New Fluctuation-Correlation Theorems in Exactly Solvable Models in Non-Equilibrium Statistical Mechanics
| Project/Area Number |
26400405
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Chuo University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 非平衡統計力学 / 可積分系 / 揺動 / 時空相関関数 / ランダム行列理論 / 確率論 / 確率過程 / 行列式 / 相互作用多粒子系 / 対称関数 / 行列式点過程 / 行列式過程 / テータ関数 / アフィン・ルート系 / 非衝突ブラウン橋 / １成分プラズマ模型 / ガウス自由場 / 対数ガス / 可積分相互作用粒子系 / 行列式点過程・行列式過程 / 楕円関数・ヤコビテータ関数 / 無限次元SLE / 臨界現象・フラクタル現象 / 可積分確率過程 / マルチンゲール / 整関数 / 共形不変性 / 可解確率過程 / ダイソン模型 / ベッセル過程 / SLE / 揺動と相関 |
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| Outline of Final Research Achievements |
Stochastic properties of non-equilibrium statistical mechanics models are described by spatio-temporal correlation functions. Non-colliding diffusive particle systems are known as the special cases in which all spatio-temporal correlation functions are given by determinants controlled by correlation kernels. The purpose of the present study is to clarify the mathematical structure of such exactly solvable non-equilibrium models called determinantal processes (DP). We have established a new notion of determinantal martingale representation (DMR). In probability theory a martingale is the general name of processes describing time evolutionary fluctuations. We have proved that if a system has DMR, then it is DP. This theorem reveals a new relationship between fluctuation and correlation in no-equilibrium. Using this new framework, we have reported elliptic-functional extensions of the systems and interesting relationship between DPs and plasma models as well as Gaussian free fields.
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| Academic Significance and Societal Importance of the Research Achievements |
統計物理学は物性基礎論として物質科学に対して理論的根拠を与える。また、相転移・臨界現象の統計力学的研究は、場の理論、特に光や電子などを記述する量子場を無限粒子系として定式化する手法を与える。また、統計物理学は確率論や表現論など数学分野の研究を刺激し、数学と物理の共進化を促している。本研究は、未解決問題が山積する非平衡統計物理学において、厳密かつ詳細に解析的計算を実行できる系を扱い、数学的に整備された精密な研究結果を導出することを行ったものである。そのため数学的普遍性を有することが期待されるが、実際、プラズマモデルや自由ガウス場といった他の物理系との非自明な関係性を明らかにすることができた。
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Report
(7 results)
Research Products
(55 results)
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[Presentation] Elliptic Dyson Models2017
Author(s)
Makoto Katori
Organizer
Workshop on Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics
Place of Presentation
Erwin Schroedinger Institute (ESI), University of Vienna
Related Report
Int'l Joint Research / Invited
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[Presentation] Complex martingales and determinantal processes2016
Author(s)
Makoto Katori
Organizer
New Approaches to Non-equilibrium and Random Systems: KPZ Integrability, Universality, Applications and Experiments
Place of Presentation
Kavli Institute for Theoretical Physics, University of California, Santa Barbara
Year and Date
2016-02-29
Related Report
Int'l Joint Research / Invited
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[Book] 例題から展開する力学2017
Author(s)
森山 修、香取 眞理
Total Pages
166
Publisher
サイエンス社
ISBN
9784781913995
Related Report
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