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Energy fluctuations in canonical ensemble
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[QUOTE="lampCable, post: 5697942, member: 579025"] [h2]Homework Statement [/h2] 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] i.e. [/I]if the thermal energy is lower than the gap between the energy levels? If so, explain why. [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] 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]i.e.[/I] when the number of particles is so big that the fluctuations are in principle zero, how can this be the reason? [/QUOTE]
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Energy fluctuations in canonical ensemble
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