Hanging Chain

The upper end of a hanging chain is fixed while the lower end is attached to a mass M. The (massless) links of the chain are ellipses with major axes l + a and minor axes and can place themselves only with either the major axis l – a, or the minor axis vertical. Figure 1.1 shows a four-link chain in which the major axes of the first and fourth links and the minor axes of the second and third links are vertical. Assume that the chain has N links and is in thermal equilibrium at temperature τ.

a) Find the partition function.

b) Find the average length of the chain.

Figure 1.1

SOLUTION

a) Let the number of links with major axis vertical be n, the number of horizontal major axis links will then be N – n. The total length of the chain is then

(1)

The energy of the system, is also a function of n since

(2)

where we let E_1,2 = Mg (l ± a). The partition function

(3)

where g_n = N!/[n!(N – n)!] is the number of possible configurations with n major axis vertical links.

b) The average energy can be found from (3):

(4)

where β = 1/τ. Therefore,

(5)

The average length is

(6)

We can check that, if M = 0,

At τ → 0, ⟨L⟩ → N (l + a) (lowest energy state). At τ → ∞, ⟨L⟩ → Nl + NβMga² = L₀ + NβMga²_.