Let us consider some trajectory ϕ(r,t) from time t = ti to t = tf. The rate of entropy production S is defined to be the logarithm of the ratio of forward probability to the backwards probability:
where PF[ϕ] is the probability functional of obtaining a particular trajectory and PB[ϕ] is the probability functional of obtaining the same trajectory going backwards in time (see image below). In another word, entropy production is a measure of time-irreversibility.
The angle brackets above indicates averaging over all noise realizations.
Entropy production for active model B
Active model B is a model for phase separation in active matter, which is a derivative of the active model B+ (see previous post). The active model B dynamics is given by:
Jd is the deterministic current and Λ is a Gaussian white noise with zero mean and delta function correlation:
The deterministic current is given by:
where μ is the chemical potential, which consists of the passive/equilibrium part: δF/δϕ and the active part: . The rate of entropy production for this dynamics can be shown to be given by (Nardini, Fodor, Tjhung, van Wijland, Tailleur, Cates, PRX, (2017)):
where the time integral is interpreted as Stratonovich. ϕt is the density field at time t and μt is the chemical potential at time t. The reason for the Strotonovich choice is follows:
When we are computing entropy production, basically, we are comparing the trajectory going forward in time and the trajectory going backwards in time.
Now the time is discretized into t = 0, dt, 2dt, 3dt, …. That means the trajectory is also split into segments. The way we split the trajectory is done in the Stratonovich way. That means the value of ϕ at time t means the value of ϕ mid-point of [t, t+dt]. i.e. ϕ(r,t+dt/2). If we discretize the trajectory in `Ito’ way, the time-reversed of that trajectory is not identical (see drawing below).
However, when we perform the simulations, we do it in `Ito’ way. i.e. we start from ϕ(r,0), then we update the time to get ϕ(r,0), ϕ(r,dt), ϕ(r,2dt), ϕ(r,3dt), ….