Derivation of expected value of a function from master equation Glauber Model (Ising Model)

Question

Let $\sigma =(\sigma_1, \sigma_2,...,\sigma_N)$ where $\sigma_i =\pm \sigma_i$. Let also $\sigma^i =(\sigma_1, \sigma_2,...,-\sigma_i, ...,\sigma_N)$. Given the master equation:

\begin{equation} \frac{d}{dt}P(\sigma,t) = \sum_{i=1}^N\left[ w_i(\sigma^i)P(\sigma^i,t) - w_i(\sigma)P(\sigma,t)\right] \end{equation}

where $w_i(\sigma)$ is the transition rate of $\sigma_i \rightarrow -\sigma_i$ that depends on the whole configuration $\sigma$. I want to find

\begin{equation} \frac{d}{dt}\left< f(\sigma)\right> = \sum_{i=1}^N\left< \{f(\sigma^i) - f(\sigma)\}w_i(\sigma)\right> \end{equation}

So far I've got to: ($ \left< f(\sigma)\right> = \sum_\sigma f(\sigma)P(\sigma,t)$)

\begin{equation} \frac{d}{dt}\left< f(\sigma)\right> = \sum_\sigma\sum_{i=1}^N\left\{ f(\sigma)w_i(\sigma^i)P(\sigma^i,t) - f(\sigma)w_i(\sigma)P(\sigma,t) \right\} \end{equation}

Then:

\begin{equation} \frac{d}{dt}\left< f(\sigma)\right> = \sum_{i=1}^N\left\{ \sum_{\sigma}\left[f(\sigma)w_i(\sigma^i)P(\sigma^i,t)\right] - \left<f(\sigma)w_i(\sigma)\right> \right\} \end{equation}

Now I don't see how that first member of the rhs is going to turn into $\left< f(\sigma^i) w_i(\sigma)\right>$ (&) which is what I would need at this point. I consider the possibility I'm not attacking the problem the right way, so (&) didn't need to be true, but I still can't figure out what I am missing. Any help or ideas would be extremeley apreciatted.

Thanks,

PD: $f(\sigma)$ and $w_i(\sigma)$ are completely arbitrary

Answer 1

Since $(\sigma^i)^i = \sigma$, the transformation $\sigma\mapsto\sigma^i$ is an involution. In particular, changing the summation variable from $\sigma$ to $\tau = \sigma^i$, you get $$ \sum_\sigma f(\sigma) w_i(\sigma^i) P(\sigma^i,t) = \sum_{\tau} f(\tau^i) w_i(\tau) P(\tau,t), $$ which is just the expectation of the function $\sigma\mapsto f(\sigma^i) w_i(\sigma)$ with respect to the measure $P(\cdot,t)$, as you want.