Fluctuations, response, and the thermodynamics in small nonequilibrium systems

现将通过本研究项目取得的重要成果简要总结如下。在小型非平衡系统中，状态很容易被设定为远离平衡。已知波动-响应关系实质上被破坏。从数学上讲，这并不令人惊讶，因为在这种状态下，缺乏详细的平衡属性。在这些日子里，当实验中刚刚观察到这种违反时，自然产生的问题是，对于远离平衡的状态是否没有规则。在这种情况下，我们发现，对涨落-响应关系的破坏与能量耗散作为一个简单而美丽的等式联系在一起。如果考虑到平衡态之间的转变，热力学第二定律即使在小系统中也是有效的。此外，在我们以前的研究中，我们提出了热力学定律的一个扩展，以便它可以应用于稳态之间的转变。根据这个公式，…此外，我们还提出了非平衡条件下的有效力概念。此外，我们还总结了我们目前对基于扩展热力学的唯象描述的理解。纳米流体系统是观察到非平凡流变行为的最简单的例子。我们提出了一个关于流变性质的优雅的理论框架。作为一个例子，我们计算了剪切稀化指数为2/3。这一结果将为理解更复杂体系的流变性质提供一个起点。玻璃体系统表现出奇异的非线性响应。与此奇点相关的涨落性质也颇为奇怪。为了阐明这种转变的机制，我们提出了一种新的理论，即遍历破裂转变对应于鞍点连接分叉。发展了围绕分岔点的涨落理论，计算了表征动力学事件临界涨落的临界指数。较少

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