# Conceptual meaning of Thermal States

Thermal states are generally defined as

$$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$

What are some physical statements one can make about them? A system in thermal equilibrium is in a thermal state? The thermal state minimizes the free energy?

These are just two things I already know. Is there anything else worth mentioning?

For example I read that the thermal state maximizes entropy. But a thermal state is not maximally mixed. How can that be? I thought only maximally mixed states, have maximal entropy for a given energy.

Also I noticed that for temperature $$T=0$$ we can't have a thermal state. Only pure states occur for $$T=0$$. Why is that?

• Statistical Mechanics is essentially an entire subject devoted to the study of thermal states – By Symmetry Jan 14 at 10:16
• In terms of entropy, thermal states maximise entropy subject to the constraint that the expectation value of the energy is fixed. Without this constraint, the totally mixed state maximises entropy. – By Symmetry Jan 14 at 10:19