28 February 2000 Ž . Physics Letters A 266 2000 394–399 www.elsevier.nlrlocaterphysleta Computer simulation and mode-coupling theory analysis of time-dependent diffusion in two dimensional Lennard–Jones fluids Sarika Bhattacharyya a , Goundla Srinivas a , Biman Bagchi a,b, ),1 a Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India b Department of Chemistry, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 13 September 1999; received in revised form 14 December 1999; accepted 12 January 2000 Communicated by M. Porkolab Abstract Ž . Ž. Self-consistent mode-coupling theory MCT has been used to calculate the time-dependent diffusion coefficient, Dt , in dense, two dimensional Lennard–Jones liquid, using detailed two particle pair and direct correlation functions. The well-known t y1 behavior of the long time tail of velocity correlation function is found to be latent at higher densities due to the dominance of the microscopic term at short and intermediate times. The calculated values are in good agreement with Ž. Ž . simulated Dt . At finite times, D t is found to exhibit interesting dependence on the density and the size of the system. Ž. Time-dependent diffusion can be used to discuss diffusion limited reactions in terms of Dt , with consequences which, however, are vastly different from those obtained with time independent diffusion coefficient. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 61.20.Ja; 61.20.Lc; 66.10.Cb Ž . Diffusion in two dimensional 2-D systems is a subject not only of great academic interest but also of practical relevance. Many important chemical and biological processes occur in systems which are best described in two dimensions. However, theoretical study of such dynamical processes faces the follow- wx ing paradoxical situation. Theoretical 1 and com- wx puter simulation 2 studies predict that the diffusion ) Corresponding author. Fax: q 91-80-3341683. Ž . E-mail address: bbagchi@sscu.iisc.ernet.in B. Bagchi . 1 Also at the Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur, Bangalore. coefficient diÕerges in two dimensions. The question then naturally arises: How to describe diffusion lim- w x ited processes in two dimensions 3–8 ? We show in this Letter that the paradoxical situation can be easily avoided if the relevant problems are discussed in a non-Markovian theory with time-dependent diffusion Ž. coefficient, Dt . The latter can be calculated quanti- tatively by the mode-coupling theory. We further show that the agreement of the theory with molecu- lar dynamics simulations is excellent both at short and long times. The classic computer simulation study of Alder and Wainwright showed that the long time diffusion coefficient diverges in 2-D due to the existence of 0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 00 00073-6