本篇論文主要在討論量子系統在量子純態的環境下， 系統熱力學第二定律及漲落定理與時間的關係。本篇論文以西元2016年\space Eiki Iyoda ，Kazuya Kaneko\space ， Takahiro Sagawa\space 的文章$^{(24)}$為主軸來討論上述的問題。假設整體系統由系統\space (S)\space 和量子純態\space (B)\space 的環境所構成並且彼此之間有交互作用\space (I)\space ，接著整體系統經由么正演化後，利用\space Lieb-Robinson Bound\space 和\space Eigenstate-Thermalization Hypothesis(ETH)\space 兩個重要的概念來討論，系統的熱力學熵及漲落定理隨著時間變化的關係。在論文的第五章中，數值模擬了整體系統為一維硬核玻色子系統的情況，並且根據不同局部交互作用力的大小，來分析系統的熱力學熵和漲落隨著時間的變化。而模擬的結果顯示出，當系統內局部的交互作用力越大時，短時間內系統的熱力學熵會滿足熱力學第二定律，但隨著時間越長後，系統熵的變化會開始隨著時間做震盪。同時，當局部的交互作用力越大時，短時間內系統的漲落定理(熱漲落)會成立，但隨著時間的增長，系統就會開始不滿足漲落定理，並且顯示出強烈的量子漲落。

參考文獻

(1)Clausius, R. (1865). The Mechanical Theory of Heat with its Applications to the Steam Engine and to Physical Properties of Bodies. London John van Voorst, 1 Paternoster Row. MDCCCLXVII.

(2)Loschmidt, Sitzungsber. Kais. Akad. Wiss. Wien, Math. Naturwiss. Classe 73, 128142 (1876)

(3)Evans DJ, Searles DJ. 1994. Equilibrium micro-states which generate 2nd law violating steady-states. Phys. Rev. E 50:1645 1648)

(4)Evans DJ, Searles DJ. 2002. The fluctuation theorem. Adv. Phys. 51:1529 1585

(5)Searles DJ, Evans DJ. 2004. Fluctuations relations for nonequilibrium systems. Aust. J. Chem. 57:1119 1123

(6)Jarzynski C. 1997. Nonequilibrium equality for free energy dierences. Phys. Rev. Lett.

(7)Jarzynski C. 1997. Equilibrium free-energy differences from non-equilibrium measurements: A master-equation approach. Phys. Rev.

(8)Jarzynski C. 2000. Hamiltonian derivation of a detailed fuctuation theorem.J. Stat. Phys.

(9)Crooks GE. 1998. Non-equilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90:1481 1487

(10)Crooks GE. 1999. Entropy production fluctuation theorem and the non-equilibrium work relation for free energy differences. Phys. Rev. E 60:2721 2726

(11)G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995).

(12)J. Kurchan, J. Phys. A 31, 3719 (1998).

(13) J. C. Maxwell, Letter to P. G. Tait, 11 December 1867 in Life and Scientic Work of Peter Guthrie Tait, C. G. Knott (ed.), Cambridge University Press, London, p.213
(1911).

(14) L. Szilard, Z. Phys. 53, 840(1929), English transla- tion by A. Rapport and M. Knoller, Behavioral Science 9:301, 1964.

(15) C. E. Shannon, Bell Sys. Tech. Journal 27, 379 (1948).

(16)R. Landauer Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development ( Volume: 5, Issue: 3, July 1961 )

(17)Sagawa, T. Ueda, M. Second law of thermodynamics with discrete quantum feedback control. PRL 100, 080403 (2008)

(18)Nature Physics 11, 131139 (2015) doi:10.1038/ nphys3230Lett. 100, 08040 (2008).

(19)E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics, 391, 1965

(20) Von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. ISBN 3-540-59207-5

(21)Sagawa, T. Second law-like inequalities with quantum relative entropy: An introduction. Lectures on Quantum Computing, Thermodynamics and Statistical Physics 8, 127 (2012).

(22)Von Neumann, J. Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics. Euro. Phys. J. H 35, 201 (1929); English translation in Preprint at <http://http://arxiv.org/abs/1003.2133>(2010).

(23)Linden, N., Popescu, S., Short, J. S. Winter, A. Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009).

(24)Eiki Iyoda, Kazuya Kaneko, Takahiro Sagawa Emergence of the informationthermodynamics link and the fuctuation theorem for pure quantum states arXiv:1603.07857v2 (2016)

(25)Lieb, E. H. Robinson, D. W. The Finite Group Velocity of Quantum Spin Systems. Commun. Math. Phys. 28, 251-257 (1972).

(26)Hastings, M. B. Koma, T. Spectral Gap and Exponential Decay of Correlations. Commun.arXiv:math-ph/0507008(2006)

(27)Mark Srednicki(1994)."Chaos and Quantum Thermaliz-ation". Physical Review E. 50 (2),888

(28)Deutsch, J.M. (February 1991). "Quantum statistical mechanics in a closed system". Physical Review A. 43 (4): 20462049

(29)Biroli, G., Kollath, C. L'auchli, M. L. Eect of Rare Fluctuations on the Thermalization of Isolates Quantum Systems. Phys. Rev. Lett. 105, 250401 (2010).

(30)Mori, T.Weak eigenstate thermalization with large deviation bound. Preprint at http://arxiv.org/abs/1609.09776(2016).

(31)Tameem Albash, Daniel A. Lidar, Milad Marvian, Paolo Zanardi Fluctuation theorems for quantum processes PHYSICAL REVIEW E 88, 032146 (2013)

(32)Jorge Kurchan A Quantum Fluctuation Theorem arXiv:cond-mat/0007360 (2000)

(33)Peter Talkner, Peter H¨anggi The Tasaki-Crooks quantum fluctuation theorem arXiv:0705.1252 (2007)

(34)A Sharp Fannes-type Inequality for the von Neumann Entropy,arXiv:quantph/0610146 (2006)

(35)Hai Tasaki , On the local equivalence between the canonical and the microcanonical distributions for quantum spin systems arXiv:1609.06983 (2016)

(36)Rigol, M., Dunjko, V. Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854-858 (2008).

(37)Rigol, M. Breakdown of Thermalization in Finite One-Dimensional Systems.Phys. Rev. Lett. 103, 100403 (2009).