Microcanonical Ensembles of a three level system: - Consider a three-level system with

N

total distingishable particles with each level consisting of

N_(i)

total particles and

\\\\sum_(i=1)^3 N_(i)=N

, with the corresponding energies given by

\\\\epsi lon_(i)

(you may assume that

\\\\epsi lon_(1)

).\ (i) Using the microcanonical ensemble show that the entropy

S

can be expressed as,\

S=-k_(\\\\beta )N\\\\sum_(i=1)^3 p_(i)log(p_(i))

\ where

p_(i)

is the probability associated with finding the particle in state

i

. Comment on the similarity of this expression to the Shannon entropy discussed in class.\ (ii) Calculate the internal energy

E(T,N)

.\ (iii) Calculate the heat capacity

C(T)

.\ (iv) Plot it and analyze the

T->0

and

T->\\\\infty

limits. Explain the physics associated with these limits.