Steady thermodynamic fundamental relation for the interacting system in a heat flow

TL;DR

This paper extends equilibrium thermodynamics to non-equilibrium steady states for a Van der Waals gas in heat flow, establishing a fundamental relation with a specific set of state parameters.

Contribution

It introduces a non-equilibrium thermodynamic formalism for the Van der Waals gas, defining state parameters that describe steady heat flow conditions.

Findings

Internal energy retains its equilibrium form in non-equilibrium states.

The non-equilibrium state is characterized by five parameters, including rescaled Van der Waals constants.

The formalism links state parameters to heat exchange with the environment.

Abstract

There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of non-equilibrium systems in steady states. We have made such an extension for an ideal gas in a heat flow [Ho\l{}yst \emph{et al.}, J. Chem. Phys. 157, 194108 (2022)]. Here we investigate whether such a description exists for the system with interactions: the Van der Waals gas in a heat flow. We introduce the parameters of state, each associated with a single way of changing energy. The first law of non-equilibrium thermodynamics follows from these parameters. The internal energy $U$ for the non-equilibrium states has the same form as in equilibrium thermodynamics. For the Van der Waals gas, $U(S^*, V, N, a^*,b^* )$ is a function of only 5 parameters of state (irrespective of the number of parameters characterizing the boundary conditions): the entropy $S^*$, volume $V$, number of particles $N$, and the rescaled Van der Waals parameters $a^*$, $b^*$. The state parameters, $a^*$, $b^*$, together with $S^*$, determine the net heat exchange with the environment.