Energy fluctuations in canonical ensemble

[lampCable]

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?

 

[DrClaude]

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?

That is the fluctuation in the total energy of the system of N particles. The fluctuation in the energy of a single particle in that system doesn't depend on the number of particles and is exactly what is given by the Boltzmann distribution.

 

[lampCable]

Okay, I think I understand. So when we talk about fluctuations in energy of a single particle in the system, we essentially talk about fluctuations in speed of that particle?

 

[DrClaude]

Okay, I think I understand. So when we talk about fluctuations in energy of a single particle in the system, we essentially talk about fluctuations in speed of that particle?

Only for a free particle without any internal structure. Otherwise, energy is distributed among all degrees of freedom.

 

[lampCable]

I see. But the Boltzmann distribution is only related to the translational degrees of freedom, so if we talk about fluctuations in say rotational energy then how is that described?

 

[DrClaude]

I see. But the Boltzmann distribution is only related to the translational degrees of freedom, so if we talk about fluctuations in say rotational energy then how is that described?

I'm not talking about the Maxwell-Boltzmann distribution of speed, but the Boltzmann factor used to calculate the probability in the canonical ensemble.