# Detailed Balance Violation and Fokker-Planck Equation

Suppose I have a system with N sites, and each site can be modified (M) or anti-modified (A). Transitions between these two states are in part random, and in part auto-regulated by recruitment of At each update, each site $$i$$ is either, with probability $$1 − \alpha$$, set to an M- or an A-state randomly. Or with probability $$\alpha$$, two other sites are chosen, and if these are in the same state, then the state of $$i$$ is set to this state. The model is parametrized by the positive feedback to the noise ratio $$F = \alpha /(1 − \alpha)$$.

It is easy to write for $$m = M/N$$ and $$\mathrm{d}m = 1/N$$, $$\frac{\mathrm{d}m}{dt} = (W_+ (m)-W_-(m))\mathrm{d}m + \xi$$ where $$\xi$$ is the stochastic noise, and $$W_+(m) = \alpha(1 − m)m^2 + (1 − \alpha)(1 − m)$$ and $$W_−(m) = \alpha m(1 − m)^2 + (1 − \alpha)m$$. In terms of a matrix, we can write a tridiagonal matrix whose diagonal and off diagonal elements are given by, $$G(M,M + 1) = W_+(m)$$, $$G(M,M − 1) = W_−(m)$$ and $$G(M,M) = 1 − G(M,M − 1) − G(M,M + 1)$$ where $$m = M/N$$ and the population evolves as $$P(M') = \sum_{M} G(M,M')P(M)$$ However, assuming stochastic events are small, I construct a master equation $$\partial_t P(m,t) = W_−(m + dm)P(m + dm,t) + W_+(m − dm) × P(m − dm,t) − [W_+(m) + W_−(m)]P(m,t)$$. Expanding this up to second order in $$\mathrm{d}m$$, in analogy with Kramers H A 1940 Brownian motion in a field of force and the diffusion model of chemical reactions I compare it a generic Fokker-Planck equation (1D) for motion in a potential

$$\partial_t P = − \partial_m J = − \partial_m[-\mu(m)\frac{\mathrm{d}U}{\mathrm{d}m}P - \frac{\mathrm{d}D(m)P}{\mathrm{d}m}]$$ where the first term is the drift in and the second is the diffusive term, and $$J$$ is the current. Collecting the diffusion and drift into an effective potential, $$− \partial_m[-\mu(m)\frac{\mathrm{d}V}{\mathrm{d}m}P - D(m)\frac{\mathrm{d}P}{\mathrm{d}m}]$$

and comparing with the master equation above after expanding it to second order, I was able to find expressions for effective $$V(m)$$ . Also I was able to solve for its fixed points.

Now, if I set $$J=0$$, and solve for a $$P_0 (m)$$, the expression for $$\ln(P_0(m)) \approx -V(m)$$

This brings me to my questions: The way I understand it the case $$J=0$$ implies the existence of detailed balance, and the distribution is an equilibrium distribution. However, for non-zero J, but zero divergence what I get are non equilibrium stead-state solutions. The way I understand it is that the given system does violates detailed balance, then why is it that I can get $$\ln(P_0(m)) \approx -V(m)$$ where $$V(m)$$ is the effective potential I have defined combining drift and diffusion terms. Perhaps the entire analysis is a bit flawed, since the system violates detailed balance to begin with. Or I guess $$P_0(m)$$ is a non-equilibrium stead state distribution?