aa r X i v : . [ phy s i c s . g e n - ph ] S e p The idea of vortex energy
V.E. Shapiro ∗ This work formulates and gives grounds for general principles and theorems that question theenergy function doctrine and its quantum version as a genuine law of nature without borders ofadequacy. The emphasis is on the domain where the energy of systems is conserved – I argue thatonly in its tiny part the energy is in the kinetic, potential and thermal forms describable by ageneralized thermodynamic potential, whereas otherwise the conserved energy constitutes a wholelinked to vortex forces, and can be a factor of things like persistent currents and dark matter.
PACS numbers:01.55.+b general physics02.90.+p mathematical methods in physics05.90.+m statistical physics, thermodynamics and nonlinear dynamical systems
The motivation and scope
The physics of phenomena is chiefly perceived throughthe interactions given by the energy function in line withthe principles of holonomic mechanics and equilibriumthermodynamics. This brilliantly unifying guideline cre-ated by Euler and Lagrange has found way in all poresof physics, and interpretations have spread out as if theguideline is a genuine universal law of nature. But withno outlined borders of adequacy, the law is a default be-lief, a source of circular theories and fallacies.In this regard, worth recalling the forces called circu-latory or vortex with all their cumulative impact beyondthe energy function pattern that can be huge, as exposedsince 19th century, e.g. [1,2] and the byword “dry wa-ter” stuck to viscosity-neglect hydrodynamic studies asinadequate, see Feynman lectures [3]. Also since 19thcentury, e.g. [4,5], the failure of the pattern was exposedin mechanics and other fields due to the reaction forcesof ideal non-holonomy, performing no work on the sys-tem, as is the case of rigid bodies rolling without slippingon a surface. Recall also a general symmetry argumentprovoked by H -theorem of Boltzmann and showing thefundamental reversibility Loschmidt’s paradox [6] on theway to conform the real world with the energy functionpattern.The physics of nowadays for all that, now in line withthe quantum mechanics claimed as more genuine thanthat of classical mechanics, further spreads out the con-viction in the genuine energy law with no outlined bor-ders of adequacy. And it has a commanding influenceon both fundamental and applied research. This com-mon trend has various sophisticated possibilities to fallinto the same trap of circular theories, which in my judg-ment occurs here and there mainly due to playing withconcepts of entropy and energy.Indeed, “You should call it entropy, for two reasons. Inthe first place your uncertainty function has been usedin statistical mechanics under that name, so it already ∗ Electronic address: [email protected] has a name. In the second place, and more important,no one really knows what entropy really is, so in a de-bate you will always have the advantage” as John vonNeumann remarked on in another connection [7]. Curi-ously enough, it no less concerns what energy really is.This note will try to sketch my approach to it and theperspective on the energy perceptions I came to on thisway; here it is through formulating and giving groundfor relevant general principles and theorems. They rec-tify and develop the idea of energy duality claimed withinsufficient argument in [8] in connection with the strongvortex effect of high frequency fields, and point clear tothe conserved energy linked to vortex forcing which iscomplementary to all forms of energy function edifice.
The “energy cake” dilemma and equal footingtheorem
The established consistent pattern of the world aroundis basically relaxation to roughly recurrent trends. It im-plies the ubiquity of irreversible forces as the generalizedforces whose infinitesimal work depends on the path ofsystem motion rather than just its instant state. An im-portant mechanism is a manifold back-reaction of media.May one then refer the irreversible forcing to the aver-aging of irrelevant variables of a conservative many-bodysystem given by a microscopic Hamiltonian and randominitial conditions? The answer to this widespread cue isno [8]. The cue misleads in the question of both statisticaland dynamical (asymptotic over fast motion) averaging– there is no way to come to the irreversible behaviorsfrom the formalism of energy functions unless resort tothe inexact reasoning residing in the averaging methodsand truncations irreducible to the separation by canoni-cal transformations.At the same time the perception of myriad of outer in-fluences even treated as time-varying Hamiltonian inter-actions is inevitably via smoothing which barely complieswith the exact separation given by canonical transforma-tions, hence, contributes to the irreversible forcing alongwith arbitrariness in modeling the trends. This is like eatcake and have it: On the one hand, the irreversible forces,unlike reversible, cannot be derived from a Hamiltonianor effective potential. On the other, insofar as the truephysics of phenomena is perceived through the interac-tions given by energy functions, so should be the physicsof irreversible phenomena.The “energy cake” dilemma formulated above is inher-ent to the perception and it imports fundamental inex-actness in reasoning in terms of energy function. Thereis no other way to account for that but to integrate theenergy formalism with a tentative (statistical) measureof energy blur/relaxation rates. This element pertains toboth classical and quantum mechanics descriptions. Theuncertainty principle of the latter is related to the pos-tulated discreteness of energy transfer, has nothing to dowith the dilemma, and the integration in point puts bothdescriptions on an equal footing.Many phenomena in radiation and rays, superconduc-tivity and other fields are commonly referred to as inde-scribable classically, which might be in one’s rights withinsome specific context; as for unconstrained assertions, itis to be questioned since contradicts the above theoremof equal footing. The same concerns the ideas of quan-tum computing claimed beyond classical physics, if theirprinciples appear to be true.Naturally, each of the two mechanics integrated withthe element of diffusion/relaxation has a niche where it ishandier depending on interest to discrete or continuoussides of phenomena. Even for such irreversible phenom-ena as highly deep cooling of matter by hf resonance fieldsboth ways have led to its independent prediction, see [9],and we found the classical way direct, free of any linkageto the uncertainty principle defined by Planck constant.The point is not so much that this constant is the same forany nature of canonically conjugated variables, it is thatthe physics of phenomena can be perceived through anyself-sufficient construction and that one can’t see throughits wall unless allowing for a dual of the formalism withrespect to wider frameworks.
The entrainment theorem
The inexactness of the trends prescribed by an energyfunction unfolds generally not only diffusion-like but alsoexponentially, able to radically change the system’s state,its stability and fluctuations. The vortex forces act so.The Gibbsian thermodynamics and the theory of general-ized thermodynamic potential [10,2] commonly acceptedin the study of phase transitions, transport through barri-ers, etc. abstract away from that. The generalized poten-tial of a system relaxing in steady conditions to a densitydistribution ρ st connects to it by ρ st ( z ) = N e − Φ( z ) , N − = Z e − Φ d Γ (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. The analogous ap-proach to systems under high frequency fields is in termsof the picture where the hf field looks fixed or its effectis time-averaged. In all this, Eq. (1) can be viewed asmerely redefining the distribution ρ st in terms of func-tion Φ, which is suitable for the notion of the entropy ofsystem states, whereas taking this function as the energyintegral of reversible motion provides the physical basisof the theory, but implies rigid constraints.Commonly, the constraints are reasoned as detailedbalance (of transition probabilities between each pair ofsystem states in equilibrium) within the framework of au-tonomous Fokker-Planck equations under natural bound-ary conditions by means of division of the variables andparameters into odd and even with respect to time re-versal, with a reserve on factors like magnetic field. Thislogic, however, is model-bound and ill-suites unsteadyconditions. Also it begs a question since detailed balanceis in leading strings, for the reserve is not universal, e.g.,breaks down in nuclear processes.A different approach to outlining the overall domainof exactness was suggested in [8] and will be developedhere. As in general, the basis is in keeping to invarianceunder transformations of variables. Obviously, Φ( z ) to bethe integral of reversible motion must be invariant underunivalent transformations z → Z , of Jacobian | det { ∂Z k ( z, t ) /∂z i }| = 1 (3)where i, k run all components of z and Z , for then notonly ρd Γ is invariant (being a number) but also d Γ is.The environment as a fluctuation/dissipation sourcefor the system causes another invariance. Connecting Φto the system’s energy function implies scaling this func-tion in terms of environmental-noise energy level. Theenergy scale set so must vary proportionally with the en-ergy 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 of that the limit ofweak background noise poses as a structure peculiar-ity – transition to modeling of evolution without re-gard to diffusion. The entrainment constraint then keepsits sense as the weak irreversible-drift limit graspedvia the scenarios of motion along the isolated pathsin line with d’Alembert-Lagrange variational principle.Thereat, however, the principle still allows for the idealnon-holonomic constraints that violate the invariance ofΦ( z ). The invariance therefore necessitates the domainof entrainment free of that, termed ideal below.We have reasoned about Φ( z ) (1), but the reasoningholds for any one-to-one function of ρ st . In unsteady con-ditions for the systems describable by a time-dependentdensity distribution ρ ( z, t ), the adequacy of energy func-tion formalism requires the entrainment ideal also. Thearguments used above for the systems of steady ρ st ( z )become there applicable with univalent transformationsof ρ ( z, t ) into t -independent distribution functions.The converse is also true: the behaviors governed by adressed Hamiltonian H ( z, t ) imply the entrainment idealand the existence of a density distribution ρ ( z, t ). As thevelocity of underlying motion, ˙ z = ˙ z ( z, t ), is constrainedby ˙ z = [ z, H ] with [ , ] a Poisson bracket, the divergence div ˙ z = div [ z, H ] = 0 and div ( ˙ zf ) = − [ H, f ] for anysmooth f ( z, t ). It implies ∂ρ/∂t = [ H, ρ ] (4)which determines ρ ( z, t ) from a given initial distributionand the natural boundary conditions preserving the nor-malization and continuity, for all other constraints areembodied in H . In no way the solution to (4) ceases toexist as unique, non-negative and not normalizable overthe phase space of z where H ( z, t ) governs the behaviors.The entrainment ideal there takes place since the solu-tion turns to ρ ( H ) in the interaction picture where H is t -independent. This completes the proof.Thus, the necessary and sufficient conditions where theenergy function doctrine is duly adequate to the evolutiondescribed by distribution functions come down to the en-trainment ideal. This theorem lays down the overall do-main of energy function adequacy sought for. 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.Remark 1. The trend of entrainment ideal can be de-prived of evidential force already in the close vicinity ofthe ideal. In steady condition this can be not so muchdue to extremely long observation times as due to idealnonholonomy, for the diffusion and irreversible drift arethen enhanced hugely. To see it, suffice to bear in mindthat the ideal nonholonomy not only reduces the numberof degrees of freedom relative to the number of general-ized coordinates, but also gives rise, see [5], to equilib-rium states and also steady-motion states that are notisolated but form manifolds of one or more dimensionsand asymmetry of secular-equation determinants.Remark 2. The evolution of density distribution ρ ( z, t )of system states from a given initial ρ ( z,
0) gives by it-self no insight into the matter of entrainment even insteady conditions, unlike, say, their multitude from var-ious initial distributions. The trend of evolving then toone and the same shape of ρ means relaxation with themean (“drift”) irreversible forces a factor. For the relax-ation to steady motion, such forces are of vortex type inΓ as their forcing toward the steady motion and againstit differ in sign. They effect both the steady and thetransient shapes of ρ . Thereby the energy integral of themotion ceases to exist with time if set initially, beingdisrupted by the drift vortex forces along with diffusion.These forces act generally stronger than diffusion, multi-plicatively, and cannot be compensated by conservative forces. Unstable are then as minima of Φ( z ) (1) as anypoints shifted from them, and the motion states can ap-pear quite apart from the minima, in defiance of the the-ory of phase transitions based on generalized potentials. The “energy - energy function” dualism and theconserved energy linked to vortex forcing
Let us refine on concepts. The issue of energy we areraising relates to the systems of finite degrees of freedomthat interact with the environment whose influences ofshort correlation time are accounted for via the notionof entrainment introduced above. The systems are as-sumed describable by a smooth evolution of the densitydistribution ρ ( z, t ) of phase space states z , a set of contin-uous variables z = ( x, p ) with the generalized coordinates x = ( x , . . . x n ) and conjugated moments p = ( p , . . . p n )of proper n taken in neglect of the constraints breakingthe energy function formalism; z may include countablesets of normal mode amplitudes of waves in continuousmedia. The smoothness of ρ means ∂ρ/∂t = − div ( vρ ) (5)with vρ the 2 n -vector flux of phase fluid at z , t . Eq. (5)turns into the evolution equation of ρ ( z, t ) under nat-ural boundary conditions with v treated as operator on ρ ( z, t ) that accounts for all constraints on the phase flows;in neglect of all nonlocal and retarded constraints, v isgenerally a t -dependent field divergent in z .For the evolution of ρ modeled by an equation of form ∂ρ/∂t = [ H, ρ ] + I (6)where H = H ( z, t ) is now, unlike in Eq. (4), an arbitrarysmooth function taken for a Hamiltonian, and I embodiesall other interactions, we have I = − div [( v − ˙ z ) ρ ]with ˙ z = [ z, H ] the local velocity of Hamiltonian phaseflow and I a canonical invariant. The invariance of I holds as in as off the entrainment ideal. Indeed, a canon-ical (univalent) transformation z → Z implies not onlythe invariance of ρ and Poisson brackets but also theconstraint ∂Z ( z, t ) /∂t = [ Z, G ] with G a function of z, t .Hence, on transforming Eq. (6) we get( ∂ρ/∂t ) Z = ( ∂ρ/∂t ) z − [ G, ρ ]where the term [
G, ρ ] is to be united with [
H, ρ ] of (6),for just so a Hamiltonian is to change. Thus, Eq. (6) inthe new variables differs by its r.h.s. changing so[
H, ρ ] + I → [ H + G, ρ ] + I (7)with I to be held invariant. This completes the proof.It should be underlined that the invariance of I is notunconditional but under canonical transformations andreflects the fact of proceeding in modeling from a Hamil-tonian. Since it is governed by a Hamiltonian only in theentrainment ideal, only there I reduces to an invariantPoisson bracket; whereas the irreducibility of I this waybeyond the ideal exposes I of Eq. (6) as the source of irre-versible, hence, vortex forcing that breaks the invarianceof I for any choice of H ( z, t ).Let us now turn to the concept of energy within theframework under study. At that, while the x , p of ρ ( z, t )is a set of phase space variables, the principle of virtualwork on the system and the law of energy conservation,which are to be taken as prime as so the material worldis perceived, are formulated in terms of isolated pathswith x and p the functions of t . Naturally, we treat anyconceivable isolated paths as abstraction of the kineticsof ρ , so the integrable correspondence between the twodescriptions is to imply on the principles of continuityand causality. The n components ( v n +1 , v n +2 , . . . v n ) ofthe actual phase flux at z , t act then as the generalizedforce conjugated to x and the scalar product v n +1 δz + v n +2 δz + . . . v n δz n (8)represents the virtual work on the system irrespectiveof whether this sum is reducible to the variation of ascalar function or not. Accordingly, for the generalizedcoordinates taken without restricting the generality inthe geometric conditions not involving time explicitly, thedensity power on the phase fluid comes down to the scalarproduct ( v n +1 v + . . . v n v n ) ρ. (9)In particular, the energy of the system is conserved aslong as the integral of density power (9) over the wholephase volume holds zero, Z ( v n +1 v + . . . v n v n ) ρd Γ = 0 . (10)This criterion bears by itself no relation to the entrain-ment ideal and shows up in both entrained and non-entrained systems and as under steady constraints (au-tonomous Eq. (5)) as unsteady.Where the energy of system is conserved, there its en-ergy measure exists in strict sense. So, the criterion (10) outlines the existence domain of the energy measure. Itincludes the whole existence domain of the energy mea-sure in the entrainment ideal, which is obviously where v is a t -independent divergent-free function of z , but canextend fairly far beyond it - however far in principle bothin drift and diffusion terms of v and whether they are re-tarded and t -independent or not.Consider in this light the conditions of energy conser-vation in the systems governed by autonomous Eq. (6).The branch I there acts on a par with [ H, ρ ] in keepingthe circulation and transformations of conserved energy,and this takes place as within as beyond the scope ofentrainment ideal. This fact means that the energy cir-culating in conditions beyond the ideal is indescribablein terms of energy function. We called it vortex form ofenergy. The energy exchange between degrees of freedomis then not via detailed balance, involves vortex forcing.At that, the stationary conditions of ideal non-holonomycan be viewed as a particular case of such form of energycirculation.Thus, unlike the conventional kinetic and potential en-ergies and the thermal energy as the chaotic variant ofkinetic energy within the generalized thermodynamic po-tential, the form of conserved energy circulating in thenon-entrained systems is irreducible to a scalar functionof system states. This form of energy is integral – relega-tion of its parts to conventional is not uniquely defined,which shows the vortex form of energy as complementaryto the energy of conventional forms. This is what we callthe energy - energy function dualism. It has nothing todo with the particle - wave dualism in quantum mechan-ics. The notion of energy quantum levels as the quanta ofphysical substance related to specific system states losesthen its strict sense, and so does the transfer of energyvia energy quanta.The ability of vortex forces to radically, cumulativelychange the system’s state, stability and fluctuations, asis the case of systems under high frequency fields, andthe ubiquity of vortex forcing also under the restrictionsof system energy conservation conveys a suggestion thatthe vortex form of conserved energy can be a factor ofpersistent currents and also puzzles like dark matter. 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