- #1

AlexTab

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**Summary::**Find the ratio of the number of particles on the upper level to the total number in the system.

Consider an isolated system of ##N \gg 1## weakly interacting, distinct particles. Each particle can be in one of three states, with energies ##- \varepsilon_0##, ##0## and ##\varepsilon_0##. The energy of entire system is ##E##.

The temperature is defined as ##\displaystyle T = \frac{\partial S}{\partial E}##.

I need to find the ratio of the number of particles on the level with ##\varepsilon_0## to the total number ##N## in low temperature conditions ##T \ll \varepsilon_0##.

I start with finding the statistical weight of each state of the system as ##\displaystyle W = \frac{N!}{n_{-\varepsilon_0}! n_0! n_{\varepsilon_0}!}##, where ##n_{-\varepsilon_0}##, ##n_0## and ##n_{\varepsilon_0}## are the numbers of particles in the corresponding states. Then I can find the system entropy ##S = \ln W##.

Obviously, the number of particles at each next level is much lower than at the previous one.

It seems to me that the above should be enough to solve the problem, but I don't understand how to use these facts. Also there is a problem with the number of particles, because we don't know ##n_{-\varepsilon_0}## and ##n_{\varepsilon_0}##, only ##N## and ##\displaystyle n_{\varepsilon_0} -n_{-\varepsilon_0} = \frac{E}{\varepsilon_0}## are given.