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研究生： 黃明彥
Huang, Ming-Yen
論文名稱： 雙粒子糾纏態之特例下的Szilard引擎
Special case of two particle entangled state on Szilard engine
指導教授： 周忠憲
Chou, Chung-Hsien
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： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 49
中文關鍵詞： 馬克士威妖 、Szilard引擎 、熱力學第二定律
外文關鍵詞： Szilard engine, the second law of thermodynamics
相關次數： 點閱：68  下載：10 
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    • 在熱力學的發展過程中，馬克士威妖在對熱力學第二定律的討論中扮演了重要的角色。而Szilard引擎是其中一種廣為討論的馬克士威妖的思考實驗。這個模型可以幫助我們討論關於在測量過程中熵的減少和資訊的流出的關係。此外，在整個循環過程中所吸收的熱及做出的功也可以被計算。不過，對於量子的Szilard引擎，有一個問題。如果一開始的狀態並不是處於熱平衡態，而是處於糾纏態。對整個熱力學過程有怎樣的差別呢?
    在這篇文章中，我會引入量子版本的Szilard引擎。並且考慮幾種不同的雙粒子糾纏初始態。在用到無限位能井的本模型中，我將會說明吸收的熱跟流出的資訊並無關係，但依然遵守熱力學第二定律。我也將會在量子Szilard引擎中引入一些特例的初始態。這些態具有類似於費米子或是波色子的性質，並可以用以分析全同粒子在引擎中的效應。

    In the development of thermodynamics, Maxwell's demon plays an important role in discussing the second law of thermodynamics. Also, Szilard engine is one of the widely discussed thought experiment for Maxwell's demon. This model helps us to find the relation between the entropy decrease and information ow in detection process. Besides, heat and work transfer in processes in Szilard engine could be clarified. But there is a problem in a quantum Szilard engine: Assuming the initial state in Szilard engine is not the thermalized state, but the entangled state. What's the difference between those thermal processes?
    In this thesis, I will investigate the quantum version of the Szilard engine and consider several cases of two-particle entangled initial states with different information flow. In the model of infinite quantum well, I will show that there is no relation between the absorbed heat and information ow. However, the second law of thermodynamics still holds. I'll also use some special initial states to analyze the identical particle effect which mimics the bosonic and fermionic properties of the quantum states in the quantum Szilard engine.

    List of Table v List of Figures vi 1 Introduction to Szilard engine 1 1.1 Maxwell's demon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Szilard engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Model for Szilard engine with measuring error rate . . . . . . . 3 1.3 Landauer's principle and information erasure . . . . . . . . . . . . . . . 5 1.4 Some basics in quantum informaiton . . . . . . . . . . . . . . . . . . . 6 1.4.1 Positive operator valued measurement (POVM) . . . . . . . . . 7 1.4.2 Entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Quantum consideration of Szilard engine . . . . . . . . . . . . . . . . . 9 1.6 Main issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Isothermal Multi-particle Szilard engine 13 2.1 The Szilard process and the work extraction . . . . . . . . . . . . . . . 13 2.2 Two-particle example: e ect of indistinguishability . . . . . . . . . . . 16 2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Single-particle quantum Szilard engine 19 3.1 Inserting the partition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Szilard engine with two-particle entangled initial states 27 4.1 A direct product state as the initial state . . . . . . . . . . . . . . . . . 28 4.2 Not-aligned entangled initial state . . . . . . . . . . . . . . . . . . . . . 30 4.3 Permutationally symmetric entangled initial state . . . . . . . . . . . . 33 4.4 Second law of thermodynamics in these examples . . . . . . . . . . . . 37 4.5 Concurrence of these three states . . . . . . . . . . . . . . . . . . . . . 38 4.5.1 Not-aligned entangled initial state . . . . . . . . . . . . . . . . . 39 4.5.2 Aligned entangled initial state . . . . . . . . . . . . . . . . . . . 39 4.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.6.1 Not-aligned entangled initial state . . . . . . . . . . . . . . . . . 40 4.6.2 Aligned entangled initial state . . . . . . . . . . . . . . . . . . . 42 5 Discussion and Summary 45 5.1 E ect of insertion process . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Method of my work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Special cases of bosons, fermions and classical particles . . . . . . . . . 46 5.3.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.3 Direct product states . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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