### What is heat?

$\newcommand{\diff}{\mathrm{d}}$

## 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.

## References

Thermodynamics and an Introduction to Thermostatistics; Second Edition 1985: John Wiley & Sons Inc.; New York. Callen, Herbert B.

Modern Thermodynamics 1998: John Wiley & Sons Ltd.; Chichester. Kondepudi, D. K.; Prigogine, I.

Rational Thermodynamics 1968: McGraw-Hill Book Company; New York. Truesdell, C.

On a Scale-Invariant Model of Statistical Mechanics and the Laws of Thermodynamics 2016: ASME. J. Energy Resour. Technol. 138(3): 032002-032002-12. Sohrab, S. H.

Non-equilibrium thermodynamics 1984: Courier Dover Publications, Inc.; New York. DeGroot, Sybren Ruurds; Mazur, Peter.

Thermodynamics of irreversible processes – past and present 2012: Eur. Phys. J. H, 37, 139-236. Müller, Ingo; Weiss, Wolf.

Irreversibility and dissipation in microscopic systems – Tesis Doctoral 2013: Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Atómica, Molecular y Nuclear. Roldán, Édgar.

A New Perspective on Thermodynamics 2010: Springer; New York. Lavenda, Bernard H.

### Instantaneous electromagnetic interactions

Newtonian gravity introduced a model of instantaneous direct interactions among massive particles. This model was latter replicated by Coulomb for charged particles. Those models have been traditional named the action-at-a-distance model; although, this name is misleading and generated unending polemics among physicists and philosophers about how a particle can act at a distance in a vacuum over distant particles. A better name for this model is direct-particle-interaction. Maxwell electrodynamics and General Relativity introduced an alternative model of contact-action, where particles don't interact directly but by means of a mediator. The former theory uses fields, whereas the latter uses a concept of curved spacetime as mediator. In the contact-action model a particle emits a signal —e.g. a photon— which propagates through the media —e.g. the electromagnetic field— until reaching a far particle, which then feels the force or interaction of the first particle. Since the maximum possible speed is the speed of light, interactions are retarded in this model. The common belief during almost a century has been that the contact-action model is accurate and that instantaneous interactions have been disproved. Nothing more far from reality! In fact, partially due to the defects of the contact-action model, partially due to set of modern experiments, the models of instantaneous interactions are seeing a renaissance in the specialized literature. It is worth to revisit this topic and clarify some misunderstandings in the literature. I will limit to electromagnetic interactions, but the material discussed here can easily ported to gravitation. We will start with the original Coulomb potential at point $$x$$ 'generated' by another charge $$e$$ placed at $$r$$ $\phi(x, t) = K \frac{e}{|R(t)|} = K \frac{e}{|x - r(t)|} .$ This potential is instantaneous because depends on the position of the 'source' particle at present time $$t$$. Now we will expand the position of the 'source' charge around its position at some early time $$t_0$$ $r(t) = r(t_0) + v(t_0) (t-t_0) + a(t_0) (t-t_0)^2 / 2 + \cdots$ and assuming that this charge is non-accelerating at the initial time $$t_0 = (t - |R(t_0)|/c)$$ we obtain the next potential $\phi(x, t) = K \frac{e}{|R(t_0)| - v(t_0) R(t_0)/c} ,$ which is evidently the scalar Lienard & Wiechert potential. The vector Lienard & Wiechert potential can be obtained in the same way if we start from the instantaneous potential $A(x, t) = K \frac{ev}{|R(t)|} = K \frac{ev}{|x - r(t)|} .$ Note that the Lienard & Wiechert potentials have been derived under the approximation of particles being in inertial initial states, $$a(t_0) = 0$$. This means that Lienard & Wiechert potentials aren't complete and this explains why they have to be complemented by adding reaction-radiation potentials to the equations of motion for curing such issues like non-conservation of energy on accelerating particles. Not only those additional potentials are obtained ad hoc but the resulting improved equations of motion are still subjected to criticism due to non-physical behaviors. Note that the $$a(t_0) = 0$$ approximation also explains why the Lienard & Wiechert potentials for a moving charge can be obtained from the potentials for a charge at rest $$A=0$$ and $$\phi = \phi_\mathrm{Coulomb}$$ applying the Lorentz transformations between the frame where the particle is at rest and the frame where the particle is moving with velocity $$v$$. The Lorentz transformations can be applied because the frames are inertial, i.e., the particle is non-accelerating. In fact some derivations explicitly assume that the charge is moving with "with uniform velocity $$v$$ through a frame $$S$$". This same approximation is also the reason why the quantum field theory assumes as the only physicall admisible states those of free particles, i.e. non-accelerating. This is picturesquely described in Feynman diagrams
The diagram consider particles in free motion, until a virtual photon is emitted and absorbed and both electrons change their state of motion to a new inertial state. Quantum field theory only can rigorously describe the initial and the final states —before and after the interaction— but not cannot provide an accurate description of what happens during the interaction. The Lienard & Wiechert potentials have been traditionally associated to retarded interactions because positions and velocities of the 'source' particles are evaluated at early time $$t_0$$. However, I have shown how they can be derived from instantaneous potentials that are function of positions and velocities at present time $$t$$. This urge to consider what is the origin of the myth of retarded interactions. For such goal I will start again with the Coulomb potential without any lost of generality, because the application to the vector potential $$A$$ is straightforward, $\phi(x, t) = K \frac{e}{|x - r(t)|}$ which will be rewritten as $\phi(x, t) = K \int \frac{\rho(y,t)}{|x - y|} \mathrm{d}y$ using a pure state electron density in position space $$\rho(y,t) = e\delta(y - r(t))$$. Now we can use the equation of motion to relate the present density to a previous density $\phi(x, t) = K \int \exp[L(t-t_0)] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .$ $$L$$ in the above expression is the Liouvillian. We can now see clearly which is the equivalence between instantaneous and retarded potentials. A kernel $$1/|x - y|$$ evaluated at present time $$t$$ is identical to a modified kernel evaluated at retarded time $$t_0$$ $\left\{ \frac{1}{|x - y|}\right\}_t = \left\{ \exp[L(t-t_0)] \frac{1}{|x - y|}\right\}_{t_0} .$ If we replace the full Liouvillian by its free part $$L^\mathrm{free} = - v \nabla$$ we obtain the kernel of the Lienard & Wiechert potentials $\exp[L^\mathrm{free}(t-t_0)] \frac{1}{|x - y|} = \frac{1}{\kappa |x - y|} ,$ with $$\kappa = 1 - v(x-y)/|x-y|c$$. Note that the denominator of the kernel being linear in space variables implies that the power series expansion of the exponential vanishes identically after the linear term in the Liouvillian. In the introduction we derive the Lienard & Wiechert potentials by neglecting acceleration and higher order terms. We can now confirm that the Lienard & Wiechert potentials are an exact consequence of the free component of the full Liouvillian, this free component of course describe inertial particles. The interaction Liovillian will introduce acceleration and higher-order corrections to the Lienard & Wiechert potentials. Corrections to the Lienard & Wiechert potentials will be explored elsewhere, now we want to identify some flaws have remained unnoticed in the electromagnetic literature during decades. Let us start with the ordinary 'wave' equation for the scalar potential $\square \phi = -4\pi K \rho$ Note that the right-hand-side contains the instantaneous charge density, not the density at early times. Now the ordinary literature integrates the equation and obtains the approximated potential $\phi = K \int \rho(t',y) \frac{\delta(t-t'-|x-y|/c)}{|x-y|} \mathrm{d}y \, \mathrm{d}t'$ If we integrate first on position and then on time we obtain the Lienard & Wiechert potential. If we integrate on time then we would obtain $\phi(x, t) = K \int \exp[L^\mathrm{free}(t-t_0)] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .$ However the standard literature gives the wrong result $\phi(x, t) = K \int \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y ,$ with a retarded density $$\rho(y,t_0)$$ which is the origin of the myth of retarded interactions. This discrepancy is due to the standard literature performing the integration of the delta function without careful analysis of the functional dependences of the argument of the delta function on the variable of integration. The correct integration is as follows. First we let $$s \equiv t' + |x-y|/c - t$$ be the new variable of integration. We have $$\mathrm{d}s / \mathrm{d}t' = 1 + \mathrm{d}|x-y|/c\mathrm{d}t'$$ and $\phi = K \int \rho(t',y) \frac{\delta(s)}{|x-y| (\mathrm{d}s / \mathrm{d}t')} \mathrm{d}y \, \mathrm{d}s$ Much care has to be taken on evaluating the term $$\mathrm{d}s / \mathrm{d}t'$$; on a first attempt we could assume that $$|x-y|$$ doesn't depend on time, which implies $$\mathrm{d}s / \mathrm{d}t' = 1$$ and recover the incorrect expression for $$\phi$$ with a retarded density. The subtle issue is that $$|x-y|$$ doesn't depend on time only outside the path of the 'source' particle, but in this trivial case the potential is identically zero. Within the particle path the term $$|x-y|$$ depends on time via the density $$\rho(y,t') = e\delta(y - r(t'))$$. Therefore, it is better to leave the term $$\mathrm{d}s / \mathrm{d}t'$$ in the integral without evaluating it when performing the integration on $$s$$ and use $$y = r(t')$$ at then end, when integrating on space coordinates. With this rigorous method we will obtain $\phi(x, t) = K \frac{e}{|x - r(t)|} ,$ in full agreement with the mechanical result. A pair of final remarks. First, I have focused on retarded potentials but it is possible to obtain the advanced potentials when integrating the equation of motion taking some future time as baseline $$\rho(y,t) = \exp[L(t-t_F)] \rho(y,t_F)$$; there is no violation of causality because the equations of motion are deterministic and can be integrated both backward and forward in time. I have also focused on electromagnetic interactions but the same arguments can be applied to gravitation resulting on instantaneous potentials $$h_{\mu\nu}$$.

### General equation for non-equilibrium reversible-irreversible coupling (kinetic formulation)

Equations used by scientists and engineers can be classified into two larger groups: the first group includes deterministic and time-reversible equations, whereas the second includes irreversible and stochastic equations. Chemical kinetics equations belong to the second group, whereas the Hamilton equations of classical mechanics belong to the first group. Usually scientist work with equations of only one group, but sometimes it is needed to study situations that require elements of both classes at once; in such situations we need a new kind of equation that combines determinism, stochasticity, time-symmetry, and irreversibility all at once. Take the phenomenological equation for Brownian motion for the average momentum of a particle contained in a fluid, as illustration $\frac{\mathrm{d} p}{\mathrm{d}t} = F_\mathrm{ext} - \frac{\zeta}{m} p .$ The term $$F_\mathrm{ext}$$ denotes the external deterministic and time-symmetric force, whereas the second term in the right-hand-side describes the irreversible force associated to friction with the fluid. This friction term has the characteristic form of kinetic equations. Since we are looking for a unified description of both terms, a first approach could be to write down the expression for $$F_\mathrm{ext}$$ using the Hamiltonian formalism and then to obtain a similar expression for the irreversible term. Using the thermodynamics of moving bodies we find $\frac{\partial S}{\partial p} = - \frac{v}{T} ,$ which helps us to rewrite the phenomenological equation of motion in a more unified fashion $\frac{\mathrm{d} p}{\mathrm{d}t} = - \frac{\partial E}{\partial x} + T\zeta \frac{\partial S}{\partial p} .$ This is the approach taken by the people behind the theory named GENERIC —acronym for General Equation for Non-Equilibrium Reversible-Irreversible Coupling—, which they write for generic state variables $$n$$ as $\frac{\mathrm{d} n}{\mathrm{d}t} = L \frac{\partial E}{\partial n} + M \frac{\partial S}{\partial n} .$ Here $$L$$ and $$M$$ are two matrices associated to Hamiltonian flow and friction respectively, in fact $$M$$ is named the friction matrix. The idea of extending the Hamiltonian formalism to irreversible phenomena is very interesting, but unfortunately most irreversible process are not so simple as the Brownian motion and the linear relationship between the generator of time-translations for dissipative processes and the gradient of entropy is lost. For instance for chemical reactions the generator is an exponential with the linear form only valid close to chemical equilibrium. For this reason expressions involving a dissipative potential $$\Xi$$ have been proposed lately for GENERIC $\frac{\mathrm{d} n}{\mathrm{d}t} = L \frac{\partial E}{\partial n} + \frac{\partial \Xi}{\partial (\partial S/\partial n)} .$ Since chemical kinetic equations are known to describe well complex nonlinear phenomena, we will consider an alternative approach. Instead taking the Hamiltonian formalism as base to rewrite the friction term $$(\zeta/m) p$$, we will take the kinetic formalism as base to rewrite the deterministic term $$F_\mathrm{ext}$$, obtaining the next reformulation of the Brownian motion equation $\frac{\mathrm{d} p}{\mathrm{d}t} = \left( K - \frac{\zeta}{m} \right) p .$ Instead writing the concrete expression for $$K$$, we will write directly the general equation for abstract state variables $$n$$ $\frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n$ and the relationship of the generator of time translations $$\Omega$$ with the traditional formulation of GENERIC $\Omega_{\alpha\beta} = L_{\alpha\beta} \frac{\partial E}{\partial n_\beta} \frac{\partial}{\partial n_\beta} + M_{\alpha\beta} \frac{\partial S}{\partial n_{\beta}} \frac{\partial}{\partial n_\beta} .$ This is not the only possible way to write it. Another possibility that resembles a dissipative generalization of the Poisson brackets of classical mechanics is $\Omega_{\alpha\beta} = \frac{\partial E}{\partial n_\alpha} L_{\alpha\beta} \frac{\partial}{\partial n_\beta} + \frac{\partial S}{\partial n_\alpha} M_{\alpha\beta} \frac{\partial}{\partial n_\beta} .$ But all the equations written up to now are deterministic. There are different ways to introduce fluctuations. A heuristic approach using a Fokker & Planck equation was used in the original paper of GENERIC, whereas a systematic approach based in projection operator techniques was used latter. I am exploring a novel approach to fluctuations. The starting point is the assumption that at some initial time $$t^0$$ the state is fully known $$n = n^0 = \overline{n}$$, but fluctuations transform the state into a stochastic variable $$n = \overline{n} + \delta n$$ at posterior times
We have to combine the pair of equations for the stochastic trajectories and the conditional average $\frac{\mathrm{d} \overline{n}}{\mathrm{d}t} = \overline{\Omega} \overline{n}$ $\frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n .$ The result I got after some algebra is $\frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n - \epsilon (n - \overline{n}),$ where $$\epsilon$$ is a positive infinitesimal. Formally, this expression resembles the Zubarev equation used as foundation for an extension of statistical mechanics to irreversible processes, the physical interpretation is however different.

### Researchgate: Are you kidding?

I lack a Researchgate account, but I noticed that Researchgate has created a fake profile about me where they are archiving works from mine whereas miss-attributing one of them to inexistent coworkers. My paper published on the International Journal of Thermodynamics is miss-attributed to two inexistent coworkers Juan Ramon and Callen Casas-Vazquez, when I am the only author.

I tried to join up to correct this blatant error, but due to lacking an institutional email, my request wasn't processed automatically but followed a manual verification procedure. I provided links to my published works and links to the works already archived by Researchgate. Moreover, during the process, the software automatically found some other works from mine.

Today I received a rejection letter:
Dear Juan Ramón González Álvarez,

Thank you for your interest in ResearchGate. Unfortunately we were unable to approve your account request.

I accepted the rejection, because it is their site and their policies, but I mentioned to them it makes little sense to negate me an account whereas archiving my works on a fake profile with inexistent co-workers. I requested Researchgate to delete my profile and the works from their archive. I just received the next funny reply:
Thanks for getting in touch. When browsing ResearchGate you might come across a profile or publications in your name. This is most likely an author profile.

Author profiles contain bibliographic data of published and publicly available information. They exist to make claiming and adding publications to your profile easier.

If you don't have a ResearchGate profile yet, click on the ?Are you Gonzlez lvarez?? button on the top right-hand side of the page to be guided through the sign-up process. Once you've created an account, you'll be able to manage and edit all of the publications on your profile.

Kind regards,

Ben
RG Community Support

Therefore, Researchgate archives my works and automatically generates a profile about me without my permission, miss-attributes one of my works to inexistent co-workers, rejects my request to join, doesn't solve the miss-attribution issue and finally suggests me to join to edit by myself the profile.

Are those guys kidding or it is just plain incompetence?

Update

Finally Researchgate has deleted the fake accounts of the inexistent coworkers Juan Ramon and Callen Casas-Vazquez, changed the profile about me to one new profile with my full name Juan Ramón González Álvarez, cleaned it, and offered me to join them:
Obviously you have proved your credentials as a researcher now, I would be happy to activate your account and assign these publications to your account, should you choose that option.

Kind regards,

Thomas
RG Community Support