600E+008 500E+008 N=32 4.00E+008 300E+008 2.00E+008 1.00E+008 0.00E+000 048121620242832
0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 0.00E+000 1.00E+008 2.00E+008 3.00E+008 4.00E+008 5.00E+008 6.00E+008 N = 32 k
When n is even, the weight is maximum atk=N2, k=N/2 =N!/N/2) When n is odd the maximum is at k= n2+ 1 As n increases, the maximum becomes sharper The weight for k=N/4 is WkeNA=N!/I(N/4)!(3N/4)! The ratio of the two weights r(n)=WkeN?/WkeN/4 is equal to (N4)!(3N4)!/|(N2)2 N|4816322566.0x1023 R(N)1.52.5715713.5x10142.6x103e+2
When N is even, the weight is maximum at k = N/2, Wk=N/2 = N! / [N/2)!]2 . When N is odd, the maximum is at k = N/2 1 As N increases, the maximum becomes sharper! The weight for k = N/4 is Wk=N/4 = N! / [(N/4)! (3N/4)!] | N | 4 8 16 32 256 6.0 x 1023 |R(N) | 1.5 2.5 7.1 57.1 3.5 x 1014 2.6 x 103e+22 The ratio of the two weights R(N) Wk=N/2 / Wk=N/4 is equal to (N/4)! (3N/4)! / [(N/2)!]2
Therefore, for a macroscopic molecular system (N-102), there are dominating configurations so that the system is almost always found in the dominating configurations, i e Equilibrium Dominating Configuration: Equilibrium Configuration
Therefore, for a macroscopic molecular system ( N ~ 1023 ), there are dominating configurations so that the system is almost always found in the dominating configurations, i.e. Equilibrium Dominating Configuration: Equilibrium Configuration
To find the most important configuration, we vary (ni to seek the maximum value of w. But how One-Dimensional Function: F(x)=x df/dx=0 Ⅹ
To find the most important configuration, we vary { ni } to seek the maximum value of W. But how? One-Dimensional Function: F(x) = x2 dF/dx = 0 X
Two-Dimensional Case: for instance, finding the minimum point of the surface of a half water melon F(x, y) OF/0x=0, aF/Oy=0 Multi-Dimensional Function: F(X, x2,.,X, GF/0x;=0,1=1,2,,n
Two-Dimensional Case: for instance, finding the minimum point of the surface of a half water melon F(x,y). F/x = 0, F/y = 0. Multi-Dimensional Function: F(x1 , x2 , …, xn ) F/xi = 0, i = 1,2,…,n