Does SuperKamiokande Observe Levy Flights of Solar Neutrinos?

The paper utilizes data from the SuperKamiokande solar neutrino detection experiment and analyzes them by diffusion entropy analysis and standard deviation analysis to evaluate the scaling exponent of the probability density function. The result indicates that solar neutrinos are subject to Levy flights. Subsequently, the paper derives the probability density function, represented as the Fox H-function, and the governing fractional diffusion equation for solar neutrino Levy flights.

💡 Research Summary

The paper investigates whether the Super‑Kamiokande solar‑neutrino detector records signatures of Lévy flights, a type of anomalous diffusion characterized by heavy‑tailed step‑length distributions. Using publicly available Super‑Kamiokande I (1996‑2001) and II (2002‑2005) data, the authors construct time series of the 8 B and hep neutrino fluxes, both in 5‑day and 45‑day averaging windows. They apply two complementary statistical tools: Diffusion Entropy Analysis (DEA) and Standard Deviation Analysis (SDA).

DEA evaluates the Shannon entropy S(t) of the diffusion process generated by the time series. Assuming a scaling form p(x,t)=t^{‑δ} f(x t^{‑δ}), the entropy grows linearly with ln t, S(t)=A+δ ln t, and the slope δ is extracted from a log‑entropy plot. SDA examines the variance σ²_x(t)=⟨x²⟩∝t^{2H} and determines the Hurst exponent H from a log‑log variance plot.

The empirical results are: for the 8 B channel, δ≈0.88 (DEA) and H≈0.66 (SDA); for the hep channel, δ≈0.80 and H≈0.36. Both channels deviate from the Gaussian benchmark (δ=H=0.5), with the hep channel showing sub‑diffusive behavior (H<0.5) and the 8 B channel indicating super‑diffusive, long‑range correlations. The authors interpret these non‑Gaussian scaling exponents as evidence that solar neutrino flux fluctuations are governed by Lévy flights.

To substantiate this claim, they develop a theoretical framework based on a continuous‑time random walk (CTRW) with a Lévy jump‑length distribution λ(x)∼|x|^{‑1‑α} (0<α<2) and an exponential waiting‑time distribution Φ(t)=τ^{‑1}e^{‑t/τ} with finite mean τ. In Fourier space the jump distribution is approximated by λ̂(k)=exp(‑σ^α|k|^α). Combining this with the Laplace transform of the waiting‑time density yields the propagator ˆp(k,s)=1/(s+K_α|k|^α), where K_α=σ^α/τ. Inverting the transforms leads to a probability density function expressed as a Fox H‑function: \