Equilibrium thermodynamics describes the energy exchange of a body with its environment. Here, we describe the global energy exchange of an ideal gas in the Coutte flow in a thermodynamic-like manner. We derive a fundamental relation between internal energy as a function of parameters of state. We analyze a non-equilibrium transition in the system and postulate the extremum principle, which determines stable steady states in the system. The steady-state thermodynamic framework resembles equilibrium thermodynamics.

Topics
Steady state process, Ideal gas, Entropy, Newtonian mechanics, Nonequilibrium thermodynamics, Thermodynamics, Equations of fluid dynamics, Fluid flows, Laminar flows, Navier Stokes equations

There is mounting evidence that the exchange of energy of a macroscopic steady state system with its environment can be described in an equilibrium thermodynamic-like fashion. It has been considered in the 1990s of the 20th century for systems with chemical reactions that may constantly occur in time.1–3 Such a steady state situation goes beyond equilibrium thermodynamics which excludes any macroscopic flows and currents.4 Around a similar time, the thermodynamic-like approach was introduced for systems with shear-flow5–8 and is still under development.9–12 Another class of macroscopic systems for which the attempt to introduce thermodynamic description has been undertaken is systems with heat flow.13–17 

The above approaches to formulate steady state thermodynamics focus on chemical reactions and shear flow, both in homogeneous temperature profiles or systems with heat flow without chemical reactions and macroscopic flows. If steady state thermodynamics is ever formulated on general grounds, it is now at its inception. About two decades ago, Oono and Paniconi18 and Sasa and Tasaki19 postulated a thermodynamic-like description based on a general footing. However, because the descriptions were postulated, they require validation and further investigation on the eventual limitation. In particular, it is not clear whether the nonequilibrium entropy defined by the integral of the local entropy density over the volume of the system14 or rather the excess heat-based entropy18 should be used to construct the principles of steady-state thermodynamics. These two entropies are not equivalent.16 It shows that the steady state thermodynamics is far from complete, and the core fundamental questions still remain open.14,17 It is worth mentioning stochastic thermodynamics, which focuses on thermodynamic notions for the system on the level of individual trajectories.20,21 Here we focus on a reduced, macroscopic description of a steady state system.22 

Allowing various macroscopic constant fluxes drives the system from equilibrium to a steady state and opens up phenomena that eludes equilibrium thermodynamics. Take for example a quiescent liquid in a uniform gravitational field in equilibrium. This situation assumes a uniform temperature and no macroscopic flows, which, together with the equations of state, determine the thermodynamic parameters at each point of the system. Allowing a vertical or horizontal heat flow radically changes the situation. The flow of heat combined with the gravitational force can cause regular mass movement, either because the denser liquid is at the top and the thinner is at the bottom or because gravity unevenly acts on areas with different horizontal densities. This unstable situation causes a mass movement in the atmosphere, oceans, planets, and stars.23 The core feature of this phenomenon is the coupling between the heat flow and the mass flow. However, the question arises whether some reduced description for non-equilibrium steady states also emerges from hydrodynamics. Can it take the shape of equilibrium thermodynamics, in which the system’s behavior is described by only a few parameters and the rules of extremum containing only these parameters?

Here we present a thermodynamic-like description of a system with heat and mass flow coupling. We consider ideal gas in shear flow with a dissipative temperature profile shown schematically in Fig. 1. We introduce the first law for this system that describes different ways of exchanging the system’s energy with its environment. We also consider a movable wall as a thermodynamic constraint in the system. We introduce the extremum principle that determines the position of the wall. We show that there is a critical shear in the system above which the equilibrium position of the internal wall becomes unstable, and the system undergoes a nonequilibrium second order phase transition. We give a complete thermodynamic-like description of this steady state system in which the position of the movable wall is a thermodynamic constraint. For the vanishing shear flow, the steady state extremum principle directly reduces to the minimum principle in the corresponding problem in thermodynamics.

FIG. 1.
Ideal gas in shear flow. The gas is confined by the immobile bottom wall and by the upper wall moving with velocity Vw. There is also a thermally insulating and moving internal wall at vertical position zw that divides the gas into two subsystems. Shearing motion causes energy dissipation in the system, which changes the temperature profile. The heat flows out from the system through the confining walls at temperature T0. Colors schematically show the temperature profile (red - hotter, blue - colder).
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Ideal gas in shear flow. The gas is confined by the immobile bottom wall and by the upper wall moving with velocity Vw. There is also a thermally insulating and moving internal wall at vertical position zw that divides the gas into two subsystems. Shearing motion causes energy dissipation in the system, which changes the temperature profile. The heat flows out from the system through the confining walls at temperature T0. Colors schematically show the temperature profile (red - hotter, blue - colder).

FIG. 1.
Ideal gas in shear flow. The gas is confined by the immobile bottom wall and by the upper wall moving with velocity Vw. There is also a thermally insulating and moving internal wall at vertical position zw that divides the gas into two subsystems. Shearing motion causes energy dissipation in the system, which changes the temperature profile. The heat flows out from the system through the confining walls at temperature T0. Colors schematically show the temperature profile (red - hotter, blue - colder).
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Ideal gas in shear flow. The gas is confined by the immobile bottom wall and by the upper wall moving with velocity Vw. There is also a thermally insulating and moving internal wall at vertical position zw that divides the gas into two subsystems. Shearing motion causes energy dissipation in the system, which changes the temperature profile. The heat flows out from the system through the confining walls at temperature T0. Colors schematically show the temperature profile (red - hotter, blue - colder).

Close modal
We consider ideal monoatomic gas between two parallel walls at walls’ temperature T0, as shown in Fig. 1. The upper wall moves with speed Vw moving in x-direction. The system is described by irreversible thermodynamic equations: the continuity equation, Navier–Stokes equation, energy balance and two equations of state.24 We assume that the system is translationally invariant in x and y directions, therefore the fields depend only on z coordinate. In particular, equations of state are given by,
(1)
(2)
with pressure pz⁠, volumetric particle number density nz⁠, temperature Tz⁠, Boltzmann constant kB, and the volumetric energy density, uz⁠. The velocity field also depends on z coordinate only and is oriented in the direction determined by the moving wall,
(3)
Such a velocity profile is called planar Couette flow.
We describe the system within linear response theory.24 Therefore, the system is locally described by equation of states (1) and (2) supplemented with mass, momentum and energy conservation. We neglect bulk viscosity. We assume that the shear viscosity coefficient η does not depend on the local state of the gas, so the viscosity does not depend on local density and temperature. The translational symmetry for the pressure, pr=pz⁠, and for the velocity given by Eq. (3) reduces the z-component of the Navier–Stokes equation,
(4)
to the formula, dpz/dz=0⁠, while the x-component of the Navier–Stokes equation gives 0=ηd2vz/dz2⁠. Therefore the pressure in the system is homogeneous,
(5)
and, using the boundary conditions v0=0 and vL=Vw⁠, there is a shear flow in the system,
(6)
The continuity equation,
(7)
with mass density ρz and the above velocity field is satisfied automatically for any density profile. The energy balance equation is given by,
(8)
where ∇v̊ijs=∇ivj+∇jvi−δij23∇⋅v/2 denotes the symmetrized and traceless velocity gradient matrix, the colon is the double contraction, δij denotes the Kronecker delta, um = u/ρ is the mass density of the internal energy and κ is the heat conductivity coefficient. Utilizing the shear velocity profile (6) and the translational invariance of the internal energy density, umz⁠, the energy balance equation reduces to the equation for the temperature profile
(9)
The above equation has the following form, 0=κd2dz2Tz+λ⁠, where λ is the volumetric heating rate,
(10)
We also introduce the dimensionless parameter
D≡ηVw2κT0.
It measures the effects of the energy dissipation rate with respect to the heat flow effects.
We first consider the shear flowing system shown in Fig. 1, but without the internal wall. In this situation, the boundary conditions are that on the upper and lower surface and the temperature profile is given by,
(11)
Number of particles in the surface area A of the translationally invariant system is determined by the particle density, N=A∫0Ldznz⁠, which with the mechanical equation of state (1) and the above temperature profile gives
(12)
where
(13)
Combination of equations of state (1) and (2) gives the following internal energy density uz=3pz/2⁠. Total internal energy in the system is given by
(14)
where we used the fact, that the pressure in the system is constant. Utilizing Eq. (12) in the above equation we get,
(15)
We determine the kinetic energy due to macroscopic motion, Ek=A∫0Ldzmnzv2z/2⁠, where m is a mass of a single ideal gas molecule. We obtain
(16)
where
(17)
Utilizing Eqs. (15)–(17) we obtain the relation between the kinetic and internal energy,
(18)

We consider transitions between steady states in the system. The system is in one steady state at time ti, after which we slightly change one or more control parameters that appear in (or are related to) the governing equations. For example, we slowly modify the velocity of the wall, the position of the upper wall (volume V) or the external temperature T0. In particular, we increase the velocity of the wall by Vwt=Vw+t−tidVw/tf−ti for ti < t < tf, which modifies velocity of the wall from Vw to Vw + dVw. This induces a time dependent hydrodynamic field. After time tf the system reaches another steady state close to the previous one due to a small change in the control parameters. We focus on steady, nonequilibrium states with a constant number of particles in the system.

Previously we have analyzed the energy balance for an ideal gas in a heat flow16 during the transition between steady states. The considerations necessary for Couette flow go along the same line. We do not repeat them but give a sketch. We monitor the change of the energy between times ti and tf,
dU=Utf−Uti=∫titfdtdUtdt,
where
(19)
ρr,t is the density and umr,t is the internal energy density per unit mass at position r and time t. We represent the time derivative of the above integral by
(20)
The first term in the above equation appears as a result of the motion of the boundary. vbr,t is the velocity of the boundary at given time and position r. Next, we utilize the internal energy balance Eq. (8). However, we use the following equivalent form24 
(21)
where the pressure tensor P in the system is given by
Pij=pδij+Πij,Π=−2η∇v̊s,
and Jq = −κ∇T is the heat flux. Using Eq. (21) in (20), we obtain expression for dUt/dt with several terms. The two of these terms produce
∫∂V(t)d2rn̂⋅vbr,t−vr,tρr,tumr,t,
which vanishes due to two reasons. On the wall, normal components of the velocities are equal, n̂⋅vb=n̂⋅v because the wall determines the system’s surface. On the surface of the system perpendicular to the wall, the above integral vanishes because the system is translationally invariant. All other terms lead to the following contributions,
(22)
with the excess heat defined by
(23)
where n̂ is the vector normal to the surface of volume V, with the work related to the expansion of the gas defined by
(24)
and the excess dissipation defined by,
(25)
where ∇vij=∇ivj is the gradient velocity matrix. The subscript st in the above equations means that the quantity should be taken at a steady state that corresponds to the control parameters at time t. The first term in the internal energy balance (22), đQ⁠, is the heat on top of the steady state heat exchanged during the transition between steady states.18 The second term in (22) is the volumetric work performed by that wall (more specifically, by the perpendicular component of the force of the wall) when the system is compressed. The third term đD is the energy dissipation in the system on top of the steady state dissipation. In the above representation, we also used the fact that in a steady state, the heat flowing out of the system is equal to the dissipation due to shear in the fluid,
(26)
Because the pressure in a steady state is spatially uniform, the work from Eq. (24) for slow transitions between steady states reduces as follows, đW≈−p∫titfdt∫Vtd3r∇⋅v⁠. The integrals of the divergence with the Gauss theorem reduce to the change of the volume, and the work reduces to the volumetric work that also appears in equilibrium thermodynamics,
đW=−pdV.
To determine the exchange of the kinetic energy, we proceed along the same line as for the internal energy above. Thus we use the kinetic energy balance equation,24 
(27)
Analysis of the change of the kinetic energy dEk=Ektf−Ekti=∫titfdtdEktdt leads to the conclusion that during the transition between steady states, the kinetic energy changes according to the following equation,
(28)
where
đWw=∫titfdt∫∂Vtd2rn̂⋅Π⋅v−Πstt⋅vstt.
According to the above equations, the kinetic energy may change in two ways. đWw is the work performed by the upper wall on top of the steady work performed to keep the shearing steady state. It may be called the excess shear work of the wall. This effect has been recently discussed by Baranyai.12 
The differential of the total energy, E = U + Ek follows from Eqs. (22) and (28) and is given by,
(29)
The total energy changes as a result of the excess heat đQ⁠, volumetric work −pdV, and the excess shear work đWw⁠. Contrary to the energy balance Eqs. (22) and (28), there is no excess dissipation đD defined by Eq. (25) in the above total energy balance equation. It follows that the excess dissipation đD describes an internal transfer of internal and kinetic energy. During the steady state transition, part of the kinetic energy changes to internal energy through the kinetic energy dissipation that occurs on top of the steady state dissipation.
By introducing
(30)
we obtain the internal energy balance (22) in the following form,
(31)
The above formula holds in the space of the control parameters, which are temperature T0, the volume of the system V, and the speed of the moving wall Vw. We do not change the number of particles N. The above formula has the same form as the internal energy of ideal gas without the macroscopic kinetic energy but with volumetric heating.15,16 Because the pressure in a shear flow system is constant, Eq. (5), and due to the ideal gas mechanical equation of state, Eq. (14), we obtain,
(32)
We notice the following two facts. First, the internal energy balance in the transition between steady states given by Eq. (31) is very similar to the equilibrium thermodynamics expression dUeq = đWeq − pdV. We see that the differential đP plays the role of the heat differential in equilibrium thermodynamics. Second, the expression for pressure (32) is also the equilibrium ideal gas pressure mechanical equation, p = 2Ueq/3V. The above two similarities between steady state and the equilibrium thermodynamic differentials suggest that there exists the integrating factor T* for đP differential,
(33)
where nonequilibrium (effective) entropy is given by the following fundamental equation for the internal energy,
(34)
where s0 is a constant such that without the shear flow we obtain the equilibrium fundamental relation, while the nonequilibrium temperature
(35)
The pressure is determined by p=−∂US*,V/∂V⁠. Whether the differential đP has integrating factor (is exact) can also be shown directly by proving that the differential đP=dU/T*U,V,Vw+pU,V,Vw/T*U,V,VwdV+0⋅dVw has equal derivatives, ∂1/T*/∂V=∂p/T*/∂U and that the differential does not explicitly depend on Vw.25 
It is worth noting that the nonequilibrium temperature T* has the interpretation of the average (density-weighted) temperature because it is also given by
(36)
To derive the above formula, we first use Eqs. (34) and (35) to obtain T* = 2U/3NkB. We then eliminate energy utilizing Eq. (32) to get T* = pV/NkB and use Eq. (1) integrated over the volume of the system. Therefore, it is justified to call T* the nonequilibrium temperature.
Similarly, it is reasonable to call S* the nonequilibrium entropy for the following reason. S* is defined as the potential of the differential đP⁠. It means that during a transition between steady states for which the nonequilibrium entropy is constant, the differential đP = 0. The physical interpretation of the latter condition follows from the use of Eq. (30), which gives đQ + đD = 0. Utilizing Eqs. (23), (25), and (26) in the latter formula, we obtain the equivalence of đP = 0 with the condition,
(37)
According to the above equation, two mechanisms of the change of the internal energy cancel each other. The total heat flowing from the system during the transition equals the dissipation due to the viscous friction inside the system. The outflow of heat equilibrates volumetric heating of the friction. In this sense, the steady state system is “adiabatically insulated,” i.e., without the net transfer of heat.
Now we focus on the system with the internal wall shown in Fig. 1. The wall splits the system into two parts, upper 1 and lower 2. In a steady state, the velocity field in the system is the shear flow given by Eq. (6). Therefore the velocity of the internal wall is Vws=Vwzw/L⁠, where zw is the position of this wall. The wall is adiabatically insulating, which changes one of the boundary conditions. There is no heat flow through the wall. Without the wall the temperature profile would be given by the symmetric parabola, Eq. (11), but the insertion of the adiabatic wall in the middle of the system leads to the situation with half of the parabolic profile. Using this observation, we conclude that from the perspective of the heat flow equation, we can use the solution without the internal wall to determine the energy of subsystem 2 as follows, U2T,A,L2,Vw,2u=UT,A,2L2,2Vw,2u/2⁠, where Vw,2u is the speed of the upper wall in subsystem 2 and L2 is the vertical length of the subsystem. Because the change to the moving reference frame does not change the internal energy, we can deduce the internal energy of the upper system in a similar manner. The internal energy in both subsystems is given by,
UiT,A,Li,Vw,iu−Vw,il=12UT,A,2Li,2Vw,iu−Vw,il for i=1,2,
where Vw,iu is the speed of the upper wall in subsystem i, and Vw,il is the speed of the lower wall in subsystem i. We have Vw,2l=0 and Vw,2u=Vw,1l⁠.
Using the properties of the equations of state for ideal gas we can show that the differentials
(38)
have integrating factors in agreement with Eq. (33),
(39)
On the other hand, the analysis of the energy balance similar to the one presented in Sec. VI leads to the conclusion that đP1/2 = đQ1/2 + đD1/2 is the sum of excess heat and excess dissipation in each subsystem. The differentials đQ1/2 and đD1/2 are determined by a direct application of formulas (23) and (25) with the replacement of the volume of integration with the volume for a particular subsystem.

Until now, we have discussed the energy balance independently for each subsystem. But it is particularly interesting to consider a perpendicular motion of the internal wall. As follows from the dynamics, the normal component of the pressure tensor on the wall in a steady state comes solely from hydrostatic pressure. The wall may move vertically only due to the pressure difference, p1 ≠ p2. Its natural position is where these pressures are balanced. Therefore the knowledge of the effective entropy S* and volume for each subsystem is sufficient to determine the force acting on the wall. The wall’s horizontal velocity enters the problem indirectly. The shear velocity plays the role of energy dissipation. The speed of the wall appears in the energy balance Eq. (9) in the source term. It thus plays the role of volumetric heating, λ≡ηVw/L2⁠.

The ideal gas with volumetric heating and an internal wall has already been considered previously.15 Such a system exhibits continuous nonequilibrium phase transition. Similarly, the system with macroscopic kinetic energy shown in Fig. 1 will exhibit the phase transition as well for the following reason. Let’s focus on the position of the internal wall zw and upper wall velocity, Vw,2u⁠, keeping the external walls temperature T0, the total volume of the system, L1 + L2 = L, and other parameters constant. Number of particles in both subsystems is equal, N1 = N2. Because the hydrostatic pressure gives the sole force normal to the walls, the motion of the internal wall can be determined by the fundamental relations (34) for each subsystem. The pressure follows from pi=piUi,Vi,Ni⁠. Moreover, the internal energy in steady state is determined by Eq. (15). For the same value of volumetric heating, the system with volumetric heating considered in Ref. 15 and the system with shear flow considered in this paper have the same temperature, density and pressure profiles. This reasoning leads to the conclusion that from the perspective of the position of the internal wall, the steady state behavior of both subsystems is the same. The speed of the upper wall for the system with kinetic energy must be related to the volumetric heating λ from Ref. 15 by λ≡ηVw/L2⁠.

The appearance of a phase transition can be explained as the result of the competition of two effects. The first effect is decrease of pressure due to expansion of the subsystems and is of equilibrium nature. The opposite effect is due to the dissipation of energy, which acts as volumetric heating of the system. In shear flow, the larger the volume of the subsystem, the larger the energy dissipated, the more significant the increase of its temperature profile and the increase of the pressure in the subsystem. As a result, the effect of shear is that the expansion of a subsystem increases its pressure. For sufficiently large shear flows, the second effect dominates.

Because there is one-to-one correspondence, we conclude that once the upper wall’s speed increases, the shear flow volumetrically heats the system. For the value, λL2/4κT0 ≈ 4.553 44 taken from Ref. 15, which with the use of λ≡ηVw/L2 gives equivalently the condition for the speed of the upper wall ηVw2/4κT0≈4.55344⁠, the central position of the wall ceases to be stable. Above the critical speed, the wall spontaneously chooses one of the two stable positions and moves away from the center. As we show below, it is possible to introduce an extremum principle that determines the spontaneous position of the internal wall. Let’s consider an additional external force Fw perpendicular to the wall. In a steady state it is determined by the pressure difference, Fext=−Ap2−p1⁠. The force performs work on the system, đWext=−Ap2−p1dzw⁠. Because the spontaneous direction of the wall is toward the equilibration of pressures, the system reaches “equilibrium” for the position of the wall zw, for which the work done by the external force is negative,
(40)
The above phenomenological fact we use to construct the thermodynamic-like extremum principle. When the upper wall does not move vertically, the work of the external force is related to the volumetric work in both subsystems,
(41)
This equation, by utilizing the internal energy balance given by Eq. (38) and the nonequilibrium entropy (39), is expressed as follows,
(42)
Equivalently, it is given by
(43)
where
(44)
and
(45)

Equation (43) puts us in the same position as in equilibrium thermodynamics: if đP12 has an integrating factor and the corresponding potential, đP12=T12*dS12*⁠, then the condition of minimal work (40) by means of expression (43) follows the condition of the minimum of the energy for “adiabatically insulated” system defined by S12*=const surface. This surface is considered in the space of T0, V1, V2, Vw, because we keep the number of particles N1, N2, and other parameters constant. Let’s notice that expressions (40)–(45) for the case of no shear flow (no heating), Vw = 0, reduces to the problem of finding the position of the wall that separates two homogeneous ideal gases. Thus, in equilibrium thermodynamics, đP⁠, becomes the heat differential. The nonequilibrium situation has been discussed earlier16 and it suggests the minimum principle as follows.

The position of the wall is determined by the minimum of total internal energy, U12, as a function of parameters of states S1*,V1,S2*,V2⁠, for the constraint
(46)
The latter condition denotes steady “adiabatic insulation.” In this principle there appears parameter r. Let’s apply the above principle to verify whether it gives a proper position of the wall and to find the meaning of r parameter. We apply the above principle with the constraint of total volume of the system fixed,
(47)
The total energy is given by
(48)
where US1*,V1 follows from the fundamental relation (34). Minimization of the above energy with constraints (46) and (47) allows us to use S1* and V1 as two independent parameters and minimize the function US1*,V1+US12*−S1*/r,V−V1⁠. Its first derivatives lead to the following necessary conditions for the minimum of the function,
(49)
It proves that the necessary condition for the minimum in the presented extremum principle leads to equality of pressures. The first of the above equations gives the interpretation of r parameter. It is the nonequilibrium temperature ratio. Notice that when r = 1, the entropy becomes additive [Eq. (46)] and the minimum principle reduces directly to the equilibrium minimum principle.

In the above reasoning the minimum principle follows from the fact, that the internal energy balance is determined by dUi = đPi − pidVi with homogeneous pressure that is solely determined by the internal energy and volume, pi=piUi,Vi⁠. Both assumptions also hold when transport coefficients depend on parameters of states.17 In particular, if the viscosity or heat conductivity depends on density or temperature. For this reason, the derived minimum energy principle is also valid beyond the regime of Onsager’s linear response theory.24 

It is also worth noting that the inequality (40) which determines the direction of spontaneous motion of the wall, with the use of Eqs. (29) and (41) applied to each subsystem is expressed by
đWext=dE12−đQ12−đWw12,
with energy differential, dE12 = dE1 + dE2, excess heat đQ12 = đQ1 + đQ2 and excess work performed by wall, đWw12 = đWw1 + đWw2⁠. The above formula suggests an approach to identify the quantity that is minimized in the system by subtraction from total energy all other terms (apart from work related to the change of the constraint) that are present in the energy balance equation (here, excess work performed by the wall and heat) which describe the ways the system exchanges energy with the environment. Such a way could lead to a mnemotechnic rule to identify the physical quantity to generalize the second law in other situations than considered in this paper (e.g., with the exchange of particles).

In this article, we investigate whether there is a description similar to equilibrium thermodynamics for out-of-equilibrium steady states. We consider this problem in the context of the Couette flow, where there is a mass current (velocity field), energy dissipation, and a continuous flow of heat. Each feature independently excludes the theory of equilibrium thermodynamics and its tools.

Since, in general, it is still unclear that such a simple, equilibrium thermodynamic-like description of nonequilibrium steady states is possible, we have chosen the most straightforward possible system, which we believe includes the coupling of mass flow and heat current, i.e., an ideal gas in shear flow. We show that nonequilibrium entropy exists, which describes how the system gains energy through excess heat or dissipation. In addition, the nonequilibrium entropy allows us to construct the principle of minimum energy, which leads to the proper state of the system after removing the constraint, which is the force acting on the wall in the system. It proves the existence of a description of a system with an out-of-equilibrium heat and mass flow, which practically takes the form of equilibrium thermodynamics and reduces to the principle of minimum energy in equilibrium thermodynamics when the shear flow disappears.

It is worth mentioning that the thermodynamics of the flowing system have been considered previously with the use of thermostated simulations.9–11 A numerical technique with a thermostat cannot be used directly to confirm relations derived in our paper, e.g., Eq. (22), because at the paper’s core is the energy balance in the system without local heat sources, which are not related to the dissipation due to shear. However, one could repeat calculations in our paper and consider the effects of the thermostat. It would produce an additional term in Eq. (22) and require the redefinition of the excess heat because the thermostat adds a volumetric heating term to the energy balance equation. A similar effect we have considered in our previous paper.16 The approach presented in our paper applies to the global energy flow from a given volume of the system during the transition between steady states. If the “given volume” would be the simulation cell, this approach could be helpful to generalize such simulation techniques as multiparticle collision dynamics26 to control the energy balance on the level of individual cells.

It is also worth noting that the description formulated in our paper uses the non-equilibrium entropy defined as the potential of the excess heat differential. This entropy differs from the non-equilibrium entropy defined as the integral of the equilibrium entropy density over the volume.16 Our result suggests that the non-equilibrium entropy defined as the excess heat differential is the one that should be utilized to construct a thermodynamic-like description for non-equilibrium steady states.

The thermodynamic-like description introduced in this paper leads to further questions inspired by equilibrium thermodynamics. Particularly interesting are the procedures for measuring non-equilibrium state parameters, effective temperature and entropy, and measuring excess heat, dissipation, and work the wall performs. It is worth noting that the obtained result goes along the line of recent experiments of Yamamoto et al., who extended calorimetry for the measurements of the excess heat of sheared fluids.27 Another issue is the possibility of developing the description of interacting systems, where the fundamental relationship for van der Waals gas with heat flow can be introduced. It is also interesting to ask about Maxwell’s identities, which in equilibrium thermodynamics appear very practical. Finally, it is fascinating to check whether such a description exists for other systems with coupled heat flow and mass current in the atmosphere and the chemical industry.

P.J.Z. would like to acknowledge the support of a project that has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 847413 and was a part of an international co-financed project founded from the program of the Minister of Science and Higher Education entitled “PMW” in the years 2020–2024; Agreement No. 5005/H2020-MSCA-COFUND/2019/2. We express our gratitude to Natalia Pacocha for her contribution in creating the figure.

The authors have no conflicts to disclose.

Karol Makuch: Conceptualization (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Konrad Giżyński: Methodology (supporting); Writing – review & editing (equal). Robert Hołyst: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). Anna Maciołek: Methodology (supporting); Writing – review & editing (equal). Paweł J. Żuk: Methodology (supporting); Writing – review & editing (supporting).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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