Theoretical Studies Of Statistical Mechanics In Mesoscopic Complex Systems

With the development of nanotechnology and life science, theoretical and experimental research of mesoscopic complex systems has become the frontier domain of scientific research. On the one hand, fluctuation in mesoscopic systems reaches a quite remarkable degree which makes dynamic properties of them notably different from macroscopic ones, and thus revealing the effects of fluctuation in mesoscopic systems is one of the key problems people should do in mesoscopic statistical mechanics research; on the other hand, real mesoscopic systems will show some interesting dynamics due to the cooperation of fluctuation and complex factors of the system in which coupling, delay and other complexity really exist, which raises new problems and challenges of research into mesoscopic systems. In this dissertation, we employ mesoscopic statistical mechanics methods to explore the interaction mechanism between fluctuation and complexity of mesoscopic systems. Main research includes these following two parts:Spatiotemporal dynamics of coupled neuronal systems Coupled neuronal systems encode external stimulation signal into action-potential sequences (spike trains), and spatial synchronization as well as temporal coherence of neuronal spike trains are crucial for neural coding and transmission of information across the neuronal systems. And as we know, in coupled neuronal systems, fluctuation and information transmission delays are unavoidable and have physiological significance. Therefore, we investigate spatiotemporal dynamics of coupled neuronal systems in dependence on the fluctuation and two types of delayed coupling: We find that for the former case, delay can dramatically enhance temporal coherence and spatial synchronization of the fluctuation-induced spike trains. In addition, there exists an optimal delay, at which dynamics of the coupled neuronal system reaches a most ordered state, which is both synchronized in space and periodic in time, demonstrating an interesting resonance phenomenon with delay. For the latter case, however, we can not achieve a similar spatiotemporal ordered state, but the neuronal dynamics exhibit interesting synchronization transition with time delay from zigzag fronts of excitations to dynamic anti-phase synchronization, and further to clustered chimera states which have spatially distributed anti-phase coherence separated by incoherence. Furthermore, we also show these findings robust to the change of fluctuation and topology of the coupled neuronal network. Finally, we make some qualitative analysis to illustrate the numerical results, finding that the interesting phenomena are due to the locking between the delay time and the inherent spiking period of the neuronal system under the effects of fluctuation.Logical stochastic resonance in logic gate and genetic regulatory network In the last few years, some scientists have yield reliable logical response in single mesoscopic logic gate and genetic regulatory networks, by considering the cooperation between fluctuation and the inner nonlinearity mechanism. They raised the concept of logical stochastic resonance similar to stochastic resonance. As we know, in computation device, logic gates are often coupled to work for logical computation, so it is very useful to investigate logical stochastic resonance in coupled logic gates. In coupled logic gates system, we focus on the cooperation between coupling and fluctuation, and study the effects of system size and coupling strength on logic stochastic resonance, finding that (1) coupling can enhance logical stochastic resonance, that is to say, coupling can make the system yield reliable results in larger fluctuation range; (2) with the increase of the system size, the corresponding fluctuation range in which the coupled logic gates can produce reliable output increases until certain value; (3) the effect of coupling strength on the reliability of coupled logic gates is non-monotonic, which means that as the coupling strength increases, the reliability of the system increases first but then decreases fast, and when the system size reaches some certain value, the logical function of the coupled logic gates is fully destroyed.In the biological logic gate built by the gene switch model of the bacteriophage? , we investigate the cooperative mechanism of fluctuation and delay. We find that proper delay can enhance logical stochastic resonance, which means that delay effect can make the biological logic gate remain reliable in a wider range, and there exists a optimal delay time at which the system show the best reliability.