Problem Set 0

Problem Set 0

DQ 0 -- Due Tuesday, January 18, by 5pm

In our Discussion Session on January 12, we watched simulations of a gas of N=8 particles and a gas of N=1024 particles, and came to the conclusion that fluctuations are reduced the bigger N becomes. Here you will do a simple calculation that will illustrate this. Instead of looking at particle positions, as we did in the simulations, here you will consider the total energy of the gas.
Let ε_i = (1/2)mv_i² be the kinetic energy of particle i. Let us assume that this energy is described by a probability distribution P(ε). The average energy of a single particle is then <ε>, and the standard deviation σ_ε of the energy of a single particle is given by σ_ε² = <(ε - <ε>)²> = <ε²> - <ε>². The standard deviation is a measure of the width of the probability distribution about its average.
The total energy of the gas is then E = Σ_i ε_i. We will assume that the particles are identical and weakly interacting, so that the ε_i can be regarded as independent, identically distributed, random variables (i.e. there are no correlations among the ε_i).
With that assumption, find a relation between the average total energy <E> and the average single particle energy <ε>, and a relation between the standard deviation of the total energy σ_E and the standard deviation of the single particle energy σ_ε. Show that the relative fluctuation σ_E/<E> vanishes as N goes to infinity.

Note: If you want to play with the simulations we did in the Discussion Session, you can find the programs on the website of the text by Gould and Tobochnik, available here. Just click on the link for Chapter 1 and you will see the link to the simulation programs.

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Pictures -- Due Thursday, January 20, by 5pm.

Upload a head shot of yourself to Blackboard, Zoom, and Slack, as described at the bottom of the Communications page. This will count 3 points towards the Participation part of your class grade (the same points as each Discussion Question).