我們透過解析法和朗之萬動力學模擬研究了非均勻溫度場中的過阻尼布朗動力學。在一維系統中，單軸溫度分佈和單勢阱陷阱可以產生單峰或雙峰穩態分佈，這些分佈可以用有效位勢來表徵。當有效勢具有雙勢阱結構時，我們推導出躍遷速率的一般近似公式。結果表明，躍遷速率越過有效勢壘遵循通常的 Kramers 指數行為。我們也考慮了雙勢阱的情況，所有理論和近似公式與數值模擬結果一致。

另一個主題是非均勻溫度場下的蘭道爾極限。這是一個具有時間相關勢的一維非平衡系統。我們的目標是找出與蘭道爾極限相關的有效溫度𝑇̃，即𝑘𝐵𝑇̃ln 2。結果表明，𝑇̃ 可能無法表示為空間溫度分佈的簡單加權平均值。在某些情況下，𝑇̃ 可能會超出整個溫度範圍。這種偏差可能源自於熱流沿溫度梯度所引起的耗散。我們進一步推廣了蘭道爾極限，應用 Jarzynski 恆等式，並以有效位勢來表徵系統。

在二維系統中，我們研究了粒子通量場的空間行為。特別地，我們給出了無粒子通量的條件以及粒子通量旋度消失的位置。此外，我們推導出了高維零通量條件下有效勢的顯式表達式。研究發現，單個非均勻溫度場可以誘導布朗粒子發生平均旋轉運動，這是由於通量場中兩對渦旋強度的不對稱所造成的，較強的渦旋決定了平均粒子旋轉的主導方向。與雙溫度系統的布朗旋轉子不同，我們提出了一種更簡單的方法來實現布朗粒子的轉動，即使轉動速率低於雙溫度系統。

We study overdamped Brownian dynamics in non-uniform temperature fields analytically and by Langevin dynamics simulations. In a one-dimensional system, a uniaxial temperature profile and a single-well trap can give rise to either unimodal or bimodal steady-state distributions, which can be characterized by an effective potential. We derive the general approximate formulas for the transition rate when the effective potential possesses a double-well structure. The results show that the transition rate over the effective barrier follows the usual Kramers' exponential behavior. We also consider the double-well potential case, and all theoretical and approximate formulas are consistent with numerical simulations.

Another topic is the Landauer limit under a non-uniform temperature field. This is a one-dimensional non-equilibrium system with a time-dependent potential. We aim to find the effective temperature $\tilde{T}$ associated with the Landauer limit, which is $k_B\tilde{T}\ln 2$. The results show that $\tilde{T}$ may not be expressed as a simple weighted average of the spatial temperature profile. In some situations, $\tilde{T}$ can exceed the entire temperature range. This deviation arises from the dissipation induced by heat flow across the temperature gradient. We further generalized the Landauer limit by applying the Jarzynski equality to characterize the system using an effective potential.

In a two-dimensional system, we studied the spatial behavior of the particle flux field. In particular, we give the conditions that there is no particle flux and the locations where the curl of the particle flux vanishes. In addition, we derived an explicit expression for the effective potential under the zero-flux condition in higher dimensions. It is found that a non-uniform temperature field can induce a mean rotational motion for the Brownian particle, which is caused by the asymmetry in the strength of two vortex pairs in the flux field, and the stronger vortex pair determines the dominant direction of mean particle rotation. Different from the Brownian rotator of the two-temperature system, we propose a simpler method to realize the Brownian particle rotation even if the rotation rate is lower than the two-temperature system.

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