在這篇論文中，我們介紹了我們的非微擾重整化理論，並且將其應用於研究在強耦合下的量子布朗運動的量子熱力學。我們應用相干態表示下的高斯積分和Feynman-Vernon影響泛函對熱庫的所有狀態自由度嚴格地求跡，並解析地解出了布朗粒子在平衡態和非平衡態的約化密度矩陣。我們研究了熱庫對布朗粒子的約化哈密頓量和約化分布函數的重整化效應，指出了熱庫與布朗粒子的線性耦合將產生重整化哈密頓量中的粒子對交互作用(pairing interaction)。更重要的是，我們發現了在強耦合下，系統-熱庫耦合將引發熱庫本身的狀態發生不可忽略的變化，必須將熱庫的變化也納入考量才能求得正確的約化分布函數。此外我們也研究了在非平衡態中，布朗粒子的非馬可夫動力學和其內能與熵的時間演化，並從中討論了系統-熱庫耦合對溫度的重整化效應。透過使用正確的重整化哈密頓量和約化分布函數，我們解決了文獻中關於熱容在不同定義下的矛盾結果、以及異常的負熱容等爭議。

In this thesis, we introduce our non-perturbative renormalization theory and apply it to the study of quantum thermodynamics of quantum Brownian motion under strong coupling. Utilizing Gaussian integrals in the coherent state representation and the Feynman-Vernon influence functional, we rigorously trace out all states of the reservoir and analytically solve the reduced density matrix of the Brownian particle in both equilibrium and non-equilibrium states. We examine the renormalization effects of the reservoir on the reduced Hamiltonian and the reduced partition function of the Brownian particle, highlighting how the linear coupling between the reservoir and the Brownian particle induces pairing interactions in the renormalized Hamiltonian. Importantly, we find that in strong coupling, the system-reservoir interaction leads to significant changes in the state of the reservoir itself, necessitating the consideration of these changes to obtain the correct reduced partition function. Furthermore, we study the non-Markovian dynamics of the Brownian particle and the time evolution of its internal energy and entropy in non-equilibrium states, which allowed us to discuss the renormalization effects of the system-reservoir coupling on the temperature. By utilizing the correctly renormalized Hamiltonian and reduced partition function, we resolve discrepancies in the literature regarding the heat capacity under different definitions, as well as the issue of anomalous negative heat capacity.

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