statistical mechanics - Steady state and thermodynamic equilibrium

What is the difference between a system being in a steady state and thermodynamic equilibrium? Can a system be in steady state but not in thermodynamic equilibrium and vice-versa? Do they have different definitions on the basis of the master equation?

Answer

To illustrate the difference take two blocks in contact with each other, with block A at a higher temperature than block B initially. As soon as we bring A in contact with B, energy will flow from A to B, the temperature of A will decrease, and the temperature of B will increase. Given enough time, the two blocks will reach the same temperature, and the energy flux will cease. They are said to be in equilibrium.

Now take our system and instead pin one end of block A at temperature $T_1$, connect the other end to block B, and pin the other end of B at $T_2 < T_1$. Note that at the interface there is initially a temperature difference $T_1 - T_2$. Once again energy will flow from A to B through that interface. However we have fixed the temperature of either end of the two block system. You will see that over time, given that the two blocks have the same temperature-independent thermal conductivity, a linear temperature profile will appear with slope $\frac{T_1-T_2}{L}$ where $L$ is the end to end distance. Since there's still a nonzero temperature gradient everywhere in the system, energy is still flowing from the $T_1$ end to the $T_2$ end. However, the bulk properties (temperature) are not changing with time at any point in the system. This system is said to be in a steady state condition, but it is not in thermal equilibrium since there is a spatial temperature variation.

You can apply this to problems in fluid mechanics. Water can be flowing through a pipe at steady state, but not at (chemical) equilibrium.

To sum up, for a given driving "force" (not a true force of course) - temperature for thermal, pressure for mechanical, chemical potential or concentration for chemical - if that quantity doesn't change with time at any point in the system, then your system is at steady state. If additionally, it doesn't vary spatially either (it is uniform throughout), then it is in equilibrium.

This ignores microscopic fluctuations, of course.