Thermal Fluctuations

In the previous section we have demonstrated that the ensemble average internal energy of a system of N two-level particles is the same when computed according to the canonical or microcanonical ensembles, as long as N is sufficiently large. Nevertheless, the internal energy fluctuates among different replicas in the canonical ensemble while remains fixed among different members of the microcanonical ensemble. The goal of this section is to demonstrate that such inherent difference between the two representations does not contradict the equivalency of ensembles. The reason for this is that the relative size of the fluctuations becomes vanishingly small in the limit when N is sufficiently large. Consider the average square fluctuation $(\delta E)^2$ of the internal energy $E$ in the canonical ensemble,

$$(\delta E)^2 = \{\hat{\rho} (\hat{H}-E)^2\} = \{ \hat{\rho} (\hat{H}^2 -2 \hat{H} \bar{E} + \bar{E}^2 ) \}.$$ (62)

Eq. (62) can be simplified according to

$$(\delta E)^2 = \{ \hat{\rho} \hat{H}^2 \} - \bar{E}^2.$$ (63)

Substituting Eq. (11) into Eq. (63) we obtain

$$(\delta E)^2 =\sum_k p_k E_k^2 -(\sum_k p_k E_k)^2,$$ (64)

and since $Z= Tr\{ e^{-\beta \hat{H}} \}$ and $\hat{\rho} = Z^{-1} e^{-\beta \hat{H}}$,

$$(\delta E)^2 =\frac{\text{Tr} \{ \hat{H}^2 e^{-\beta \hat{H}} \}}...... \{ \hat{H} e^{-\beta \hat{H}} \})^2}{(\text{Tr} \{ e^{-\beta \hat{H}} \})^2 }.$$ (65)

Therefore,

$$(\delta E)^2 =\frac{1}{\text{Tr} \{ e^{-\beta \hat{H}} \} } \frac......al \text{Tr} \{ e^{-\beta \hat{H}} \}}{\partial \beta} \Bigg )_{N,V} \Bigg )^2,$$ (66)

and

$$(\delta E)^2 =\frac{1}{Z} \frac{\partial^2 Z}{\partial \beta^2} \......ial \beta^2} \Bigg )_{N,V} = - \frac{\partial E}{\partial \beta} \Bigg )_{N,V}.$$ (67)

Note that these energy fluctuations are vanishingly small, in comparison to the total internal energy $E$ of the system,

$$\frac{\sqrt{(\delta E)^2}}{E} = \frac{\sqrt{-\frac{\partial E}{\partial \beta}}}{E} \propto \frac{1}{\sqrt{N}},$$ (68)

whenever $N$ is sufficiently large. For instance, when $N \sim 10^{23}$ the energy fluctuations are extremely small in comparison to $E$.
 

Note: As a by-product of this calculation we can obtain a remarkable result. Namely, that the rate at which energy changes due to changes in the temperature is determined by the size of the energy fluctuations $(\delta E)^2$. In order to obain this result, consider that according to Eq. (67)

$$(\delta E)^2 = -\frac{\partial E}{\partial T} \Bigg )_{NV} \frac{\partial T}{\partial \beta} = \frac{\partial E}{\partial T} \Bigg )_{N,V} k T^2.$$ (69)

Therefore,

$$(\delta E)^2 = C_v k T^2,$$ (70)

where

$$C_v \equiv \frac{\partial E}{\partial T} \Bigg )_{N, V},$$ (71)

is the heat capacity of the system, i.e., the property of the system that determines the rate of energy change due to changes in the temperature.