## Energy transfer or state quantity?

Everyone has an intuitive conception of heat as something related to temperature, but a rigorous and broadly accepted scientific definition of heat is lacking despite several centuries of study. Callen defines heat as the variation in internal energy \( E \) has not been caused by work \[ đQ = \diff E - đW . \] We find here the first anacronism. Heat is represented by an «

*inexact differential*» (symbol đ) 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 mass flow 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 and, in fact, they do not write down the state space used for their expression. The same defficiency is found in the irreversible formulations, as we will see below.

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

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

*net working*». 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.

On the opposite side we find to Sohrab, who proposes to abandon inexact differentials by upgrading the concept of heat to state function through the definition \( Q=TS \), which yields \[ Q = E + pV - \mu N . \] Differentiating this we obtain an expression for an exact differential \( \diff Q \), exact in the thermodynamic space, because heat \( Q \) as defined by Sohrab is function of state. There are, however, issues with his approach. The differential is \[ \diff Q = \diff E + p \diff V - \mu \diff N + V \diff p - N \diff \mu , \] but the intensive parameters \( p \) and \( \mu \) are not state functions, and the Gibbs & Duhem expression cannot be used here to get rid of their respective differentials; therefore, functional expressions \( p = p(E,V,N) \) and \( \mu = \mu(E,V,N) \) have to be introduced to relate the differentials \(\diff p \) and \(\diff \mu\) to the state space. Sohrab never takes this needed step.

It is common to switch to a local formulation in term of fluxes and densities in the irreversible formalisms. Callen proposes a definition of heat flux given by the internal energy flow \( J_E \) minus a chemical contribution \[ J_Q = J_E - \mu J_N , \] whereas Kondepudi & Prigogine propose \[ J_Q = J_E - (\mu + Ts_m) J_N , \] and DeGroot & Mazur use \[ J_Q = J_E . \] Not only we find here three different definitions for the fluxes, but they are associated to different concepts 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 \[ \frac{\diff q}{\diff t} = - \nabla J_Q , \] whereas Kondepudi & Prigogine modify this expression by adding a source term \[ \frac{\diff q}{\diff t} = - \nabla J_Q + \sigma^\mathrm{heat} . \] This heat source contains different contributions including what the authors call the «

*heat of reaction*» associated to chemical reactions taking place inside the system.

The inconsistencies are obvious now. If heat is, as Callen emphasizes, «

*only a form of energy transfer*», then it makes no sense to talk about the production of heat inside a system. If we consider that heat can be produced or absorbed then it cannot be exclusively identified with a transfer of energy. This confusion is specially amplified in the engineering literature, where the term «

*heat transfer*» is used routinely. Summarizing, if heat is only a transfer, like most thermodinamicists claim, then heat transfer is an oxymoron. If heat can be produced or absorbed, then heat transfer has to be considered just another mode of change of heat.

Even if we consider heat only as transfer, and rework the existing thermodynamic formalisms to eliminate any heat source term, this does not completely eliminate the inconsistencies. Consider the flow of internal energy \( J_E \), this is a quantity introduced to express changes in energy due to flow through boundaries; a given quantity of energy initially in system \( A \) flows to the adjacent system \( B \) and this flow is represented by \( J_E \). The same happens with other flows such as the flow of mass \( J_m \). In fact, Callen

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

*state variable*. However, the heat flow \( J_Q \) does not represent a flow of heat, but a

*flow of energy*, which invalidates his general flow definition, because heat is not a state variable in his formalism. Heat is not stored in system \( A \) and flowing to system \( B \) through a boundary, but 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 \( Q \stackrel{wrong}{=} J_Q \). This criticism is also addressed to myself, because I also contributed with a proper definition of \( J_Q \) for open systems. I now retract from such work.

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.

## Relativistic heat

We will ignore now the issues reported in the previous section and will take, as granted, a traditional expression \( đQ \) for a system at rest. The question we want to bring to this section is, which is the heat for a system moving with velocity \( \boldsymbol v \)?

If you ask Planck, Einstein, von Laué, Pauli, or Tolman the heat \( đQ' \) for the moving system is given by \[ đQ' = \frac{đQ}{\gamma} , \] with \( \gamma \) the Lorentz factor, whereas Ott, Arzeliés, and Einstein (again) propose the alternative expression \[ đQ' = \gamma \, đQ . \] It is important to mention how Møller, in the first edition of his celebrated textbook on relativity, used the Planck expression, whereas changed to using the Ott expression in late editions. More recently Landsberg et al. introduced still another expression \[ đQ' = đQ . \] 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 \( đQ \) and those that claim that heat has to be generalized to a four-vector \( đQ^\mu \) quantity for a proper relativistic treatment.

The conclusion for this section is the lack of consensus on what is relativistic heat or how it behaves under Lorentz transformations.

## Microscopic heat?

Traditionally, heat has been relegated to the macroscopic classic domain; however, there is a increasing interest in last decades to extend thermodynamic concepts to mesoscopic and microscopic domains. Once again we will ignore all the debate and issues regarding the macroscopic definition of heat and will focus on answering which microscopic version will yield the traditional expression \( đQ \).

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 \[ đQ = \mathrm{Tr} \{H \diff\rho\} , \] which suggested to some authors to take \( \{H \diff\rho\} \) as the «

*microscopic definition*» of heat. This identification is not aceptable. 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, the amount of heat needed to cook an egg does not decrease for a well-informed chef. Atoms do not care about our knowledge!

Moreover, this «

*microscopic definition*» is not microscopic at all, and it would be better considered mesoscopic, because the definition combines microscopic elements such as the Hamiltonian of a system of particles, with macroscopic elements as the parameters that define the 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 the dissipative and random components of work \[ đQ = ( \boldsymbol F^\mathrm{Diss} + \boldsymbol F^\mathrm{Rand} ) \diff \boldsymbol x , \] which after formal manipulations yields —typos and sign mistakes in Roldán's work are corrected here— \[ đQ = \diff \left( \frac{1}{2} m \boldsymbol v^2 + \Phi^\mathrm{ext} \right) - đW , \] with \( \Phi^\mathrm{ext} \) the external potential energy and what he calls «

*microscopic work*» being given by \[ đW = \frac{\partial \Phi^\mathrm{ext}}{\partial\lambda} d \lambda . \] 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. The total energy of a system, \( E_\mathrm{tot} = K + E \), is the sum of bulk kinetic energy \( K \), and internal energy \( E \). Roldán is defining the

*internal*energy of the Langevin particle as the sum of its

*bulk kinetic*energy plus the

*external*potential energy, which is clearly incorrect. In the second place, his definition of work is invalid as well. 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. In fact, mechanical work is just associated to the variation on positions, this is the reason why work is given by the product of forces and displacements. 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. It is amazing that despite those strong mistakes his work was nominated as an outstanding Ph. D. thesis by the Universidad Complutense of Madrid.

## Heat from first principles

We start with the mechanical expression for the internal energy \( E \) of a system and compute the infinitesimal variation \[ \diff E = \boldsymbol F \diff \boldsymbol x . \] This is a standard mechanical result. Now, assuming only \( pV \) work for simplicity, we rewrite this result as \[ \diff E = -p \diff V . \] This continues being a purely mechanical expression. \( p \) is what authors call the «

*microscopic or instantaneous pressure*». This quantity is not the pressure used in thermodynamics, but \( p^\mathrm{thermo} = \langle p\rangle \). The thermodynamic internal energy also corresponds to an average \( E^\mathrm{thermo} = \langle E\rangle \). Splitting the instantaneous quantities into average plus deviation components and reorganizing the result yields \[ \diff \langle E\rangle = -\langle p\rangle \diff V - [ (\delta p)\diff V + \diff (\delta E) ] . \] Comparing this result with classical thermodynamics allows us to identify heat like minus the term between square brackets, \[ \diff \langle E\rangle = -\langle p\rangle \diff V + đQ . \] We can further obtain explicit expressions for the heat if we write an explicit expression for the average internal energy. The internal energy for a nonrelativistic system can be shown to be given by \[ E = C_V T + \Phi , \] 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 \( \Phi \) the potential energy; this expression for the energy is exact and \( T \), the instantaneous or microscopic temperature, would not be confused with the thermodynamic temperature, which is given by the average \( T^\mathrm{thermo} = \langle T\rangle \). Taking the average of the mechanical energy and differentiating it, we obtain \[ \diff \langle E\rangle = C_V \, \diff \langle T\rangle + \frac{\partial\langle \Phi\rangle}{\partial V} \diff V . \] which yields \[ đQ = C_V \, \diff \langle T\rangle + \left[ \frac{\partial\langle \Phi\rangle}{\partial V} + \langle p\rangle \right] \diff V , \] or, in thermodynamic notation, \[ đQ = C_V \, \diff T^\mathrm{thermo} + L_V \diff V , \] with \( L_V \) the «

*latent heat*», another unfortunate name. Let us now write this latent heat \[ L_V = \frac{\partial\langle \Phi\rangle}{\partial V} + \langle p\rangle , \] and compare it with expressions found in the literature. Lavenda gives \[ L_V = \frac{\partial\langle E\rangle_0}{\partial V} - \left\langle\frac{\partial E}{\partial V}\right\rangle_0 . \] A priori the first terms seem to disagree; however, the partial derivative \( \partial\langle \Phi\rangle / \partial V \) in my general expression is equivalent to \( \partial\langle E\rangle / \partial V \) in this context, because the kinetic term is maintained constant by virtue of differentiating at constant temperature. Thus, the only difference is that his average \( \langle E\rangle_0 \) is restricted to an ensemble average, where the subindex zero means he is using «

*unperturbed probabilities*» \( \pi_j^0 \) associated to a canonical distribution. It is worth to mention that the canonical distribution has only approximated validity; e.g. the distribution is only valid for large systems without long-range correlations, whereas my approach continues to work with more general averages, including time averages taken over a single physical system.

The real discrepancy is on the second terms of our expressions for the latent heat. My expression contains a general average of the microscopic pressure \( p \), whereas Lavenda gives a unperturbed canonical average of a partial derivative \( \partial E / \partial V \). But what variable is left constant during the partial differentiation? Close inspection reveals that this variable is the thermodynamic temperature \( T^\mathrm{thermo} \), which enters in his unperturbed probabilities through the canonical distribution.

We find a inconsistency here, because the mechanical energy levels \( E_j \) do not depend functionally on the thermodynamic temperature —temperature is only a parameter for the canonical ensemble—, which makes his partial derivative mathematically undefined and physically meaningless. The expression given by Lavenda has to be corrected to \[ L_V = \sum_j E_j \frac{\partial\pi_j^0}{\partial V} = \frac{\partial}{\partial V} \sum_j E_j \pi_j^0 - \sum_j \frac{dE_j}{dV} \pi_j^0 \] or, what is the same, \[ L_V = \frac{\partial\langle E\rangle_0}{\partial V} - \left\langle\frac{\diff E}{\diff V}\right\rangle_0 = \frac{\partial\langle E\rangle_0}{\partial V} + \langle p\rangle_0 . \] I restricted the discussion to \( pV \) work for simplicity; of course, the same general arguments and corrections apply to general work \( \chi \diff \lambda \) and latent heat \( L_\lambda \).

## 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 corresponding alternative notation will be given in another part.

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