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I understand that energy of a system in a canonical ensemble is given by the Boltzmann distribution and the temperature is fixed by the heat bath. If the temperature is a measure of kinetic energy, shouldn't fixing temperature cause the energy to remain constant?

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The temperature is the measure of the average kinetic energy.

In the canonical ensemble, different from the microcanonical ensemble, the total energy $E= E_\text{kin} + V = p^2/2m+V(x)$ fluctuates. The average kinetic energy is given by $$\left\langle E_\text{kin} \right\rangle = \frac{\int d^3x\,d^{3}p E_\text{kin} e^{-E/k_B T}} {\int d^3x\,d^{3}p e^{-E/k_B T}} = \frac{\int d^{3}p E_\text{kin} e^{-E_\text{kin}}/ k_B T} {\int d^{3}p e^{-E_\text{kin}/k_B T}} = \frac{3}{2}k_B T .$$ For $N$ particles, we have the result $$ \left\langle E_\text{kin} \right\rangle = \frac{3}{2} N k_B T.$$

The fluctuations in the kinetic energy are described by the standard deviation $\delta E_\text{kin} $. They describe the effect that the kinetic energy is not fixed but fluctuates due to the presence of the reservoir. Statistical mechanics determines the fluctuations to be given by $$ (\delta E_\text{kin})^2 = \left\langle (E_\text{kin} -\tfrac32 k_B T)^2 \right\rangle =\frac{\int d^{3}p (E_\text{kin} -\tfrac32 k_B T)^2 e^{-E_\text{kin}}/ k_B T} {\int d^{3}p e^{-E_\text{kin}/k_B T}} = \frac{3}{2} (k_B T)^2 .$$ For $N$ particles, we have the result $$ \left\langle E_\text{kin} \right\rangle = \frac{3}{2} \sqrt{N} k_B T.$$

Thus we have that the kinetic energy is a stochastic variable. It is fluctuating and given by $$E_\text{kin} = \tfrac32 Nk_B T \left(1 \pm \sqrt{\tfrac2{3N}} \right)\,.$$ For large systems ($N\to\infty$) the fluctuations are small and thus the kinetic energy of the system is well described by the average.

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The bath is assumed to be so large that the small amount of energy it exchanges with the system is a negligible fraction of its total energy, and so results in negligible fluctuations in its average energy, which determines the temperature.

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  • $\begingroup$ I think the OP is asking about the system, not the bath. $\endgroup$ – Aaron Stevens Mar 8 '18 at 2:42
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The relation $$\langle E \rangle = \langle E_{kin} \rangle = \frac 3 2 N k_B T$$ is valid only for an ideal gas. Notice the $\langle \cdot \rangle$s, which means that we are considering average quantities.

However, the answer to

shouldn't fixing temperature cause the energy to remain constant?

is yes, if the number of particles $N$ is large enough.

That is because it can be shown that in general the relative energy fluctuation satisfies (see for example this answer of mine):

$$\frac{\sqrt{\langle E^2 \rangle - \langle E \rangle ^2}}{\langle E \rangle} \equiv \frac{\delta E}{\langle E \rangle} \propto \frac 1 {\sqrt N}$$

which means that

$$\lim_{N \to \infty} \frac{\delta E}{\langle E \rangle}= 0$$

i.e., in the limit of large system size fixing the temperature is equivalent to fixing the energy.

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