Wiggling Wire

A wire of length L and mass per unit length ρ is fixed at both ends and tightened to a tension f. What is the rms fluctuation, in classical statistics, of the midpoint of the wire when it is in equilibrium with a heat bath at temperature τ? A useful series is

SOLUTION

First solution: Consider the midpoint of the wire P fixed at points A and B (see Figure 1.1). let be the deviation of the wire from the line segment AB. Then, in equilibrium, the wire will consist of segments AP' and P'B. To find ⟨s²⟩, we will have to find the minimum work W_min to change the shape of the wire from APB to AP'B:

(1)

Figure 1.1

Using a standard formula for the probability of fluctuation

(2)

we obtain

(3)

So,

(4)

This answer can be easily generalized for an arbitrary point x along the wire:

(5)