1. Irreversibility may be measured by the amount of entropy production.

2. Let us look at a modified version of our previous example:

Consider the two subsystems as separate systems. Now System I is at 300 K and System II is at 400 K. Let each be brought into contact with a reservoir at 350 K.

If we now compute the entropy change of all the four subsystems put together, we will get the same answer as before.

3. Now, instead of bringing System I from 300 to 350 K in one step, consider bringing it in two steps: 300 to 325 K, and then from 325 to 350 K. Similarly, let System II be taken through two steps — from 400 to 375, and then from 375 to 350. If we now compute the entropy change of the two subsystems, and four thermal reservoirs, we get a lower figure.

4. We can easily see where this argument is going: if we heat up the cold system in many, many steps, and cool down the hot system in many, many steps, the entropy increase for the universe approaches zero.

5. Not only that. Focus your attention on subsystem I. After we take it up from 300 to 350 K through this infinitesimally slow process (in which it comes in contact with many, many different thermal reservoirs), we can bring it back down from 350 to 300 K through a similarly slow process. After going through this cycle, the system is back to its original state. And so are the infinite number of reservoirs back to their initial states!

6. It is in this sense that this infinitesimally slow process is reversible.

7. Equilibrium state: No memory of history. A one-mole piece of copper at 350 K is no different from another one-mole piece of copper [we are disregarding its internal microstructure in making this assertion]. The first piece may have come from solidification of one mole of copper, while the second piece might have reached this state by having two half-mole pieces reach thermal equilibrium — precisely the example we looked at.

9. Admissible (or, permissible) states, and impermissible states.

9. Equilibrium states represent the limit of permissible states. If the system moves from one equilibrium state to another, the process is bound to be reversible. If it jumps from deep inside the space of permissible states to the equilibrium “surface”, it is irreversible.

10. Spontaneous process: Time’s arrow. A fundamental new feature that thermodynamics brings to science. Other ‘laws of physics’ (in mechanics, electromagnetism, etc) don’t care about time’s direction.

11. Spontaneous processes are irreversible — they increase entropy. Since we see all kinds of irreversible stuff around us, one worries about “heat death”!