Consider a one-dimensional Ising system in a fluctuating magnetic field (B), so that [ Q_{N}(s, T) sim int_{-infty}^{infty} d B sum_{left{sigma_{i} ight}} exp left{-frac{beta N

Consider a one-dimensional Ising system in a fluctuating magnetic field \(B\), so that

\[
Q_{N}(s, T) \sim \int_{-\infty}^{\infty} d B \sum_{\left\{\sigma_{i}\right\}} \exp \left\{-\frac{\beta N B^{2}}{2 s}+\sum_{i=1}^{N}\left[\beta \mu B \sigma_{i}+\beta J \sigma_{i} \sigma_{i+1}\right]\right\}
\]

with \(\sigma_{N+1}=\sigma_{1}\). Note that when the system is very large (i.e., \(N \gg 1\) ) the typical value of \(B\) is very small; nevertheless, the presence of this small fluctuating field leads to an order-disorder transition in this one-dimensional system! Determine the critical temperature of this transition.