Research and Analysis of Branched Flow

Branched flow is a universal wave propagation phenomenon which steps across different scales and systems, from nanometer scales to huge universe scales. This occurs primarily because, as waves propagate through complex environments, even minor random variations within the environments can cause the waves to diverge, leading to extreme fluctuations in intensity. There are many branched flows in natural environments. For example, branched flow, which arises in electron waves refracted by residual disorder in high-mobility semiconductors and by deformation potentials in pure metals and semiconductor materials, governs the electrical resistivity in these materials from 0 K to room temperature. The concept of branched flow offers a new perspective on the collisions between electrons and moving bumps, highlighting how electrons exchange momentum and energy with these bumps while demonstrating the ability to surpass perturbative states. This perspective accurately reflects actual events rather than providing statistical measurements. Another very important phenomenon brunched flow phenomenon is ocean tsunami waves. Subsurface earthquakes and large coastal landslides can generate high-energy surface ocean waves with wavelengths ranging from tens to hundreds of kilometers. In shallow water, these waves can propagate at speeds of hundreds of kilometers per hour relative to their wavelength. The speed of tsunami waves is proportional to the square root of the depth of the ocean, with underwater mounds acting as focusing lenses and depressions serving as diffusing lenses. For waves far from the coast, even minor refractions caused by changes in ocean depth can accumulate, leading to significant variations in wave height characterized by branching shapes. If the locations of distant tsunami branches can be accurately predicted, life-saving forecasts can be made before the destructive energy reaches the shore. In addition to these phenomena, there are other examples, such as sound waves, light waves, and so on. Branching flows also contribute to the formation of voids and filaments in the structure of the universe. We hope that this paper can provide research value in terms of numerical, experimental, and especially mathematical representation.