Schematic plot of a system with a movable wall.

Fundamental Relation for Gas of Interacting Particles in a Heat Flow

Robert Holyst, Karol Makuch, Konrad Gizynski, Anna Maciolek, Pawel J. Zuk

Entropy, 2023


There is a long-standing question of whether it is possible to extend the formalism of
equilibrium thermodynamics to the case of nonequilibrium systems in steady-states. We have made
such an extension for an ideal gas in a heat flow. Here, we investigated whether such a description
exists for the system with interactions: the van der Waals gas in a heat flow. We introduced a steadystate
fundamental relation and the parameters of state, each associated with a single way of changing
energy. The first law of nonequilibrium thermodynamics follows from these parameters. The internal
energy U for the nonequilibrium 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 five parameters of state (irrespective of the
number of parameters characterizing the boundary conditions): the effective entropy S*, volume V,
number of particles N, and rescaled van der Waals parameters a*, b*. The state parameters, a*, b*,
together with S*, determine the net heat exchange with the environment. The net heat differential
does not have an integrating factor. As in equilibrium thermodynamics, the steady-state fundamental
equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.

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