WeiShan Ng
- 36
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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?
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?