# Mathematical formulation of the concept of temperature

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?

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