Reference no: EM132525288
Statistical Physics and Thermodynamics
Statistical Mechanics: Ergodicity and Entropy
Part 1 Ergodicity and Stationary Phase Space Densities
We consider a 1-D classical harmonic oscillator with the Hamiltonian
H (p,q) = 1/(p2 + q2)
a) The energy of the one-particle system is conserved, i.e. H(p, q) = U with U being a constant. Is this an ergodic system, i.e., given some initial condition p0, q0, does the system visit all points in phase space with energy U?
Hint: No calculations required. How does the curve of constant energy and an orbit (q(t), p(t)) in phase space look like?
b) Again, the energy is conserved, i.e. H(p, q) = U . Write down the microcanonical probability density.
Hint: No calculations required. You should be able to write down the result.
c) Now, we allow fluctuations in the energy. The average energy is given by H(q, p) = U . Calculate the stationary phase space density of the 1-D classical harmonic oscillator given in equation (1) from Liouville's equation, i.e. solve
L ρ(p, q) = 0, (2)
where L is the Liouville operator given in the lecture. Express the constants in terms of U .
Hint: Use the separation Ansatz for ρ(q, p) and the conditions limq→∞ ρ = limp→∞ ρ = 0 and dpdq ρ(q, p) = 1.
d) Consider the 2-D harmonic oscillator
H(p, q) = 1/2(px2 + py2 + x2 + y2) (3)
For a fixed energy, the motion of the system is uniquely determined by the initial conditions (p(0), q(0)) = (p0, q0). Can you think of an initial condition that will generate a trajectory which clearly will NOT be ergodic?
Hint: No calculations required. Think of an initial condition, explain what kind of a curve it would generate in phase space and think about how the energy surface in phase space looks like.
Part 2: Concepts of Entropy and Temperature
We consider the random discrete motion of n particles on a lattice with N sites.
Each of the N lattice sites can be empty or occupied by one particle. If the lattice site is empty, its energy is zero. If it is occupied by one particle, its energy is ε. So the total energy of the system is given by E = nε.
a) Write down the number of possible microstates Γ(p) for a given probability p = n/N and show that for large N 1 (such that the probability p is constant), it becomes
Γ(p) = e-N (p ln p+(1-p) ln(1-p)). (4)
b) Use the result in equation (4) to compute the temperature dependence of the probability p, the total energy E and the entropy S, i.e. find the functions p(T ), E(T ) and S(T ) for N 1.
c) Discuss the two temperature limits T → ∞ and T → 0 for your results from b)?
Part 3 The Most Probable Probability
Suppose you have been hired by a fast-food company to analyze their international sales. You visit their Burger restaurants all over Europe, and determine that, on average, people are paying 8 € for a burger.
Burger Price
A Beef 5 €
B Chicken 7.5 €
C Fish 12 €
D Tofu 12.5 €
After you return, your supervisors ask about the probabilities of a customer ordering each of the four value meals. In other words, they want to know p(A), p(B), p(C) and p(D). You are horrified to realize that you did not keep the original data, and there is no time to repeat your trip. You have to make the best estimate of the probabilities p(A), p(B), p(C) and p(D) consistent with the two things you do know:
p(A) + p(B) + p(C) + p(D) = 1 (5)
5p(A) + 7.5p(B) + 12p(C) + 12.5p(D) = 8 (6)
There are a range of values of the probabilities that are consistent with what you know. However, you decide to pick the one that maximizes the uncertainty S
S = -p(A)lnp(A) - p(B)lnp(B) - p(C)lnp(C) - p(D)lnp(D). (7)
Find the probabilities p(A), p(B), p(C) and p(D) that maximize S using Lagrange multipliers. For this, first find the probabilities in terms of the Lagrange multipliers. Then, use the condition in equation (5) to express one of the Lagrange multipliers in terms of the other one. Finally, use the condition in equation (6) to find the value of the unknown Lagrange multiplier numerically, for example using Newton's method.