An exploratory study toward a foundation of nonequilibrium statistical mechanics based on the fluctuation theorem

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K18737
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Osaka University
• Principal Investigator: Morita Takehiko 大阪大学, 理学研究科, 教授 (00192782)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Keywords: ゆらぎの定理 / 非平衡定常状態 / 熱力学形式 / エルゴード理論 / 大偏差原理

Outline of Final Research Achievements

The aim of the project was to get a clue to establish non-equilibrium thermodynamic formalism based on the so-called 'Fluctuation Theorem' by careful investigations into statistical properties of chaotic dynamical systems. Now it turns out that I need to spend more time to study various kinds of limit theorems for dynamical systems. Therefore, I have to say that we are still in our way and we have just arrived at the threshold of the main part of the problem. But fortunately, the attempt enables us to obtain a new method for showing some limit theorems for dynamical systems via thermodynamic formalism and analytic perturbation of transfer operators and an idea to formulate sample-wise limit problems for random dynamical systems by introducing the notion of their direct products.

Free Research Field

エルゴード理論

Academic Significance and Societal Importance of the Research Achievements

力学系の極限定理に対する転送作用素の解析的摂動による接近法は強力な方法ではあるが、扱う対象となる極限定理の多様性という点では不十分という感があった。本研究ではこれまであまり取り扱われなかった形態の極限定理についても踏み込んだことは意義がある。ランダム力学系の標本毎極限問題において直積力学系を定式化することによって、必ずしもノイズが独立でない場合にも適用可能な枠組みを構築したことの意義は大きい。