在量子力學的描述下，一個封閉系統的動力學完全其哈密頓算符(Hamiltonian)決定，如此動力學之特性為保持機率總和不與時變。然而，當我們想要暸解一個量子系統如何達成熱平衡時，描述系統亂度常用的物理量-熵(entropy)，將不隨時間演化。如此矛盾的結果可歸因於封閉系統的特性，因此，這邊將會把討論的範疇延伸到開放式系統(open system)，在此討論下系統的動力學將不再由哈密頓算符決定。
本論文先討論熱平衡熵(thermal equilibrium entropy)在近代物理中，如何正確的和系統模態數(mode number/counting)做連結。模態數的定義會直接引響到後續熱力學量的特性，尤其是巨觀性(extensive)的一致性，在文內會深入比較著名不同的定義下的結果。
在第四和五章會討論熵在開放式系統下的動力學，由於目前鮮少有實際可解析的例子，我們提出來一個熱力學簡化後的定性的模型，稱之為"刺蝟形安排系統"(hedgehog model)，此系統由數環境自旋和一個中心自旋構成，我們只關心中心自旋和環境對其影響的動力學 。此系統熵在時域上解析，此外，不同的系統參數可構成分類為不同的解析族，解析族可由對應的熱力學極限曲線描述，代表著其解析解有普世特性，最終，這顯示我們更應思考熱平衡在量子力學下的表述。

Under the description of quantum mechanics, the dynamics of an isolated system is fully determined by Hamiltonian, which preserves the probability as a result. However, when one tries to understand how a quantum system approaches thermal equilibrium, entropy, as a mean to characterize the level of disorder, would stay constant in time. Such a paradox can be attributed to the assumption of isolated system at the beginning, hence, it is necessary to consider open system, where the dynamics is no longer determined by Hamiltonian.
The thesis discusses how the thermal equilibrium entropy relates to mode counting in modern physics perspective, here we would compare three different ways of counting and look into thermodynamics quantities derived from each method, especially those expected to be extensive.
Next we’d study the dynamics in the open quantum system, it is rare to see the existence of exact solvable example. Here we propose hedgehog model that quantitatively describes the thermalized process in general. Numerous number of spin units and central spin comprise the global system but we only focus on central spin and how environment influences the central spin. This model features the analytic solution in time domain, in addition, systems with different parameters can be categorized into different groups, each group is corresponding to a unique thermodynamics limit curve, this indicates the universal property of solution. In the end, this model implies we have to take a deeper thought about the meaning of thermal equilibrium in view of quantum mechanics.

摘要
Abstract
目錄
1. Introduction…………………………………………………………………………….3
2. The formalism of open quantum system………………………………..6
2.1. Terminology……………………………………………………….….……………..8
2.1.1. Isolated, closed and open system……………………….……………..8
2.1.2. Environment, reservoir and heat bath…….………….……………...8
2.2. Density matrix……………………………………………………….……………..9
2.3. Lindblad equation…………………………………….………….……………..12
2.3.1. Dynamical mapping…………………………….…………….……………..12
2.3.2 Kraus decomposition………………………………………….………..…..13
2.3.3 Lindblad form…………………………………………………….……………..15
2.4. Example………………………………………………………..…….……………..16
3. Entropy and mode counting in equilibrium system…………….…..19
3.1. Entropy as mode counting….………………………………….……….…..20
3.2. Quantum entropy in thermal equilibrium….……………..………....24
3.3. Examples for three mode counting method….………….………...29
3.3.1. Simple harmonic oscillator….…………………………………………...29
3.3.2. Binary system….……………………………………………………….……..34
3.4. Numerical demonstration….……………………………………..………..35
4. Hedgehog spin model…………………………………………………………...39
4.1. Introduction and physical implication………………………………....39
4.2. Exact solution to the hedgehog model ……………………….……..43
4.3. Multiplicity function………………………………………………………..…..37
5. Hedgehog model and open quantum system…………………….…..56
5.1. Reduced density matrix of central spin…………………….…….…..56
5.2. Rescaling and thermodynamics limit…………………………………..62
5.3. The relaxation dynamics……………………………………………..……..67
5.4 Thermalization of x+ state…………………………………………………..70
5.5 Conclusion…………………………………………………………………………..74
Appendices
A. Supplemental calculation of hedgehog model……………….……..77
A.1. The hedgehog model with magnetic field…………………………..78
A.2. Side spin in the hedgehog model………………………………..……..87
B. Useful relation during calculation………………..………………………..91
C. Short discussion on relaxation in hedgehog model………..……..93
Bibliography………………………………………………………………………………95

(Selected)
1. R. K. Pathria & Paul D. Beale Statistical Mechanics, 3rd edition Academic Press(2011)
2. J. Loschmidt, J. Sitzungsber. der kais. Akad. d. W. Math. Naturw. II 73,128(1876)
3. R. P. Feynman & F. L. Vernon The theory of a general quantum system interacting
with a linear dissipative system Ann. Phys.(N.Y.) 24, 118
4. Mark Srednicki Chaos and quantum thermalization Phys. Rev. E 50, 888(1994)
5. A.,O., Caldeira & A.,J., Leggett Path integral approach to quantum Brownian motion
Physics A 121,587-616(1983)
6. Heinz-Peter, Breuer Non-Markovian generalization of the Lindblad theory of open
quantum systems Phys. Rev. A 75, 022103(2007).