Random Consolidations and Fragmentations Cycles Lead to Benford

A process which constantly alternates between minuscule random consolidations and tiny random fragmentations within a large set of quantities is found to converge to the logarithmic after sufficiently many such cycles. Randomness in selecting the particular quantity to be fragmented, as well as randomness in selecting the two particular quantities to be consolidated, is essential for convergence. Surprisingly, fragmentation could be performed either randomly via say a realization from the continuous Uniform on (0, 1), or deterministically via any fixed split ratio such as say 25% - 75%, and Benford's Law emerges in either case.