EUROPHYSICS LETTERS zyxwvutsr Europhys. Lett., zyxwvuts 2 (31, pp. 167-174 (1986) 1 August 1986 Relationship between Classical Diffusion, zyx 1/0 Noise and the Motion of a Quantum Particle. T. SCHNEIDER, M. P. SOERENSEN(*), E. TOSATTI(**) and M. ZANNETTI(***) IBM Zurich Resarch Laboratory - 8803 Riischlikon, Switzerland (received 9 December 1985; accepted in final form 3 May 1986) PACS. 05.40. - Fluctuation phenomena, random processes, and Brownian motion. PACS. 71.555 - Localization in disordered structures. PACS. 72.70. - Noise processes and phenomena. Abstract. - The mapping of classical diffusion onto a quantum system is used to explore the connections between diffusion phenomena in random media, zyxw llw noise and the motion of a quantum particle in a random potential. Much work has been devoted to the subjects of l/o noise [l, 21, classical diffusion in random media [3-71 and localization [8-111. These research fields developed rather inde- pendently. In this letter, we establish relationships by mapping the diffusion problem of a classical particle in a random environment onto an associated quantum system. It describes the motion of a quantum particle in a random potential, so that localization might occur. In doing so, we use the connection between the Langevin, describing classical diffusion, and the Fokker-Planck equations. The latter is then reduced to an imaginary-time Schrodinger problem, defining the Hamiltonian of the associated quantum system [12-141. We consider two distinct cases: i) A classical particle trapped in the random medium. Here, the ground state of the quantum analog is exponentially localized. ii) Diffusive motion of the classical particle in the random environment. In this case, the mean-square displacement, corresponding to the variance of the ground state in the associated quantum system, diverges. Thus, the ground state is not exponentially localized. Moreover, the excitation spectrum is gapless at the bottom. These properties, in the classical analog, for sublinear diffusion, lead to the generation of l/o noise. In recent years, various interesting models, exhibiting such a sublinear diffusion below the upper critical dimension, have been proposed [3-71. By invoking dynamic scaling, the long-time behaviour of classical diffusion (*) Permanent address: Laboratory of Applied Mathematical Physics, Technical University of (**) Permanent address: International School for Advanced Studies, Strada Costiera 11, 1-34100 (***) Permanent address: Dipartimento di Fisica, Universitti degli Studi di Salerno, 1-84100 Denmark, Building 303, DK-2800 Lyngby. Trieste, Italy. Salerno, Italy.