#### Energy transfer or state quantity?

Callen defines heat as the variation in internal energy \( E \) has not been caused by work \[ \dbar Q = \diff E - \dbar W . \] We find here the first anacronism. Heat is represented by an «

*inexact differential*» (symbol \(\dbar \)) because heat is not a state function in the thermodynamic space.

Kondepudi & Prigogine suggest the alternative definition \[ \diff Q = \diff E - \diff W - \diff_\mathrm{matter} E . \] Not only a new mechanism of interchange of energy associated to changes in the composition \( N \) produced by a mass flow with surrounds is introduced, but exact differentials are used because the classical thermodynamics space has been extended with time as variable. Their \( \diff Q \) has to be interpreted in the sense of \( \diff Q(t) \), albeit Kondepudi & Prigogine do not explain how the state space has to be extended. Do they mean \( (E,V,N,t) \) or \( (E(t),V(t),N(t)) \)? Something else?

Truesdell tries to abandon inexact differentials by just working with rates \[ \mathfrak{Q} = \dot{E} - \mathfrak{W} , \] here \( \mathfrak{Q} \) is what he calls «

*heating*» and \( \mathfrak{W} \) the «

*net working*»; the dot denotes a time derivative. But this didn't solve anything, because the issue reappears when one want to compute \( \diff E \) without being forced to use time as variable. What is more, even using time, we would be carrying up expressions like \( \mathfrak{Q} \diff t\).

A similar inconsistency is found in the work of Müller & Weiss, when they write down the rate of change of energy of a body as the contribution of what they call «

*heating*» \( \dot{Q} \) and «

*working*» \( \dot{W} \). Again this kind of notation is ambiguous and looks as the time derivative of state quantities \( Q \) and \( W \) that do not really exist in their formalism.

On the opposite side we find to Sohrab, who proposes to abandon inexact differentials by introducing a new concept of heat \( Q=TS \) as the product of temperature \( T \) and entropy; upon differentiation, \[ \diff Q = T \diff S + S \diff T . \] There are, however, issues with his approach because \( T \) and \( S \) cannot be both variables of state at same instant, and the Gibbs & Duhem expression cannot be used here to get rid of the undesired differential, like in the tradittional approach. The mixed quantity defined by Sohrab lives somewhat between the state spaces \( (S,V,N) \) and \( (T,V,N) \).

It is common to switch to a local formulation in term of fluxes and densities in the irreversible formalisms. Callen

*defines*a generic flow \( J_G \) through \( J_G = dG/dt \), with \( G \) being any extensive

*state variable*. Callen then proposes a heat flux given by the internal energy flow \( J_E \) minus a chemical contribution weighted by the chemical potential \( \mu \) \[ J_Q = J_E - \mu J_N . \] Not only this heat flow concept does not match his generic definition of flow because \( J_Q \) does not represent a flow of heat, but a

*flow of energy*. Heat is not stored in system \( A \) before flowing to system \( B \) through a boundary; there is only energy flowing and people calling heat to part of that energy flow. In practice, authors act like if the terms heat and heat flux are interchangeable, which is so inconsistent as pretending that \( E \stackrel{wrong}{=} J_E \). If this was not enough, not everyone agrees with (5), and whereas Kondepudi & Prigogine add a molar entropy contribution \( s_m \) to the chemical potential \[ J_Q = J_E - (\mu + Ts_m) J_N , \] DeGroot & Mazur use a plain \[ J_Q = J_E . \] Not only we find here three different definitions for the flux, but each introduces fundamental changes to the concept of heat. For instance, in the formalism of DeGroot & Mazur, the rate of change of heat per unit of volume \( q \) is exclusively due to flow through the boundaries \[ \frac{\diff q}{\diff t} = - \nabla J_Q , \] but the choice by Kondepudi & Prigogine forces us to modify this expression by adding a source term associated to the «

*production of heat*» \[ \frac{\diff q}{\diff t} = - \nabla J_Q + \sigma^\mathrm{heat} . \] This source term contains different contributions, including what the authors call the «

*heat of reaction*» generated by chemical reactions taking place inside the system.

The inconsistencies are obvious now. This confusion is amplified in the engineering literature, where the term «

*heat transfer*» is used routinely. If heat is, as Callen emphasizes, «

*only a form of energy transfer*» through the boundaries, then it makes no sense to talk about the production of heat inside a system, whereas the term heat transfer is an oxymoron. If we consider that heat can be produced or absorbed inside the system, then heat cannot be exclusively identified with a mechanism of transfer of energy. Even if we consider heat only as transfer of energy, and rework the existing thermodynamic formalisms to eliminate any heat source term from equations, this does not completely eliminate the inconsistencies. This criticism is also addressed to myself, because I also contributed with a definition of \( J_Q \) for open systems. I now retract from such work.

#### Relativistic heat

We will ignore now the issues reported in the previous section. The question we want to bring to this section is, which is the heat for a moving system if \( \dbar Q \) is the heat for a system at rest?

If you ask Planck, Einstein, von Laué, Pauli, or Tolman the heat \( \dbar Q' \) for the moving system is given by \[ \dbar Q' = \frac{\dbar Q}{\gamma} , \] with \( \gamma \) the Lorentz factor, whereas Ott, Arzeliés, and Einstein (again) propose the alternative expression \[ \dbar Q' = \gamma \, \dbar Q . \] It is important to mention how Møller, in the first edition of his celebrated textbook on relativity, used the Planck expression, but replace it by the Ott expression in late editions. More recently Landsberg et al. introduced still another expression \[ \dbar Q' = \dbar Q . \] Thus, heat can decrease, increase, or be a Lorentz invariant depending on whom you ask.

Related to this, there are further discussions between authors that claim that relativistic heat is a scalar \( \dbar Q \) and those that claim that heat has to be generalized to a four-vector \( \dbar Q^\mu \) quantity for a proper relativistic treatment.

The conclusion for this section is the lack of consensus on what is correct concept of heat to be used in a relativistic context or how this heat behaves under Lorentz transformations.

#### Microscopic heat?

Traditionally, heat has been relegated to the macroscopic classic domain; however, there is increasing interest in last decades to extend thermodynamic concepts to mesoscopic and microscopic domains. We will ignore all the debate and issues reported in the former sections and will focus on answering what concept of heat at microscale corresponds to the traditional expression \( \dbar Q \).

Most authors start from the statistical mechanics expression for the average internal energy of a system \[ \langle E \rangle = \mathrm{Tr} \{H\rho\} , \] with \( \mathrm{Tr} \) denoting a quantum trace or the classical phase space integration, \( H \) the Hamiltonian, and \( \rho \) the statistical operator or the classical phase space density representing mixed states. Differentiation of this expression gives \[ \diff \langle E \rangle = \mathrm{Tr} \{\rho \diff H\} + \mathrm{Tr} \{H \diff\rho\} , \] so macroscopic heat is identified with the second term \[ \dbar Q = \mathrm{Tr} \{H \diff\rho\} , \] which suggested to some authors to take \( \{H \diff\rho\} \) as the «

*microscopic definition*» of heat. This identification is open to debate. The first problem is that the definition is based in a density operator or phase space density that is associated to our ignorance about the microscopic state of the system. Standard literature claims that heat is related to changes on the probabilities of state occupations, but this claim is difficult to accept because it would suggest that heat varies with our level of knowledge about a system. Indeed, if we know the positions and velocities of particles (e.g., in a computer simulation), then the phase space density is given by a product of Dirac delta functions \( \rho = \delta_D(\boldsymbol x- \boldsymbol x(t))\delta_D(\boldsymbol v- \boldsymbol v(t)) \) and it is easy to verify that \( H \diff\rho = 0 \) in this case. However, atoms do not care about our knowledge!

A second problem is that, this «

*microscopic definition*» is not microscopic at all, and it would be better considered mesoscopic, because it is combining microscopic elements such as the Hamiltonian of a system of particles, with macroscopic elements as the parameters that define the Gibbsian ensembles; indeed, the thermodynamic temperature associated to the canonical ensemble is not a microscopic quantity.

Roldán, based in former work by Sekimoto, proposes an alternative expression for microscopic heat. He starts with Langevin dynamics \[ m \frac{\diff \boldsymbol v}{\diff t} = \boldsymbol F^\mathrm{sist} + \boldsymbol F^\mathrm{diss} + \boldsymbol F^\mathrm{rand} , \] then he associates heat with dissipative and random components of work \[ \dbar Q = ( \boldsymbol F^\mathrm{diss} + \boldsymbol F^\mathrm{rand} ) \diff \boldsymbol x , \] which after formal manipulations yields —typos and sign mistakes in his work are corrected here— \[ \dbar Q = \diff \left( \frac{1}{2} m \boldsymbol v^2 + \Phi^\mathrm{ext} \right) - \dbar W , \] with \( \Phi^\mathrm{ext} \) the external potential energy and his «

*microscopic work*» being given by \[ \dbar W = \frac{\partial \Phi^\mathrm{ext}}{\partial\lambda} \diff \lambda . \] Roldán claims to «

*recover the first law of thermodynamics in the microscopic scale*». This is not true. First, what he calls internal energy is not an internal energy, but the total energy of the system. In the second place, his definition of work is invalid. Work is not given by the variation of energy maintaining constant the position. It is impossible to do \( pV \) work on a system maintaining intact the positions of particles, for instance. Finally, what he considers a microscopic approach is not microscopic at all, but mesoscopic; precisely the dissipative and random forces in Langevin dynamics are obtained from averaging the microscopic forces over a heat bath distribution that describes the bath only in a macroscopic sense.

#### Heat from first principles

After this basic review of the difficulties and inconsistencies with usual thermodynamic literature, our role will be to rigorously identify heat from a fundamental approach. We start with the mechanical expression for the internal energy \( E^\mathrm{micr} \) of a system and compute the infinitesimal variation \[ \diff E^\mathrm{micr} = \boldsymbol F^\mathrm{ext} \diff \boldsymbol x . \] This is a standard mechanical result with \( \boldsymbol F^\mathrm{ext} \) the forces from the surrounds. Note that the macroscopic internal energy \( E \) used in thermodynamics corresponds to taking an average over the mechanical expression \( E = \langle E^\mathrm{micr} \rangle \).

Now, we will split the mechanical motions of the particles into two modes: a collective mode that produces changes associated to a parameter \( \lambda \) that describes some property of the whole system, plus individual modes \( \boldsymbol s \) that describe changes on particle positions are not measured by this parameter. We will take as parameter the volume \( V \) of the system; this choice is motivated by simplicity, the generalization to other parameters is straighforward. The split is given by \[ \diff \boldsymbol x = \frac{\partial \boldsymbol x}{\partial V} \diff V + \frac{\partial \boldsymbol x}{\partial \boldsymbol s} \diff \boldsymbol s . \] Introducing this back into (20) yields \[ \diff E^\mathrm{micr} = -p^\mathrm{micr} \diff V + \boldsymbol F^\mathrm{ext} \frac{\partial \boldsymbol x}{\partial \boldsymbol s} \boldsymbol s . \] This continues being a purely mechanical expression. \( p^\mathrm{micr} = - \boldsymbol F^\mathrm{ext} {\partial \boldsymbol x}/{\partial V} \) is what authors call the «

*microscopic or instantaneous pressure*». The macroscopic pressure \( p \) used in thermodynamics is again given by an average \( p = \langle p^\mathrm{micr} \rangle \). Since the first term in the above equation is a microscopic generalization of the \( pV \) work used in thermodynamics, we can associate the second term with a microscopic generalization of thermodynamic heat \[ \dbar Q^\mathrm{micr} = \boldsymbol F^\mathrm{ext} \frac{\partial \boldsymbol x}{\partial \boldsymbol s} \diff \boldsymbol s . \] We recover here a concept of heat as changes in energy associated to modes of motion that do not produce change in the mechanical parameters that describe the system as a whole. Note that, unlike the conventional wisdom, heat here is not associated to ignorance; we can utilize a complete description of atomic motion. We can obtain further expressions for the heat if we write explicit expressions for \( E^\mathrm{micr} \). The internal energy for a nonrelativistic system can be shown to be given by \[ E^\mathrm{micr} = C_V T^\mathrm{micr} + \Phi^\mathrm{micr} , \] with \( C_V \) being what thermodynamicists call the «

*heat capacity*» at constant volume —an unfortunate name if one insists on considering heat only as transfer of energy— and \( \Phi^\mathrm{micr} \) the interaction energy; this expression for the energy is exact and \( T^\mathrm{micr} \), the instantaneous or microscopic temperature, would not be confused with the thermodynamic temperature which is evidently given by \( T = \langle T^\mathrm{micr} \rangle \).

Differentiating energy and using the split (21) we obtain for the

**microscopic heat**\[ \dbar Q^\mathrm{micr} = C_V \diff T^\mathrm{micr} + \left[ \frac{\partial\Phi^\mathrm{micr}}{\partial V} + p^\mathrm{micr} \right] \diff V + \frac{\partial\Phi^\mathrm{micr}}{\partial \boldsymbol s} \diff \boldsymbol s . \] We can now split each one of the microscopic quantities into an average term plus a deviation from the average; for instance, for the interaction energy \( \Phi^\mathrm{micr} = \langle \Phi^\mathrm{micr} \rangle + \delta \Phi^\mathrm{micr} = \Phi + \delta \Phi^\mathrm{micr} \); and use this to obtain an expression for the classic thermodynamic heat \( \, \dbar Q \) plus microscopic corrections \( \, \dbar (\delta Q^\mathrm{micr}) \) \[ \dbar Q = C_V \diff T + \left[ \frac{\diff\Phi}{\diff V} + p \right] \diff V , \] \[ \dbar (\delta Q^\mathrm{micr}) = C_V \diff (\delta T^\mathrm{micr}) + \left[ \frac{\partial(\delta\Phi^\mathrm{micr})}{\partial V} + \delta p^\mathrm{micr} \right] \diff V + \frac{\partial(\delta\Phi^\mathrm{micr})}{\partial \boldsymbol s} \diff \boldsymbol s . \] Note that the fact that the macroscopic average of the interaction energy does not depend on microscopic variables has been used to transform the partial derivative into a total derivative in (26). Taking the average of (24) we can write the macroscopic heat like \[ \dbar Q = C_V \diff T + \left[ \left( \frac{\partial E}{\partial V} \right)_T + p \right] \diff V , \] that is just the expression for the macroscopic heat found in the classical thermodynamic literature, the term within square brackets being what thermodynamicians call the «

*latent heat*» —another unfortunate name— and denote by \( L_V \). Now let us compare our expression with those found in the literature. For instance, Lavenda gives in his study of the «

*Microscopic Origins of the Carnot–Clapeyron Equation*» \[ L_V = \left( \frac{\partial\langle E^\mathrm{micr}\rangle_0}{\partial V} \right)_T - \left\langle \left( \frac{\partial E^\mathrm{micr}}{\partial V} \right)_T \right\rangle_0 . \] The subindex zero means he is using «

*unperturbed probabilities*» \( \pi_n^0 \), associated to a canonical distribution, to compute the averages. Our expression \[ L_V = \left( \frac{\partial E}{\partial V} \right)_T + p = \left( \frac{\partial\langle E^\mathrm{micr}\rangle}{\partial V} \right)_T + p \] is not limited to ensemble averages and works beyond the scope of the canonical ensemble. Ignoring this detail, the real discrepancy is on the second terms. (30) contains a general average of the microscopic pressure \( p = \langle p^\mathrm{micr} \rangle \), whereas Lavenda uses the next unperturbed canonical average \[ \left\langle \left( \frac{\partial E^\mathrm{micr}}{\partial V} \right)_T \right\rangle_0 = \sum_n \pi_n^0 \left( \frac{\partial E_n^\mathrm{micr}}{\partial V} \right)_T .\] We find a inconsistency here, because the mechanical energy levels \( E_n^\mathrm{micr} \) do not depend functionally on the thermodynamic temperature —temperature is only a macroscopic parameter for the canonical ensemble—; this makes his partial derivative mathematically undefined and physically meaningless.

#### Perspectives

The concept of heat presented here has been derived from first principles, one assumption I have made is that the kinetic energy can be expressed like \( C_V T \), whereas this is exact in the non-relativistic domain, it remains to be evaluated if this expression can be maintained in the relativistic regime —apart from residual \( mc^2 \) terms, of course—. I can guarantee something now, however, and it is that a four-component heat concept is unneeded. Thus, relativistic heat will be a scalar.

I have used inexact differential notation for the sake of familiarity with standard thermodynamics literature. A way to avoid the term «

*inexact differential*» and the corresponding alternative notation will be given in another part.

#### Acknowledgement

I thank Prof. Bernard H. Lavenda for interesting discussions.

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