Canonical ensemble and the size of the system

Question

In classical derivations of the Canonical ensemble, we can deduce that at equilibrium, the energy of the system ($\epsilon$) distributes as: $$ P(\epsilon) \propto e^{-\epsilon/kT} $$ which is the Boltzmann distribution, where $T$ is the temperature of both, the reservoir and the system (see Concepts in Thermal Physics, section 4.6 Canonical ensemble).

Here my question: If we have two systems, $A$ and $B$, so $A$ is twice as large as $B$ and both are at equilibrium with the same reservoir, their energies should be identically distributed. But, how can we reconcile that the probability of measuring an extensive variable, the energy $\epsilon$, does not depends on the size of the system?

Answer 1

The distribution of the energy in the canonical ensemble is not $$ P(\epsilon) \propto e^{-\epsilon/kT}, \tag{1} $$ but $$ P(\epsilon) \propto g(\epsilon)e^{-\epsilon/kT}, \tag{2} $$ where $g(\epsilon)$ is the value of the density of the states at the energy $\epsilon$. Formula ($1$) refers to the distribution of the energy of single microstates, but, at each energy, there are $g(\epsilon)$ microstates.

In all the formulae above, the energy $\epsilon$ is meant to be the extensive energy of a finite-volume system in the canonical ensemble. Stressing the correct form of the energy dependence of the energy distribution is the key ingredient to understanding its behavior with the system size.

Formula $(1)$ has a maximum at $\epsilon=0$, independently of the temperature. In contrast, formula $(2)$ is the product of an exponentially increasing function by an exponentially decreasing function. It has a maximum at some energy $\epsilon_{max}$ scaling linearly with the system size for systems admitting a thermodynamic limit.

A the same thermodynamic limit, the average energy per particle (or per unit volume) coincides with the limit of the corresponding normalized $\epsilon_{max}$. The behavior of average quantities of macroscopic systems is extremely close to the thermodynamic limit. However, distribution of energies implies fluctuations around the maximum value. The analysis with the size of the variance of the (extensive) energy distribution shows that its width generally (*) increases as the square root of the system size, implying that the relative fluctuations vanish at the thermodynamic limit.

Summarizing, for systems large enough that the thermodynamic limit dominates their behavior,

1. the distribution of energy ($2$) is peaked at an extensive energy $\epsilon_{max}$ linearly scaling with the system size;
2. the distribution is so sharply peaked around its maximum that its average value is indistinguishable from $\epsilon_{max}$;
3. fluctuations increase with the system size, but their relative value can generally be (*) ignored for macroscopic systems.

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(*) The only exception to the vanishing of the relative fluctuations is the critical behavior, where fluctuations dominate the behavior of the system and relative fluctuations diverge.