We were taught the following.

Consider three systems $A$, $B$ and $C$, which have exactly two independent variables each, $(X,Y)$, $(X’,Y’)$ and $(X’’,Y’’)$.

Thermal equilibrium for $A$ and $B$ is defined by bringing them in some communication (they are still physically separate) so that the new state has exactly three variables out of $X,Y,X’,Y’$ are independent so that each is an explicit function of the other three.

Zeroth law was defined by saying that thermal equilibrium of $A$ and $B$, $A$ and $C$ implies the thermal equilibrium of $B$ and $C$.

Now it was stated in the class that there was some mathematical notion of temperature coming out of this and it was shown that there existed functions such that $$f_{AB}(X,X’,Y’)=f_{AC}(X,X’’,Y’’).$$ This I can prove.

But then it was somehow shown that there exist functions $h_{B}(X’,Y’)$, $h_{C}(X’’,Y’’)$ $\eta (X)$ and $\xi (X)$ such that $$f_{AB}=h_{B}\cdot \eta +\xi,$$ and $$f_{AC}=h_{C}\cdot\eta +\xi$$ so that, $$h_{B}=h_{C}.$$ These in turn can be characterised as the temperatures of the systems $A$ and $B$ which are equal as $A$ and $B$ are in thermal equilibrium.

Can you show how this came about?

  • $\begingroup$ Do you know where this argument comes from? It looks very interesting. $\endgroup$ – doetoe Jul 26 at 9:44
  • $\begingroup$ I don’t know any exact source... $\endgroup$ – Atom Jul 26 at 9:45

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