Stochastic Dynamics in Systems with Two Thermal Baths

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題名:	Stochastic Dynamics in Systems with Two Thermal Baths
作者:	林于軒;Lin, Yu-Syuan
貢獻者:	物理學系
關鍵詞:	隨機動力學;Stochastic Dynamics;Time Irreversibility;Langevin Equation;Nonequilibrium steady state;Fokker-Planck equation;Detailed Balance Violation
日期:	2025-06-16
上傳時間:	2025-10-17 11:52:37 (UTC+8)
出版者:	國立中央大學
摘要:	我們研究了對處於空間非均勻溫度場中的布朗粒子的隨機動力學。分析具有溫度不連續與外加位能共存環境下的朗之萬方程(Langevin equation)和福克-普朗克方程(Fokker-Planck equation)，我們提出了在開放邊界中的時間不可逆性以及離散時間數值模擬中違反詳細平衡的現象。我們的理論架構修正了傳統方法，引入有效溫度分佈，成功地修正了連續時間下福克-普朗克分析的預測，與數值模擬之間的差異，尤其在那些標準理論假設失效的溫度不連續處附近。我們透過使用歐拉法(Euler Method)進行數值模擬，展示了在週期性邊界條件下非平衡穩態通量的出現。這與熱力學中廣泛討論的布提克-蘭道爾馬達(Buttiker–Landauer motor)相似。這項研究的一個關鍵發現是離散時間模擬中存在的記憶效應，特別是在具有陡峭溫度不連續處，粒子會保留前一步的信息——這一效應在連續時間的福克-普朗克方程表述中並不存在。我們證明，透過在相應的福克-普朗克方程中以有效平滑化的溫度分佈取代原始的不連續溫度分佈，可以很好的解釋這些記憶效應，從而得到與模擬數據相似的機率分佈。此外，我們研究了在二維系統中的慣性動力學，並計算了從非平衡穩態(Non-equilibrium steady state)到有效平衡的慣性誘導現象。透過對穩態概率分佈(Probability Steady State Distribution)、熵產生(Entropy Production)、熱通量(Heat Flux)和角動量(Angular Momentum)的分析推導，研究慣性效應如何促進空間維度間的有效能量交換，使不同方向的有效溫度趨於其算術平均值。這種收斂表現為機率分佈傾斜角的特徵變化，在極限下接近45度 ($\tan\phi =1 $)。數值模擬證實了這些理論預測，同時也發現了歐拉法在低阻尼區域的模擬極限，在此區域傳統歐拉方法變得不穩定，且能量不守恆，儘管溫度梯度仍然存在，但需要採用高階方法才能準確捕捉系統向平衡態行為的過渡。;In this study, we investigate the stochastic dynamics of a Brownian particle, focusing on nonequilibrium steady states induced by temperature inhomogeneity or anisotropy. We first consider systems with sharp temperature discontinuities under external potentials. Our analysis shows that time-irreversibility, or detailed balance breakdown, is evident in discrete-time simulations. We introduce modified effective temperature profiles that reconcile Fokker-Planck predictions with numerical results, especially near thermal interfaces where standard assumptions fail. A key finding is the emergence of time irreversibility in discrete-time simulations with temperature discontinuities in open systems—a phenomenon absent in continuous-time formulations. The effective temperature profiles in our modified Fokker-Planck approach incorporate a memory effect, a key source of time-irreversibility. We demonstrate that any discontinuous temperature interface inherently violates detailed balance, highlighting the system’s nonequilibrium nature. By replacing discontinuous profiles with smoothed, effective ones, we obtain probability distributions that closely match simulation data, resolving discrepancies with theoretical predictions. We further examine inertial dynamics in two-dimensional systems under anisotropic harmonic potentials. The system is coupled to anisotropic heat baths, leading to unequal random kicks along different axes. In the large-inertia limit, the system exhibits effective thermal equilibrium. Analytical results for steady-state distributions, entropy production, heat currents, and angular momentum reveal that strong inertia enhances energy exchange between spatial dimensions, causing effective temperatures to converge toward their arithmetic mean. This convergence is reflected in a characteristic 45-degree tilt of the probability distribution. Simulations confirm these results but also expose numerical challenges in low-damping regimes, where higher-order integrators are needed to capture the system’s approach to equilibrium-like behavior despite persistent anisotropy.
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