1.5 The inverse temperature as an integrating factor

The phrase ”integrating factor” origins from the theory for solving differential equations. The factor \(1/T\) is called an integrating factor for \(\delta Q\), since \(dS = \delta Q / T\) is a ”total differential”.
A simple example may illustrate this:
Let

 \begin{equation*} F(x , y) = x^{2} y \quad , \end{equation*}(1.6)

thus

 \begin{equation*} dF(x , y) = 2 x y dx + x^{2} dy \quad . \end{equation*}(1.7)

We search for solutions \(F(x , y) = const.\), but only know the deviation

 \begin{equation*} 0 = dF = 2 x y dx + x^{2} dy \quad , \end{equation*}(1.8)

and after transformation

 \begin{equation*} dy/dx = - 2y/x \quad . \end{equation*}(1.9)

The four equations are equivalent to some extend, but we lost the factor 1/x in the last equation.
For the solution \(0 =2 y dx + x dy\) it is hard (impossible) to find a function with \(dG(x,y) = 2 y dx+ x dy\)
(try?!?). We first have to multiply with the factor \(x\).
Same as in the above example only after multiplying with the integrating factor \(1/T\) a total differential is found

 \begin{equation*} dS = \delta Q / T \quad . \end{equation*}(1.10)


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© J. Carstensen (Stat. Meth.)