Homework Set # 5 - Due October 9

1. Consider a system that can be in any of the following 5 states: 0, +, -, ++, and - - with energies E(0)=0, E(+)=E(-)=ε, E(++)=E(- -)=2ε. If the energy E of the system is such that ε-δ<E<ε+δ, (with δ<<ε), find the probabilities P(0), P(+), P(-), P(++) and P(- -)

   a) in the microcanonical ensemble;

   b) in the canonical ensemble.

   In both cases plot the probability P(ε) vs ε. Provide the actual numerical value of each probability and check that they are normalized as expected.

2. In class we saw that the entropy is related to the probability distribution by the relationship S=-k ln<P_i> (Eq. 3.3.11) and we said that for k=1 this expression can be applied to any probability distribution.

   1. Using the expression given above calculate S for the probability distribution P(i)=δ_ij for i=1,2,...,M.

   2. Using the expression given above calculate S for the probability distribution P(i)=1/M for i=1,2,...,M. Notice that this distribution provides the maximum possible value of S (S is maximized with respect to all the probabilities) and it is called the best unbiased estimation in which all the outcomes are equally likely.

   3. Now use the expression for S to calculate the probabilities P_i for a dice which is loaded such that 6 occurs twice as often as 1. Provide the probabilities for the 6 faces of the dice. Hint: Construct an expression for the entropy S in terms of the probabilities for each face (Eq.3.3.11 with k=1) and include the known constraints on P_i using Lagrange multipliers (see Section 3.2) and use the fact that S has to be a maximum with respect to each one of the P_i.

3. Problem 3.5

4. Problem 3.20

5. Problem 3.21 (do only part (a) in the classical case).