Transition from normal to non-normal distributions of an electromagnetic field in a disordered time-varying cavity

We investigate the statistical properties of electromagnetic field strength in locally disordered and time-varying cavities. Our findings demonstrate that in the majority of cases, temporal disorders in part of the cavity lead to an approximate normal distribution of the electric field strengths at some instants throughout the cavity. However, the stability of this normal distribution significantly deteriorates as time approaches extremely large values. The observation results suggest that this phenomenon is primarily attributed to the exponential increase in energy gain caused by the time interface. By employing the concept of disordered space-time crystals and Lyapunov's theorem, we establish the requisite conditions for achieving a stable normal distribution---the boundedness condition of electric field strength. At long times, the exponential increase in energy resulting from the time interface violates this boundedness condition, leading to an exponential increase in the variance of electric field strength. Consequently, it becomes challenging to observe an approximate normal distribution of electric field strength over extended periods of time. Employing the generalized central limit theorem, we demonstrate that the electric field strengths will ultimately evolve into a non-normal L\'evy skew alpha-stable distribution. Furthermore, we discuss potential experimental schemes based on the transmission-line metamaterials that are anticipated to yield future experimental observations of these phenomena. This study elucidates the statistical properties of transient signals in disordered time-varying systems, thereby paving the way for further exploration into the wave dynamics of analogous systems.