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The revealed integral energy shows the vortex nature of quantum realm Shapiro, V. E. Apr 25, 2014

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The revealed integral energy shows the vortex nature of quantum realmV.E. Shapiro∗(Dated: April 25, 2014)We assert the existence and ubiquity of the energy form complementary to the energy given bythe function of states of interacting systems, substantiate its principles and criteria. This form iscritical to stability but not covered by the thermodynamics and entropy laws and the trends ofBrownian motion and phase transitions based on detailed balance. It is of vortex nature related torelaxation, and gives rise to all quantum realm with its super currents and stability against statecollapse. We elucidate hardened beliefs detrimental to search for new states of matter and energy.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 a 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 of states. This energy concept has found waysin all porous of physics but is limitted. Taking it forgranted leads to circular theories and fallacies.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 theory bridgedto Lagrangian mechanics has a commanding influence onboth fundamental and applied research, and it furtherspreads the conviction in the genuine law of energy givenby functions of system states with no limits of validity. Itmight look just the way things are but narrows the reachof thought about the stable matter in equilibrium and itsenergy measure in vast realm of nonholonomy, relaxation,diffusion. This is what the present work is about. It isbased on the principle of energy duality we introduced in[5] and now clarify. We begin with outlining the validitydomain of the traditional energy concept, then introducea dual energy measure, dwell on things it opens up, andrest our case on quantum physics.∗Electronic address: vshapiro@triumf.caThe existence domain of energy function conceptsLet us think of energy concepts 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]. Its basis is in keeping with invariance under2transformations of variables. On doing so the energy in-tegral of reversible motion implies the invariance underunivalent 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 changes inmoving frames, this constraint can hold only for the sys-tems entrained - carried along on the average for everysystem’s degree of freedom with the environment causingirreversible 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 a smooth distribution ρ(z, t) exists,∂ρ/∂t = [H, ρ] (4)and 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 the distribution function of system statescome down to the entrainment ideal.This theorem of [5] lays down the overall domain ofdesired energy function validity. It includes the systemsisolated or in thermodynamic equilibrium, as well as en-trained in steady or unsteady environments generally ofnon-uniform temperature or indescribable in tempera-ture terms so long as the diffusion, irreversible drift andideal 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. Suchand more ordered pattern formation of conserved-energymacromotion beyond the ideal, stable against chaoticbackground in stationary outer conditions, is the gist ofenergy duality concept presented below.The missed form of energyThe systems will be defined as describable by a smoothevolution of the density distribution function ρ(z, t) ofphase space z. z is a set of continuous variables z =(x, p) - the generalized coordinates x = (x1, . . . xn) andconjugated moments p = (p1, . . . pn) taken in neglect ofthe constraints breaking the energy function formalism;z may include sets of normal mode amplitudes of wavesin media. We shall not discuss relativistic specifics. Thesmoothness 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 asmooth functional of ρ. Eq. (5) turns into the evolutionequation for ρ(z, t) with vˆ treated as a proper opera-tor that accounts for all constraints on the phase flowsunder the boundary conditions taken natural for the zcomponents set unbound. Generally the constraints arenon-local in z and non-anticipating in t. Let us considerthe conditions of stationary environment, when Eq. (5)is autonomous and describes relaxation of systems to astable distribution of ρ in response to perturbations.In this general approach a measurable property, andnonmeasurable are off physics, presumes conservation ofits measure judged by the solutions ρ(z, t) to Eq. (5), itsCauchy problem as a function of t in space z. In ourcase this is the conservation of system energy in outerstationary conditions. The general principles of work onthe system and the law of energy conservation, with theenergy determined by the work, are to be taken as primeas so the material world is perceived. This base fully andby far covers the basics given by the notion of energyconservation of reversible processes.3It is customary to formulate the principles of work interms of isolated trajectories z = (p, q) as functions oft without account of diffusion and retardation in vˆ in-dependent of ρ. The motion z(t) from an initial z(0)reduces then to delta-function ρ(z, t) = δ(z− z(t)) in (5)where vˆ is a function v = {vi(z)}, and is given by the setof equationsdzi/dt = vi(z), i = 1, . . . 2n. (6)But already in this mechanics the notion of energy as afunction of system states much narrows the conditions ofenergy conservation for the case. The conservation overclosed paths of system motion z(t) means vanishing∮vidzi = 0 (7)assumed summing over dummy indeces. The criterion (7)is satisfied for the ideal, for then v is to be divergence-free. However, the equilibrium states to be stable requireirreversibility in the vicinity of such paths, that is div v 6=0. Absorbing the energy means div v < 0. An archetypeexample is Rayleigh dissipation function, then div v(z) =k with k = const < 0, so the irreversible contributionto the drift v represents viscous forces linear in systemvelocity in terms of Lagrange variables of physical space.These forces disappear in the states of rest and it realizesin the minima of potential forces of the system, a pointor their set 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 principles ofwork in such conditions if we take into account that thesign of div v(z) may vary so that the work can be ir-reversibly absorbed on some parts of motion z(t) andgained on other parts. The quadratic form vdz in theregions of motion satisfying (7) where div v(z) 6= 0 thenincludes vortex forces, and the work due to them dependson the path of motion, becomes not the function but afunctional of system states. The conserved energy as de-termined through the work over the whole path of motionis thus a functional embodying vortex forces. For thesystems described by a retarded vˆ, Eqs. (7) are integrod-ifferential and the existence conditions for such integralenergy measure diversify.Certainly, the phenomenon of motion in point isnegated if we associate the energy conservation with thework on an imaginable Hamiltonian system, rather than areal, physical system determinable by Eq. (5) of measure-able parameters. The possibility to overrun the narrowframework of energy as a function of system states muchwidens further with account of diffusion, which meansnonlocality in z of action vˆ on ρ(z, t). For the generalconditions of modeling within Eq. (5) wherediv(vˆρ) = (div vˆ)ρ+ vˆ gradρ, (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.For vˆρ modeled linear in ρ and not retarded, the limit(div vˆ)ρ = 0 in the volume corresponds to the ideal oftotal, over z, self-compensation of ireversible drift forcesand diffusion, whereas, the stability requires such condi-tions in their vicinity to be violated with predominance ofviscous forces; clearly the domain of that is limited. Dif-fusion contributes significantly to irreversibly gaining theenergy as this creates mass new possibilities for changingthe sign of (div vˆ)ρ, hence, for the equilibrium states ofsystem motion and its energy beyond customary notionsof reversible physics phenomena.So, the utmost wide energy measure of stable systemsin the outlined general conditions includes the integralenergy that embodies the vortex forces of drift and dif-fusion. It extends much beyond the ideal, which is a lowdimension limit where (div vˆ)ρ = 0, depending on thecharacter of irreversible drift and diffusion. The energyof equilibrium system off the ideal aria represents its ρstate 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 (5) acquire the form∂ρ/∂t = [H, ρ] + I (9)where H = H(z, t) is, unlike in (4), an arbitrary Hamiltonfunction, 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 a Hamiltonian governingthis relative motion. The canonical invariance of ∂ρ/∂t−[H, ρ] in (9) follows and, hence, of the I term whatever itsfunctional form may be. This formulation generalizes ourtheorem IV in [5a]. It follow important consequencies.In the domain of ideal, I reduces to a [H, ρ]-like Pois-son bracket since the evolution is then to be governed by4a dressed Hamiltonian. Beyond the ideal, the entrain-ment theorem implies that I is not reducible to a [H, ρ]-like bracket and cannot keep invariance, hence, both the∂ρ/∂t − [H, ρ] and [H, ρ] of (9) cease to be invariant inthe process of actual evolution for any choice of 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 this 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 vor-tex forces along with strong forces of effective potential.Thereat, the quasi-steady states, fluctuations and stabil-ity of systems at resonances appear quite different fromwhat the theories of quasi-energy and generalized ther-modynamic potential prescribe. Various essential effectsof this kind, their general features and methods of anal-ysis were elucidated in our work cited in [5a].The extended equilibrium 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 inan immencely wider area of conditions. The conclusionis immediate from the equations (5) describing generallyirreversible kinetics with ∂ρ/∂t 6= 0 and allows for thestates of fast motion. However, it relates to the nonequi-librium statistical physics associated with an irreversibletransfer of energy rather as a contraposition, for the mat-ter concerns the physics of systems in equilibrium statesin outer stationary conditions when the energy is con-served. 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 ofbehaviors with respect to time translation from the timeof perturbation, and it is inherent as in the states of restas in the states of steady motion. The difference betweenthe two displays itself in comparing the temporal corre-lations between the pictures of steady distributions of ρfor the system. Just determining these correlations or thepower spectra corresponding to them for these or thoseforms of motion gives one the approach to the phenom-ena in point, including the region of rest-motion tran-sition. Generally, the approach within the framework ofEq. (9) does not give one the entire range of desired corre-lations, for it governs the trends already smoothed overfrequently alternating influences on the system, hence,generally insufficient for descriptions of all correlationsof system states at different moments of time. The lat-ter are described by more detailed kinetics and energymeasures. We shall 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 systems is then conserved with the branchI acting 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 roughly t-independent. It is clearthat the conditions of such energy circulation and trans-formations in systems include the whole domain of ideal,but are not limited by it at all and can stretch beyondthe ideal vastly.Whereas in the domain of ideal (whether Eq. (12) holdsor not) the energy is a function of system states given bya dressed Hamiltonian with its potential and kinetic en-ergy, both of regular and chaotic origin, the conservedenergy of systems beyond the ideal includes or comprisesthe energy of a different form, complementary to all thosetypes given by Hamiltonian functions, for it representsboth the reversible and irreversible drift forces and diffu-sion in their integral 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.On the macromotion and vortex energy criteriaLook 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 states. The same is for a number5of interacting charged particles between reflecting walls.Various thoughtforms of this and other kind of directedBrownian motion we presented in [9], lots of interestingis in [10], and it resonates to the stochastic approach,e.g. review [11]. On writing this paper we found morerecent material of relevance also treated within the tra-ditional concept of energy function, see reviews [11,12].Let us formulate a general criterion related to our energyduality principle 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 (no summing over i on the right)〈z˙i〉 = N∫[zi,H]e−βH(dΓ/dzi) = 0 (16)by virtue of natural boundary conditions for ρst(z).The theorem 〈z˙i〉 = 0 holds not only for Gibbs but forother statistics of ρst, a function of z via H(z). It is inline with presented above from a more general perspec-tive and shows no place for a stable macromotion statein stationary conditions within the framework of gener-alized thermodynamic potential constrained by Eq. (2).Correspondingly the states of macromotion, e.g., persis-tent currents are thus a Litmus test of conserved vortexenergy. A distinctive feature of the phenomenon is itsrobustness as the stability of macromotion state is withrelaxation a factor and with reversion in response to per-turbations. Thereby the paradigm of Brownian motioncaused by eternal chaos as non-directional extends to thatof directional.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 isalso treated as a state given by an energy function. Themacromotion is then attributed to spontaneous symme-try breaking, topological defects and what-not, which isproblematic as it implies the conditions (13) to be some-how miraculously restored. Anyhow, in the end one facesthe above theorem banning a stable macromotion withinthis beaten path down-the-line.In contrast to the essence of pattern formation as aprocess that makes the Cauchy problem of kinetic equa-tion (9) [even its quasi-static (∂/∂t → 0, not just omit∂ρ/∂t) limit (12)] the corner stone of the theory of en-ergy, as we do, the theory of phase transitions in questionmakes the boundary value problem imposing the evolu-tion trend given by the energy function the corner stone.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)6Therein 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(ϕ−H)/ΘdΓ, (19)Ak =∫(−∂H/∂ak)e(ϕ−H)/ΘdΓ (20)withϕ = −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 ϕ(a,Θ) as the Helmholtz freeenergy determining the work of forces A = {Ak} and alsothe expression Θ∂ϕ/∂Θ as the binding energy function.The entropy function, which is introduced in pure ther-modynamics as S(a,Θ) =∫(δQ/Θ) by postulating theexistence of the integrating multiplier of δQ with 1/Θ,amounts by Eqs. (19), (21) toS = −∂ϕ∂Θ=E − ϕΘ. (22)So, all physics of Gibbsian thermodynamics is given onthe base of energy function ϕ. 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.The latter assertion is to be common to any entropya function of system states treated not only on the baseof first law but also on the base of any its generalizationwithin the framework of entrainment ideal. Indeed, theentropy function and the generalized potential must thencommute, for the ideal entrainment holds where this po-tential for the system is its energy integral. The violationof entropy conservation law would mean that the entropyis not a function of parameters entering in the potentialfor the case. Moreover, the assertion is also true in sta-tionary conditions where the law of energy conservationholds beyond the entrainment ideal, which is the area ofvortex energy, for the opposite would then mean the ex-istence of the energy integral of the system there. As tothe conservation law of entropy in conditions where theenergy of system is not conserved, the entropy functionagain cannot be related to the energy of system, for suchnotion 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 of vortex energy, es-pecially as the very screening arises due to the diffusionand relaxation of gradient of charge-particle density un-der Coulomb field perturbations. The stability criterion(13) then transits into that where f comprises both re-versible and irreversible drift forces, which is accessiblefor measurements.A comment on surface phenomena. The equilibrium7thermodynamics 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. Suchis the case for our example of Brownian particles andchances are for many systems with surfaces, interfaces,dislocations, domain walls, e.g., bear on superconduct-ing topological insulators. Recall also the instability ofelectron fluid suggested by Vlasov [10] by analogy withthe physics of capillary waves going back to Stokes andRayleigh [16] - the attraction of surfaces particles to thebulk of fluid gives a negative contribution to the potentialenergy of ripple wave motion on the susrface of fluid, sosuch states can evolve into a steady ripple that transportsmass and charges. Our point here - there is no other wayfor the phenomenon to exist as robust in equilibrium butto imply stabilization due to stored vortex energy.On the philosophy of energy dualityThe energy principles of interacting systems set forthrepresent a consistent causality approach to the conser-vation of energy via ability of systems to produce work.It departs from the conventional energy concept. Bothenergy concepts, conventional and set forth, rest on suchability, but conventional proceeds from the notion of workdetermined by the function of system states. This mea-sure corresponds to the systems behaviors within theframework of boundary value problem imposing the con-ditions of ideal. Unlikely, we proceed from the notion ofwork 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 equilibrium areisolated, determined on gratings, for the transitions be-tween each pair of states are determined by balance of re-versible drift forces and a separate balance of irreversibleforces of drift and all diffusion terms. Meanwhile, theseparation of balances should not be postulated, followsfrom nowhere and does not cover all effect of irreversibleforcing which generally matters no less to provide sta-bility. The system’s states are then not isolated, haveopen vicinities. So it is important for the energy mea-sure of forcing able to produce work to include the forcesof whole drift and diffusion.A self-sustained action of irreversible forcing, with dif-fusion or not, is behind this unconventional energy mea-sure we came to and called integral vortex or just vor-tex. For the first time the energy measure incorporatesthe irreversible forcing. Its effect can be crucial in clear-ing the hurdles of potential barriers in outer autonomicconditions, so the stable states of macromotion can takeplace.Just as important from a general physics standpoint,the vortex and standard energies are complentary formscomprising the total energy measure in equilibrium.Thereat, we proceed from physics as the science perceiv-ing all phenomena exclusively through the notion of en-ergy and work on the basis of cause-effect relations. Thenotion of entropy has then no relation to the vortex en-ergy form and adds no physics to the energy duality inpoint, as shown above. The standard energy form is be-hind the states of rest and emerges as a low-dimensionallimit in the parameter space of vortex form generatingthe states of macromotion. And it is all about stabilityin small and finite with regard to the vortex forcing as thevery issues of equilibrium states and inseparable balanceare resolved through it. The issue of stability is criti-cal for our conclusions, and the criteria we formulated interms of kinetics are with an eye on it.Since the vortex energy is essentially integral and ofvortex nature, it characterizes behaviors that do not mea-sure up to the principles of equilibrium statistical me-chanics and the first and second laws of thermodynamics.Thereat, the existence domain of matter stability can sig-nificantly extend or shrink compared with the predictionsof theory of energy as a function of system states. Alsothe revealed vortex energy provides a wider insight intoBrownian motion as its unidirectionality becomes allow-able. An unidirectional demon besides Maxwell’s mayfit for equilibrium. These principles, together with theobserved stability of material world, mean the ubiquityof stored vortex energy, while negating the vortex energyform would essentially narrow the physical perception ofstable equilibrium systems and structures on the base ofenergy.By all these reasons the vortex energy form is impor-tant both to general and applied research. It is evident,for example, that the theory of relativity being bound tothe order of things given by Lagrangian dynamics maynot comply to the physics behind the complementary vor-tex energy. Also and since this energy form generates(see below) the quantum physics, it effects various fields,up to astrophysics with its puzzles of black holes, darkmatter and dark energy.The vortex nature of quantum physicsThe concept of vortex energy we have established andclarified above within the framework of classical physicsdescription does not rely on any postulates of quan-tum physics, its particle-wave notion, quantization, en-ergy transfer by quanta. However, the two physics arenot complementary but competing. Whereas in outerstationary conditions the quantum concept admits su-per currents in and of itself, from the standpoint of ourcomplementary energies all kinds of stable macromotionstates then emerge due to the irreversible kinetics. Thesame is true for all other non-classical features and can8be formulated as the following energy theorem or princi-ple of the classical footing of quantum physicsThe observable phenomena treated quantum are exclu-sively of vortex nature - pertain to the domain of vortexenergy form.Indeed, treating phenomena quantum implies thattheir existence domain is describable by a wave functionψ(z, t) of system state whose evolution is governed by aquantum Hamiltonian Hˆ, an Hermitian operator havingterms of variables that do not commutate. With the ob-servable values of Hˆ defined in this quantum ideal as theaverages〈Hˆ〉 =〈ψ∣∣∣Hˆ(zˆ, t)∣∣∣ψ〉(23)and the same rule for the observales 〈Aˆ〉 of Hermitian op-erators Aˆ associated with the observable system’s prop-erties, the eigen-spectrum of Hˆ represents the observableenergy measure of system. But the non-commutativeterms make this energy measure different from that ofgiven by a classical Hamiltonian. Correspondingly, thequantum phenomena acquire unusual features unexplain-able if to adhere to the principles of Lagrangian mechan-ics, hence, should fall in conditions of energy conservationinto nothing else but the category given by the vortex en-ergy measure.The quantum limit in question assumes that the mea-surements perturb the system in a away of transitionsonly between its pure states. Beyond, the quantum phe-nomena should also fall into the vortex-energy categorysince the states of quantum system are then assumed tobe a mixture of pure states each given a statistical weightin the sense used in the classical physics. The proof isimmediate while reasoning in terms of density matrix aswell as the Feynman’s path integral approach.So, all phenomena attributed to quantum physics rep-resent classical irreversible phenomena. To attribute thequantum phenomena to an uninterpretable sort of re-versible processes, constrained with no arrow of time inthe sense of Eq. (2), turns out to be wrong if to adhereto the basic energy principle of physics unifying this sci-ence of realm. This our point of view is in marked con-trust with the mainstream of fundamental quantum the-ory. The theory, proceeding with research via Koopman-von Neumann approach [18,19], path integral [20] andother methods, is presently in intensive works of manyresearchers, e.g. [21,22,23]. For all that, to find real, onthe basis of energy measure for physical systems, classi-cal routes of quantum phenomena, one then needed tocome up to or rely on the general idea of complementaryenergy measures [5] substantiated in the present work.The principle of footing we claim is not an equal foot-ing, for the existence domain of the quantum conceptis unknown in view of a fundamental problem of quan-tum mechanics - how to protect a quantum state froma measuring apparatus against the quantum state col-lapse. The stabilization of states is required, and we seeno other way based on the notion of energy to that but toidentify it with the vortex energy. Without the latter alsothe concept of entanglement and nonlocality inherent inwave functions hardly becomes sustainable physically.We do not augment validity to the quantum conceptby resorting to the concept of entropy and a semiclassicalapproach to the irreversible kinetics, but acquire it exclu-sively due to this kinetics. The principle of complemen-tary energies with its vortex energy covers all aspects ofthat in a self-sustained way in terms of measurable kinet-ics adequate to the reality of inseparable balance betweenthe reversible and irreversible forcing.Thereby, the established principles of integral vortexenergy inherent in stable systems unravel the inventionsof quantum mechanics and place them on a tangibleground to test and unveil the potential in search for newstates of matter and energy.[1] R. Feynman, R. Leighton, M. Sands, The FeynmanLectures on Physics, Vol. 2 Chapter 40, 2006.[2] Yu.I. Neimark, N.A. Fufaev, Dinamika NegolonomnihSystems. Nauka, Moscow, 1967. (Dynamics of Nonholo-nomic Systems. Providence, Am. Math. 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)Principles of statistical mechanics:the energy duality. Phys. Lett. A 372 (2008) 6087-6093;b)The idea of vortex energy. arXiv: 1109.2605 Sep 2011;1305.5108 Jul 2013; c)Unconventional energy of stablesystems. In: http://circle.ubc.ca/handle/2429/46075 Feb2014.[6] V.E. Shapiro, Cooling of matter by a highy frequencyresonance field. Zh. Eksp. Teor. Fiz. 70 1463-1976 (1976).[7] R. Graham and H. Haken, Generalized thermody-namic potential for Markoff systems in detailed balance,Z. Physik 243 289-302 (1971).[8] F.G. Gustavson, On constructing formal integrals of ahamiltonian system near an equilibrium point. Astron. J.,71 670-686 (1966).[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.(Statistical Distribution Functions) Nauka, Moscow, 1966.[11] L. de la Pen´a-Auerbach and A.M. Cetto, Does quan-tum mechanics accept stochastic support? Found. Phys.12 (1982) 1017-1037.[12] R.D. Astumian and P. Hanggi, Brownian motors,Physics Today 55 (2002) 3339.[13] Bo Sun, D.G. Grier, A.Y. Grosberg, Phys. Rev. E. 82,021123 (2010).[14] L.D. Landau, E.M. Lifshitz, Statistical Physics. Perg-9amon, New York, 1980.[15] D.C. Brydges and Ph.A. Martin, Coulomb systems atlow density: a review, J. Stat. Phys. 96 (1999) No 5/6,1163-1330.[16] H. Lieb and J. Lebowitz, The constitution of mat-ter: existence of thermodynamics for systems composedof electrons and nuclei. Adv. in Math. 9 316-398 (1972).[17] Rayleigh, The Theory of Sound. London Macmillan,Vol. 2, 1896.[18] B.O. Koopman, Proc. Natl. Acad. Sci. USA 17 (1931)315-318.[19] J. von Neumann, Ann. Math. 33, (1932) No. 3, 587;33 (1932) No. 4, 789.[20] R.P. Feynman, Rev. Mod. Phys. 20 (1948) No 2, 367-387.[21] A.A. Abrikosov, E. Gozzi, D. Mauro, Geometric de-quantization. Ann. Phys. 317 (2005) 24-71.[22] D. Bondar, R. Cabrera, R. Lompay, M. Ivanov, H. Ra-bitz, Operational dynamic modeling transcending quan-tum and classical mechanics. Phys. Rev. Lett. 109 (2012)19.[23] N. Buric´, D.B. Popovic´, M.Radonjic´, S. Prvanovic´,Hamiltonian formulation of statistical ensembles andmixed states of quantum and hybrid systems. Found.Phys. 43 (2013) 1459-1477.


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