論文 / 著書情報
Article / Book Information
題目(和文)

一列縦隊拡散の巨視的揺らぎの理論における厳密解

Title(English)

Exact solution in the macroscopic fluctuation theory for single-filediffusion

著者(和文)

守屋宏紀

Author(English)

Hiroki Moriya

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12307号,授与年月日:2023年3月26日,
学位の種別:課程博士,
審査員:笹本 智弘,齋藤 晋,村上 修一,西田 祐介,藤澤 利正

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12307号,
Conferred date:2023/3/26,
Degree Type:Course doctor,Examiner:,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis種別(和文)要約

Type(English)

OutlinePowered by T2R2 (Science Tokyo Research Repository)

（博士課程）Doctoral Program論 文 要 約ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ

系・コース：
Department of, Graduate major in

学生氏名：
Student’s Name

物理学
物理学
守屋 宏紀

系
コース

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

指導教員（主）：Academic Supervisor(main)（理学笹本智弘）

指導教員（副）：Academic Supervisor(sub)要約（自由形式）
Thesis Summary

This thesis examines the long time statistics of the fluctuations or large deviations for thetime-integrated current passing through the origin in one-dimensional single-file systems, including thesymmetric simple exclusion process (SEP), the reflecting Brownian motion (RBM), and theKipnis-Marchioro-Presutti (KMP) model.
For this purpose, the macroscopic fluctuation theory (MFT) is applied to these models. The MFT postulates
the action functional for diffusive models and its saddle point equations so-called the MFT equations
give the optimal path of the density field and its conjugate response field. The MFT equations must be
solved with the annealed or quenched initial conditions depending on whether we allow the initial statefluctuates or not. However, the MFT equations for the interacting models have prevented detailed analysis
for a long time due to their non-linearity and mixed time boundary conditions.
In this thesis, we present the general scheme to solve the MFT equations. Through a non-linear
transformation that can be regarded as the generalization of the canonical Cole-Hopf transformation, theMFT equations turn out to be mapped into classically integrable partial differential equations (PDEs)
to which the inverse scattering method (ISM) can be applied. This approach opens a new way to study thelarge deviations, and thus enables us to derive the first exact solution of the time-dependent MFT equationsfor SEP under the annealed initial condition. Additionally, we obtain exact and compact formulas for theoptimal density profile and the response field which produce a required fluctuation, both at initial andfinal times. Using these results, the large deviation function of the current can be computed and it
coincides with the formula obtained previously by microscopic calculations. This provides the first
analytic confirmation of the validity of the MFT for an interacting model in the time-dependent regime.
Later, the analysis of the quenched initial condition is also conducted. The problem amounts to solving
the boundary equations with respect to the final profile.
In the case of the SEP and the KMP model, we analyze the boundary equation through the perturbation method
and we confirm the perfect agreement between the final profiles which are responsible for the second
cumulant, obtained by the conventional perturbative method in MFT and the ISM. In the case of the SEP
starting from the uniform mean density, we derive the conjectural new formula for the sixth cumulant,
which has never been achieved within the framework of the previous approaches.

備考 : 論文要旨は、和文 2000 字と英文 300 語を 1 部ずつ提出するか、もしくは英文 800 語を 1 部提出してください。
Note : Thesis Summary should be submitted in either a copy of 2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800
Words (English).
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Attention: Thesis Summary will be published on Tokyo Tech Research Repository Website (T2R2).