# Equilibrium between two or more system

A system is said to be in a state of equilibrium if the thermodynamical variables are time-independent.

Now, What it meant for two or more systems to be in thermodynamical equilibrium?

In Kardar's book, He said, If two system $$A$$ and $$B$$ are in equilibrium with each other then this implies that their coordinates are connected by the constraint equation $$f_{AB}(A_1,\cdots,B_1,\cdots )=0$$ Is that the definition? If not what then How one can conclude this?

• What is $f_{AB}$? – GiorgioP Feb 20 at 10:09
• It's a constraint function that connects the coordinates of two systems. – Young Kindaichi Feb 20 at 10:55
• I have difficulty understanding what constraint means in this context. I would say that for example, the two temperatures must be equal. But I would not call it a "constraint". Maybe some more context could help. – GiorgioP Feb 20 at 12:37

## 1 Answer

Take the paradigmatic example of two systems that only can exchange Energy. It is easier to discern the meaning of this constraint on it. Thermodynamics tells us that these two systems have the same temperature at equilibrium.

$$T_1=T_2$$

Statistical physics tells us that you can derive this principle by saying that the equilibrium state is the more likely one. Moreover, this macroscopic state turns to be the one that maximizes the number of microstates or the entropy (see Kardar or Pathria for a detailed explanation):

$$\partial_{E_1}S_1(E_1,N_1,V_1)=\partial_{E_2}S_2(E_2,N_2,V_2)$$

This expression could be rewritten giving rise to one of these constraint equations that link the variables of both systems:

$$\partial_{E_1}S_1(E_1,N_1,V_1)-\partial_{E_2}S_2(E_2,N_2,V_2)=0$$