Studies Of Mechanism And Method On Mesoscopic Statistical Mechnincs

Systems with tiny dimensions have gained growing attention with the rapid progresses in physics, chemistry and life science. Examples of such systems include magnetic domains in ferromagnets (which are typically smaller that 300 nm), quantum dots and biological molecular machines (which range in size from 2 to 100 nm), solidlike clusters that are important in the relaxation of glassy systems (whose dimensions are a few nanometers), etc. Since the characteristic size of these systems is much smaller than the one of macroscopic systems considered in classical statistical mechanics, they are named as"mesoscopic systems"or"small systems". In general, the dissipation energy rate of mesoscopic systems is also much smaller than the one of macroscopic systems. For instance, the typical dissipation energy rate of steam engines is more than 10³J/s, but the one of mesoscopic systems is below 10-18J/s , which is about 10-1000kBT/s. As a result, the influence of thermal fluctuation will be significant, and will affect the dynamics and thermodynamics. On the other hand, a mesoscopic system contains only a small amount of molecular, so the influence of internal noise can not be neglected, too. These fluctuations will lead to observable and significant deviations from the system's average behavior. Therefore, the rules and methods in classical statistical mechanics may be not suitable for such systems and need to be developed. In this thesis, we have studied the following two kinds of statistic problems which are relevant to the delayed mesoscopic systems:Stochastic Thermodynamics and Fluctuation Theorems of Delayed SystemsIn recent years, nonequilibrium thermodynamics of small systems has gained extensive attention. Of particular interests are the stochastic thermodynamics and fluctuation theorems. It has been found that, in the mesoscope systems without delay, the second law isΔS_tot≥0, whereΔS_totdenotes the total entropy change along a stochastic path and <·> stands for average over the path ensemble. Meanwhile,Δstot fulfills the integral fluctuation theorem <e^-ΔS_tot> stot= and the detailed fluctuation theorem p(ΔS_tot)/p(-ΔS_tot)= e^ΔS_tot. Herein, by using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s ( t )can be well-defined in a similar way as that in a system without delay. Since the presence of time delay brings an additional entropy flux into the system, the conventional second lawΔS_tot≥0no longer holds true. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functionalη[χ(t)] . We show that the total dissipation functional R =Δs +η, whereΔsdenotes the system entropy change along a path, obeys R≥0, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem e^-R= 1andp(R)/p^-R = e^R also hold true.Minimum Action Method for Delayed SystemsThe realistic dynamical systems are often subject to weak random perturbations. It has been a common sense that the weak noise can produce a profound effect on the long time dynamics by inducing rare but important events. For instance, weak noise may result in transitions between metastable sets of deterministic dynamical system, which can be related to a large number of interesting phenomena in physics, chemistry and biology such as nucleation processes, chemical reactions, and biological switches, etc. Brute force simulation of these events by the differential dynamical equation is difficult because of the huge disparity between the time step which must be used to perform the simulations and the time scale on which the rare events occur. The Freidlin-Wentzell theory of large deviations provides the right framework to search the rare events quickly. Several methods are proposed for this purpose, such as, the original minimum action method, the multiscale minimum action method, the adaptive minimum action method, the geometric minimum action method, etc.Herein, we expended the minimum action method to study the rare events in delayed mesoscopic systems. To show the effect of delay, we consider a modified version of the Maier-Stein model with linear delayed feedback as an example. By an analysis using small delay approximation, we find that, as the delay time increases, there is a threshold above which the MLP undergos a symmetry breaking bifurcation via a transverse instability. The bifurcation is verified by numerical simulation. What's more, the dependence of transition probability between metastable states on delay time shows distinct difference below and above the threshold. Our results indicate that time delay can dramatically influence the mechanism of noise-induced transition in delayed mesoscopic systems.