Calculate relative fluctuations in the energy of the system about the mean value in canonical ensemble.

In statistical mechanics, the canonical ensemble is used to describe a system that is in thermal equilibrium with a reservoir at a constant temperature $T$. The system exchanges energy with the reservoir, and its energy fluctuates around a mean value. To quantify the fluctuations in energy, we calculate the relative fluctuations in the energy of the system about its mean value.

Let us begin by considering the probability distribution for the energy in the canonical ensemble. The probability $P(E)$ of the system having energy $E$ is given by the Boltzmann distribution:

$$
P(E) = \frac{1}{Z} e^{-\beta E}
$$

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