Equation of State for a Far-from-equilibrium Thermodynamic System with Emergent Scales at Steady-state

One of the hallmarks of soft-condensed matter systems is their ability to exhibit emergent order - spanning a wide range of length and time-scales - when driven out-of-equilibrium. Some of these patterns emerge and disappear at sub-nanosecond scales while some are large enough that they can be physically measured. We present a field theoretic formalism by defining the Lagrangian density as a function of a generic thermodynamic scalar. Our definition of the thermodynamic Lagrangian density involves two components, the internal work or the coherent part which gives rise to emergent order, and the internal dissipation or the incoherent part which acts as the internal sink. The salient feature of this formulation is that it takes into account the spatial and temporal gradients of the generic thermodynamic scalar as the system is driven out-of-equilibrium. On minimizing the action and solving the Euler-Lagrange equations, we obtain a generalized thermodynamic equation of state.