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## Homework Statement

Consider an ensamble of particles that can be only in two states with the difference ##\delta## in energy, and take the ground state energy to be zero. Is it possible to find the particle in the excited state if ##k_BT=\delta/2##,

*i.e.*if the thermal energy is lower than the gap between the energy levels? If so, explain why.

## Homework Equations

## The Attempt at a Solution

We calculate the partition function which becomes ##Z = 1+e^{-\delta/k_BT} = 1+e^{-2}## and so the probability for finding a particle in the excited state is ##P(excited) = \frac{e^{-\delta/k_BT}}{Z} = \frac{e^{-2}}{1+e^{-2}} \approx 0.12##. So we can therefore expect to find particles in the excited state.

Since the thermal energy ##k_BT## is too small to put particles in the excited state there must be something else going on. The solution to the problem says that it is due to energy fluctuations for a system in thermal contact with a reservoir of constant temperature. Now, it is possible to show that the fluctuations in the energy in the canonical ensamble is $$\frac{\Delta E}{E} \propto \frac{1}{\sqrt{N}},$$ where ##E## is the energy, ##\Delta E## is the standard deviation in ##E## and ##N## is the number of particles. But in the thermodynamic limit,

*i.e.*when the number of particles is so big that the fluctuations are in principle zero, how can this be the reason?