33-765 — Statistical Physics
Department of Physics, Carnegie Mellon University, Spring Term 2020, Deserno
Problem sheet #4
12. The origin and limitations of classical error propagation (5 points)
Consider a collection of random variables X= (X1, X2,...,Xn), from which we calculate a function of interest, F(X).
Assume we know all expectation values hXiiand even all covariances Cij := Cov(Xi, Xj) = (Xi− hXii)(Xj− hXji).
1. Taylor-expand Fto second order around hXi. Now take the average and show how hF(X)idiffers from F(hXi).
2. For the special case of n= 1 and a convex F, show that your result is consistent with Jensen’s inequality!
3. The variance of Fis given by σ2
F=[F(X)− hF(X)i]2. Simplify this by replacing Fwith its first order Taylor
expansion. Show further that if all Xiare uncorrelated, you end up with the “standard” formula for error propagation!
Note: if the random variables
Xi
are correlated, the more general erro r propagation formula is not necessarily much more
complicated. For inexplicable reasons, this hardly ever seems to be taught. Well, now you know. You’re welcome.
13. Generalized geometric and arithmetic mean (3 points)
Let {xi}i=1...N be a set of Npositive real numbers and {pi}be a probability distribution. Prove the following inequality
between a generalized arithmetic mean and a generalized geometric mean:
N
X
i=1
pixi≥
N
Y
i=1
xpi
i.(1)
14. Pumping gas (4 points)
When Alice needs to go to the gas station, she always purchases gasoline for a fixed amount of money. When Bob needs to get
gas, he always fills up the whole tank. Considering that gas prices fluctuate, show that these two strategies differ economically!
Try to estimate how much better the cheaper strategy is.
Hint: assume that whenever Alice or Bob go to the gas station, the “price per mile”,
pi
, is a random variable with some unknown
distribution. Calculate the total fuel cost of Bob after
N
visits to the gas station, and the total number of miles Alice reaches
after
N
visits. Then calculate the effective average price per mile after
N
visits for Alice and Bob. Now remember Jensen.
15. Work done by a moving piston—valuable lessons from kinetic theory (8 points)
In class we studied the pressure exerted by an ideal gas onto a hard wall within the framework of kinetic theory. Here we want
to extend these thoughts and contemplate what happens if the wall (let’s now call it a “piston”) moves against the gas.
1. Assume the piston moves towards the ideal gas with a constant velocity u. Using kinetic theory, and contemplating the
choice of a clever frame of reference, show that the pressure Pp, which the gas exerts onto the piston, is given by
Pp=P(1 + ˜u2)1 + erf ˜u
√2+r2
π˜ue−˜u2/2=P1 + 2r2
π˜u+ ˜u2+O(˜u3),(2)
where Pis the pressure of the idea gas, v= 1/√βm is the roo t mean square velocity of the particles in one direction, and
˜u=u/vis the scaled piston velocity. This yucky expression comes from an integral which you can use MATH EM ATIC A
to solve and expand. I care much more about you being able to explain carefully, what is the correct integral to start with.
2. Calculate the rate ∆E/∆tat which the piston adds energy to the gas, and show that it vanishes in the limit u→0.
3. We (usually) do not care ho w long a piston moves, but by how much. Calculate the total energy change that h appens when
a piston compresses the gas by a volume ∆V, while moving at velocity u. Show that in the limit u→0this reduces to
the well-known expression ∆E=P∆V(which—great Scott!—we have thereby revealed to be an approximation).
4. If the piston performs harmonic oscillations with amplitude aand frequency ω, show that to lowest order the time-aver-
aged rate of energy change is ∆E/∆t=W aω 2/√2πv, where W=P∆Vis the equilibrium work done in one stroke.
5. In real life pistons don’t ever move infinitely slowly. But then, we usually pretend that they do. Can we really? Estimate
whether ∆E=P∆Vis a good approximation for the pistons in a car engine, which runs at about 3000 rpm!
