoff. As external forcing increases, scale-free statistics progressively break down and the system enters a strongly driven regime that can support reservoir computation. We implement the Freedericksz model as a reservoir for delayed prediction of time-series data and show that performance depends on physically interpretable parameters including elastic coupling, dielectric anisotropy, and applied electric field strength. Importantly, optimal performance occurs away from the regime exhibiting the strongest scale-free organization, indicating that scale-free dynamics alone are not sufficient for optimal computation. Together, these results show that scale-free organization can arise in non-stationary continuous-field dynamics without explicit event segmentation, while establishing Freedericksz liquid-crystal dynamics as a tunable model for relating changes in collective dynamical regime to physical computation.