Consider a one-dimensional Ising magnet in a fluctuating magnetic field, h. The particular model to be considered has the partition function                               with SN+1 = S1. Note that...

Consider a one-dimensional Ising magnet in a fluctuating magnetic field, h. The particular model to be considered has the partition function

  with SN+1 = S1. Note that when this system is large (i.e., N→ ∞), the typical size of h is very small. Nevertheless, the presence of this small fluctuating field leads to an order disorder transition for this one-dimensional system.

(a)   Integrate out the fluctuating spins with h held fixed and thereby determine the reversible work function for h, A(h: β, N).

(b)   Show that below some temperature (i.e., above some value of β, N) becomes a bistable function h.

(c)   Derive an equation for the critical temperature below which this system exhibits the phenomenon of symmetry breaking.