Problem 1 [10 points]
In our first lecture 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 εi = (1/2)mvi2 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>=∫dε P(ε) ε, and the standard deviation σε of the energy of a single particle is given by σε2 = <(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 = Σi εi. We will assume that the particles are identical and weakly interacting, so that the εi can be regarded as independent, identically distributed, random variables (i.e. there are no correlations among the εi). In this model, since the εi are independent random variables, then E is also a random variable (physically, you can think that the total energy can fluctuate because there can be energy input from some surrounding thermal bath, rather than E being fixed by energy conservation if the system was in perfect isolation from its surroundings)
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 σE and the standard deviation of the single particle energy σε. Show that the relative fluctuation σE/<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.