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研究生： 張昕
Hsin Chang
論文名稱： Virtual Potentials in Electric Circuit and Motion of Brownian Gyrator
指導教授： 陳永富
Yung-Fu Chen
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 74
中文關鍵詞： 隨機過程 、虛擬位能井 、回饋控制 、布朗旋轉子 、非平衡穩態動力學
外文關鍵詞： Stochastic Process, Virtual potential, Feedback Control, Brownian Gyrator, Non-equilibrium Steady State Dynamics
相關次數： 點閱：8  下載：0 
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    • 受熱擾動影響的電路，其隨機行為類似於高粘度流體中的微觀粒子。藉由外加回饋控制技術(feedback-controlled technique)，我們通過實驗在電路中產生可調虛擬位能的布朗動力學(Brownian dynamics)。當回饋速度夠快，電路雜訊在虛擬位能井(virtual potential)中的行為與在真實的位能井中沒有區別。實驗結果顯示電路版本的回饋阱對於研究非平衡系統是個簡單、有效且可編程的方案。而這樣的方法理論上可以擴展到多自由度的系統。通過回饋耦合兩組電阻電容電路，便可觀察到布朗旋轉子(Brownian Gyrator)的現象。

    布朗旋轉子的製備是二維布朗粒子在諧和位能(harmonic potential)中，並受到來自不同方向且具備不同溫度的隨機力撞擊。我們將諧和位能替換為其他類型的位能，並通過模擬研究其非平衡穩態動力學(non-equilibrium steady state dynamics)。在非諧位能的情況下，我們發現此系統機率流(probability flow)不沿著機率等高線流動。甚至在可線性近似的位能低點附近也存在違反此特徵的情況。此現象意味著機率流的方向不垂直於機率梯度，且粒子的熵隨穩態軌跡而改變。

    Electric circuits affected by thermal agitation are analog to confined Brownian particles in a high-viscosity fluid. Here we experimentally demonstrate an effective technique of generating tunable potentials for the Brownian dynamics in an electric circuit, realized by external controlled feedback. The thermal fluctuation undergoes equivalent Brownian dynamics in the authentic potentials as long as the feedback is fast enough to react for designed potentials. The results show that the electric-circuit version of feedback-trap provides a simple, effective, and programmable scheme to study non-equilibrium system. The generation can in principle be expanded to systems with multi degrees of freedom. The phenomenon of the Brownian gyrator can be observed by coupling two RC circuits with a feedback function instead of a coupling capacitor.

    A Brownian gyrator features a two-dimensional random particle under a harmonic potential, while random thermal kicks are of distinct temperatures along the two orthogonal axes. We replace the harmonic potential by other types of potentials, and we investigate the non-equilibrium steady states through simulation studies. In contrast to the Brownian-gyrator case, where the probability flow circulates along the probability contour, the results for the non-harmonic cases do not follow this signature. Moreover, the violation of this feature exists even near the potential minima, where the harmonic approximation work. This implies that the probability flow is not perpendicular to the probability gradient and the entropy of the particle changes along with the steady state trajectory.

    Abstract i Contents ii List of Figures iv List of Tables v Glossary vi Acronym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction 1 2 Stochastic Thermodynamics 3 2.1 One-Dimensional Brownian Motion . . . . . . . . . . . . . . . . . . . 4 2.1.1 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Overdamped Approximation . . . . . . . . . . . . . . . . . . . 9 2.1.3 Johnson-Nyquist Noise . . . . . . . . . . . . . . . . . . . . . . 12 2.1.4 Analogous Brownian Particle in Electric Circuit . . . . . . . . 12 2.2 Brownian Gyrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . 20 2.3 Virtual Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Experimental Studies of Virtual Potentials in Electric Circuit 27 3.1 Experimental Setup of Feedback Control . . . . . . . . . . . . . . . . 28 3.2 Virtual Harmonic Potential . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Discrete Delay Equation . . . . . . . . . . . . . . . . . . . . . 33 3.2.3 Discrete Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Virtual Double Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 Static Double Well . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.2 Stochastic Resonance in Time-varying Double Well . . . . . . 41 4 Numerical Studies of Motion of Brownian Gyrator 44 4.1 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Expression of Probability Flow . . . . . . . . . . . . . . . . . . . . . 47 4.3 Harmonic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.1 Numerical Result . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.2 Analytical Calculation of Probability Distribution and Flow . 50 4.4 Two-Dimensional Double Well . . . . . . . . . . . . . . . . . . . . . . 52 4.4.1 Numerical Result . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4.2 Entropy Changing Rate . . . . . . . . . . . . . . . . . . . . . 52 4.5 Isotropic Quartic Potential . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5.1 Numerical Result . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5.2 Parity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Conclusion 57 Bibliography 58

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