%0 Book Section %1 statphys23_0279 %A Isobe, M. %A Petrosky, T.Y. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K autocorrelation disk driven dynamics event function hard long mode-counpling molecular self-consistent statphys23 tail theory time topic-3 velocity %T Long time tail of the velocity auto-correlation function in two-dimensional hard disk fluid %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=279 %X More than thirty five years ago, Alder and Wainright discovered a slow power decay $t^-d/2$ ($d$:dimension) of the velocity auto-correlation function (VACF) in moderately dense hard sphere fluids by the event-driven molecular dynamics (EDMD) simulation 1. In two dimensional (2D) case ($d=2$), transport coefficient (e.g. the diffusion constant D) derived by the time-correlation function in linear response theory show a logarithmic divergence, which is called ``2D long time tail problem'' 2. Since the longitudinal sound wave propagates at the speed of sound in the system, the maximum correlation time obtained by EDMD with periodic boundary conditions is limited by the system size (i.e. particle number). Therefore, we must essentially perform a large scale computer simulation to solve this problem. In this paper, we revisit the ``2D long time tail problem'' to perform a large scale EDMD simulation with one million hard disks by using a modern efficient algorithm 3 systematically. As the most remarkable result, we show that the decay of VACF in moderately dense fluids is slightly faster than ($1/t$)1,4, which seems to agree with the prediction of self-consistent mode-coupling theory 5 in the long time limit ($1/(tłnt)$). We will discuss the relationship between the obtained numerical results and theoretical predictions 4-6. 1) B.J. Alder and T.E. Wainwright, Phys. Rev. A1, 18 (1970).\\ 2) Y. Pomeau and P. Résibois, Phys. Rep. 19C, 63 (1975).\\ 3) M. Isobe, Int. J. Mod. Phys. C 10, 1281 (1999).\\ 4) J.R. Dorfman and E.G.D. Cohen, Phys. Rev. Lett. 25, 1257 (1970).;M.H. Ernst, E.H. Hauge and J.M.J. van Leeuwen, Phys. Rev. Lett. 25, 1254 (1970).\\ 5) K. Kawasaki, Phys. Lett. 34A, 12 (1971).;T. E. Wainright, B. J. Alder and D. M. Gass, Phys. Rev. A 4, 233 (1971).\\ 6) T. Y. Petrosky, Foundation of Physics 29, 1417 (1999); ibid., 1581 (1999).\\

@incollection{statphys23_0279, abstract = {More than thirty five years ago, Alder and Wainright discovered a slow power decay $\sim t^{-d/2}$ ($d$:dimension) of the velocity auto-correlation function (VACF) in moderately dense hard sphere fluids by the event-driven molecular dynamics (EDMD) simulation [1]. In two dimensional (2D) case ($d=2$), transport coefficient (e.g. the diffusion constant D) derived by the time-correlation function in linear response theory show a logarithmic divergence, which is called ``2D long time tail problem'' [2]. Since the longitudinal sound wave propagates at the speed of sound in the system, the maximum correlation time obtained by EDMD with periodic boundary conditions is limited by the system size (i.e. particle number). Therefore, we must essentially perform a large scale computer simulation to solve this problem. In this paper, we revisit the ``2D long time tail problem'' to perform a large scale EDMD simulation with one million hard disks by using a modern efficient algorithm [3] systematically. As the most remarkable result, we show that the decay of VACF in moderately dense fluids is slightly faster than ($\sim 1/t$)[1,4], which seems to agree with the prediction of self-consistent mode-coupling theory [5] in the long time limit ($\sim 1/(t\sqrt{\ln{t}})$). We will discuss the relationship between the obtained numerical results and theoretical predictions [4-6]. 1) B.J. Alder and T.E. Wainwright, Phys. Rev. A1, 18 (1970).\\ 2) Y. Pomeau and P. R\'esibois, Phys. Rep. 19C, 63 (1975).\\ 3) M. Isobe, Int. J. Mod. Phys. C 10, 1281 (1999).\\ 4) J.R. Dorfman and E.G.D. Cohen, Phys. Rev. Lett. 25, 1257 (1970).;M.H. Ernst, E.H. Hauge and J.M.J. van Leeuwen, Phys. Rev. Lett. 25, 1254 (1970).\\ 5) K. Kawasaki, Phys. Lett. 34A, 12 (1971).;T. E. Wainright, B. J. Alder and D. M. Gass, Phys. Rev. A 4, 233 (1971).\\ 6) T. Y. Petrosky, Foundation of Physics 29, 1417 (1999); ibid., 1581 (1999).\\}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Isobe, M. and Petrosky, T.Y.}, biburl = {https://www.bibsonomy.org/bibtex/273905fd0054151e15b55097b25f0b399/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {3f0f7370e45ce92227bf0e91cb31ffea}, intrahash = {73905fd0054151e15b55097b25f0b399}, keywords = {autocorrelation disk driven dynamics event function hard long mode-counpling molecular self-consistent statphys23 tail theory time topic-3 velocity}, month = {9-13 July}, timestamp = {2007-06-20T10:16:15.000+0200}, title = {Long time tail of the velocity auto-correlation function in two-dimensional hard disk fluid}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=279}, year = 2007 }