NSF Award
0501315
TRANSPORT AND FLUCTUATIONS ON SMALL TIME AND SPACE SCALES <br/><br/>The proposed project is a theoretical and numerical study of fundamental aspects of transport and fluctuations on small time and length scales, and comprises two problems in far-from-equilibrium statistical mechanics: I. The fluctuations of work and heat in a non-equilibrium stationary state, possibly far from equilibrium and, II. The transport of mass, momentum and energy (temperature) on the nanometer length scale and picosecond time scales. <br/>I. The work on the first problem involves a study of the fluctuations of work and heat in non-equilibrium stationary states. The Conventional Fluctuation Theorem (CFT) gives a simple asymptotic relation for the ratio of the probability to find a particular value of the entropy (heat) of a certain magnitude a to the probability to find a value -a. Crucial assumptions for the CFT, which is a refinement of the Second Law of Thermodynamics, appear to be either deterministic or stochastic dynamics as well as a very chaotic (Anosov-like) behavior of the system. Recently, a different kind of system was studied, based on a Langevin model for Brownian motion in a harmonic potential or for electrical circuits not in equilibrium. Here a new, Extended Fluctuation Relation (EFR) was found for heat fluctuations, possibly due to the presence of mixed deterministic and stochastic dynamics, while the work still satisfies the CFT. Since the EFR has so far only been seen in these two Langevin based models, in contrast to the CFT, which is known to hold for a large class of systems, the main aim is to investigate the generality of the EFR, its connection with the CFT and its implications for a more general theory of non-equilibrium stationary state fluctuations. <br/>II. The second problem will be attacked using a new transport theory based on Green?s functions, as first developed by Kincaid for self-diffusion. Green's functions are used not only to avoid an expansion in gradients, whose coefficients tend to diverge, but also to capture both the short and long time behavior of the system. Apart from self-diffusion, the theory has so far only been applied to heat diffusion in a dense, single component fluid. In the self-diffusion case it could be analytically proved that the very short and long time behavior were well described by the theory, whereas for the heat conduction case there is no proof but good agreement with numerical simulations. These simulations also showed that this Green's function theory far outperforms traditional theories (hydrodynamics and the telegrapher's equation) for processes occurring on the very short time and length scales of fractions of picoseconds. Nevertheless, the hydrodynamical equations, appeared to give surprisingly good results on somewhat larger timescales of about two picoseconds. This could imply a much wider applicability of hydrodynamics than expected. A study of a binary fluid mixture is intended to further investigate this. In particular, an analytical investigation into the approach to hydrodynamics for longer times and a numerical check of the short time behavior for this system will be undertaken. <br/>The intellectual merit of the project lies mainly in the elucidation of several fundamental questions in nonequilibrium statistical physics, but also in the possible importance for the design of stable nano-structures, since the EFR predicts, in general, a probability ratio of heat absorbtion to heat production by the Brownian particle or the resistor considerably in excess of that predicted by the CFT, while the Green's functions allow a detailed description of systems on small space and time scales. <br/>In a broader context: it is important that scientists learn how to present a clear and generally understandable lecture. The art of transferring knowledge should be part of the scientific educational process. Training students, post-docs and younger colleagues in this has always taken place here during their research. A program in which high school students and science teachers will be taught by example and special seminars, is planned to be organized by the Rockefeller University Outreach Program.
