Publications
Peer-reviewed articles and preprints
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Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang InterfacesT. Iwatsuka, Y. T. Fukai, and K. A. Takeuchi, Physical Review Letters 124, 250602 (2020).The nonequilibrium steady state of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality class has been studied in-depth by exact solutions, yet no direct experimental evidence of its characteristic statistical properties has been reported so far. This is arguably because, for an infinitely large system, infinitely long time is needed to reach such a stationary state and also to converge to the predicted universal behavior. Here we circumvent this problem in the experimental system of growing liquid-crystal turbulence, by generating an initial condition that possesses a long-range property expected for the KPZ stationary state. The resulting interface fluctuations clearly show characteristic properties of the 1D stationary KPZ interfaces, including the convergence to the Baik-Rains distribution. We also identify finite-time corrections to the KPZ scaling laws, which turn out to play a major role in the direct test of the stationary KPZ interfaces. This paves the way to explore unsolved properties of the stationary KPZ interfaces experimentally, making possible connections to nonlinear fluctuating hydrodynamics and quantum spin chains as recent studies unveiled relation to the stationary KPZ.
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Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational FormulaY. T. Fukai and K. A. Takeuchi, Physical Review Letters 124, 060601 (2020).We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.
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Kardar-Parisi-Zhang Interfaces with Inward GrowthY. T. Fukai and K. A. Takeuchi, Physical Review Letters 119, 030602 (2017).We study the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) interfaces growing inward from ring-shaped initial conditions, experimentally and numerically, using growth of a turbulent state in liquid-crystal electroconvection and an off-lattice Eden model, respectively. To realize the ring initial condition experimentally, we introduce a holography-based technique that allows us to design the initial condition arbitrarily. Then, we find that fluctuation properties of ingrowing circular interfaces are distinct from those for the curved or circular KPZ subclass and, instead, are characterized by the flat subclass. More precisely, we find an asymptotic approach to the Tracy-Widom distribution for the Gaussian orthogonal ensemble and the Airy1 spatial correlation, as long as time is much shorter than the characteristic time determined by the initial curvature. Near this characteristic time, deviation from the flat KPZ subclass is found, which can be explained in terms of the correlation length and the circumference. Our results indicate that the sign of the initial curvature has a crucial role in determining the universal distribution and correlation functions of the KPZ class.
Others
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Ph. D. Thesis: Experimental and Numerical Investigation of Universal Fluctuations in Out-of-Equilibrium Interface GrowthY. T. Fukai, Ph. D. Thesis, the University of Tokyo, 2018.The concept of scale-invariance-associated universality has been successfully extended to describe out-of-equilibrium phenomena with fluctuations, becoming an essential concept to explore universality in out-of-equilibrium systems. In this dissertation, the author aims to extend knowledge about the concept itself, by studying out-of-equilibrium interface growth phenomena associated with the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class, a prototypical universality class for out-of-equilibrium systems.
The dissertation is devoted to the following two subjects: The first is the fluctuation of interfaces in the KPZ class with curved initial conditions, and the second is the fluctuation of a growing phase-boundary interface formed near the critical point of a nonequilibrium phase transition of the directed percolation (DP) universality class.
The first study is motivated by recent theoretical and experimental ndings which indicate that interfaces in the KPZ class show distinct statistical properties depending on their geometries, or equivalently, the initial conditions. Though the statistical properties have been intensively studied for special cases such as flat and circular interfaces, knowledge fo more general cases is still limited. To experimentally investigate interfaces with general initial conditions, the author constructed an experimental setup that enables investigation of the growth of the turbulent state of an electrically driven liquid-crystal film from arbitrarily designed initial conditions. With the experiments as well as numerical simulations of a cluster growth model, we revealed statistical properties of the interfaces with initial conditions with a shape of a circular ring, which naturally generalize those of the flat and circular interfaces. We further discuss the theoretical representation of the height distribution for locally parabolic initial conditions by a conjectural formula called the variational formula, which was found to be consistent with the experimental and numerical results we obtained.
The second study is inspired by empirical observations implying that models showing the DP-class transition also shows the growth of the phase-boundary interfaces with the KPZ-class fluctuation far from the critical point. To elucidate interface fluctuation near the critical point where the universality of the DP class arises, the author numerically investigated the interface growth process with the Langevin equation which describes the DP-class transition. We found a crossover connecting a novel interface fluctuation characterized by the DP-class universal exponents and the KPZ-class fluctuation.
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Master's Thesis: KPZ界面の初期条件依存普遍性に対する実験的・数値的研究 (Experimental and Numerical Approaches to Initial-Condition-Dependent Universality of KPZ Interfaces)Yosuke Fukai, Master’s Thesis, the University of Tokyo, 2016.界面がゆらぎながら成長する過程は、幅広い例でみられる典型的な非平衡現象である。このような過程においては、多くの例で自発的にスケール不変性が現れ、臨界現象と同様に、対称性や保存則などの大域的な性質のみで決まる「普遍クラス」として分類できることが知られている。
近年、ゆらぐ界面の普遍クラスの一つである、1 + 1 次元Kardar-Parisi-Zhang (KPZ)
普遍クラスに属するいくつかのモデルの厳密解や、液晶電気対流を用いた実験において、ゆらぎ分布関数・空間相関関数などの性質が、界面の初期条件に依存するということが明らかとなった。例えば、直線から成長する界面（直線界面）と、一点から成長する界面（円形界面）では、異なる普遍的なゆらぎ分布・空間相関関数を示す。これは、KPZ 普遍クラスが初期条件に依存する「サブクラス」に分類できる、ということを示している。
この論文ではまず、「何が属する普遍サブクラスを決めるのか」という問いに関して、直線界面・円形界面の中間と考えられる、有限曲率初期条件からの成長過程を、液晶電気対流を用いた実験・クラスター成長モデルを用いた数値シミュレーションで調べた。この結果、正の有限曲率初期条件のもとでは、ゆらぎ分布関数・空間相関関数・時間相関関数が直線界面サブクラスから円形界面サブクラスへのクロスオーバーを示すこと、その典型的な時間は曲率が小さくなるにつれて長くなることがわかった。一方、負の有限曲率初期条件のもとでは、曲率の大小にかかわらず直線界面の属するサブクラスの性質しかみられなかった。これは単に曲率の有無がサブクラスを決定するわけではない、ということを示唆している。
また、KPZ 普遍クラスの指数は、決定論的な時空カオスに対する摂動でもみられることが知られている。この論文では「決定論的なゆらぎにも普遍サブクラスの性質が現れるか」という問いのもとで、時空カオスに対する摂動の対数を界面成長過程として解析し、初期摂動に依存した普遍的な分布関数・空間相関関数などの性質が現れることを数値的に明らかにした。これは時空カオスの摂動のゆらぎが、KPZ 普遍クラスに現れるランダム行列理論などの確率的な理論でよく説明できることを意味する。
... and more in the future, hopefully!
Last modified: 27 / Apr / 2020
