The diffusion in the quantum Smoluchowski equation |
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| Affiliation: | 1. Institute of Physics, University of Silesia, 40-007 Katowice, Poland;2. Institute of Mathematics, Polish Academy of Sciences, 40-007 Katowice, Poland;3. Institute of Mathematics, University of Silesia, 40-007 Katowice, Poland;4. Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany;1. Lehrstuhl für Theoretische Physik I, Institut für Physik, Universität Augsburg, Universitätsstr. 1, 86135 Augsburg, Germany;2. Dpto. Física Atómica Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Av. Reina Mercedes s/n, 41012 Sevilla, Spain;1. Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan, ROC;2. Institute of Physics, Academia Sinica, Taipei, Taiwan, ROC;1. Università degli Studi di Torino & Collegio Carlo Alberto, Department of Economics and Statistics, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy;2. Université de Bourgogne Franche-Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16, route de Gray, 25030 Besançon Cedex, France;3. Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland |
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| Abstract: | A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit. |
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