A Note on the Entropy Force in Kinetic Theory and Black Holes
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug A Note on the Entropy Force in Kinetic Theory and Black Holes
Rudolf A. Treumann and Wolfgang Baumjohann International Space Science Institute, Bern, Switzerland Space Research Institute, Austrian Academy of Sciences, Graz, Austria Geophysics Department, Ludwig-Maximilians-University Munich, Germany,
Correspondence to : [email protected]
Abstract.
The entropy force is the collective effect of inhomogeneity in disorder in a statistical many particle system. Wedemonstrate its presumable effect on one particular astrophysical object, the black hole. We then derive the kinetic equations ofa large system of particles including the entropy force. It adds a collective therefore integral term to the Klimontovich equationfor the evolution of the one-particle distribution function. Its integral character transforms the basic one particle kinetic equationinto an integro-differential equation already on the elementary level, showing that not only the microscopic forces but the holesystem reacts to its evolution of its probability distribution in a holistic way. It also causes a collisionless dissipative term whichhowever is small in the inverse particle number and thus negligible. However it contributes an entropic collisional dissipationterm. The latter is defined via the particle correlations but lacks any singularities and thus is large scale. It allows also for thederivation of a kinetic equation for the entropy density in phase space. This turns out to be of same structure as the equationfor the phase space density. The entropy density determines itself holistically via the integral entropy force thus providing aself-controlled evolution of entropy in phase space.
Keywords.
Collisionless reconnection, electron scale current sheets, solar wind
About thirty years ago, Prigogine (1980) attempted a microscopic theory of entropy assuming that, by some quantum process,seeds of entropy could be generated. Such a hypothetical process would, in the early universe, possibly lay down the directionof time. Unfortunately, so far, such microscopic sources of entropy have not been confirmed. It seems that they can hardly beexpected because quantum uncertainty itself is a stochastic process, which by its own nature does not contain any direction.It is hard to believe that it could lead to entropy production if not aided by some kind of dissipative interaction. Entropy is athermodynamic concept, which by itself requires an underlying dynamics, which allows for the presence of many states that asystem consisting of many subsystems, components, particles would be able to occupy.More recently, it has been speculated (Verlinde, 2011) that that kind of a mesoscopic entropy in quantum string theory couldcause gravity to emerge from the action of a quantum entropic force as a gradient of entropy generated in string interactions,intended to provide a physical basis for the so-called modified Newtonian gravity, which proposes that Newton’s law shouldbe corrected on the large scales to eliminate the problem of dark matter in astronomy. rom a completely different point of view, the idea of an entropy force has been picked up in the discussion of maximumentropy methods in prediction theory (Wissner-Gross & Freer, 2013; Kappen, 2013) in open systems where the probabilisticversion of entropy depends on space and time, propagates into the future and, thus, has a finite gradient in space and time, whichis interpreted as force. It apparently is capable of allowing, based on maximization of entropy, predicting the time evolutionof the system, an interesting and possibly far-reaching predictive concept. In a sufficiently small closed system, it necessarilymust describe the evolution of entropy towards a finite thermal state of maximum entropy. Recently, entropy forces have alsobeen applied to molecular dynamics in proteins ((Keul et al., 2018) and the references therein). Spatial smallness is required bycausality to enable synchronization. Therefore, the concept applies to the universe on the cosmological timescale only in orderto allow for homogenization of entropy. On any local scale, the entropy produced in the classical system represents a localizedexcess in entropy. If not artificially confined, this excess tends to expand and affect its environment. This necessarily generatesa local entropy force, a classical force that should not be mixed up with the above-mentioned entropic force in string systems.This force follows from the first law in thermodynamics: dE = T dS − P dV (1)which relates the three different forms of energy E , pressure P V , and entropy
T S . Gradients in energy, pressure, and tem-perature are known to be forces, and the gradient of volume causes dispersion, flows, and forces, for instance in chargedsystems.In a similar vein, a gradient in entropy corresponds to a collective macroscopic effect as the entropy tends to expand andmaximize. This is a purely macroscopic effect indeed because, similar to density/volume and pressure, the entropy S is definedonly for macroscopic systems, consisting of a large number of subsystems, to which finite temperature and density can beassigned and which occupy a finite volume. In the first law, it is only the energy E that maintains its meaning also in themicroscopic world down to only one particle, to which assigning temperature makes no sense. The entropy potential U = T S indeed is not just a thermodynamic potential; it is also a real potential always being positive and thus repulsive. The entropyforce is then given as its gradient: F = −∇ U (2)as usually taken negative. It consist of two parts, a thermal force − S ∇ T , which is of no interest here, and the genuine entropyforce: F S = − T ∇ S (3)This might look trivial; however, it is not, as we will demonstrate below with a particular example: the black hole.However, before proceeding, we recall that, since both T and S are positive definite, the entropy force is repulsive in thedirection negative to the gradient of entropy. This means that an accumulation of entropy at some location, if not artificiallyconfined to a box, will act outward. Adopting an interpretation of entropy as disorder, which by no means is generally justified,thus implies that disorder tends to infect its external region. It has the tendency to expand. ith temperature T in energy units, the entropy S has no dimension. Moreover, the product of temperature and entropy is ascalar function with the dimension of a potential. For scalar temperature, i.e., at temperature isotropy, S is also a scalar. Underconditions of anisotropic temperature, the inverse temperature becomes a vector (Nakamura, 2009), and thus, S becomes avector as well (more generally, both become tensors). In the interest of simplicity, we do not consider this case in the following.This entropy force does not depend on particle mass or charge, at least not explicitly. Mass is contained in temperature andenergy, but there is no explicit reference to it in the definition of the entropy force. Thus, for a given temperature, all particlesindependent of their properties will be subject to the same entropy force. In this sense, the entropy force is a general mechanicalforce seeking to restore smoothness in disorder on a higher level of disorder, completely independent of which kind of particleshave contributed to the inhomogeneity in disorder.By its nature, the above entropy force is a long-range force. It does not compete with the Coulomb force on the shortscales. From the first law, one realizes that the main force it competes with is the pressure gradient. Both are proportional tothe temperature, which thus drops out when comparing the forces. Both are proportional to the density gradient. Thus, in thepresence of changing volume, the entropy force adds to or diminishes the effect of the pressure force. Let us turn to our example, the Schwarzschild black hole, which we chose for demonstration because of its simplicity andcleanness compared to Kerr or charged Nordstrom black holes. Black holes are known to carry entropy (Bekenstein, 1972,1973; Bardeen et al., 1973; Hawking, 1971, 1975, 1976). More correctly, since the interior of the black hole is not accessible,it is the black hole horizon that carries an entropy. This came as a surprise, as it implies that the horizon possesses a temperatureand therefore must be considered as a macrosystem, which occupies a large number of states. Microscopically, this puzzle hasnot been resolved until today, even though a large number of attempts have been put forward to elucidate the internal structureof the horizon (see(Brout et al., 1995; Jacobson, 1995; Doran et al., 2008; Padmanabhan, 2009; Taylor & Tabachnik, 2013)and several others). Such considerations were based on the Bekenstein–Hawking entropy and the Hawking radiation of ablack hole (Hawking, 1975), which is attributed to its finite entropy and thus finite temperature. It implies the existence of athermodynamic for the black hole horizon with the implication that the horizon physics involves a very large number of statesthat can be occupied. Jacobson (1995) extended this concept to horizons in general in order to develop a thermodynamics ofgravitational horizons from which he found that Einstein’s gravitational field equations formally play the role of equations ofstate. This concept was reviewed and extended subsequently by Padmanabhan (2009) to speculate about the general importanceof horizon physics in general relativity and cosmology, suggesting that all the physics is holographically contained in thephysics of horizons.Schwarzschild black holes are in the first place classical objects. However, their entropy includes the quantum nature ofmatter at the horizon (cf., e.g., (Brout et al., 1995)), which is induced by the sharpness of the horizon and indicates that blackholes are not purely classical. Let us ask what the entropy force related to the presence of the horizon would be.
A Schwarzschild black hole of mass M has energy M c , radius R S = 2 GM/c , and spherical surface A BH = 4 πR S . Formingthe ratio of the total black hole energy and the Schwarzschild radius yields: F S = M c /R S = c / G = 6 . × N (4)a constant that has the dimension of a force and that we call the Schwarzschild constant. This is a universal constant, whosevalue is independent of any property of the black hole, a force. This force is the Planck force, which so far has not been givenany physical meaning. We prefer to call it the Schwarzschild constant as the black hole is the only place where it naturallyarises. The entropy of the black hole horizon is the (dimensionless) Bekenstein–Hawking entropy: S BH = A BH λ P = πR S c G ~ (5)with λ P the Planck length, and its corresponding black-body radiation temperature is the Hawking temperature: T BH = k B T H = ~ c / πGM (6)here given in energy units. This yields trivially the Bekenstein–Hawking energy E BH = T BH S BH = M c of the horizon,just half the black hole energy. For a classical black hole, the horizon has no width, suggesting an infinite gradient whencrossing it. In order to obtain the force, the relevant distance taken for the gradient is the diameter R S of the black hole,yielding for the modulus of the outward directed entropy force: F BHS ∼ T BH S BH R S = c G = F S ≈ . × N (7)a quarter of the Schwarzschild constant. A more precise calculation would correct for the numerical factor, which, however,is of order O (1) . Formally, the Schwarzschild constant, and thus also the entropy force at the horizon, is in fact a very strongforce. Its presence, if real, at the horizon is rather surprising. It suggests that the physics of what happens inside the black holeis not really well enough understood.Forming the ratio of the gravitational force F BHG = − GmM/R S any particle of mass m experiences when approaching andtouching the black hole horizon to this horizon entropy force, one obtains: (cid:12)(cid:12)(cid:12) F BHG /F BHS (cid:12)(cid:12)(cid:12) ∼ m/M (8)For any massive black hole and any normal mass particle, this is a small number. A light particle m ≪ M will barely overcomethis repulsion when hitting the horizon. To overcome it, it requires the collision of two black holes of nearly equal mass ∼ M , which would make the ratio m/M → M /M ≈ . Collisions of such nearly-equal mass black holes have onlyrecently been detected by the spectacular observation of gravitational waves.At its horizon, the entropy force of a massive black hole M ≫ m compensates by far for the black hole’s gravitationalattraction on m . This is a consequence of the enormous sharpness of the entropy gradient at the horizon where the entropy isrestricted to the surface of the horizon only, whose width is not precisely given, but as generally assumed, is of the order ofa few Planck lengths λ P only. This force is remarkable only at the horizon itself when the mass m gets into contact with thehorizon. It will not be susceptible at some larger finite distance.This follows from the fact that there is no known classical entropy field that would allow the entropy force to extend a distanceahead of the black hole into the surrounding space and is conjectured from the complete absence of any radial dependence ofthe force outside the horizon. In this picture, the entropy gradient is felt only locally across the horizon of the black hole whenthe particle touches it, an instant that is never seen or experienced by an external observer for whom the time the particleapproaches the horizon stretches out to infinity. The particle, however, does in fact experience the presence of the black holeand, assuming that it remains intact having survived the enormous attraction during its inward spiraling motion, in its properframe at proper time, really touches the horizon and wants to cross it. Shortly before this instant, however, the horizon entropycomes into play and stops the particle.The gravitational force on the particle of mass m (assuming it retains its mass till reaching the horizon, which is certainlynot the case) would overcome the entropic force still only at a small fraction of the radius given by: ∆ rR S ≈ (cid:18) M ⊙ M (cid:19) (cid:18) mM ⊙ (cid:19) ≈ × − (cid:18) M ⊙ M (cid:19) (9)where on the right, we assumed a proton. For instance, this distance for a proton and a M = 10 M ⊙ massive black hole isof the order of only ∆ r ∼ − m, deep inside the submicroscopic domain, though ten orders of magnitude larger than thePlanck length. If it applies, then it would cause accumulation of matter in a film of roughly this width only.The classical picture does not inform about the microscopic physics going on when this happens. Elucidating the real physicsrequires a quantum electrodynamic calculation for instance along the paths drawn by Hawking when calculating the black-body black hole radiation. Referring to Hawking’s results implies that the horizon will be surrounded by a dilute and thin radialdust film of newly- and continuously-created virtual particles, which sustain and support the weak Hawking radiation whentunneling into reality. The radial extension of this dust film implies a softening of the radial entropy gradient corresponding toa finite radial extension of the action region of the entropy force. It would be this radial domain where the light particle in itsinward spiraling motion becomes trapped and retarded and is ultimately stopped and prevented from entering the interior ofthe black hole. What happens to the entropy inside the black hole would be important to know (cf. (Hamilton, 2018) for a discussion ofHawking radiation inside black hole geometry) in order to resolve the puzzle. It is rather improbable that the entropy would beconstant inside the black hole, as this would require that the interior is in thermal equilibrium, which it is certainly not when eing under the conditions of collapsing matter under gravitational attraction. One might speculate that towards deeper inside,the entropy decreases with decreasing radius, because the surface decreases as ∼ ( r/R S ) . Assigning a Hawking temperatureto each shell of such a radius, the corresponding Hawking temperature would increase only as ∼ R S /r . Thus any entropy-force potential should decrease towards the interior like ∼ r/R S . The outer black hole horizon becomes the black hole shell ofmaximum entropy, the radius where the black hole entropy maximizes. The interior entropy force, the gradient of the potential,remains constant throughout the entire interior volume of the black hole with the exclusion of the singularity.Notably, this entropy force points towards the interior of the hole, i.e., towards the singularity. It thus adds to the alreadyexisting gravitational acceleration being felt throughout the entire interior. By pointing inside towards decreasing radius r , itwould push any existing massive particle that made it across the horizon into the singularity up in energy. This probably meansthat classically, no massive particle can make it across the horizon. It can only be non-massive radiation that crosses inward:photons and gluons, the massless bosons of electrodynamics and chromodynamics. Whether massive particles like electronsand quarks can indeed tunnel across the horizon remains a question that cannot be answered in the realm of classical physics.Admittedly, these considerations are rather speculative as long as the evolution of entropy with increasing radial distancefrom the horizon towards inside and also outside the black hole has not been microscopically inferred. In any case, the questionof the horizon representing a sharp surface remains a question that probably only quantum gravity can give an ultimate answerto, as it must proceed on scales close to the Planck scale λ P . This is not our concern here. Naked isolated black holes are invisible except for their weak and so far inaccessible Hawking radiation. The question whyblack holes, which are embedded into surrounding matter, when accreting become visible at all is comparably easy to answer.Any mass flow approaching the horizon before encounter feels the gravitational field of the black hole, spirals in, accelerates,heats up, becomes partially transformed into radiation, and starts radiating violently. General relativity indicates that thisprocess stretches time to infinity. Hence, even though the matter starts as material particles that cannot be digested by the blackhole, because matter cannot pass the horizon due to the barrier the classical entropic force provides for massive particles, theemitted radiation is visible for long. The latter is a well-known fact, and though radiation from accreting black hole suspectshas been observed for decades already, the observational proof has only very recently been given when a particular black holesignature could be resolved in radio emission.During inward spiraling, the matter irradiates, which happens for all the matter that consists of much smaller mass particles(gaseous clouds, dust, stars) than the black hole. The total mass of the matter that hits the horizon at each instant is much lessthan M . The radiation produced in this process may be considered isotropic because there is no remarkable beaming until thematter becomes charged during gravitational compression, heating, and ionization. It then via a dynamo process generates itsproper magnetic field, which tangentially surrounds the black hole horizon and funnels the irreducible to radiation chargedparticles into two approximately symmetric jets. This self-generated magnetic field provides the outflow channel for the escapeof light not irradiated charged matter, which the entropy force rejects from crossing the horizon. Matter accreted by a massiveblack hole does not arrive at the very horizon as matter, but as charge and massless radiation: mostly photons, possibly gluons, epending on the strength of the gravitational force (i.e., on the mass M of the black hole whether or not it would be largeenough to overcome gluon confinement, which is rather unbelievable).The fraction of radiative energy that hits the black hole and gets trapped consists of photons. These are massless and donot feel the entropy force when encountering the horizon, because the entropy force acts on finite mass particles only. Thephotons feel, however, the gravitational black hole potential, which is acting on their energy and thus gravitationally deflectstheir orbits. The photon paths become spirally warped until they ultimately hit the horizon. When this happens, the photonsmake it across the horizon and enter the interior of the black hole. Looked at from the outside, this takes infinitely long again.It is an open question as to what happens to them inside the horizon, whether or not they collapse, and whether a singularityforms at all if only photons are available. We do not ponder about those interesting questions here.The implication is that the mass influx into the hole proceeds via irradiation of matter as radiative inflow, not as matter inflow.The black hole is fed by photons. Feeding a black hole with small portions of matter in this view proceeds via transformationinto radiation. Those parts of matter that do not transform into radiation, protons and the required neutralizing electrons,become expelled along the newly-formed magnetic funnels into space in the form of jets before touching the horizon. The jetseither disperse in interaction with distant matter or become part of cosmic radiation. The process of how radiation may tunnelacross the horizon is answered by the quantum electrodynamics of this process including the positive entropy potential drop atthe horizon the radiation passes when hitting the black hole, which however barely affects the uncharged and massless photons.Another question concerns the merging of two equal mass massive black holes. This case is of substantial interest becauseit has been observed in the first detections of gravitational radiation. If there is just a small mass difference, then probablythe two almost equally-strong forces would produce a deformation of the horizons at contact, causing a bubble to evolve,like in the encounter of two soap bubbles. Merging of the horizons takes place at the circumference where the gradients ofthe entropy become tangential. The holes would start here to merge until the horizon encompasses both holes, with a trappedbubble forming in its common interior and thus becoming invisible to the external observer.Finally, what happens when asking for the mysterious planckions, Planck particles of mass M ∼ − kg ( ∼ GeV),which may have been created in the Big Bang and are believe to be Planck scale black holes? According to Hawking radiationtheory, they should have evaporated in a Planck time of ∼ − s already after production, though it is not clear whether atthe Planck scale, one can speak at all of black holes, as inside the planckion, quantum gravity necessarily comes into play,and Hawking’s quantum electrodynamical calculations should become invalid. Looked at from the outside, a planckion is itsown horizon and thus is fuzzy because its radius and diameter equal spatial uncertainty. Assuming that one still could speakabout their surface, entropy, and entropy force, their entropy would be of the order of S ∼ O (1) , while the entropy force wouldremain huge, equal to the Schwarzschild constant, outrunning the gravitational attraction force. Does this mean that planckionswould neither radiate, nor be able to merge, keeping one another at a distance and in larger numbers causing some crystal-liketexture? In this case, they could have survived (cf., e.g., (Carr et al., 2016) for contras) since the Big Bang and accumulatedin agglomerations like clusters of galaxies where they could well serve (see (Garny et al., 2016) for pros) as a dark mattercandidate. So far, we just discussed the effect of the entropy force on a particular object: Schwarzschild black holes in astrophysics.We now turn to the general kinetic problem of the microscopic evolution of the particle distribution in many-particle physics.We restrict solely to classical systems, i.e., to systems that are described on the microscopic level by the classical Liouvilleequation, respectively its Klimontovich (Klimontovich, 1967) equivalent in Equation (12) given below, and its hydrodynamicgeneralization (Gerasimenko et al., 2011).Liouville’s equation describes the evolution of the microscopic phase space density in N -dimensional phase space. On theclassical elementary level of indistinguishable point charges, which have some properties like mass m a , possibly some charge e of different sign, and are distributed over a spatial volume V with volume element d q and the momentum volume of element d p can be described alternatively (Klimontovich, 1967; Gerasimenko et al., 2011) by an exact known phase space density: N ma ( p , q , t ) = N a X i =1 δ (cid:0) p − p ai ( t ) (cid:1) δ (cid:0) q − q ai ( t ) (cid:1) (10) ≡ N a X i =1 δ (cid:0) x − x ai ( t ) (cid:1) which simply counts the number of particles of sort a in the entire 6 D -phase space volume, such that it is normalized as: N a = Z d p d q N ma ( p , q , t ) (11)For a constant particle number, the time dependence is implicit in the particle trajectories p ai ( t ) , q ai ( t ) such that integrationhas to be performed along all of them. One may note that the microscopic phase space density N ma is otherwise dimensionless.This is seen from the definition of the delta-functions, which in the integration over phase space simply count numbers, whichof course means that the normalization to space and momentum is implicit to them. Later, we will make the normalizationmore explicit, as this will be required by reference to the entropy.Since the assumption is that the particles are classical, then in the absence of any particle sources or losses, the particlenumber in phase space is conserved along all the dynamical trajectories of the particles under their mutual, as well as externalforces. In this case, the continuity equation of the particles, i.e., the microscopic Liouville equation in the N a -particle 6d-phasespace (Klimontovich, 1967), reads simply: ˙ N ma ≡ ∂ N ma ∂t + p m a · ∇ q N ma + d p dt · ∂ N ma ∂ p = 0 (12)Of course, here, ˙p = F is the total force that acts on the particles at their location q = q ai ( t ) and thus on the phase spacedensity, and the two last terms together constitute the Poisson bracket [ . . . ] in the Liouville equation, which in N a -phase space,the 6d-phase space that within, the N a particles perform their trajectories, is a tautology.The entropy force can be compared with other more conventional forces. Let, for simplicity, the total force F = F Q + F S bethe sum of the entropy force and of another potential force F Q = −∇ q U , where U ( q , t ) is the force potential. The entropy forcejust adds the potential T S ( q , t ) to the force potential U . Note that the entropy-force potential is always positive, as already ade use of above, because there are neither negative temperatures (Dunkel and Hilbert, 2014) (The absence of negativetemperatures is immediately clear from the definition of the temperature T as proportional to the mean ensemble averagedsquare of the momentum fluctuations T ∼ h ( δ p ) i of all particles in the volume, which clearly, is a positive definite quantity.Negative temperatures would require imaginary momenta or particle mass. There are no candidates for such particles, thoughexperiments of neutrino oscillations provide negative mean square masses, which, however, are interpreted differently.), norare there negative entropies. Clearly, for strong forces acting on the particles and weak entropy gradients, the entropy forceis negligible. This might be the usual case. On the other hand, if on the large scale the inter-particle forces compensate, theentropy force will remain because there is no obvious counterpart that could compensate it. For instance, when dealing withelectrostatic interactions only in the absence of any external fields and forces, the microscopic force F mQ ( q , t ) = P a e a E m ( q , t ) is the Coulomb force acting on the charges e a = ae with a = + , − in the microscopic electrostatic field E m ( q , t ) obeyingMaxwell–Poisson’s equations: ∇ q · E m = 1 ǫ X a ρ mae ( q , t ) , E m = −∇ q Φ me ( q , t ) (13)with electrostatic potential Φ me , and thus, U = P a e a Φ me . On the microscopic level in phase space, the microscopic electricspace charge (not the charge density) of species a is: ρ mae ( q , t ) = e a Z d p N ma ( p , q , t ) , Z d qρ mae = e a N a (14)It simply counts all charges in the total volume not relating them to the spatial volume V a yet. Summing over all species a , thetotal space charge is obtained. On average, it will be zero. The charges are moving, and there is a microscopic current: j mae ( q , t ) = e a m a Z d p p N ma ( p , q , t ) = e a v a ( q , t ) N a (15)with v a ( q , t ) the average velocity of particles in group a . It gives rise to an internal magnetic field, which, in the electrostaticapproximation, is relativistically small and is thus neglected (e.g., (Klimontovich, 1967)), though this is not completely correct,because in a linear theory of fluctuations, it should be taken into account.Electrostatic interactions have been the subject of exhaustive investigations in the literature. Here, they serve only as anotherforce field against which the entropy force can be compared. The striking difference is that for the entropy force, no field isgenerated because there is no entropy charge comparable to e a and, hence, no singularity that would act as the source of theentropy field. In other words, the entropy field is, in contrast to the electric field, not related to field equations and thus lacks afield theory. Disorder lacks any elementary source not being a field, at least in classical physics.The entropy of a system entering the first law of total energy conservation is an integral quantity. In order to refer to it on theelementary level of the microscopic kinetic equation in 6d-phase space, one has to return to its microscopic definition as thephase space average of the probability distribution.It is convenient to define an entropy phase space density by referring to Gibbs–Boltzmann’s definition of entropy throughthe probability density. Entropy density will then be obtained by integrating out the momentum space coordinates in theusual way. In the definition of the phase space density of entropy, we will at this point not yet make the assumption that the hase space volume is constant, but include the spatial dependence as well. This is advantageous because it allows makinguse of phase space densities. Integrating out the volume can be done at a later stage. With this in mind, the “microscopicBoltzmann entropy phase-space density” of species a becomes (A number of other definitions or generalizations of entropydifferent from Boltzmann–Gibbs have been put forward in the near past (Rényi, 1970; Wehrl, 1978; Tsallis, 1988; Treumann,1999; Treumann et al., 2004; Treumann & Jaroschek, 2008; Treumann & Baumjohann, 2014), the physical, not the statisticalmeaning of which is not entirely clear. Though the theory could be extended to include those, we will neither refer to, nor usethem in this note.): S maB ( p , q , t ) = − log N ′ ma ( p , q , t ) , N ′ ma = N ma /N a (16)where the phase space density has been normalized to the total number N a of particles of species a . This makes the argumentof the logarithm smaller than one, of which the negative sign takes care. A definition like this leans on Boltzmann’s proposal.It is incomplete on the microscopic level because the entropy is a collective quantity, which is obtained by integrating overmomentum space with the microscopic phase space density N a as the weight. One thus has as microscopic N -particle entropyphase-space density in ( p , q ) -space: S ma ( p , q , t ) = −N ma ( p , q , t ) log N ′ ma ( p , q , t ) (17)This microscopic phase-space density of the entropy is not the entropy itself, which is a function solely of the space coordinates.The N a -particle entropy of sort a is explicitly obtained by the integration of (17) over the entire momentum phase space: S ma ( q , t ) = Z d p S ma ( q , p , t ) (18) ≡ Z d p ′ d q ′ δ ( q − q ′ ) S ma ( q ′ , p ′ , t ) and is always positive, as is easily seen from the definition of the microscopic phase space density, a positive quantity, and theabove choice of entropy. Summing over all particle species a then gives the total entropy. Moreover, because information istransported via some field, for instance the electromagnetic field, the time under the integral in Equation (17) is the retardedtime t R = t − | q − q ′ | /c where c is the velocity of signal/information transport between the particles at locations q and q ′ inthe real-space subspace of the 6d-phase space (Wheeler &Feynman, 1945; Wheeler & Feynman, 1949). In conventional kinetictheory, retardation is neglected because c is the velocity of light, and the distances between particles are usually less than ct .This is also assumed in the following.There is a direct correspondence between this real space microscopic entropy density and the real space charge density ρ m ( q , t ) . Both enter the force term via taking the spatial gradient. The difference is that for the electric charge density, this steppasses through the electric field E m , which is generated by the space charges. Repeated again, for the entropy, there is no suchfield, nor field equation in classical physics. Entropy is not a charge of some entity and thus does not generate a field. Takingits spatial gradient directly provides the force that acts on the particle at location q .
10 Kinetic Equation with Entropy Force
The entropy force acting on species a is the negative gradient of Equation (18). This leads to a repulsive force, independentof any charge. It adds to the potential U in Klimontovich’s equation. Thus, taking it into account in Equation (12), it becomesclear that it does not affect the particle number and thus does not imply any important change in the microscopic N a -particlephase space density N ma . The main interest is in its effect on the one-particle kinetic phase space distribution function f a ( x , t ) .This is defined through the ensemble-averaged N a -particle phase space density: N a V a f a ( x , t ) = (cid:10) N ma ( x , t ) (cid:11) (19)where h . . . i indicates the ensemble average, and explicitly for the one-particle distribution: f a ( x a , t ) = V a Z f N d x a . . . d x aN a Y b = a d x b . . . d x bN b (20) (cid:10) N ma ( x , t ) (cid:11) = N a Z δ ( x − x a ) f N Y a d x a . . . d x aN a (21)with V a ≡ V the spatial volume occupied by the indistinguishable particle sort a . f N is the N -particle distribution function,and the integration is with respect to all indistinguishable particles N − , but one, the particle with coordinates x a , as hasbeen defined by Klimontovich (1967). In fact, the distribution function f N is not explicitly given. It can be resolved on the wayof sequentially stepping up the ladder from the one-particle distribution function to higher order distribution functions, whichdepend on one, two, three, or more indistinguishable particles.Before proceeding to rewriting the Klimontovich Equation (12), it is necessary to investigate what happens to the entropywhen performing the ensemble average implied in the former equations. We do actually not need the entropy itself, rather itsspatial gradient, i.e., we need the spatial gradient of the entropy-phase space density (17). For this, we have: −∇ q S ma = (cid:16) N ′ ma (cid:17) ∇ q N ma (22)The entropy force is the integral of the gradient of Equation (18) over the primed phase space: F aS ( q , t ) = T ∇ q Z d p ′ d q ′ δ ( q − q ′ ) × (23) × N ma ( p ′ , q ′ , t ) log N ′ ma ( p ′ , q ′ , t ) Reduction of the entropy-force term in Klimontovich’s equation requires performing the ensemble average of the term: F aS ( q , t ) · ∂∂ p N ma ( q , p , t ) (24)In order to do so, we need to consider different groups of particles such that: F S ( q , t ) = T X b ∇ q Z d p ′ d q ′ δ ( q − q ′ ) × (25) × N mb ( p ′ , q ′ , t ) log N ′ mb ( p ′ , q ′ , t ) his produces formally the entropy force contribution to the Klimontovich equation: F S · ∂∂ p N ma = T ∇ q X b Z d p ′ d q ′ δ ( q − q ′ ) · (26) · ∂∂ p (cid:28) N ma ( p , q , t ) N mb ( p ′ , q ′ , t ) log N ′ mb ( p ′ , q ′ , t ) (cid:29) where h . . . i indicates the ensemble average, and we have used Equation (22). The momentum differentiation affects only termscontaining the phase space density. This leads to the appearance of the logarithmic term on the right and introduces a third-order correlation term. The ( N − -particle ensemble-averaged term provides problems because it contains the logarithm ofthe phase space density. In a somewhat severe approximation, we may assume that the logarithm is a slowly-varying function.Its argument is smaller than one such that it can be expanded, which yields: (cid:28) N ma ( p , q , t ) N mb ( p ′ , q ′ , t ) log N ′ mb ( p ′ , q ′ , t ) (cid:29) (27) ≈ (cid:28) N ma ( p , q , t ) N mb ( p ′ , q ′ , t ) (cid:16) N ′ mb ( p ′ , q ′ , t ) − (cid:17)(cid:29) This generates the ensemble-averaged Klimontovich equation: ∂ h N ma i ∂t + p m a · ∇ q h N ma i − T ∇ q X b = a Z d x ′ δ ( q − q ′ ) (28) · ∂∂ p D N ma ( x , t ) N mb ( x ′ , t ) E = − D C Sa ( x , t ) E The average purely entropic collision term on the right collects the third-order correlations: D C Sa ( x , t ) E = T ∇ q X b = a Z d x ′ δ ( q − q ′ ) · ∂∂ p D N ma ( x , t ) N mb ( x ′ , t ) N ′ mb ( x ′ , t ) E In these expressions, we have, for simplicity of writing, only included the entropy force term. One trivially adds the microscopicCoulomb or any other force term to this if required (Note, however, that adding the microscopic gravitational force causesproblems because it remains uncompensated (As in any kinetic theory, this is an important difference between gravitation andany other force. It implies that in kinetic theory gravitation can only be included consistently in a general relativistic quantumgravitation where gravitation is balanced by quantum fluctuations.).). The entropy force term resembles the latter, but lacks acharge singularity. This is replaced by the spatial derivative of the delta-function, which appears under the integral.The main difference is that already on this very basic level, the presence of the entropy force contributes a purely entropicdissipative term hC Sa ( x , t ) i , which has been transferred to the right in the above expression. This term arises due to the genera-tion of entropy in the system. It is a three-particle correlation term, as will become clear below. It is caused by the logarithm inthe entropy, the continuous growth of entropy in a many-particle system. Whether it can be neglected as being of higher orderis a subtle question. It causes collisionless dissipation in the presence of entropy. Since this effect is non-collisional, whenneglecting particle collisions, one must take care whether its neglect is allowed. Below, we show that, however, dissipation isproportional to the inverse particle number N − a and can in most cases for very large numbers of particles be neglected. he next step in this theory is to relate the last equation to the one-particle distribution function defined in Equation (19).Following Klimontovich (1967), this is achieved via considering the fluctuations: δ N ma ( x , t ) = N ma ( x , t ) − D N ma ( x , t ) E (29)When ensemble-averaged, these deviations from the mean phase-space density vanish, and we have: D N ma ( x , t ) N mb ( x ′ , t ) E = D N ma ( x , t ) ED N mb ( x ′ , t ) E (30) + D δ N ma ( x , t ) δ N mb ( x ′ , t ) E We can now make use of the definition of the one-particle distribution function f a ( x , t ) by Klimontovich (1967). Define theparticle density n a = N a /V a to obtain: D N ma ( x , t ) N mb ( x ′ , t ) E = n a n b h f a ( x , t ) f b ( x ′ , t ) (31) + g ab ( x , x ′ , t ) i + δ ab δ ( x − x ′ ) n a f a ( x , t ) Here, g ab ( x , x ′ , t ) is the two-particle correlation function, which results from the ensemble-averaged product of the fluctuations δ N m of the phase space density in the last term on the right in Equation (30). The three terms in the expression (31) are of thesame structure as in the ordinary one-particle kinetic theory (Klimontovich, 1967). One may note that the last term, which islinear in the distribution function, simply becomes absorbed in the convective term in the kinetic equation. In non-relativistic,theory it just causes a translation. We can immediately write down the one-particle kinetic equation including the entropy force.One must, however, take care of to which terms the gradient and momentum operations apply. This yields the result: ∂f a ∂t + p m a · ∇ q f a − T ∇ b q X b n b Z d x ′ (32) × δ ( q − q ′ ) · ∂∂ p h f a ( x , t ) f b ( x ′ , t ) i = (cid:16) G S (x ,t ) ab − D C S (x ,t ) a E(cid:17) ( x , t ) which, when integrating over the primed spatial coordinate, simplifies to: ∂f a ∂t + p m a · ∇ q f a − T ∂∂ p f a ( q , p , t ) (33) ·∇ q X b n b Z d p ′ f b ( q , p ′ , t ) = (cid:16) G S (x ,t ) ab − D C S (x ,t ) a E(cid:17) ( x , t ) In this expression on the right, the term G ab results from the two-particle correlation term g ab ( x , x ′ , t ) . It corresponds to whatin kinetic theory is understood as direct particle collisions. The last expression contains the integral over the primed momentumspace. Only the distribution f b depends on this integration. It is therefore convenient to define the number density ρ a of species a as: ρ a ( q , t ) = n a Z d p f a ( q , p , t ) (34) nd the last expression just includes the global entropy force term: ∂f a ∂t + p m a · ∇ q f a − T (cid:18) ∇ q X b ρ b ( q , t ) (cid:19) · ∂f a ∂ p (35) = (cid:16) G S (x ,t ) ab − D C S (x ,t ) a E(cid:17) ( x , t ) The sum is over all particle components, implying the total number density. Thus, the entropy force term simply adds toany other potential force term in the kinetic equation. This is true already to first order in the expansion of the logarithmicterm in the definition of the entropy. In the case of charged particles, such a force is the Coulomb force or the Lorentz force,when including magnetic fields. The difference is, however, that this force term does not depend on charge while acting onthe microscopic particle phase space distribution. It resembles the gravitational force, but does not contain its inverse squaredependence on the inter-particle distance. This is advantageous as it releases from the necessity of compensation. On the otherhand, the new force term introduces another non-linearity contained in the density, which itself is the integral of the distributionfunction.The entropy force resembles a pressure force on the kinetic level. With zero right-hand side in the kinetic equation, itconserves particle number. This is a rather simple result, which, of course, could have been anticipated, without reference toany complicated derivation from first principles as done here, by adding a macroscopic entropy force to the force terms.The ensemble-averaged term (cid:10) C Sa (cid:11) is a purely entropic lowest order (cid:0) in the smallness of N ′ ma ) dissipation term for which,in conventional kinetic theory, no equivalence arises. This term is, however, small and thus negligible, as will be shown in thenext section.In the collisionless kinetic theory of forces between particles, any non-collisional dissipation term caused by particle in-teractions via their fields yields correlations, which can be neglected, respectively discussed away by comparing dissipationand collisionless scales. The entropic dissipation term instead remains because it is not caused by particle collisions, norwave–particle interactions. There is no entropy source field that leads to the correlations between particles. Rather, it is theinhomogeneity in the macroscopic disorder that is responsible for the fluctuations and the appearance of the dissipative en-tropic correlations between the fluctuations leading to the dissipation term. Hence, this term remains even under completelycollisionless conditions. Once disorder exhibits spatial structure, it will always be present. In the next section, we provide theexplicit versions of these two terms. In order to complete the theory, one needs to express the two dissipative terms in the final kinetic Equation (32). The collisionterm G ab is of the same structure as the Coulomb collision term (Klimontovich, 1967; Treumann & Baumjohann, 2017). It addsto the latter: G Sab = T X b n b ∇ q · Z d x ′ δ ( q − q ′ ) ∂∂ p g ab ( x , x ′ , t ) (36) n the particular case that the correlation g ab does not contain any singularity, this expression has no singularity at q = q ′ otherthan that in the derivative of the delta-function, which replaces q ′ → q in the correlation function g ab when the integration iscarried out. The remaining expression becomes: G Sab ( q , p , t ) = T ∂∂ p · X b n b Z d p ′ ∇ q g ab ( q , p , p ′ , t ) (37)The presence of the entropy force thus contributes to the collision term via the particle correlation function. Clearly, dueto the strong Coulomb force at short distances, the particle interaction on the short scales is dominated by the Coulombforce. However, at distances larger than the Coulomb collision length, the entropic collisional interaction remains. In a chargecollisionless plasma, for instance, the Coulomb term causes charge screening felt inside the Debye sphere and eliminates themicroscopic electric field between the charges on scales larger than the Debye length λ D . On such scales, entropic dissipationmight enter the scene. Since it is proportional to temperature T and also number density ρ , it has the character of a collisionalcontribution of the pressure in the inhomogeneities of the entropy.One may take notice that the spatial gradient operator can be taken out of the integral in the last expression. This allowswriting: G Sab ( q , p , t ) = T ∇ q · ∂∂ p G Sa ( q , p , t ) (38)where we introduced the entropic correlation integral: G Sa ( q , p , t ) = X b n b Z d p ′ g ab ( q , p , p ′ , t ) (39)We will return to the discussion of this correlation integral below, because it contains the most interesting effect introduced bythe entropy force.In general, the correlation will contain contributions from the forces that depend on local charges. This leads from the fieldequations to singularities in the correlation function, as the example of the Coulomb force suggests. The simplified form of thecorrelation term in the dissipation function retains the form in Equation (36) and must be explicitly spelled out. Exchangingthe derivatives, this can be written as: G Sab = T ∂∂ p · X b n b ∇ q Z d x ′ δ ( q − q ′ ) g ab ( x , x ′ , t ) (40)Let us now turn to the remaining term (cid:10) C Sa (cid:11) . From Equations (27) and (28), one realizes that it contains the product ofthree microscopic phase space densities before taking the ensemble average. This complicates its calculation. In analogy toEquations (30) and (31), it requires the introduction of higher order correlations. Formally, this is quite simple as it has beenpioneered by Klimontovich (1967) how one would have to deal with it in this case. We need the third-order ensemble average, hich becomes in the same way as (31): D N ma N mb N mc E = n a n b n c f abc ( x , x ′ , x ′′ )+ δ ab n a n c δ ( x − x ′ ) f ac ( x , x ′′ )+ δ ac n a n b δ ( x − x ′′ ) f ab ( x , x ′ ) (41) + δ bc n a n c δ ( x ′ − x ′′ ) f ac ( x , x ′′ )+ δ ab δ bc δ ( x − x ′ ) δ ( x − x ′′ ) f a ( x ) For convenience, we dropped the common variable t . The two- and three-particle distribution functions f ab , f abc read: f ab ( x , x ′ ) = f a ( x ) f b ( x ′ ) + g ab ( x , x ′ ) f abc ( x , x ′ , x ′′ ) = f a ( x ) f b ( x ′ ) f c ( x ′′ ) (42) + f a ( x ) g bc ( x ′ , x ′′ ) + f b ( x ′ ) g ac ( x , x ′′ )+ f c ( x ′′ ) g ab ( x , x ′ ) + g abc ( x , x ′ , x ′′ ) With their help, the dissipation function can be constructed. However, its structure simplifies substantially because in our case,on the left in the first line in Equation (41), the microscopic phase space densities of index b and c are identical, and only the f ab contributes. Hence, the ensemble average becomes: D N ma N mb N ′ mc δ bc δ ( x ′ − x ′′ ) E = n a n b f ab ( x , x ′ ) 1 N b (43)Therefore, only the two-particle distribution function f ab would be relevant in the determination of the dissipative term, i.e.,in the first line in Equation (42). This is, however, the same as what we already used in Equation (31). To this result, one hasto apply the operation of space and momentum differentiation. Hence, the collisionless dissipative term contributed by theentropy force is of the same kind as the collision term G Sab it contributes, though being of a different sign. In other words, thetwo terms would cancel to first order if there were not the normalization to particle number N b . Since N b ≈ N a ≫ , we findthat the collisionless dissipation due to the entropy force hC Sa i ≪ G Sab is small and can be neglected in comparison with theentropic collision term.Thus, it is the collisional correlation term G Sab ( x , t ) (37) that is retained as a long-range collisional dissipation introduced bythe presence of the entropy force. It adds to the Coulomb collisions and might become important on the large scales much largerthan the Debye scale or any other inter-particle interaction scale. this important result suggests that the entropic dissipation is amesoscale, respectively macro effect. We should, however, point out here that we are still dealing with a non-relativistic theory.At large scales, transport and propagation of information, respectively entropy, cannot be neglected anymore, and the theoryhas to given a covariant relativistic formulation.
16 Kinetic Equation for Fluctuations
The remaining problem is the behavior of fluctuations. These are defined as deviations in the one-particle phase space distri-bution f a from its mean “equilibrium” value ¯ f a as: δf a = f a − ¯ f a , δf a = 0 (44)The evolution equation of the fluctuations is obtained from Equation (33) via subtracting the averaged kinetic equation: ∂ ¯ f a ∂t + p m a · ∇ q ¯ f a − T X b n b Z d p ′ × (45) × ∇ b q · ∂∂ p h ¯ f a ( q , p , t ) ¯ f b ( q , p ′ , t ) + δf a δf b i = G Sab
The mean collision term on the right contains all the contributions of the correlations of the mean and fluctuating quantities.Being interested only in linear fluctuations and assuming that the collisions are weak enough to not contribute to the evolutionof fluctuations, we drop this term in the following. Subtracting from the complete kinetic equation, the fluctuations obey thenon-collisional equation: ∂δf a ∂t + p m a · ∇ q δf a − T X b n b Z d p ′ × ∇ b q · ∂∂ p h δf a ( q , p , t ) ¯ f b ( q , p ′ , t ) i = (46) T X b n b Z d p ′ ∇ b q · ∂∂ p h ¯ f a ( q , p , t ) δf b ( q , p ′ , t ) − δf a δf b i This expression still contains the average δf a δf b of the squared fluctuations. If this is a constant on the fluctuation time scale,then the equation can be rescaled. In linear theory, it would be neglected to first order and taken into account to second orderin a quasi-linear approach.Again, carrying out the integration with respect to p ′ , the last expression simplifies to: ∂δf a ∂t + p m a · ∇ q δf a − T ∂δf a ∂ p · ∇ q (cid:18) X b ¯ ρ b ( q , t ) (cid:19) (47) = T X b = a n b Z d p ′ ∇ b q · ∂∂ p h ¯ f a ( x , t ) δf b ( q , p ′ , t ) − δf a δf b i where in the term on the right-hand side, we retained the fluctuation in the distribution function, not replacing it with the densityfluctuation δρ b ( q , t ) for the obvious reason that this is an equation for the fluctuations in the distribution function itself. Theterm on the right couples all fluctuations in the different particle components b = a to ¯ f a .The linear theory is still complicated by the fact that it contains the sum over the particle correlations. Here, one must includeall particles contained in the medium. Moreover, we have written here only those terms that result from the inclusion of theentropy force. To these terms, one must add the electromagnetic force terms. Since the electromagnetic and entropy forces uperimpose, this does not produce any additional mixing, but simply adds those common and well-known terms that takecare of the electromagnetic interactions. In this sense, the formal theory is complete. The ranges of the two different forcesacting on the particle populations are vastly different because the entropy force is a collective force, which does not originatefrom any elementary charge. There is no singularity of the entropy that could give rise to an entropy field. The search for suchsingularities is outside classical physics. Having defined the entropic phase space density in Equation (17,) the question arises how it possibly evolves in phase space.Since the entropy is given as the phase space integral with respect to the entropic phase space density, an always positivequantity, this question is not senseless. Once knowing its evolution, the entropy can be calculated by integration. Moreover,if an equation for the phase space density can be obtained, its entropic momentum should yield an evolution equation for theentropy, which, essentially, in the long-term limit should be the fundamental thermodynamic laws, while in the short term, itshould give the evolution equation of entropy with time. In order to construct the entropic phase space equation, we multiplyEquation (12) by − log N ′ ma to obtain: ∂ t S ma + h H N a , S ma i − n ∂ t N ma + h H N a , N ma io = 0 (48)The term in the braces vanishes identically, yielding ultimately: ∂ t S ma + h H N a , S ma i = 0 (49)We thus find the almost trivial result that the microscopic entropic phase-space density S ma itself satisfies the Liouvilleequation, i.e., the continuity equation for the entropic phase space density in the phase space. It thus evolves in the microscopicparticle phase space like a dissipationless fluid. This is just another expression for the fact that (classically), there are nomicroscopic sources of entropy, nor is there any entropic field. Entropy is just disorder in the particles.However, the microscopic entropic density in phase space is not entirely independent of any disorder. It acts back on itselfvia the integral entropy force term contained in the above Hamiltonian H N a . It provides an entropy potential contribution U S = T S to the Hamiltonian with S , the momentum space integral of the entropy phase space density S ma {N ma } , which itselfis a function of the phase space density N ma . Though the structure of the kinetic equation for the entropy density S ma remainsthe same as that of the phase space density N ma , both containing the entropy force term and becoming integro-differentialequations, the phase space density is determined by the integral entropy density through the entropy force. This force isobtained by adding up all contributions over all phase space. This shows that both the kinetic equation for the particle densityand the kinetic equation for the entropy density in phase space must be solved together as both are intimately related.One can interpret this result in the way that the holographic reaction of the integral entropy on the evolution of the phasespace density of the entropy appears like an elementary entropy source. This is not unsatisfactory, because it can hardly beexpected that a microscopic source of entropy would exist as there are no entropy charges and no entropy fields in the world, t least not classically. Instead, the elementary source arises from the non-linear self-interaction of the entropy. This is a ratherimportant conclusion in that, presumably, little will be changed when including quantum effects in a quantum mechanicaltreatment, making the transition to quantum statistical mechanics. In this note, we included the force that an inhomogeneous entropy might exert on the dynamics of particles in phase space.This is not an obvious step. It is a purely collective effect. So far, any such force has not yet been included in the dynamics oflarge numbers of particles and kinetic theory.Collective effects of this kind are known from ponderomotive forces and pressure forces. However, the inclusion of a separateponderomotive force or a pressure force on the microscopic level is not necessary. Ponderomotive forces are taken care of bythe interaction of the electromagnetic field with the particles. They arise from correlations. Similarly, the pressure force, whichis directly proportional to the gradient of the particle density on any level, is already included in the evolution of the phasespace density.The entropy force is different in the sense that it is not obvious that it is given by the gradient of density. The entropy forcetakes care of the chaotic disorder that is produced by the dynamics itself. Neither the pressure, nor the temperature accountfor it. Entropy is a function of the phase space density, not its moment like pressure and temperature. It is the cause of chaoticexpansion of the phase space in the course of the dynamics. It therefore gives rise to a collective effect felt on the level ofthe microscopic phase space density. In many processes, this effect may be very small and negligible. This will depend onscales. Elucidating those effects requires investigation of particular cases. The present work presents the theory on which suchattempts must be based.We derived the basic kinetic equation for the one particle phase space density including the global entropy force. Sincethis force is a global integral one, it has no microscopic source, while it affects the dynamics of the one-particle phase spacedensity through the interaction Hamiltonian. This gives rise to the construction of a kinetic equation for the entropic phasespace density, which turns out to be of similar structure, like the Liouville (Klimontovich) equation for the phase space density.There is, however, a fundamental difference in that the entropic kinetic equation is self-referential. It determines the dynamicsof the entropy density in phase space by reference to the integrated entropy of the entire system itself. This is a very importantfinding because it shows that the entropy density in phase space evolves holographically. It is determined by itself. It takes careof its own evolution, which in addition depends on the evolution of the matter phase space density. This can be interpreted as aself-regulation of the evolution of entropy on the microscopic level, which takes care of the total produced entropy. Accordingto this finding, any many-particle system that in the course of its dynamics generates entropy in an inhomogeneous and time-dependent way is subject to the entropy that controls its own evolution. This we feel is the most important insight we arrivedat in our analysis. Entropy generates and controls itself in this highly nonlinear way already on the microscopic level of phasespace density. We conjecture that these properties remain when including quantum effects in kinetic theory on the microscopicquantum level without the need to introduce a seed source of entropy. n order to demonstrate an effect, we in the introductory part of this note treated the astrophysical example of a Schwarzschildblack hole. This led us to the definition of the Schwarzschild constant, a universal constant, which is essentially the Planckforce, which so far had been formally postulated without finding a physical interpretation. Its meaning lies in the Schwarzschildconstant as the entropy force at the black hole horizon. Some of its implications we have discussed briefly. The entropy forceat the horizon, being independent of charge, is of only one sign. It causes a repulsive force. This is similar to the gravitationalforce, though counteracting it. It thus in large space may compete with the gravitational attraction, where it may become kindof an anti-gravity. Investigation of its effect on the universal expansion might also be of interest, as also the role it may play inprimordial black holes and planckions, which may have been generated in the early universe, the Big Bang, and if surviving,possibly due to the anti-gravity action of the entropy force, could provide all or part of the mysterious dark matter on the scalesof clusters of galaxies. Acknowledgement.
This work was part of a brief Visiting Scientist Programme at the International Space Science Institute Bern. We ac-knowledge the interest of the ISSI directorate as well as the generous hospitality of the ISSI staff, in particular the assistance of the librariansAndrea Fischer and Irmela Schweitzer, and the Systems Administrator Saliba F. Saliba.
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