Problem set 5 due Monday June 25 by 2:45 pm in class or in the mailbox.
Problem 1. Entropy and Boltzman distribution
(a) You want to produce a dice with n faces such that the a priori information someone has on the outcome
of a dice throw is minimal.
You can choose the probabilities of outcome p1 , p2 , · · ·, pn of each of the faces of the dice.
The information function is given by
I(p) =
n
X
pi log pi
i=1
What values of p will satisfy your demands ?
Hint : Don’t forget the constraint
Pn
i=1
pi = 1.
(b) Consider a physical system consisting of N particles. Each particle can be in one of n physical states, of
energies Ei . The physical system is completely described by the proportions pi = Ni /N of particles that are in
the i-th state. In the thermodynamic approximation, the number N of particles considered is very big, and we
treat the proportions pi are though they were real numbers (not necessarily rationals).
If we let this physical system evolve at fixed energy (so the mean energy, measured by the temperature T , is
constant), we know it will converge towards an equilibrium state that maximizes the entropy (entropy somehow
measures disorder, lack of information).
To find the physical state of equilibrium, we thus want to solve the following optimization problem.
We’re considering n variables p1 , p2 , · · ·, pn .
Let the entropy be given by
S(p) = −
n
X
pi log pi
i=1
We want to find the P
maximum point of S, when the proportions satisfy the relations
n
number of particles) and i=1 pi Ei = T (fixed temperature).
Pn
i=1
pi = 1 (fixed
Show that pi = Z −1 ecEi where Z and c are two real numbers.
Problem 2.
Consider the circle C given by the equation x2 + y 2 = r2 , and the line L given by the equation ỹ = ax̃ + b.
We want to find a point x on the circle C, and a point x̃ on the line L such that the distance d(x, x̃) is minimal.
Hint : Minimizing d(x, x̃)2 = (x − x̃)2 + (y − ỹ)2 is way easier.
To understand what is happening, don’t hesitate to do drawings ! Apart from intersection points if any, you
should find that at critical points of the Lagrangian, x and x̃ are both proportional to a certain vector (what
direction is it pointing towards ?).
To conclude that one of the critical point is actually a minimum point, use convexity of the Lagrangian at
that point (again don’t hesitate to do drawings to help you see which critical point is a minimum point). A neat
drawing can also replace advantageously lengthy words explanation and save you precious time.
Problem 3.
(a) Find maxima and minima of
1
f (x, y) = x3 + y 3 − 3x − 3y
on the domain D defined by x ≥ 0, y ≥ 0 and x + y ≤ 3.
Hint : Due to what the function f is, convexity/concavity is out of question to argue that critical points
are minima/maxima. Use compactness instead : argue that f has a minimum and a maximum. Find them by
computing the values taken by f at all the critical points.
Problem 4.
Consider the quadratic form q(x, y) = 3x2 − 4xy.
(a) Find the maxima and minima of q on the disk D = {(x, y), x2 + y 2 ≤ 1)}.
(b*) Use what you did in (a) to diagonalize q i.e. to write q as a weighted sum of two squares.
Hint : Think of the proof of the spectral theorem. Guess a nice formula for q and show it is q through direct
computation.
(c) Use interpretation of Lagrange multipliers to find an approximation of the maximum value taken by q on
the disk D̃ = {(x, y), x2 + y 2 ≤ 1.01)}.
Problem 5.
Find the function x(t) that minimizes the integral
Z 1 2
ẋ
+ xet dt
2
0
with boundary conditions given by x(0) = 0 and x(1) = −1.
2