Quantum Hydrodynamics: Mixed States, Dissipation, and a New Hybrid Quantum-Classical Approach IRENE BURGHARDT, 1 KLAUS B. MØLLER, 2 GE ´ RARD PARLANT, 3 LORENZ S. CEDERBAUM, 4 ERIC R. BITTNER 5 1 De´partement de Chimie, Ecole Normale Supe´rieure, 24 rue Lhomond, F-75231 Paris cedex 05, France 2 Department of Chemistry, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark 3 Laboratoire de Structure et Dynamique des Syste`mes Mole´culaires et Solides, Universite´ Montpellier II, F-34095 Montpellier, France 4 Theoretische Chemie, Universita¨t Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany 5 Department of Chemistry, University of Houston, Houston, Texas 77204, USA Received 7 April 2004; accepted 26 May 2004 Published online 27 August 2004 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.20253 ABSTRACT: The formulation of quantum hydrodynamics for mixed states provides a complement, in a quantum-statistical context, to its pure-state “Bohmian mechanics” analogue. We summarize our recent work in this area, focusing on two key aspects. We first describe the formulation of a generalized hydrodynamic force, which includes the effects of thermal fluctuations and “quantum fluctuations” and thus determines the time evolution under the influence of dissipation. Second, we introduce a mixed hydrodynamic–Liouville space representation in terms of partial moments, as applied in the context of coupled electronic states and mixed quantum-classical dynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem 100: 1153–1162, 2004 Key words: Bohmian mechanics; density matrix; dissipation; mixed quantum-classical dynamics; moment hierarchy; quantum hydrodynamics “Les e´quations traditionnelles de la me´canique des fluides sont beaucoup plus proches de la the´orie quantique qu’on ne le croit ge´ne´ralement. ... La proce´dure consiste a` substituer a` la transformation de Fourier de la repre´sentation de Wigner un de´- veloppement en se´rie entie`re.” Yvon, J. Navier-Stokes et Quanta; J Phys Lett 1978, 39, L-363. Introduction W hile much recent interest and activity in the field of quantum hydrodynamics has fo- cused on recasting the time evolution of wavefunc- tions in a fluid-dynamical or “Bohmian” picture [1, Correspondence to: I. Burghardt; e-mail: irene.burghardt@ens.fr Contract grant sponsors: Centre National de la Recherche Scientifique; Danish Natural Science Research Council; National Science Foundation. International Journal of Quantum Chemistry, Vol 100, 1153–1162 (2004) © 2004 Wiley Periodicals, Inc.