## Entropy production

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:

$\mathcal{S}=\lim_{(t_f-t_i)\rightarrow\infty}\frac{1}{t_f-t_i}\left<\ln\left(\frac{P_F[\phi]}{P_B[\phi]}\right)\right>$,

where PF[ϕ] is the probability functional of obtaining a particular trajectory $\left\{\phi(\mathbf{r},t)|t\in[t_i,t_f]\right\}$ 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 $\left<\dots\right>$ 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:

$\frac{\partial\phi}{\partial t}+\nabla\cdot\left(\mathbf{J}^{d}+\boldsymbol{\Lambda}\right)=0$,

Jd is the deterministic current and Λ is a Gaussian white noise with zero mean and delta function correlation:

$\left<\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right>=2D\delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')$

The deterministic current is given by:

$\mathbf{J}_{d} =-\nabla\underbrace{\left(\frac{\delta F}{\delta\phi}+\lambda|\nabla\phi|^{2}\right)}_{\mu}$

where μ is the chemical potential, which consists of the passive/equilibrium part: δF/δϕ and the active part: $\lambda|\nabla\phi|^2$. The rate of entropy production for this dynamics can be shown to be given by (Nardini, Fodor, Tjhung, van Wijland, Tailleur, Cates, PRX, (2017)):

\begin{align} \mathcal{S} & =\frac{1}{\tau}\int_{0}^{\tau}dt\,\frac{\partial\phi}{\partial t}\circ\mu\nonumber\\ & =\frac{1}{\tau}\sum_{t}\Delta t\frac{\phi_{t+\Delta t}-\phi_{t-\Delta t}}{2\Delta t}\mu_{t},\nonumber \end{align}

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), ….