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 $f_{AB}$? Commented Feb 20, 2021 at 10:09
• It's a constraint function that connects the coordinates of two systems. Commented Feb 20, 2021 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. Commented Feb 20, 2021 at 12:37

$$T_1=T_2$$
$$\partial_{E_1}S_1(E_1,N_1,V_1)=\partial_{E_2}S_2(E_2,N_2,V_2)$$
$$\partial_{E_1}S_1(E_1,N_1,V_1)-\partial_{E_2}S_2(E_2,N_2,V_2)=0$$