When we write down the full Hamiltonian of a system in contact with a thermal bath, it is as follows:

$$H_{\text{total}} = H_{\text{system}} + H_{\text{system+bath}} + H_{\text{bath}}.$$

As our focus is only on the system Hamiltonian, we can ignore the ( H_{\text{bath}} ) term. This reduces to:

$$H_{\text{total}} = H_{\text{system}} + H_{\text{system+bath}}.$$

From this, it is clear that the system's energy levels will be affected due to its interactions with the thermal bath. However, in statistical physics, we assume that the system's energy levels do not get affected and only particle distributions change. On the other hand, by changing the system's volume (keeping the system in a constant pressure environment), the energy levels of the system do change.

Why do the energy levels remain unaffected for the system when the system is in contact with a thermal bath, and why is it the opposite when it is kept at constant pressure allowing the volume to change?

When we write down the full Hamiltonian of a system in contact with a thermal bath, it is as follows:

$$H_{\text{total}} = H_{\text{system}} + H_{\text{system+bath}} + H_{\text{bath}}.$$

As our focus is only on the system Hamiltonian, we can ignore the $ H_{\text{bath}}$ term. This reduces to:

$$H_{\text{total}} = H_{\text{system}} + H_{\text{system+bath}}.$$

From this, it is clear that the system's energy levels will be affected due to its interactions with the thermal bath. However, in statistical physics, we assume that the system's energy levels do not get affected and only particle distributions change. On the other hand, by changing the system's volume (keeping the system in a constant pressure environment), the energy levels of the system do change.

Why do the energy levels remain unaffected for the system when the system is in contact with a thermal bath, and why is it the opposite when it is kept at constant pressure allowing the volume to change?