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Unconventional Energy of Stable Systems Shapiro, V. E. Feb 17, 2014

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The unconventional energy of stable systemsV.E. Shapiro∗203-2053 West 8th Avenue, Vancouver BC v6j 1w4, Canada(Dated: February 17, 2014)Based on the established consistent pattern of steady systems and motions in eternal chaos around,we argue the existence of energy form complementary to that of physics standard - given by thefunction of states of interacting systems, formulate its principles and criteria. The complementaryform is integral, not quantized, of vortex origin. It violates the laws of thermodynamics and entropy,effects the theory of phase transitions, Brownian motion, can strongly change the borders of matterstability, and gives rise to the quantum physics with its effects and super currents in equilibrium.PACS numbers: 01.55.+b 02.90+p 05.90.+mIntroductionThe gist of physics is in the unifying quideline of en-ergy measure bound to the law of its conservation in theevolution of all things existent. The meaning of all isvague and better determined via perception of phenom-ena through the energy exchange between systems givenby the function of their states. Such is the energy con-cept of mechanics by Euler and Lagrange and that ofequilibrium thermodynamics by Gibbs and more generalstatistics. This energy concept has found ways in allporous of physics but is limitted. Taking it for grantedleads to circular theories and fallacies. A few were paidattention in mechanics and other fields ages ago.Let us recall the forces called circulatory or vortex withall their cumulative impact beyond the energy functionconcept that can be huge, and the term “dry water”coined by von Neumann stuck to viscosity-neglect hy-drodynamic studies as inadequate, see [1]. Also since the19th century, e.g. [2,3], it was exposed in mechanics andother fields the invalidity of the concept due to the reac-tion forces of ideal non-holonomy, performing no work onthe system, as is the case of rigid bodies rolling withoutslipping on a surface. Recall also a general symmetryargument provoked by the H-theorem of Boltzmann andshowing the Loschmidt’s fundamental paradox [4] of re-versibility on the way to conform the real world with theenergy function concept.Physics nowadays in line with quantum mechanics re-lating its wave states with Hamiltonian mechanics byEhrenfest theorem unifies both approaches to the mea-sure of energy, making it the megatrend both of funda-mental and applied research. Nevertheless, its presentcover of cause-and-effect relations is constrained, unre-lated to a wide class of systems that complied with theenergy duality approach I came to earlier and crystal-lized into this work. It clarifies my results [5] on the ideaof energy suggested first in [5a] in connection with thestrong vortex effect of high frequency fields, and presentsthe idea of two energy measures in a new form.∗Electronic address: vshapiro@triumf.caThe dilemma of energy function approachThe established consistent pattern of the world aroundus is basically relaxation to reccurent equilibrium statesof matter, its rest or motion. The mass character andconsistency of relaxation in all chaos around mean ubiq-uity of irreversible forces as generalized forces whose workdepends on the path of motion rather than just its edgestates. It may seem correct to refer the irreversible forc-ing to the averaging of irrelevant variables of a conserva-tive many-body system given by a microscopic Hamilto-nian and random initial conditions. This cue, however,misleads in the question both of statistical and dynamical(over fast motion) averaging.First, it is impossible to come to the irrevesrsible be-haviors from the Hamiltonian unless resorting to themethods of averaging and truncations irreducible to theseparation of veriables within the framework of canonicaltransformations. Secondly, the arising inaccuracy accu-mulates with time, which is essential for the notion ofenergy as a conservative measure of the evolution in sta-tionary conditions.Also it is important that the perception of miriad ofouter influences, even treated within the Hamiltonian dy-namics as frequently alternating interactions, is possibleonly through averaging, which practically cannot be pre-sented as the exact averaging given by canonical transfor-mations, hence, leads to irreversible influences that maysignificantly accumulate for long times.Thus, the concept of energy as a function of systemstates gives rise to the following dilemma: On the onehand, the irreversible forces, unlike reversible, cannot bederived consistently from a Hamiltonianor effective po-tential. On the other, insofar as the true physics of phe-nomena is perceived through the interactions given byenergy functions, so should be the physics of irreversiblephenomena.The formulated dilemma is inherent to the perceptionof energy exchange in terms of the energy function con-cept and brings in fundamental inexactness. There is noother way to account for the inexactness but to integratereasoning in such terms with a tentative (statistical) mea-sure of energy blur/relaxation rates. This element per-tains to both classical and quantum mechanical descrip-2tions and the integration in point makes both approacheson equal footing in the sense of fitting to reality.In this regard, referring phenomena to purely quan-tum is inevitably conditional, may be proper within somespecific context. As for unconstrained assertions, it con-tradicts the above argument. In particular, it shouldconcern the ideas of quantum computers. Also, even forsuch phenomena as extremely deep cooling of matter byhigh frequency resonant fields, both ways have led to itsindependent prediction, see [6], and showed the classicalway direct, free of linkage to the uncertainty principle asits limitation.In all, the energy concept as a function of system statesseems just the way things are, but it narrows the reachof thought about the energy exchange and constitutionof matter. The approach developed below points clearlyto that, emerges as if unrelated to the classical-quantumissue but fully covers its basics and routes. Let us firstoutline the validity domain of the traditional energy con-cept, and rest our case on that of quantum physics.The entrainment theoremLet us think of energy concept in terms of general-ized thermodynamic potential commonly accepted in thestudy of phase transitions, transport through barriersand many other things. The generalized potential of asystem relaxing in steady conditions to a density distri-bution ρst connects to it byρst(z) = Ne−Φ(z), N−1 =∫e−ΦdΓ (1)where the integral is over the volume Γ of system phasespace variables z and the reversible motion is on surfacesΦ(z) = const. (2)The properties of the system mainly depend then on thelocal properties of the minima of Φ. Also, it gives in-sight from the observed symmetries of a physical system.An analogous approach to systems under high frequencyfields is in terms of the picture where the hf field looksfixed or its effect is time-averaged. In all this, Eq. (1)can be viewed as merely redefining the distribution ρst interms of function Φ, whereas, taking this function as theenergy integral of reversible motion provides the physicalbasis of the theory, but implies rigid constraints.Commonly, going back to Boltzmann and Onsager, tomention a few, the constraints are reasoned based onmicroscopical reversibility. It corresponds to detailedbalance of transition probabilities between each pair ofsystem states in equilibrium. The balance of transitionprobalities implies system descriptions within the frame-work of autonomous Fokker-Planck equations and anal-ysis, see [7], via division the system parameters into oddand even with respect to time reversal, with a reserve onfactors like magnetic field. In so doing the logic of timereversal is model-bound, and the reserve rule can be eas-ily broken, e.g., in nuclear processes and where spin-orbitinteractions are a factor, particularly near surfaces, inter-faces, dislocations. So, a problem arises even with thismodel-bound case.A different approach to outlining the overall domainof generalized thermodynamic potential validity was sug-gested in [5] and will be shown here. Its basis is in keepingwith invariance under transformations of variables. Ondoing so the energy integral of reversible motion impliesthe invariance under univalent transformations z → Z,of Jacobian|det{∂Zk(z, t)/∂zi}| = 1 (3)where i, k run through all components of z and Z. Φ(z)(1) satisfies this condition, for then not only ρdΓ is in-variant (being a number) but also dΓ. The environmentas a diffusion/dissipation source for the system brings inanother invariance. Connecting Φ to the system’s en-ergy function implies scaling this function in terms ofenvironmental-noise energy levels. The energy scales setthis way are fixed but must vary proportionally with theenergy function in arbitrary moving frames Z = Z(z, t)to hold Φ invariant. Since the energy function changesin moving frames, this constraint can hold only for thesystems entrained – carried along on the average at anyinstant for every system’s degree of freedom with the en-vironment causing irreversible drift and diffusion.Also account must be taken where the limit of weakbackground noise poses as a structure peculiarity – tran-sition to modeling of evolution with possible irreversibledrift without regard to diffusion. This means motionalong isolated paths. The entrainment constraint thenkeeps its sense as the weak irreversible-drift limit. Suchmechanics allows for the ideal non-holonomic constraintsthat do not perform work on the system but reduce thenumber of its degrees of freedom, which violates the de-sired invariance of Φ(z). Hence, the invariance necessi-tates the domain of entrainment free of that, termed idealentrainment or just ideal below.We have discussed all conditions on Φ(z), but the rea-soning holds for any one-to-one functions of ρst. For thesystems describable by a time-dependent density distri-bution ρ(z, t), the adequacy of energy function formalismalso requires the entrainment ideal. The arguments usedabove for the systems of steady ρst(z) become applica-ble there with univalent transformations of ρ(z, t) intot-independent distribution functions.The converse is also true: the behaviors governed bya dressed (due to the environment) Hamiltonian H(z, t)imply the entrainment ideal and the existence of a den-sity distribution ρ(z, t). The velocity function z˙ = z˙(z, t)of underlying motion is then constrained by z˙ = [z,H]with [, ] a Poisson bracket, so the divergence div z˙ =div[z,H] = 0 and div(z˙f) = −[H, f ] for any smoothf(z, t). It implies, if distribution ρ(z, t) exists,∂ρ/∂t = [H, ρ] (4)3and that (4) determines ρ(z, t) from given initial con-ditions and the boundary conditions taken natural at|z| → ∞ (ρ and its derivatives vanish) to preserve thenormalization∫ρdΓ = 1, for all other constraints areembodied in H. Such solution to (4) cannot cease to ex-ist as smooth, unique and non-negative over the phasespace of z where H(z, t) governs the behaviors. The en-trainment ideal there takes place since the solution turnsinto a function ρ(H) in the interaction picture where His t-independent. This completes the proof.Thus, the necessary and sufficient conditions where theenergy function concept is duly adequate to the evolutiondescribed by distribution functions come down to theentrainment ideal. This theorem lays down the overalldomain of desired energy function validity. It includesthe systems isolated or in thermodynamic equilibrium,as well as entrained in steady or unsteady environmentsgenerally of non-uniform temperature or indescribable intemperature terms so long as the diffusion, irreversibledrift and ideal nonholonomy can be neglected.The ideal as an asymptotic limit in the parameterspace of modeling is iherent in a boundary layer andintermittency where the limit trend can be deprived ofevidential force in the close vicinity of the ideal like tran-sitions to turbulence for large Reynolds numbers.The notion of energy from first principlesAs in the above, the systems will be defined as describ-able by a smooth evolution of the density distributionfunction ρ(z, t) of phase space z. z is a set of contin-uous variables z = (x, p) - the generalized coordinatesx = (x1, . . . xn) and conjugated moments p = (p1, . . . pn)taken in neglect of the constraints breaking the energyfunction formalism; z may include infinite in n sets ofnormal mode amplitudes of waves in media. The smooth-ness of ρ(z, t) will be understood to mean∂ρ/∂t = −div(vˆρ) (5)with vˆρ the 2n-vector flux of phase fluid at z, t; vˆρ is afunctional of ρ. Eq. (5) turns into the evolution equa-tion for ρ(z, t) with vˆ treated as a proper operator thataccounts for all constraints on the phase flows under theboundary conditions taken natural for the z componentsset unbound. Generally the constraints are non-local in zand non-anticipating in t. Let us consider the conditionsof stationary environment, when Eq. (5) is autonomousand describes relaxation of systems to a stable distribu-tion of ρ in response to perturbations.In this general approach all properties of systems, in-cluding the issues of energy, are determined by the solu-tion of ρ(z, t) to Eq. (5), its Cauchy problem as a functionof t in space z. A measurable property, and nonmeasur-able are off physics, presumes conservation of its measure.In our case this is the conservation of system energy inouter stationary conditions. The general principles ofwork on the system and the law of energy conservation,with the energy determined by the work, are to be takenas prime as so the material world is perceived. This basefully covers the basics given by the notion of energy con-servation of reversible processes.It is customary to formulate the principles of work andenergy in terms of isolated trajectories z = (p, q) as func-tions of t without account of diffusion and retardationin vˆ independent of ρ. But already in such mechan-ics along paths, the notion of energy as a function ofsystem states significantly narrows the underlying base.The motion z(t) from an initial z(0) corresponds to delta-function ρ(z, t) = δ(z − z(t)) in (5) where vˆ is a functionv = {vi(z)}, and is given by the set of equationsdzi/dt = vi(z), i = 1, . . . 2n. (6)The conservation of energy on the motion over closedpaths of system z(t) means vanishing the sum over i∮vidzi = 0. (7)The criterion (7) is satisfied for the ideal, for then v is tobe divergence-free. However, the equilibrium states to bestable require irreversibility in the vicinity of such paths,that is div v 6= 0. Absorbing the energy means div v <0. An archetype example is the models like Rayleighdissipation function, then div v(z) = k with k < 0 isindependent of z, so the irreversible contribution to thedrift v represents viscous forces linear in system velocity(in terms of Lagrange variables of physical space). Theseforces disappear in the states of rest and it realizes in theminima of potential forces of the system, a point or theirset depending on the potential shape.The phenomenon of stable motion, rather than rest,in conditions of energy conservation, is considered im-possible in classics. Obviously we are talking about theequilibrium phenomena - steady motions governed by au-tonomous equations in conditions without supply of en-ergy. However, nothing contradicts to the first principlesif we take into account in (7) that on some parts of mo-tion z(t) the energy may irreversibly be gained, and onother absorbed. The quadratic form vdz in the regionsof motion satisfying (7) where div v(z) 6= 0 then includesvortex forces, and the work due to them depends on thepath of motion, being not the function but a functionalof system states. The conserved energy as determinedthrough the work over the whole path of motion is thusa functional embodying vortex forces. For the systemsdescribed by a retarded vˆ, Eqs. (7) are integrodifferen-tial and the existence conditions for the integral energymeasure diversify.Certainly the phenomenon is negated if we associatethe energy conservation with the work on an imagin-able, Hamiltonian system, rather than a real, physicalsystem, determined by Eq. (5), of measureable parame-ters along its path according to (5). The possibility tooverrun the narrow framework of energy as a functionof sysyem states widens with account of diffusion, which4means nonlocality in z of action vˆ on ρ(z, t). For thegeneral conditions within Eq. (5) wherediv(vˆρ) = vˆ grad(ρ) + div(vˆ)ρ, (8)the criterion of energy conservation (7) extends into thatthe integral in (7) is replaced by a multidimensional inte-gral of vˆρdΓ over the volume of steady phase fluid flows.In conditions of vˆ not retarded, the limit div(vˆ)ρ = 0in the volume corresponds to the ideal of total, overz, self-compensation of ireversible drift forces and dif-fusion, whereas, the stability requires such conditions intheir vicinity to be violated with predominance of vis-cous forces; clearly the domain of that is limited. Diffu-sion contributes significantly to gaining the energy andthus creates mass new possibilities for changing the signof div(vˆ)ρ, hence, for the equilibrium states of systemmotion and its energy beyond customary notions of re-versible physics phenomena.So, the utmost wide energy measure of stable systemsin the outlined general conditions includes the integralenergy that embodies vortex forces of drift and diffusion.It reduces to the ideal as a low dimension limit wherediv(vˆ)ρ = 0 and extends much beyond depending on thecharacter of irreversible drift and diffusion. The energyof equilibrium system off the ideal region represents its ρstates of motion in t, and the motion may be not necessar-ily slow since the compensation of irreversible drift forcesand diffusion is broken locally also by emerging motion.For all that, the stability of motion states corresponds tothe inseparable balance of all forcing, the reversible andirreversible drift and diffusion for the case.The canonical property of irreversible kineticsThe general kinetic equations acquire the form∂ρ/∂t = [H, ρ] + I (9)where H = H(z, t) is, unlike in (4), an arbitrary smoothfunction, if we take for the term I the expressionI = −div[(vˆ − z˙)ρ] (10)with z˙ = [z,H] the local velocity of Hamiltonian phaseflows governed by H. An important feature of presen-tation (9) noted in [5] is the canonical invariance of Iholds as in as off the entrainment ideal, i.e. stands for alldrift and also diffusion. To prove, note that a canonical(univalent) transformation z → Z implies not only theinvariance of ρ and Poisson brackets but also the con-straint∂Z(z, t)/∂t = [Z,G] (11)with G a scalar function of z, t. Herein ∂Z(z, t)/∂t isthe relative velocity of reference frame Z at (z, t), so thefunction G(z, t) plays the role of Hamiltonian governingthis relative motion. The canonical invariance of ∂ρ/∂t−[H, ρ] in (9) follows and, hence, of the I term whateverits functional form may be. This formulation generalizesour theorem IV in [5a].In the domain of ideal, I reduces to a [H, ρ]-like Pois-son bracket since the evolution is then to be governed bya dressed Hamiltonian. Beyond the ideal, however, theentrainment theorem implies that I is not reducible toa [H, ρ]-like bracket and cannot keep invariance, hence,both the ∂ρ/∂t − [H, ρ] and [H, ρ] of (9) cease to be in-variant in the process of actual evolution for any choiceof H(z, t).Abstracting of the evolution, the state of ρ at any giveninstant t = ti can be taken for ideally entrained by fit-ting. Due to this and since ρ is assumed smooth in t, theeffect of the irreducibility of I and [H, ρ] to invariants isweak for t’s close to ti. So, it may seem reasonable tojudge about their figure of merit for not small t − ti bypopular perturbation methods of dynamical systems, e.g.[8]. But such insight is insufficient and fails in the longrun beyond the domain of ideal to match the future withthe past. As the relience on such perturbation theoriesconforms to the trends of ρ in line with a dressed Hamil-tonian, it conduces to the belief in this energy functiontheory beyond its above-established rigid constraints.By virtue of the considered canonical feature the sys-tem’s behavior in externally applied fields changes gen-erally in a non-conservative way. It shows up vigor-ously in systems under fields of high frequences, partic-ularly near resonances, including parametric and combi-national. The averaged effect of hf fields in the lowest,quadratic order in field amplitude results in static vortexforces along with strong potential. Thereat, the quasi-steady states, fluctuations and stability of systems atresonances appear quite different from what the theoriesof quasi-energy and generalized thermodynamic poten-tial prescribe. Various essential effects of this kind, theirgeneral features and methods of analysis were elucidatedin our work cited in [5].The extended equilibrium notion and measurementsIn the above, along with the new notion of energy wehave come to a new notion of equilibrium states – theextention of habitual stereotype of system states of restto the equilibrium states of motion that take place in animmencely wider area of conditions. The conclusion hascome from the equations (5) describing generally irre-versible kinetics with ∂ρ/∂t 6= 0 and allows for the statesof fast motion. However, it relates to the nonequilibriumstatistical physics associated with an irreversible trans-fer of energy rather as a contraposition, for the matterconcerns the physics of systems in equilibrium states inouter stationary conditions when the energy is conserved.Is there then a derogation from stationarity?The question is not so academic as principal for themeasurements and understanding of the claimed princi-ples. The stationarity for the case is the invariance of5behaviors with respect to time translation from the timeof perturbation, and it is inherent as to the states of restas to the states of steady motion. The difference be-tween the two displays itself in comparing the temporalcorrelations between the pictures of steady distributionsof ρ for the system. Just determining these correlationsor the power spectra corresponding to them for theseor those forms of motion gives one the approach to thephenomena in point, including the rest-motion transitionregion. Generally, the approach within the framework ofEq. (9) does not give one the entire range of desiredcorrelations, for it describes the evolution smoothed overfrequently alternating influences on the system, hence,generally insufficient for descriptions of all correlationsof system states at different moments of time. Such cor-relations are described by more detailed kinetics. Weshall not dwell on that here.For the systems relaxing in stationary outer conditionsto equilibrium distributions of states close to the statesof rest, in the sense of limit ∂ρ/∂t → 0, we get in termsof Eq. (9)[H, ρ] + I = 0. (12)The energy of system is then conserved with the branch Iacting on a par with [H, ρ] in jointly keeping the circula-tion and transformations of conserved energy. To similartrends we come for the energy of stable equilibrium statesof fast motion in the picture of canonical transformationswhere the steady ρ is t-independent. The conditions ofsuch energy circulation and transformations which is con-served in systems include the whole domain of ideal, butare not limited by it at all and can stretch beyond theideal vastly.Correspondingly, whereas in the domain of ideal theenergy is a function of system states given by a dressedHamiltonian with its potential and kinetic energy, bothof regular and chaotic origin, the conserved energy of sys-tems beyond the ideal includes or comprises the energy ofa different form, complementary to all those types givenby Hamiltonian functions, for it represents both the re-versible and irreversible drift forces and diffusion in theirintegral inseparable balance.This is just the integral vortex energy of equilibriumstates in systems. It is not related to the principles of de-tailed balance and habitual trends of relaxation, which isto the minimum of energy as a function of system states,hence, the trends of stability and preference relations inphase transitions - all that given by the traditional the-ory of phase transitions and other phenomena based onthe theory of generalized thermodynamic potential.Comment on transitions rest - persistent currentsLook first at a Brownian particle on a reflecting plate.Gravity tends to press the particle down and chaotic in-fluences of the environment keep it hopping on the platein stationary conditions. For the particle charged andplaced in a field of a permanent magnet, its drift arisesacross both the magnetic and gravity force fields. Thedrift modifies but not disappears for the plate rolled intoa closed pipe or box. The energy of steady drift is con-served, hence, contains a vortex form not given by a func-tion of the system. The same is for a number of interact-ing charged particles between reflecting walls. Variousthoughtforms of this and other kind of directed Brown-ian motion we presented in [9] and lot of interesting is in[10]. On writing this paper we found more recent mate-rial of relevance treated within the traditional concept ofenergy, see reviews [11,12]. To calculate something rigor-ously is impossible, of course. Let us formulate a generalcriterion on that score.Consider a system described by a distribution func-tion in stationary outer conditions. For the case of sta-ble ideal, the system relaxes to the state of rest deter-mined by a dressed Hamiltonian H(z) bound from below,commuting with the generalized thermodynamic poten-tial Φ(z), [H,Φ] = 0, and being its monotonic function,for unambiguity. Thereat, the vanishing irreversible forc-ing on the average for every component i of variables zimplies according to (12) the constraints for each i(fi − dik∂∂zk+ . . .)ρst(z) = 0 (13)where f = {fi(z)} is the irreversible drift forces, d ={dik(z)} a symmetric non-negative definite (for stability)matrix of diffusion and ellipsis stands for the higher orderdiffusion terms of expansion of I into a series in ∂/∂z. AsI is generally an integrodifferential form in z, so the oper-ator bracket of (13) is. The constraints of (13) generalizethe conditions of detailed balance.Neglecting the higher order terms in the bracket re-duces Eq. (13) to the algebraic fluctuation-dissipationrelationsfi = −(dΦ/dH)dikz˙k (14)with z˙ = [z,H] and dΦ/dH > 0. For the distributionρst of Maxwell-Boltzmann form and general Gibbs form,dΦ/dH = β is independent of H, which reduces (14) tof = −βdz˙ = −βd[z,H]. (15)β−1 = Θ is the energy scale of absolute temperaturewhose meaning expounds the known equipartition theo-rem: for every component of z (coordinate or momen-tum) whose contribution to H reduces to a square term,say, k1(zj−k2)2 with k1 > 0 and k1, k2 independent of zjbut may depend on other components of z and t, its meanover the Gibbs statistics comes to 〈k1(zj − k2)2〉 = Θ.It is easily seen that the ρst taken a Gibbs rules outpersistent currents since for any 〈z˙i〉, a function of ziaveraged over the phase subspace off zi, one gets on in-tegrating by parts〈z˙i〉 = N∫[zi,H]e−βH(dΓ/dzi) = 0 (16)6by virtue of natural boundary conditions for ρst(z). Thetheorem 〈z˙i〉 = 0 holds not only for Gibbs but for otherstatistics of ρst, a function of z via H(z).These results, just as presented above from a moregeneral perspective, show no place for a stable macromo-tion state in stationary conditions within the frameworkof generalized thermodynamic potential. Such states arethus a Litmus test of conserved vortex energy. A distinc-tive feature of the phenomenon is robustness as the sta-bility of macromotion state is to be at least asymptotic,with relaxation a factor and with reversion in responseto perturbations. It extends the paradigm of Brownianmotion caused by eternal chaos as non-directional to thatof directional also.While any system at a certain standing can be takenvia fitting as ideally entrained, governed by an energyfunction of its states, the theories of transition from thereunder a shift of parameters to a stable macromotion bega question whenever the emerging macromotion state isagain treated as a state given by an energy function. Themacromotion is then attributed to spontaneous symme-try breaking, topological defects and what-not, whichis problematic as it implies the conditions (13) to besomehow miraculously restored. Anyhow, in the endone faces the above theorem banning a stable macromo-tion within this beaten path down-the-line. To claim thephenomenon as just quantum is not sufficient, for as inclassics this needs consistently applied principles of cor-respondence, causality and relaxation/diffusion in equi-librium to account for the transition to a stable storedenergy of vortex type.In contrast to the essence of pattern formation as a pro-cess that makes the Cauchy problem of kinetic equation(9), even its quasi-static (∂/∂t → 0, not just ∂/∂t = 0)limit (12), the corner stone of the theory of energy, aswe do, the theory of phase transitions in question makes,in fact, a boundary value problem of kinetic equationsthe corner stone since Hamiltonian dynamics is imposed.This results in the geometrization beauty of kinetics butrules out the formation intrinsic to a stable non-entrainedstate in equilibrium, hence, the macromotion and vortexenergy.Thermodynamic laws in the light of vortex energyLet us look into equilibrium thermodynamics. It pro-ceeds from the existence of internal energy E of ther-modynamic system as a function of external parametersa = {ak} and temperature Θ so that the differential dEin space (a,Θ)dE =∂E∂ΘdΘ +∂E∂akdak = δQ+ δW (17)expresses the first law by introducing the heat transferQ as the difference between the internal energy and thework on the system W defined for any processes as purelymechanical, for Θ fixed. For the processes to proceed theparameters are assumed to vary in time, but slowly - inthe quasi-static limit |d(a,Θ)/dt| → 0. Whereas Q andW may freely depend on the path chosen in (a,Θ) withδQ and δW not bound to be exact differentials, Eq. (17)implies for any cyclic process∮δQ = −∮δW. (18)Therein lays the principle of equivalence between thework and heat. The principle of first law in the form(17) is tantamount to that of (18). This being for anyclosed paths in (a,Θ), the vortex energy is thereby com-pletely excluded.Not only the first law appears to be the law of energyconservation bound to the framework of energy a functionof system states for the case, but also it implies, since thethermodynamic equilibrium is treated as a stable state,relaxation to be exactly towards the miniumum of en-ergy function of system states in terms of (a,Θ) withoutintroducing any entropy function.The second law of thermodynamics in this regard spec-ifies the equation of system state, its caloric-thermal re-lations - by assuming that the energy function is addi-tive with respect to the partition of system volume, aone-dimension external parameter, in independent smallparts. It best fits the ideal gas confined by rigid walls, isin line with Gibbs statistics of ρst and poses the energyE and forces Ak = −∂E/∂ak as the averagesE =∫He(Ψ−H)/ΘdΓ, (19)Ak =∫(−∂H/∂ak)e(Ψ−H)/ΘdΓ (20)withΨ = −Θ lnN, N =∫e−H/ΘdΓ (21)and the Hamiltonian H assumed a function of z andslowly varying parameters a but not Θ, to avoid umbigu-ity. These equations show Ψ(a,Θ) as the Helmholtz freeenergy determining the work of forces A = {Ak} andwhich shows the expression Θ∂Ψ/∂Θ as the binding en-ergy function. The entropy function, which is introducedin pure thermodynamics as S(a,Θ) =∫(δQ/Θ) by pos-tulating the existence of the integrating multiplier of δQwith 1/Θ, amounts toS = −∂Ψ∂Θ=E −ΨΘ. (22)So, all physics of Gibbsian thermodynamics is given onthe base of energy function Ψ. Also it shows entropy asnot a self-sustained notion for that matter and that thesecond law just as the first law is not reflective of vortexenergy and its trends.7The latter assertion is to be common to any entropyfunction treated not only on the base of first law but alsoon the base of any its generalization within the frame-work of entrainment ideal. Indeed, the entropy functionand the generalized potential must then commute, forthe ideal entrainment holds where this potential for thesystem is its energy integral. The violation of entropyconservation law would mean that the entropy is not afunction of parameters entering in the potential for thecase. Moreover, the assertion is also true in stationaryconditions where the law of energy conservation holds be-yond the entrainment ideal, which is the area of vortexenergy, for the opposite would then mean the existenceof the energy integral of the system there. As to the con-servation law of entropy in conditions where the energyof system is not conserved, the entropy function againcannot be related to the energy of system, for such no-tion ceases to exist then. Thus, the notion of entropy,however defined and by which statistics, does not addphysics to the vortex energy.Of various statistics linked to the second law, onlyGibbs statistics assigns to the thermodynamics the mean-ing given by the equipartition theorem. But at that,only a small area of Gibbs statistics domain fits the ther-modynamics, as particularly evident from the paragraphwith Eqs. (13)-(16). Namely, it implies H(z) to be boundfrom below and the additivity postulate to limit its long-ranged interactions, and the interactions and parametersentering into H should not depend on Θ and statisticalfactors – to preserve the very separating principle be-tween the balances of reversible and irreversible forcingand avoid ambiguity in its definition.In this light, the known Landau theorem [13], referredto as the outright ban on classical routes to persistentcurrents, should not be treated so. The theorem statesthat a closed system of interacting parts in thermal equi-librium admits only uniform translation and rotation asa whole. The proof proceeds from the system’s entropy Staken in the form of a sum∑Si where each summand Siis a function of the difference Ei − P 2i /2mi between thetotal and kinetic energy only of part i, and the argumentsand calculations do not go beyond the first and secondlaws. The theorem does make allowances for a differ-ence between the motion state of systems and ambiencein equilibrium, but the natural next step - the inferenceof the vortex energy and its importance for wider con-ditions of thermal equilibrium which razes the theoremstrength beyond thermodynamics and also the exclusiv-ity of quantum concept - was not made then a days.Let us touch upon the physics of matter stability, itselement based on Gibbsian thermodynamics for Coulombsystems. By the rigorous theory, see [14,15], and mean-field theories going back to Debye, the screening of long-range Coulomb potential 1/r between moving charges ofopposite sign at large distances r in matter makes thepotential short-ranged, so the free energy per unit vol-ume is bound below and tends to a finite limit as thesystem volume increases. Our point here is the sufficientconditions of such equilibrium states should include thestability with respect to the factor to be accounted for- the vortex energy, for the very screening arises due tothe diffusion and relaxation of gradient of charge-particledensity under Coulomb field perturbations. The stabilitycriterion (13) then transits into that where f comprisesboth reversible and irreversible drift forces, which is ac-cessible for measurements.A comment on surface phenomena. The equilibriumthermodynamics of particle systems confined or self-confined in a finite volume abstracts away of surface ef-fects, but it may not comply without vortex energy. Itfits into this group our example of Brownian particles andmany systems with surfaces, interfaces, dislocations, do-main walls. It may concern, e.g., superconducting topo-logical insulators. Recall also the instability of electronfluid suggested by Vlasov [10] by analogy with the physicsof capillary waves going back to Stokes and Rayleigh [16]- the attraction of surfaces particles to the bulk of fluidgives a negative contribution to the potential energy ofripple wave motion on the susrface of fluid, so such statescan evolve into a steady ripple that transports mass andcharges. Our point here - there is no other way for thephenomenon to exist in equilibrium as robust but to im-ply stored vortex energy for stabilization.Concluding remarks on the two energy formsThe energy concept of interacting systems set forthdevelops a consistent causality approach to the generalkinetics of realities which departs from the conventionalenergy concept. On this way, first introduced in [5] andnow substantiated, both the unconventional form of en-ergy and the class of system states in equilibrium emerge.Let us first emphasize its key elements within the classi-cal physics.Both the conventional energy concept and the pre-sented concept rest on the conservation of energy viaability of systems to produce work, but conventional pro-ceeds from the notion of work determined by the functionof system states. This measure corresponds to the sys-tems behaviors within the framework of an ideal bound-ary value problem. Unlikely, we proceed from the notionof work measurable by the evolution of distribution func-tion of system states according to the Cauchy problem ofkinetics governing the evolution. The departure is thusfrom the physics of basically predetermined world to thatof real, diverse world where nothing happens by itself butdepends on circumstances.This is like transition from the world of integers to thatof reals and deeper, being on a functional level. Withinthe framework of the law of energy conservation given bythe function of system states, the states in equilibriumare isolated, determined on gratings, for the transitionsbetween each pair of states are determined by balanceof reversible forces. Meanwhile, such balance should notbe postulated, does not cover all possibilities at all, for8the inseparable balance with irreversible forcing gener-ally matters no less to provide stability. The system’sstates are then not isolated, have open vicinities, whichis crucially important - the energy measure represents allforcing, the whole drft and diffusion. Thereby, and asfirst claimed in [5], the notion of energy represents alsothe vortex forces.A self-sustained action of irreversible forcing, with dif-fusion or not, is behind this unconventional energy wecalled vortex. Its impact can be huge in overcomingpotential barriers and cause stable states of persistentmacromotion not predicted at all by the standard theoryof energy in equilibrium. What is essential from a gen-eral physics standpoint, both energies are complemen-tary measures of energy. Resorting to entropy does notadd physical perception to all this physics, as we haveshown. The standard form of energy emerges as a low-dimensional limit of conditions in parameter space allow-able for the vortex form.The vortex energy form is behind the equilibriumstates of macromotion, while the standard form is behindthe equilibrium states of rest. And it is all about stabil-ity in small and finite with regard to the vortex forces -as the very issues of equilibrium states and inseparablebalance are resolved through it. The factor is critical forour conclusions, and the criteria we formulated in termsof kinetics are with an eye on it.The vortex energy being integral characterizes behav-iors that do not measure up to the principles of equilib-rium statistical mechanics and the first and second lawsof thermodynamics. Thereat, the existence domain ofmatter stability can significantly extend or shrink com-pared with the predictions of theory of energy as a func-tion of system states. Also the revealed vortex energyprovides a wider insight into Brownian motion - its uni-directionality becomes allowable. An unidirectional de-mon besides Maxwell’s may fit for equilibrium. Theseprinciples, together with the observed stability of mate-rial world, mean the ubiquity of stored vortex energy,while negating the vortex energy would essentially nar-row the physical perception of stable equilibrium systemsand structures on the base of energy measure.By all these reasons the vortex energy cannot be unim-portant both to general and applied research. It is ev-ident, for example, that the theory of relativity beingbound to the order of things given by Lagrangian dy-namics may not comply at all to the physics behind thecomplementary vortex energy. Also and since this energymeasure generates the quantum physics, it effects variousfields, up to astrophysics with its puzzles of black holes,dark matter and dark energy.The vortex energy generates the quantum physicsThe vortex energy measure set forth above within theframework of classical physics as the complementary en-ergy counterpart given by the function of system statesand clarified with relevant classsical physics, has nothingto do with the quantum physics, its noncommuting op-erators of variables, energy transfer by quanta, particle-wave duality, and associated quantization. However, thetwo physics are not complementary but competing.Whereas the quantum concept admits super currentsin and of itself, from the standpoint of our complemen-tary energies, all kinds of stable macromotion statesemerge due to the irreversible kinetics. The same is truefor all other non-classical features and can be formulatedas the following energy theorem of the classical footingof quantum physicsThe vortex energy gives rise to all phenomena treatedquantum: the existence domain of vortex energy effectsincludes the existence domain of all quantum physics ef-fects.So, according to this, all that phenomena attributed toquantum physics represent classical irreversible phenom-ena. Before, it was argued for that the quantum phenom-ena cannot by perceived classically. Instead, it was com-monly attributed to an uninterpretable sort of reversibleinteractions, constrained in the sense of Eq. (2). Butit turns out to be just the opposite, explorable throughthe classical irreversible kinetics and its methods. Oneused to think of quanta, say, a photon, but the clas-sical irreversible kinetics is behind that, hence, admitsobservations and classical perception up to the notion ofneutrinos and nuclear energy. It concerns quantum ther-modynamics, quantum theories of matter, quantum elec-trodynamics, relativity, gravity, etc. Thereat, we proceedfrom physics as the science perceiving all phenomena ex-clusively through the notion of energy defined by workon physical systems.The proof of this footing theorem is immediate fromthe Ehrenfest theorem of quantum-classical correspon-dence and the principle of complementary energy formswe defined above within the general framework of clas-sical kinetics. As a reminder, the Hamiltonian H(z, t)governing the classical dynamics is related by Ehrenfestto its quantum counterpart Hamiltonian Hˆ(zˆ, t), a Her-mitian operator, by equatingH(z, t) =〈ϕ∣∣∣Hˆ(zˆ, t)∣∣∣ϕ〉. (23)This quantum-mechanical average is over the wave func-tion ϕ(z, t) of quantum states whose dynamics is gov-erned by the quantum Hamiltonian. But the term aver-age here is only by name as the associated density matrixadmits negative density distributions of mixed states, i.e.provides the integration measure rather than any possibledensity measure of states.So, it is just postulated the energy measure of quantumsystem to be given by the energy given as a function ofsystem states. As for the measure of energy we definedabove by the work on the physical system within thegeneral kinetics, it comprises the complementary energyforms - the vortex energy form and its counterpart givenby the function of system states. So far as the quantum9phenomena are defined (postulated) as non-classical, be-yond the framework of physical systems governed by anypossible classical Hamiltonian functions of system states,this implies the vortex energy form to be behind that.The fact that the quantum energy states and aver-aging over them have nothing to do with the averagingover classical energy states and represents rather a way ofwording is known since the time of Wigner and von Neu-mann, and is in the intensive works of many researcherspresently, particularly via the Koopman-von Neumannapproach [17,18], e.g. [19,20]. However, all previous ef-forts known to me to find real classical routes, on the baseof energy, to the phenomena attributed to the quantumphysics failed, for one then needed to come up to thegeneral principle of the complementary energy measuresstated in [5] and elucidated in this note.It should be underlined that the footing in point is notan equal footing, for the existence domain of the quantumconcept is unknown and there is a fundamental problemwith quantum reality, how to protect a quantum statefrom a measuring apparatus against the quantum statecollapse. The stabilization of states is required, and thereis no other way based on the notion of energy to that butto identify it with the vortex energy.We do not augment validity to the quantum conceptresorting to a semiclassical approach to the irreversiblekinetics, but acquire all its strength exclusively due tothis kinetics. The principle of complementary energieswith its vortex energy covers all aspects of that in aself-sustained way in terms of measurable kinetics ade-quate to the reality of inseparable balance between thereversible and irreversible forcing. Moreover, it makesreasonable and rational to relate this or that quantumeffect directly to the structure of kinetics. This can befruitful in search for new equilibrium states of matter andsources of energy and their control.Thus, the established principles of vortex energy un-ravel the problems of quantum mechanics and place themon a tangible ground of consistent constructive unifiedphysics.[1] R. Feynman, R. Leighton, M. Sands, The FeynmanLectures on Physics, Vol. 2, 1970.[2] Yu.I. Neimark, N.A. Fufaev, Dinamika NegolonomnihSystems. Nauka, Moscow, 1967. (English transl. Dynamicsof Nonholonomic Systems. Providence, American Mathe-matical Society, 1972).[3] H.R. Hertz, The Principles of Mechanics Presented ina New Form, English transl. Macmillan, London, 1899.[4] J. Loschmidt, first published by W.Thomson in 1874,reprinted in Kinetic Theory, vol 2, 176-487, 1966.[5] V.E. Shapiro, a) Phys. Lett. A 372 (2008) 6087; b)arXiv: 1109.2605 Sep 2011; 1305.5108 Jul 2013.[6] V.E. Shapiro, Zh. Eksp Teor. Fiz. 70 (1976) 1463. (Sov.Phys. JETP 43 (1976) 763).[7] R. Graham and H. Haken, Z. Physik 243 (1971) 289;245 (1971) 141.[8] F.G. Gustafson, Astronom. Journ., 71 (1966) 670.[9] V.E. Shapiro, Noise-induced drift with broken sym-metry and classical routes to super-conductivity, PreprintTRI-PP-94-3 (Jan 1994) TRIUMF, Vancouver, Canada.[10] A.A. Vlasov, Statisticheskie Funktsii Raspredelenia.Nauka, Moscow, 1966.[11] R.D. Astumian and P. Hanggi, Brownian motors,Physics Today 55 (2002) 3339 .[12] Bo Sun, D.G. Grier, A.Y. Grosberg, Phys. Rev. E. 82,021123 (2010).[13] L.D. Landau, E.M. Lifshitz, Statistical Physics. Perg-amon, New York, 1980.[14] D.C. Brydges and Ph.A. Martin, J. Stat. Phys. 96(1999) No 5/6, 1163.[15] H. Lieb and J. Lebowitz, Adv. Math. 9 (1972) 316.[16] Rayleigh, The Theory of Sound. London Macmillan,Vol. 2, 1896.[17] B.O. Koopman, Proc. Nat. Acad. Sci. USA 17 (1931)315.[18] J. von Neumann, Ann. Math. 33, (1932) No. 3, 587;33(1932) No. 4, 789.[19] A.A. Abrikosov, E. Gozzi, D. Mauro, Annals ofPhysics 317 (2005) 24.[20] D. Bondar, R. Cabrera, R. Lompay, M. Ivanov, H.Rabitz, Phys. Rev. Letters 109 (2012) 19.


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