1、分布函數(shù)理論基礎(chǔ),引言,分布函數(shù)及其統(tǒng)計力學(xué)基礎(chǔ),定義一個函數(shù),為流體的徑向分布函數(shù)，物理意義是：與一指定分子相距r處，流體分子的局部密度與平均密度之比,分布函數(shù)的統(tǒng)計力學(xué)基礎(chǔ),The h-body correlation function,Some basic points:,in the absence of external potential, the n-body distribution function are independent of the choice of origin. The distribution function is isotropic In a unif

2、orm fluid with no external forces, all distribution functions are translational invariant.,The properties of distribution functions,Recursion relation,Normalization relation,The ideal gas limit,Limit of large separation,Other correlation function,1. Total correlation function:,2. Background correlat

3、ion function:,Revisiting the meaning of g (r),Given a molecule in the differential volume dr1 at a position r1, the probability of finding any one of the N-1 remaining molecules in dr2 at r2 is,By a variable transformation, the probability that a specific molecule is,We let the given molecule move o

4、ver the entire volume. Integration of the above equation,This equation gives the probability that a second molecule is to be found in a differential volume at a position from the central molecule,For g(2) independent of orientation, we could form a spherical shell of thickness dr at a distance r fro

5、m the central molecule,This expression gives the number of molecules in the spherical shell of thickness dr at a distance r from the central molecule. Integration over a coordination distance L gives the so-called coordination number,N(L) counts the number of molecule inside a sphere of radius L sur

6、rounding the central molecule. Namely, it is a counter of neighbor.,徑向分布函數(shù)與流體熱力學(xué)性質(zhì),1. 流體的能量,已知在正則系綜中,系統(tǒng)的能量可用下式表示,其中正則配分函數(shù)可表示為,對于N個單原子分子構(gòu)成的經(jīng)典流體,假定流體的勢能等于各分子對勢能之和,上式就是能量與徑向分布函數(shù)的關(guān)系,也稱為能量方程,流體的壓力,在正則系綜中,引入分布函數(shù):,與Virial狀態(tài)方程相比, Virial系數(shù)為,流體的化學(xué)位:,We denote the Helmholtz free energy of an N-body system as,For system with larger N, we can approximate the derivate by a difference,流體化學(xué)位,For isotropic fluids,多組份流體的熱力學(xué)函數(shù),以A,B二組分體系為例，設(shè)A,B組分的粒子數(shù)分別為NA和NB，,上式的構(gòu)型積分：,混合物的徑向分布函數(shù),流體的總能量,壓力：,分布函數(shù)的理論計算,由上述可知，求取熱力學(xué)性質(zhì)關(guān)鍵在于獲得徑向分布函數(shù) g(r)，一方面 g(r) 可由實驗獲得；另一方面可根據(jù)分子間力的知識去理論計算或采取分子模擬的方法求得,1. Kirkwood 積分方程,2. Yvon-Bor