Fluctuations, response, and the thermodynamics in small nonequilibrium systems

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I present a brief summary of important results obtained through this research project as follows.1. In small non-equilibrium systems, the states are easily set to be far from equilibrium. It has been known that fluctuation-response relations are violated substantially. Mathematically, it is not surprising because the detailed balance property is lack in such states. The natural question arising in these days when the violation has been just observed in experiments is whether or not there is no rule for the states far from equilibrium. In this situation, we have found that the violation of a fluctuation-response relation is connected with energy dissipation as a simple and beautiful equality.2. The second law of thermodynamics is valid even in small systems if a transition between equilibrium states is concerning. Furthermore, in our previous study, an extension of the thermodynamic law was proposed so that it can be applied to a transition between steady states. Based on this formula, … More we have proposed an idea of effective force under non-equilibrium conditions. Furthermore, we have summarized our present understanding of phenomenological description based on extended thermodynamics.3. A nano-fluid system is the simplest example in which non-trivial rheological behavior is observed. We have presented an elegant theoretical framework for rheological properties. As one example, we have calculated a shear thinning exponent as 2/3. This result will be a starting point for understanding of rheological properties in more complicated systems.4. Glassy systems exhibit a singular non-linear response. Related to this singularity, the fluctuation properties are also rather strange. In order to elucidate a mechanism of the transition, we have proposed a new theory in which the ergodicity breaking transition corresponds to the saddle connection bifurcation. Developing a fluctuation theory around the bifurcation point, we have calculated the critical exponents that characterize critical fluctuations of dynamical events. Less