tag:blogger.com,1999:blog-35645356523062173262017-02-12T02:59:09.128-08:00juanrgaJuan Ramón González Álvareznoreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3564535652306217326.post-50038548635675957012017-02-04T11:43:00.000-08:002017-02-05T06:13:01.905-08:00There are no negative heat capacities<link rel="icon" href="http://www.juanrga.com/favicon.ico?v=2" /> The existence of exotic systems with anomalous negative heat capacities have been claimed in the recent literature. Those negative heat capacities are responsible for all kind of peculiar thermodynamic behaviors, such as the temperature of the anomalous system being reduced when the system is heated. Close inspection shows that negative heat capacities have not been measured, neither do exist. <br><br>Classic thermodynamic stability theory states that heat capacities have to be positive [1,2]. For a system in thermal contact with a larger system, the second order virtual variation in entropy around an equilibrium state is \[ \delta^2 S = - \frac{C_V (\delta T)^2}{T^2} \lt 0 , \] with heat capacity at constant volume \[ C_V = \left( \frac{\partial E}{\partial T} \right)_V , \] and the negative sign required by stability implying that \( C_V \gt 0 \). Recall that the quantities used in thermodynamics are averages over microscopic realizations. For instance, the thermodynamic internal energy is given by \( E = \langle E^\mathrm{micr} \rangle \). <br><br>Statistical mechanics introduces a different proof of the positivity of heat capacities [3]. It starts from the expression for the energy average over a canonical ensemble \[ E = \langle E^\mathrm{micr} \rangle = \frac{\sum_i E_i^\mathrm{micr} \exp(-E_i^\mathrm{micr}/k_\mathrm{B}T)}{\sum_i \exp(-E_i^\mathrm{micr}/k_\mathrm{B}T)} , \] and gets \[ \frac{\diff E}{\diff T} = \frac{1}{k_\mathrm{B}T^2} \bigg\langle \big(E_i^\mathrm{micr} - \langle E^\mathrm{micr} \rangle \big)^2 \bigg\rangle , \] which is clearly a positive quantity. Notice that the canonical ensemble does not include the volume as variable, and the above total derivative is equivalent to the partial derivative that defines \( C_V \). <br><br>Now, astronomers claim that the above proofs are wrong and that heat capacity can be negative in 'exotic' systems [3]. Their starting point is the virial theorem that relates kinetic energy \( K = \langle K^\mathrm{micr} \rangle \) and potential energy \( \Phi = \langle \Phi^\mathrm{micr} \rangle \) for an isolated gravitational system with energy \( E = K + \Phi \) \[ 2 K + \Phi = 0 . \] Using \( K = \frac{3}{2} N k_\mathrm{B} T \) yields \[ \frac{\diff E}{\diff T} = - \frac{\diff K}{\diff T} = - \frac{3}{2} N k_\mathrm{B} . \] Their 'proof' concludes by associating \( C_V \) to this negative quantity. Astronomers and nuclear physicists not only pretend that negative heat capacities are theoretically admissible, but they also claim those capacities are routinely measured in gravitational systems and nuclear clusters [3], by plotting \( E \) vs \( T \) and then finding regions where energy decreases when temperature increases; i.e., finding regions where \( (\diff E /\diff T) \lt 0 \). A more accurate description of their claim is <blockquote>when heat is absorbed by a star, or star cluster, <b>it will expand</b> and cool down.</blockquote> I bolded the relevant part that disproves their claims. The problem is that the quantity \( (\diff E/\diff T) \) that they are measuring is not \( C_V \) because volume is not held constant during differentiation. If we write a detailed model of the energy we find that the potential energy depends on volume through the interparticle distances \[ E(T,V) = K(T) + \Phi(V) = \frac{3}{2} N k_\mathrm{B} T + \Phi(V) . \] Now, using the definition (2), we find \[ C_V = \left( \frac{\partial E(T,V)}{\partial T} \right)_V = \frac{\diff K(T)}{\diff T} = \frac{3}{2} N k_\mathrm{B} \gt 0 . \] Therefore, what astronomers and others are really measuring is not the heat capacity \( C_V \) but the abstract quantity \[ \frac{\diff E}{\diff T} = C_V + \frac{\Phi(V(T))}{\diff T} . \] This quantity can be positive, negative, or zero depending of the nature of the interactions. For systems satisfying the virial theorem this quantity is just <i>minus</i> the heat capacity \( (\diff E/\diff T) = - C_V \). <br><br><br><h4>References</h4><br><b>[1]</b> Modern Thermodynamics <b>1998:</b> <i>John Wiley & Sons Ltd.; Chichester.</i> Kondepudi, D. K.; Prigogine, I. <br><br><b>[2]</b> Critical analysis of negative heat capacity in nanoclusters <b>2007:</b> <i>EPL, 79, 43001.</i> Michaelian K.; Santamaría-Holek I. <br><br><b>[3]</b> Negative Specific Heat in Astronomy, Physics and Chemistry <b>1998:</b> <i>arXiv:cond-mat/9812172 v1.</i> Lynden-Bell, D. Juan Ramón González Álvarezhttps://plus.google.com/109139504091371132555noreply@blogger.comtag:blogger.com,1999:blog-3564535652306217326.post-14393380137095867692017-01-03T09:50:00.001-08:002017-01-14T04:53:25.623-08:00State space evolution beyond mechanicsOur starting point will be the assumption that the state of our system (biological, physical, chemical, or otherwise) at a given time \( t \) is represented by a collection of \( D \) generic coordinates joined in a vector \( \mathbf{C}(t) = (C_1(t), C_2(t), C_i(t), \ldots C_D(t)) \). Note this vector depends on time <i>implicitly</i>. <br><br>Next, we postulate the existence of a conserved property, named energy, as a function of the state variables \( E = E(\mathbf{C}(t)) \). Differentiating the energy yields \[ \frac{\diff E}{\diff t} = \sum_j \frac{\partial E}{\partial C_j} \frac{\diff C_j}{\diff t} , \] which provides an exact expression of the rate of change of the generic coordinate \( C_i \) \[ \frac{\diff C_i}{\diff t} = \left( \frac{\partial E}{\partial C_i} \right)^{-2} \frac{\diff E}{\diff t} \frac{\partial E}{\partial C_i} - \sum_{j\ne i} \frac{\diff C_j}{\diff t} \left( \frac{\partial E}{\partial C_i} \right)^{-1} \frac{\partial E}{\partial C_j} = \sum_j L_{ij} \frac{\partial E}{\partial C_j} . \] We can write the above equation in vector-matrix form \[ \frac{\diff \mathbf{C}}{\diff t} = \mathbf{L} \frac{\diff E}{\diff \mathbf{C}} = \mathbf{K} \mathbf{C} . \] This is a general equation for the deterministic evolution of any system whose state is given by a non-stochastic vector \( \mathbf{C} \). The scope of this equation of evolution is beyond mechanics because \( \mathbf{C} \) is not limited to the positions and velocities (or momenta) of particles. Note that even if we restrict the vector to \( \mathbf{C} = (\mathbf{p}, \mathbf{q}) \) the description is still more general than Hamiltonian mechanics because the equation (3) can deal with dissipative systems. <br><br><br><h4>Uncertainty and stability</h4><br>The above expressions are deterministic. To introduce fluctuations we can use a variational scheme to find the rate of the deviations from the average deterministic evolution \[ \frac{\diff (\delta\mathbf{C})}{\diff t} = (\delta\mathbf{K}) \mathbf{C} + \mathbf{K} (\delta\mathbf{C}) + \frac{1}{2} (\delta\mathbf{K}) (\delta\mathbf{C}) . \] Combining this expression with (3) yields for the random vector \( \mathbf{\widetilde{C}} = \mathbf{C} + \delta\mathbf{C} \) \[ \frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{K} \mathbf{\widetilde{C}} + \mathbf{\widetilde{f}} , \] for a fluctuation component of the rate given by \( \mathbf{\widetilde{f}} = (\delta\mathbf{K}) \mathbf{C} + (1/2) (\delta\mathbf{K}) (\delta\mathbf{C}) \). Using \( \mathbf{\widetilde{K}} = \mathbf{K} + \delta\mathbf{K} \) the rate can be rewriten in the suggestive form \[ \frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{\widetilde{K}} \, \mathbf{\widetilde{C}} - \frac{1}{2} (\delta\mathbf{K}) \Big [ \mathbf{\widetilde{C}} - \mathbf{C} \Big ] . \] <div class="separator" style="margin:2em 0;text-align:center;"><a href="https://4.bp.blogspot.com/-zkSDSl1FTc4/WHkqpYWbSVI/AAAAAAAAAaI/vU6lLEMWoicqc1MBZ0JSQHkhHWRmsjZmgCLcB/s1600/Fluctuations.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-zkSDSl1FTc4/WHkqpYWbSVI/AAAAAAAAAaI/vU6lLEMWoicqc1MBZ0JSQHkhHWRmsjZmgCLcB/s600/Fluctuations.png" /></a></div> <link rel="icon" href="http://www.juanrga.com/favicon.ico?v=2" />Juan Ramón González Álvarezhttps://plus.google.com/109139504091371132555noreply@blogger.comtag:blogger.com,1999:blog-3564535652306217326.post-65794854077457345952016-12-03T06:28:00.001-08:002017-01-28T12:24:28.888-08:00What is heat?<link rel="icon" href="http://www.juanrga.com/favicon.ico?v=2" /> \( \newcommand{\dbar}{{{}^{-}\mkern-12.5mu \diff}} \) 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. <br><br><br><h4>Energy transfer or state quantity?</h4><br>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 «<i>inexact differential</i>» (symbol \(\dbar \)) because heat is not a state function in the thermodynamic space. <br><br>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? <br><br>Truesdell tries to abandon inexact differentials by just working with rates \[ \mathfrak{Q} = \dot{E} - \mathfrak{W} , \] here \( \mathfrak{Q} \) is what he calls «<i>heating</i>» and \( \mathfrak{W} \) the «<i>net working</i>»; 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 \( \mathfrak{Q} \diff t\). <br><br>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 «<i>heating</i>» \( \dot{Q} \) and «<i>working</i>» \( \dot{W} \). 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. <br><br>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 \( (T,V,N) \). <br><br>It is common to switch to a local formulation in term of fluxes and densities in the irreversible formalisms. Callen <i>defines</i> a generic flow \( J_G \) through \( J_G = dG/dt \), with \( G \) being any extensive <i>state variable</i>. 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 <i>flow of energy</i>. 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 «<i>production of heat</i>» \[ \frac{\diff q}{\diff t} = - \nabla J_Q + \sigma^\mathrm{heat} . \] This source term contains different contributions, including what the authors call the «<i>heat of reaction</i>» generated by chemical reactions taking place inside the system. <br><br>The inconsistencies are obvious now. This confusion is amplified in the engineering literature, where the term «<i>heat transfer</i>» is used routinely. If heat is, as Callen emphasizes, «<i>only a form of energy transfer</i>» 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. <br><br><br><h4>Relativistic heat</h4><br>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? <br><br>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. <br><br>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. <br><br>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. <br><br><br><h4>Microscopic heat?</h4><br>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 \( \dbar Q \). <br><br>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 «<i>microscopic definition</i>» 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! <br><br>A second problem is that, this «<i>microscopic definition</i>» 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. <br><br>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 «<i>microscopic work</i>» being given by \[ \dbar W = \frac{\partial \Phi^\mathrm{ext}}{\partial\lambda} \diff \lambda . \] Roldán claims to «<i>recover the first law of thermodynamics in the microscopic scale</i>». 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. <br><br><br><h4>Heat from first principles</h4><br>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 \( E = \langle E^\mathrm{micr} \rangle \). <br><br>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 «<i>microscopic or instantaneous pressure</i>». 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 «<i>heat capacity</i>» 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 \( T = \langle T^\mathrm{micr} \rangle \). <br><br>Differentiating energy and using the split (21) we obtain for the <b>microscopic heat</b> \[ \dbar Q^\mathrm{micr} = C_V \diff T^\mathrm{micr} + \left[ \frac{\partial\Phi^\mathrm{micr}}{\partial V} + p^\mathrm{micr} \right] \diff V + \frac{\partial\Phi^\mathrm{micr}}{\partial \boldsymbol s} \diff \boldsymbol s . \] 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 splitting to obtain an expression for the classic thermodynamic heat \( \dbar Q \) plus corrections \[ \dbar Q = C_V \diff T + \left[ \frac{\diff\Phi}{\diff V} + p \right] \diff V , \] \[ \dbar (\delta Q^\mathrm{micr}) = C_V \diff (\delta T^\mathrm{micr}) + \left[ \frac{\partial(\delta\Phi^\mathrm{micr})}{\partial V} + \delta p^\mathrm{micr} \right] \diff V + \frac{\partial(\delta\Phi^\mathrm{micr})}{\partial \boldsymbol s} \diff \boldsymbol s . \] The expression for the macroscopic heat agrees with the classical thermodynamic literature, the term within square brackets in (26) is what thermodynamicians denote by \( L_V \), the «<i>latent heat</i>» —another unfortunate name—. 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. The expression for the deviation \( \dbar \delta (Q^\mathrm{micr}) \) is new. Using (24), the latent heat can be rewriten like \[ L_V = \frac{\diff\langle\Phi^\mathrm{micr}\rangle}{\diff V} + p = \left( \frac{\partial\langle E^\mathrm{micr}\rangle}{\partial V} \right)_T + p . \] Let us now to compare this with the literature. Lavenda gives in his study of the «<i>Microscopic Origins of the Carnot–Clapeyron Equation</i>» \[ L_V = \left( \frac{\partial\langle E^\mathrm{micr}\rangle_0}{\partial V} \right)_T - \left\langle \left( \frac{\partial E^\mathrm{micr}}{\partial V} \right)_T \right\rangle_0 . \] With the subindex zero, Lavenda means he is using «<i>unperturbed probabilities</i>» \( \pi_n^0 \), associated to a canonical distribution, to compute the averages. Our expression is not limited to ensemble averages; 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 our mechanical approach continues to work with more general averages. In what follows we will treat both averages as equivalent \( \langle\rangle = \langle\rangle_0\) for simplicity. The first terms of our respective expressions for the «<i>latent heat</i>» agree, the real discrepancy is on the second terms. My expression 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. Note as well that Lavenda claims that for «<i>gases with heat capacities that are power laws of the temperature</i>», the latent heat is given by \[ L_V = \epsilon + p , \] with \( \epsilon \) the «<i>the energy density</i>». Our equation (28) shows that it depends on the potential energy density instead. <br><br><br><h4>Perspectives</h4><br>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. <br><br>I have used inexact differential notation for the sake of familiarity with standard thermodynamics literature. A way to avoid the term «<i>inexact differential</i>» and the corresponding alternative notation will be given in another part. <br><br><br><h4>References</h4><br>Thermodynamics and an Introduction to Thermostatistics; Second Edition <b>1985:</b> <i>John Wiley & Sons Inc.; New York.</i> Callen, Herbert B. <br><br>Modern Thermodynamics <b>1998:</b> <i>John Wiley & Sons Ltd.; Chichester.</i> Kondepudi, D. K.; Prigogine, I. <br><br>Rational Thermodynamics <b>1968:</b> <i>McGraw-Hill Book Company; New York.</i> Truesdell, C. <br><br>On a Scale-Invariant Model of Statistical Mechanics and the Laws of Thermodynamics <b>2016:</b> <i>ASME. J. Energy Resour. Technol. 138(3): 032002-032002-12.</i> Sohrab, S. H. <br><br>Non-equilibrium thermodynamics <b>1984:</b> <i>Courier Dover Publications, Inc.; New York.</i> DeGroot, Sybren Ruurds; Mazur, Peter. <br><br>Thermodynamics of irreversible processes – past and present <b>2012:</b> <i>Eur. Phys. J. H, 37, 139-236.</i> Müller, Ingo; Weiss, Wolf. <br><br>Irreversibility and dissipation in microscopic systems – Tesis Doctoral <b>2013:</b> <i>Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Atómica, Molecular y Nuclear.</i> Roldán, Édgar. <br><br>A New Perspective on Thermodynamics <b>2010:</b> <i>Springer; New York.</i> Lavenda, Bernard H.Juan Ramón González Álvarezhttps://plus.google.com/109139504091371132555noreply@blogger.comtag:blogger.com,1999:blog-3564535652306217326.post-17110540067568581402016-07-26T09:15:00.000-07:002017-01-11T04:10:36.851-08:00Instantaneous electromagnetic interactions<link rel="icon" href="http://www.juanrga.com/favicon.ico?v=2" /> 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. <br><br>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! <br><br>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. <br><br>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 <i>ad hoc</i> but the resulting improved equations of motion are still subjected to criticism due to non-physical behaviors. <br><br>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 "<i>with</i> uniform <i>velocity \( v \) through a frame \( S \)</i>". <br><br>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 <div class="separator" style="margin:2em 0;text-align:center;"><a href="https://1.bp.blogspot.com/-HSJrRELFH8A/V5JbX4_dQfI/AAAAAAAAAF0/42MqYiFQRigAluVe1_FTWsvEZg-Y4gI_ACLcB/s1600/feynman_diagram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-HSJrRELFH8A/V5JbX4_dQfI/AAAAAAAAAF0/42MqYiFQRigAluVe1_FTWsvEZg-Y4gI_ACLcB/s400/feynman_diagram.jpg" width="400" height="310" /></a></div> 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 <i>during</i> the interaction. <br><br>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 \[ \phi(x, t) = K \int \exp \Big[ L(t-t_0) \Big] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .\] \( L \) 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 \[ \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{1}{|x - y|} = \frac{1}{\kappa |x - y|} ,\] 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. <br><br>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 \[ \phi(x, t) = K \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .\] 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 <i>only outside the path of the 'source' particle</i>, 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. <br><br>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 \( h_{\mu\nu} \). Juan Ramón González Álvarezhttps://plus.google.com/109139504091371132555noreply@blogger.comtag:blogger.com,1999:blog-3564535652306217326.post-59975137501916954902016-04-18T10:23:00.001-07:002017-01-11T04:03:27.786-08:00Researchgate: Are you kidding?<link rel="icon" href="http://www.juanrga.com/favicon.ico?v=2" /> 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 <a href="http://dergipark.ulakbim.gov.tr/eoguijt//article/view/1034000436/0" target="_blank">International Journal of Thermodynamics</a> is miss-attributed to two inexistent coworkers <a href="https://www.researchgate.net/researcher/2053054767_Juan_Ramon" target="_blank">Juan Ramon</a> and <a href="https://www.researchgate.net/researcher/2053135147_Callen_Casas-Vazquez" target="_blank">Callen Casas-Vazquez</a>, when I am the only author. <br /><br />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. <br /><br />Today I received a rejection letter: <br /><blockquote>Dear Juan Ramón González Álvarez, <br /><br />Thank you for your interest in ResearchGate. Unfortunately we were unable to approve your account request. </blockquote><br />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: <br /><blockquote>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. <br /><br />Author profiles contain bibliographic data of published and publicly available information. They exist to make claiming and adding publications to your profile easier. <br /><br />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. <br /><br />Kind regards, <br /><br />Ben <br />RG Community Support </blockquote><br />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. <br /><br />Are those guys kidding or it is just plain incompetence? <br /><br /><br /><h4>Update</h4><br />Finally Researchgate has deleted the fake accounts of the inexistent coworkers <q>Juan Ramon</q> and <q>Callen Casas-Vazquez</q>, changed the profile about me to one new profile with my full name <a href="https://www.researchgate.net/researcher/2053069412_Juan_Ramon_Gonzalez_Alvarez" rel="nofollow" target="_blank">Juan Ramón González Álvarez</a>, cleaned it, and offered me to join them: <br /><blockquote>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. <br/><br/>Kind regards, <br/><br/>Thomas <br/>RG Community Support </blockquote><br />I didn't reply... <br /><br /><br /><h4>Second Update</h4><br />Just for curiosity, I checked the status of the fake profile that they still maintain about me. They have changed things again. Apart from listing a set of incorrect disciplines with little to no relation to my work, now they only attribute to me a single publication, whereas my work about heat doesn't appear in my profile but appears standalone. <table width="100%"><tbody><tr><td><div class="separator" style="margin-top:2em;text-align:center;"><a href="https://1.bp.blogspot.com/-yjA6lpNaJEI/V4Uy9GaxOFI/AAAAAAAAAFU/3Sj8OdQGXzUDH0e1g_FKUvinFLlA6tPxQCLcB/s1600/Captura%2Bde%2Bpantalla%2Bde%2B2016-07-07%2B02%253A54%253A07.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-yjA6lpNaJEI/V4Uy9GaxOFI/AAAAAAAAAFU/3Sj8OdQGXzUDH0e1g_FKUvinFLlA6tPxQCLcB/s320/Captura%2Bde%2Bpantalla%2Bde%2B2016-07-07%2B02%253A54%253A07.png" width="420" height="250" /></a></div></td><td><div class="separator" style="margin-top:2em;text-align:center;"><a href="https://1.bp.blogspot.com/-nu59DQluQSE/V4Uy7RHFmzI/AAAAAAAAAFQ/WvODIKSrnIonIhE1ECEUm93uKjhjAnGiACLcB/s1600/Captura%2Bde%2Bpantalla%2Bde%2B2016-07-07%2B02%253A53%253A52.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-nu59DQluQSE/V4Uy7RHFmzI/AAAAAAAAAFQ/WvODIKSrnIonIhE1ECEUm93uKjhjAnGiACLcB/s320/Captura%2Bde%2Bpantalla%2Bde%2B2016-07-07%2B02%253A53%253A52.png" width="420" height="250" /></a></div></td></tr></tbody></table>Juan Ramón González Álvarezhttps://plus.google.com/109139504091371132555noreply@blogger.com