Unified formalism for the emergence of space in non-equilibrium description

Previous studies indicate that the expansion law in emergence of space can be derived from the first law f thermodynamics. It has been proposed a unified formulation for the expansion law applicable to a general set of gravity theories in equilibrium description. In that formulation, which is based on the first law of thermodynamics, the non-equilibrium terms are ignored. Additionally, the structure of surface degrees of freedom in that formulation deviates from the standard notion, where $N_{sur} \ne 4S$ in general theories of gravity. This motivates us to develop a new unified formulation for the expansion law by incorporating non-equilibrium terms into the first law of thermodynamics. In this work, we formulate a unified expansion law in a non-equilibrium context, utilizing the first law of thermodynamics along with the definition of an effective Misner-Sharp energy. Compared to previous generalizations of the expansion law, our formulation reconciles the basic definition of surface degrees of freedom $N_{sur} = 4S$ for a general set of gravity theories. The unified expansion law in non-equilibrium not only highlights the direct relationship between the rate of areal volume and the difference in degrees of freedom between the surface and bulk $(N_{sur}-N_{bulk})$, but also reveals an inverse correlation with the density of surface degrees of freedom. We also demonstrate that the unified expansion law is instrumental in deriving the expansion law for a general set of gravity theories, including those with higher-order curvature corrections such as $f(R)$ theories of gravity, which require a non-equilibrium description.