Sheet 3 Exercise 7 (home work) 2 points The value of N!

Advanced Statistical Physics SS 2008Exercises (7.05.2008) Sheet 3Exercise 7 (home work) 2 pointsThe value of N! often is estimated for large N with the help of approximative formulas.(a) Show that in first approximation,ln N! ≈ N · ln N − N. (1)(b) Show how one can replace (1) by the more exact “Stirling formula”:ln N! ≈ N · ln N + 1 ln(2πN) − N.2Hint: Start with the integral representation, n! = ∫ ∞x n e −x dx. Expand the logarithmof the integrand into a power series around its maximum. Then take the0three most important summands, exponentiate again, and derive the value of thecorresponding integral.(c) Discuss the accuracy of the two formulas.Exercise 8 (presentation) 2 pointsImagine an unlimited reservoir with equal numbers of red and white balls. Now take outN balls at random and arrange them arbitrarily on N places.(a) Determine the total number m ∗ of all possible red-white combinations and theirprobability. With a total number N w of white balls, how many configurations g(N w )are possible?(b) Determine the maximum of the distribution function g max (N w ) and discuss the behaviourof ln(g max)for N → ∞.ln(m ∗ )Exercise 9 (presentation) 2 pointsConsider a system of N non-interacting Ising spins in a magnetic field. Their internalenergy is given by U = −(N ↑ − N ↓ )mB. The degeneracy function g(U, N) is a binomialdistribution, which for large N can be approximated with the help of Stirling’s formula.

Derive the entropy function S(U, N) = k B ln g(U, N) in this approximation, and comparethe result with the postulates for the entropy.Are there postulates which are violated, and if so, for which energies? To study thesequestions, determine the temperature as a function of energy. What is the accessible energyrange, and which energies can be obtained in thermodynamic equilibrium? What is thetemperature at the minimum energy? Is our derivation using the Stirling approximationvalid at this energy? Try to compute the temperature at the energy minimum as a differencequotient between the ground state and the first excited state (without using the Stirlingapproximation), and show that it converges to zero in the thermodynamic limit (N → ∞).Exercise 10 (home work) 3 pointsIn combinatorial exercises combinations of k elements are built out of a defined set of ndistinguishable elements (n, k ∈ N). By the following restrictions to the combinations andtheir arrangements four different kinds of exercises can be realized.I) Repetition of the same elements is possible.II) No repetition of the same elements is allowed, all elements have to be different.III) The arrangement of the elements is accounted for (“variations”).IV) Combinations that are rearrangements of each other are considered as the same(indistinguishable ordering).The number of possible combinations N c for k elements out of a set of n distinguishableelements (k, n ∈ N, k ≤ n) then is given as follows:1) Combining II + III,N c = n · (n − 1) · · · · · (n − k + 1).2) Combining II + IV,N c =n · (n − 1) · · · · · (n − k + 1)k!=( nk).3) Combining I + III,N c = n k ,4) Combining I + IV,( ) n + k − 1N c =.k

Prove the validity of these statements, e.g. by induction with respect to k. For 4),the following relation might be useful:∑n−1( ) j + k=kj=0( ) n + kk + 1(2)