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

In summary, the passage discusses an isolated composite system consisting of two paramagnetic systems, each with its own number of spins and magnetic field. The total number of states for the composite system can be expressed as a product of the individual systems' states, resulting in a Gaussian distribution with a relative width that approaches zero in the thermodynamic limit. This suggests that in thermal equilibrium, the states of the combined system are exhausted by the states of the isolated systems.
  • #1
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?
 
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  • #2
Does it mean that the states of the combined system are the same as the states of the isolated systems?
 

1. What is meant by "exhausting the states of a system" in statistical physics?

Exhausting the states of a system refers to reaching the maximum possible number of configurations or arrangements that a system can have. In statistical physics, this concept is often used to describe the point at which a system has explored and occupied all of its available states or energy levels.

2. How does exhausting the states of a system relate to entropy?

Exhausting the states of a system is directly related to entropy, which is a measure of the disorder or randomness of a system. As a system exhausts its states, its entropy increases, and the system becomes more disordered.

3. Can a system ever truly exhaust all of its states?

In theory, a system can exhaust all of its states. However, in reality, this is highly unlikely. The number of possible states for a system is incredibly large, and it would take an infinite amount of time and energy to explore and occupy every single one.

4. How does the concept of exhausting states apply to real-life systems?

In real-life systems, exhausting states is often used as a simplifying assumption to understand and model the behavior of complex systems. For example, in thermodynamics, the concept of exhausting states is used to analyze the behavior of ideal gases.

5. Can exhausting the states of a system be reversed?

In some cases, it is possible to reverse the process of exhausting states. For example, in a thermodynamic system, the states can be exhausted by increasing the temperature and energy, and then reversed by decreasing the temperature and energy. However, for other systems, such as a closed system, exhausting states is often an irreversible process.

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