Polymer Extension

$z=\frac{⟨L⟩}{L_{\text{tot}}}$

• $⟨L⟩$ is the expected polymer length
• $L_{\text{tot}}$ is the total length of the polymer when completely stretched out

$z_{\text{1D}}=\tanh (\beta fa)$

• Polymer can either move left or right
• $\beta$ is [coldness]
• $f$ is the force applied to the stretch
• $a$ is the length of one segment

$z_{\text{3D lattice}}=\frac{2\sinh (\beta fa)}{4+2\cosh (\beta fa)}$

• Polymer can move in $\pm e_i$ direction

$⟨L_{\text{3D}}⟩=-\frac{\partial G}{\partial f}=Na\left[\coth (\beta fa)-\frac{1}{\beta fa}\right]$

• Polymer can move in any direction in 3D