Thermal state vs equibilibrium state Could someone explain what's the difference between a thermal state and an equilibrium state? Or is it even the same?
 A: Basically there are two types of equilibrium in thermodynamic context :


*

*Thermal equilibrium. It's when in system there are no heat flow :
$$ {\frac {d Q}{d t}}= 0 $$

*Thermodynamic equilibrium. No net flow of matter or energy in a system.
So for example take a look at this diffusion process of dye in a water. In this case system is in thermal equilibrium, but not in thermodynamic equilibrium, because there exist net macroscopic flow of dissolved dye particles in a vessel. (However I bet that there may be even a small heat flow too, due to dye particles absorbing water molecules kinetic energy, but this needs to be tested). So in this process, there is a matter flow, thus breaking thermodynamic equilibrium. At other cases, this type of equilibrium can be broken by energy flow. It's when in system there exist a net radiation or other means of energy transfer. For example, when you bump with a metal stick on a railroad track,- you generate sound wave which transfers vibrational energy across track, thus breaking track's thermodynamic equilibrium.
EDIT
If thermodynamic equilibrium (TE) to be formalized, then :
$$ \frac {d E}{d t} + c^2 \frac {d m}{d t}= 0 $$
There are two solutions to this :


*

*$\frac {d E}{d t} = 0\,,\,\frac {d m}{d t} = 0$ $\to$ inflow of energy and mass zero

*$\frac {d E}{d t} = - c^2 \frac {d m}{d t}\,\to$ inflow of energy equals to outflow of mass (or in reverse). 
As an example of this case imagine laser pulses heating gas effectively converting it to a plasma, at the same time pushing heated plasma out of gas container. In this case container will be in TE.

A: Thermal states are used to explain the thermal equilibrium state, where the system attains the same temperature as that of the bath and will not exchange any more heat with the bath. This situation can be explained using the quantum mechanical equivalent of the canonical ensemble (that represents possible states of the mechanical systems at thermal equilibrium at a constant temperature). That's why the density matrix is represented as Gibbs state $\rho= Z^{-1}e^{-\beta H}$.
On the other hand, equilibrium states can be thermal, chemical or mechanical equilibrium. This is a generalized context. for example, If the volume of the system is kept constant (quantum mechanically volume can be anything like the frequency of the harmonic oscillator or external magnetic field used to drive a quantum system etc.), then the system is in mechanical equilibrium, where, we can never extract any work from the system. At the same time, we can exchange heat in to/out of the system, which makes the system thermally out of equilibrium (cannot be a thermal state/Gibbs state).
A: As far as I can tell thermal state refers to a system in thermal equilibrium with a Boltzmann distribution of states i.e canonical ensemble. On the other hand I could have the micro-canonical ensemble and be in thermal equilibrium. In such a situation no name is given we just say equilibrium state. 
