If a physical tension $f$ acts on a state of extension $l_i$, its mechanical energy is $-fl_i$. The <canonical ensemble> therefore assigns that state the <Boltzmann factor>
$$
e^{-(-fl_i)/(k_BT)}=e^{fl_i/(k_BT)}.
$$
Comparison with the statistical weight $e^{\tau l_i}$ gives
$$
\boxed{\tau=\frac{f}{k_BT}},
\qquad
\boxed{f=k_BT\tau}.
$$
Equivalently, $f$ is the <thermodynamic conjugate variable> to $L$ in the <Gibbs free energy>, just as pressure is conjugate to volume.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 37A / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/37a/e/solution