## Time reversal symmetry breaking

In statistical physics we often say this system is in equilibrium and that system is out-of-equilibrium et cetera. But what is equilibrium? and what is out-of-equilibrium?

We say that a system is in equilibrium if it respects time reversal symmetry (TRS) at steady state and out-of-equilibrium if it violates TRS at steady state. For example, consider the pedestrations on Champs Elysee. They clearly constitute a non-equilibrium system because if we watch the movie backwards they will appear to walk backwards.

But now suppose we pay half the people 5€ each to walk backwards. They now become an equilibrium system because if we watch the movie backwards they look the same (statistically).

## Active field theories

Probably we are familiar with the Cahn-Hiliard equation:

$\frac{\partial\phi}{\partial t}+\nabla\cdot\left(-\nabla\frac{\delta F}{\delta\phi}+\boldsymbol{\Lambda}\right)=0$

In the equation above, ϕ is the density of the fluid. If we start from a homogenous density ϕ = -0.6, the system will phase-separate into high density ϕ = +1 (liquid) and low density ϕ = -1 (vapour) phase. In steady state t = ∞, we end up with a single big blob of liquid.

Introducing the non-equilibrium Cahn-Hiliard equation (also called active model B+)….

$\frac{\partial\phi}{\partial t}+\nabla\cdot\left( -\nabla\frac{\delta F}{\delta\phi} + \boldsymbol{\Lambda} + \lambda\nabla|\nabla\phi|^2 + \zeta(\nabla^2\phi)\nabla\phi \right)=0$

If we start from the same intial configuration, we see similar phenomena where the fluid phase-separate into high density and low density. However in the steady state, we get a boiling droplet. The droplet is in perpetual boiling state (see movie at the end), which is not possible in equilibrium because all the liquid will be evaporated.