[Stat Phy] What does exhausting the states of a system mean?

[WeiShan Ng]

I was reading the *Statistical Physics An Introductory Course* by Daniel J.Amit and need some help to understand a certain passage:

In an isolated composite system of two paramagnetic system:

System a with $N_a$ spins and a magnetic field $H_a$
System b with $N_b$ spins and a magnetic field $H_b$

The total number of states of the composite system for which system a has energy $E_a$ can be written as:
$$\begin{aligned}\Gamma_T &=\Gamma (E_a,H_a,N_a) \cdot \Gamma (E_b,H_b,N_b) \\
&= C_aC_b exp \left( - \frac{E_a^2}{2N_a\mu_B^2H^2} \right) exp \left[ -\frac{(E-E_a)^2}{2N_b\mu_B^2H^2} \right] \\ &= C_T exp \left[ -\frac{N_a+N_b}{2(\mu_BH)^2N_aN_b}(E_a-\bar{E}_a)^2 \right]\end{aligned}$$
which $\bar{E}_a$ is the value of $E_a$ at the maximum of $\Gamma_T$ and $C_T$ includes all the factors which do not depend on $E_a$.
This equation describes a Gaussian distribution whose width (standard deviation), $\Delta E_a$ is $\mu_B H \sqrt{N_aN_b/(N_a+N_b)}$ and the relative width of the distribution, $\Delta E_a/\bar{E}_a$ is
$$\frac{\Delta E_a}{\bar{E}_a} \sim N^{-1/2} \underset{N\rightarrow \infty}{\longrightarrow} 0$$
**which means that in the thermdynamic limit the states of the combined system for which $E_a=\bar{E}_a$ exhaust the states of the isolated system.**. Hence, $E_a=\bar{E}_a$ describes thermal equilibrium between the two systems.

What does exhausting the states mean?

 

[nbryan5]

Does it mean that the states of the combined system are the same as the states of the isolated systems?