Entropy and Probability

Let, s = Entropy additive

W = probability multiplicative

At equilibrium: s, W maximum

Let us consider two system of probabilities or disorder number W₁ and W₂ and entropies s₁ and s₂ .

s1 = f (W1)

(5.250)

s2 = f (W2)

(5.251)

For a single system of the two together

s = s1 + s2

(5.252)

W = W1.W2

(5.253)

∴ Finally,

(5.254)

or,

(5.255)

Differentiating w.r.t. W₁ and W₂,

(5.256)

(5.257)

Dividing Eq. (5.120) by Eq. (5.121), we get

(5.258)

or,

(5.259)

each side is a function of one variables, hence

(5.260)

or,

(5.261)