一維非簡諧振子之比熱

Abstract

一維系統的比熱可藉由兩種方法求得，用配分函數或機率密度函數來計算。在本論文中，我們針對三種不同的非簡諧位能，用數值的方法計算達熱平衡的一維非簡諧陣子之比熱。 從古典統計力學來看，藉由配分函數的方法求比熱是標準的方法；另一方面我們可由隨機變數的方式設造一機率密度函數，用這機率密度函數算比熱。這方法能給我們更多的資料，使我們更了解比熱的物理意義。 接著分別計算對稱單井、對稱雙井和非對稱單井位能的比熱，我們發現對於對稱雙井以及非對稱單井位能，低溫時的比熱有個極大值；比較此三種位能的比熱，我們用位能的曲率去解釋此極大值產生的原因。而在非對稱單井以及對稱雙井中，我們發現位能在抹一範圍其曲率為負值。 最後，我們認為當比熱對溫度的曲線有個極大值時，可能伴隨著位能曲線在抹範圍內的曲率為負值。

The specific heats of one-dimension systems are obtained with two different methods, which are achieved in terms of the partition function and the probability density function, respectively. In this thesis, we consider three kinds of anharmonic potentials and calculate numerically the specific heat of a particle moving in a potential in one dimension and also at thermally equilibrium with a heat bath. From the views of classical statistical mechanics, the method from the partition function is standard, direct and simple. In addition, we construct the probability density function of a stochastic variable. Since we can calculate < V > / T and < V > / T , this method can gives us more information. Namely, we can realize the specific heat more deeply. For a symmetric double-well potential, V , we find that at very low temperature the curve of the specific heat has a peak. In addition, we also respectively calculate the specific heats of a symmetric single-well potential, V , and an asymmetric single-well one, V . After comparing the specific heats of these three potentials, we interpret the specific heats of them from the view of the geometry of these potentials. In V and V , we find that there is a region in the potential where the potential curvatures are negative. Finally we suggest that the peak of the specific heat of a one-dimension system, once it appears, is associated with the existence of negative curvature of the potential.