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)論文要旨

Type(English)

Summary

Powered by T2R2 (Science Tokyo Research Repository)

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

系・コース：
Department of, Graduate major in

物理学
物理学

学生氏名：
Student’s Name

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)要旨（英文 800 語程度）
Thesis Summary （approx.800 English Words ）For the last few decades, the large deviations of physical observables were studied with great effort
and its importance in non-equilibrium statistical mechanics has been widely recognized both in classicalphysics and quantum physics. Unlike the typical Gaussian fluctuations in equilibrium state, fluctuations
in non-equilibrium state can exhibit totally different behavior in which new physics may be hidden. The
large deviation function is a function that characterizes such rare events away from the mean value, andit can be understood as an analogue of the thermodynamic potential at equilibrium.
This thesis is devoted to the analysis of the long time statistics of the fluctuations or large deviations
for the time-integrated current passing through the origin in one-dimensional single-file systems, inwhich moving particles kept their ordering in a narrow channel. In particular, we studied three related
models, the symmetric simple exclusion process (SEP), the reflecting Brownian motion (RBM), and theKipnis-Marchioro-Presutti (KMP) model. Among them, the SEP is intensively studied as the rest of models
can be deduced from the results of the SEP by rescaling and taking a certain limit.
For this purpose, the macroscopic fluctuation theory (MFT), developed by G. Jona-Lasinio and his
collaborators since the early 2000s, is applied to these models. The MFT postulates the action functional
at a hydrodynamic level that depends on only two transport coefficients, a diffusivity and a mobility.
Through a variational principle, 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 theannealed or quenched initial conditions depending on whether we allow the initial state fluctuates ornot. However, the MFT equations for the interacting models have prevented detailed analysis for a longtime due to their non-linearity and mixed time boundary conditions. Hence, except for the explicitly
solvable non-interacting case, properties of fluctuations have been investigated from a variety of aspects
including the stationary setting, perturbation method, the large current regime, numerical simulations,and so on.
In this thesis, we present a general scheme to solve the MFT equations with a constant diffusivity anda quadratic mobility with respect to the density. This can be achieved by constructing a non-linear andnon-local transformation using the notion of the gauge equivalence. Applying this transformation thatcan be regarded as the generalization of the canonical Cole-Hopf transformation, the MFT equations turnout to be mapped into the celebrated Ablowitz-Kaup-Newell-Segur (AKNS) equations, which are closely related
to the Nonlinear Schrödinger (NLS) equation. Since the system is a set of classically integrable partial
differential equations (PDEs), the inverse scattering method (ISM) can be applied.
The ISM is a standard method to solve the soliton equations and has a great advantage compared to theconventional approaches in MFT because the time evolution is drastically simplified in terms of thescattering amplitudes. For this remarkable property, the ISM is recently used to study the short timelarge deviations for the Kardar-Parisi-Zhang (KPZ) equation by Krajenbrink and Le Doussal and the longtime large deviations for the KMP model by Bettelheim, Smith, and Meerson, except for the SEP. Their works
are also important, however their saddle point equations are obviously classically integrable and themixed time boundary conditions are simplest, which do not fit the MFT for SEP. Moreover, since the original
KMP model is not microscopically integrable, their result of the KMP model cannot prove the validity ofthe MFT.
Application of the ISM opens a new way to study the large deviations, and thus enables us to derive thefirst exact solution of the time-dependent MFT equations for SEP under the annealed initial condition.
Additionally, we obtain exact and compact formulas for the optimal density profile and the response field
which produce a required fluctuation, both at initial and final 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 by Derrida and Gerschenfeld. 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 theconjectural 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).
注意：論文要旨は、東工大リサーチリポジトリ（T2R2）にてインターネット公表されますので、公表可能な範囲の内容で作成してください。
Attention: Thesis Summary will be published on Tokyo Tech Research Repository Website (T2R2).