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arXiv:1403.3900v2 [.flu-dyn] 20 Jun 2014 omter asta h aefnto guides Broglie- De wave the , that In says par- theory . the Bohm physical of the de- position in function probabilistic ticle wave a The only scribes particle. conditioned a is detecting collapse by its and function to wave priority the gives interpretation dualism. wave-particle Copenhagen the The of foun- concept their the in dation have interpretations Both Broglie-Bohm de theory. the and on based interpretation interpretation Copenhagen the the me- are quantum chanics of interpretations known [14]. well discipline Two this clearness understanding of sake ontological the of for other interpretations with The each compete the and experiment. one physical describing same for applied sub- are some points, in tle another one me- contradicting may quantum it chanics, of as interpretations ironic as different , sound, ontological for contribu- activ- As human triumphal of ities. areas shows technological many to tion present to up ∗ lcrncades [email protected] address: Electronic unu ehnc einn ihisbirth its with beginning Quantum rmteNwo’ ast oin ftefli n uefli va superfluid and fluid the of to laws Newton’s the From eec;dolt otxtb;vre ig uefli vacu superfluid ring; vortex tube; vortex droplet; ference; ass ude ftevre ie a rnmtatru fro a transmit can lines because vortex momenta the angular of the It Bundles drople and . of medium. masses superfluid model nonzero having peculiar a o pairs a gives solutions represents trajectories are vacuum Bohmian physical objects the these eleme along guiding Th fluid balls function the wave of potential. the vector by with magnetic multiplied the solenoidal momen with the the analogy to added by w Th being function, potential component wave potential. solenoidal the The (quantum) named equation. bat Bohmian function, complex-valued the a a is (c) define term barotropic; t modification reduction is admits The equation fluid Navier-Stokes the the ) (b) components; solenoidal Keywords: wn otrecniin nml:()tevlct srepre is velocity the (a) (namely: conditions three to Owing .INTRODUCTION I. R ucao nttt,Gthn,Lnnrddsrc,18 district, Leningrad Gatchina, Institute, Kurchatov NRC airSoe;Sh¨dne;Frdywvs aefunctio wave waves; Schr¨odinger; Faraday Navier-Stokes; nvriyo aiona ekly ekly A970 US 94720, CA Berkeley, Berkeley, California, of University .P osatnvS.Ptrbr ula hsc Institut Physics Nuclear Petersburg St. Konstantinov P. B. eateto lcrclEgneigadCmue Science Computer and Engineering Electrical of Department otxtbs ig,adothers and rings, tubes, vortex Dtd uut1,2018) 14, August (Dated: aei .Sbitnev I. Valeriy ilntcag t eoiyuls nexternal an unless in is velocity that its object change an external not an it; will unless upon rest at acts stay object will rest an at is frames: postulates that reference law The inertial first of the [37]. existence (a) 1687 read: in laws Philosophy Newton’s Natural Prin- Mathematical of in ciples published New- three first are laws They ton’s fundamen- funda- physics. most classical the from of laws from begin tal begin us to Let sense mentals. makes it question, vertical small to a subjected the surface, is for oil vibrations. which wave silicon with guiding the bath the on of bouncing role droplets Faraday the a that play found was waves It passing after droplets. far many arises the of in pattern that interference shown two an have field containing surface authors The obstacle oil [8]. an silicon slits through a moving on at bounce with which experiment from an droplets faraway is area This the mechanics. quantum from came theory Bohm role. func- active wave more the case a this plays In tion along path. detector optimal a most to a up source its from particle the hr stuh nodrt nwro this on answer to order In truth? is Where Broglie- de the of confirmation amazing, is It um h oie aitnJcb equation. Hamilton-Jacobi modified the o t otxtbs ig,adblsalong balls and rings, tubes, Vortex nt. ∗ hseuto.Mto ftevortex the of Motion equation. this f n oaigcasclds oother. to disk classical rotating one m smvn ntefli ufc.The surface. fluid the on moving ts ossso oeparticle-antiparticle Bose of consists ihi ouino h Schr¨odinger the of solution a is hich etrptnili ersne by represented is potential vector e ihtefli nege vertical undergoes fluid the with h u prtrpssisl savector a as itself poses operator tum frttn bu h etro the of center the about rotating of etdb u firttoa and irrotational of sum by sented srdcinoespsiiiyto possibility opens reduction is ;poaiiydniy inter- density; probability n; 30 Russia; 8350, A e, s, cuum: 2 force acts upon it. The is a property of lies. This vector characterizes a quantita- the bodies to resist to changing their velocity; tive measure of the vortex motion and it is called (b) the second law states: the net force applied . Vector product [~v ~ω] is perpendicu- to a body with a mass M is equal to the rate lar to the both vectors ~v and× ~ω. It represents of change of its linear in an inertial the Coriolis of the body under ro- reference frame tating it around the vector ~ω. In turn, the term v d~v ρM [~ ~ω] represents the acting on F~ = M~a = M ; (1) the fluid× in the volume ∆V . d t The term (5) entering in the Navier-Stokes (c) the third law states: for every there is equation is responsible for emergence of vortex an equal and opposite reaction. structures. The vortex tubes, rings, and balls Leonard Euler had generalized Newton’s laws are considered briefly in Sec. 2 The latter objects of motion on deformable bodies that are assumed are good models of the droplets moving on the as a continuum [16, 21]. We rewrite the second fluid surface. In Sec. 3 we transform the Navier- Newton’s law for the case of deformed medium. Stokes to the Schr¨odinger-like equation. The Let us imagine that a volume ∆V contains a fluid transformation is achieved due to introduction medium of the mass M. We divide Eq. (1) by ∆V of a material dependent parameter which substi- and determine the time-dependent mass density tutes the Plank constant. Solutions of the equa- ~ ρM = M/∆V . In this case F is understood as tion disclose interference patterns arising from the force per volume. Then the second law in the motion of the vortex balls along optimal this case takes a form: paths, the Bohmian trajectories, through an ob- v v stacle containing slits. The Schr¨odinger equation ~ dρM ~ d~ dρM F = = ρM + ~v . (2) describing flow of a superfluid medium known as d t d t d t the physical vacuum is considered in Sec. 4. Here The total in the right side can be we deal with vortex lines which arise due to rota- written down through partial derivatives: tion of the virtual electron-positron pairs. The lines can self-organize into twisted vortex bun- dρM ∂ρM = +(~v )ρM , (3) dles in a rotating cylindrical container. Kinetic d t ∂ t ∇ of these vortex bundles is sufficient to be- d~v ∂~v = + (~v )~v. (4) gin to rotate a top disk covering the container. d t ∂ t ∇ Concluding remarks are given in Sec. 5. Eq. (3) equated to zero is seen to be the conti- nuity equation. As for Eq. (4) we may rewrite II. THE NAVIER-STOKES EQUATION AND the rightmost term in detail MOTION OF VORTICES v 2 (~v )~v = [~v [ ~v]]. (5) Taking into account the ∇ ∇ 2 − × ∇ × let us rewrite Eq. (2) by omitting the rightmost As follows from this formula the first term, mul- term and specify which should be in this tiplied by the mass, is of the kinetic equation: energy. It is a force applied to the fluid element ~ ∂~v 2 F for its shifting on the unit of length, δs. The ρM +(~v )~v = P +µ ~v + . (6) ∂ t ∆V second term is acceleration of the fluid element ·∇ ! −∇ ∇ directed perpendicularly to the velocity. Let the The following of forces is presented in this fluid element move along some curve in 3D space. equation: (i) the first term is the gra- to the curve in each refers to ori- dient. It takes place when there is a differ- entation of the body motion. In turn, the vector ence in pressure across a surface; (ii) the sec- ~ω = [ ~v] is orientated perpendicularly to the ond term represents the of a Newtonian plane,∇ where × an arbitrarily small segment of the fluid (here µ is the dynamic viscosity, its units 3 are N s/m2 = kg/(m s)); (iii) F/~ ∆V is a body 0.3 0.15 force per· unit volume· acting on the fluid. So, we (a) wrote down the Navier-Stokes equation for the 0.2 0.1 m/s

viscous incompressible Newtonian fluid [29]. The rad/s

term (~v )~v in Eq. (6) can be rewritten in de- 0.1 0.05 , , tail as shown·∇ in Eq. (5). The vector ~ω = [ ~v] ∇ × in Eq. (5), named vorticity, underlies the vortex 0 0 formation. Let us consider the simplest exam- 0 0.5r 1m 1.5 ples. y

A. Helmholtz vortices max Let the force F~ be conservative. Then by ap- plying to Eq. (6) the we obtain right away the equation for the vorticity: ∂ ~ω +(~ω )~v = ν 2~ω. (7) ∂ t ·∇ ∇

Here ν = µ/ρM is the kinematic viscosity. Its di- mension is [m2/s]. It corresponds to the diffusion coefficient. For that reason the energy stored in min the vortex will dissipate in thermal energy. As a result the vortex, with the lapse of time, will dis- x appear. For supporting the vortex activity, the energy has to be supplied permanently. Trivial FIG. 1: Vortex tube oriented along the axis z: (a) Vorticity solution of Eq. (7) is seen to be ~ω = 0. ω and velocity u of a flow around the center as functions of remoteness from this center. The velocity u vanishes in the We suppose that the fluid is ideal, barotropic, center and tends to zero on infinity. (b) Cross-section of the and the mass forces are conservative [28]. With vortex. Values of the velocity u are shown in grey ranging omitted the right term, i.e., at ν = 0, the from grey (min u) to dark grey (max u). Density of the Helmholtz theorem says: (i) if fluid particles pixels represents magnitude of the vorticity ω form in any of the time a vortex then the same particles form the vortex line both the second case they are vortex rings lying in in the past and in the future; (ii) ensemble of the plane (x, y). Consider the both cases. the vortex lines traced through a closed contour forms a vortex tube. Intensity of the vortex tube is constant along its length and does not change in time. Intensity of the vortex is 1. Vortex tube of the velocity around the contour encompass- ing the vortex. From above said follows that the Let the vortex tube be oriented along the axis vortex tube (a) either goes to infinity by both z and its axis is aligned with the origin (x, y)= endings; (b) or these endings lean on walls of (0, 0). Eq. (7) rewritten in cross-section of the bath containing the fluid; (c) or these endings vortex flow reads are closed to each on other forming a vortex ring. ∂ω ∂ 2ω 1 ∂ω As for the vortex solutions of Eq. (7) it is = ν + . (8) convenient to proceed to cylindrical or toroidal ∂ t ∂r2 r ∂r ! depending on a task which is to be considered. In the first case the solutions Here we do not write a sign of vector on the top are vortex tubes oriented along the axis z. In of ω since ω is oriented strictly along the axis z. 4

Solution of this equation is as follows z Γ r 2 ω = exp . (9) 4πν t ( −4ν t ) Here Γ is the integration constant having dimen- sion [m2/s]. Components of the velocity ~v are ly- ing in the plane (x, y) along of . We have the Lamb-Oseen solution r 1 ′ ′ Γ − 2 a x v(r, t)= ωr dr = 1 e r /4ν t . (10) y c r 2πr − b Z0   FIG. 2: Helicoidal vortex ring: r0 = 2, r1 = 3, ω2 = 12ω1, Fig. 1(a) shows the vorticity ω and the velocity and φ1 = φ2 = 0. u as functions of distance from the vortex center. Because of ν > 0 the vortex decays with time. coordinates (x, y, z). And r is the radius of the Qualitative view of the vortex in its cross-section 0 tube. A body of the tube, for the sake of visual- is shown in Fig. 1(b). Values of the velocity u isation, is dashed by light grey . Eq. (11) are shown in grey from light grey (minimal ve- parametrized by t gives a helicoidal vortex ring locities) to dark grey (maximal ones). One can shown in Fig. 2. Parameters ω and ω are fre- see that the cross-section of the vortex gives a 1 2 quencies of rotation along the arrow a about the good illustration of a hurricane shown from the centre of the torus (about the axis z) and ro- top. In the center of the vortex a so-called eye of tation along the arrow b about the centre of the hurricane is well viewed. Small values of the the tube (about the axis pointed by arrow c), velocity in the eye of the hurricane are marked by respectively. Phases φ and φ have uncertain light grey. Observe that in the region of the eye 1 2 quantities ranging from 0 to 2π. By choosing the a wind is really very weak, especially near the phases within this interval with small increment, center. This is in stark contrast to conditions we may fill the torus by the helicoidal vortices in the region of the eyewall, where the strongest everywhere densely. winds exist (it is shown in Fig. 1(b) by a dark grey annular region encompassing the light grey As was mentioned above, the vorticity is max- region of the eye). imal along the centre of the tube. Whereas the velocity of rotation about this centre in the vicin- ity of it is minimal. A stream on the centre 2. Vortex ring will go in parallel to this central axis. However, the velocity grows at increasing distance from If we roll up the vortex tube in a ring and glue the centre. After reaching some maximal value together its opposite ends we obtain a vortex that depends on magnitude of the viscosity, see ring. A result of such an operation put into the Eq. 10, the velocity begins to decrease. So the (x, y) plane is shown in Fig. 2. Position of points vortex ring has a finite size. Further we shall on the helicoidal vortex ring [52] in the Cartesian return to the helicoidal vortex ring for the case is given by when the radius r1 will tend to zero.

x = (r1 + r0 cos(ω2t + φ2)) cos(ω1t + φ1),

y = (r1 + r0 cos(ω2t + φ2)) sin(ω1t + φ1), (11) 3. Vortex

z = r0 sin(ω2t + φ2). Here we shall represent a vortex ball that Here r1 represents the distance from the center can simulate the droplet. Let the radius r1 in of the tube (pointed in the figure by arrow c) to Eq. (11) tends to zero. The helicoidal vortex ring the center of the torus located in the origin of in this case will transform into a vortex ring en- 5 veloping a spherical ball. The vortex ring for the shown in Fig. 3. Motion of a small particle (SP) case r0 = 4, r1 0, ω2 = 3ω1, and φ1 = φ2 = 0 along the vortex ring takes place with a velocity drawn by thick≈ curve colored in dark green is

v = r ω sin(ω t + φ ) cos(ω t + φ ) r ω cos(ω t + φ ) sin(ω t + φ ) R,x − 0 2 2 2 1 1 − 0 1 2 2 1 1 ~vR =  vR.y = r0ω2 sin(ω2t + φ2) sin(ω1t + φ1)+ r0ω1 cos(ω2t + φ2) cos(ω1t + φ1)  . (12) v = −r ω cos(ω t + φ )  R,z 0 2 2 2   

x x 0 - 0 .... + 0 0 0 0 0 0 0 .. 0 .. 0 . ... 0 z + 0 - 0 0 2 0 ... 0 y . 0 0 0 z y 1 - + 0

FIG. 3: Helicoidal vortex ring convoluted onto the vortex FIG. 4: The vortex ball rotating about axis z with the max- ball: r0 = 4, r1 = 0.01 ≪ 1, ω2 = 3ω1, φ1 = φ2 = 0. imal velocity v 0 that is reached on the surface of the ball. The radius r0 represents a mean radius of the ball, where the velocity v 0 reaches a maximal value. The ratio ω2/ω1 = 3 was chosen with the aim in order not to overload the picture. circles centered on the axis z. We see a ball that rolls along the axis y, Fig. 4. The spherical harmonics are perfect modes in this case [12]. The velocity of SP at the initial time is vR,x = 0,

vR,y = r0ω1, vR,z = r0ω2 = 3r0ω1 (the initial point (x, y, z) = (4, 0, 0) is on the top of the III. TRANSFORMATION OF THE ball). We designate this velocity as ~v+. Through NAVIER-STOKES EQUATION t = π/ω1 the SP returns to the top position.

The velocity in this case is vR,x = 0, vR,y = r0ω1, Before we shall begin to subject the Navier-

vR = r ω = 3r ω . We designate this ve- Stokes equation to transformations it would be ,z − 0 2 − 0 1 locity as ~v−. Sum of the two opposite , instructive to recall works touching upon analo- ~v+ and ~v− , gives the velocity ~v0 = (0, 2r0ω1, 0). gies between hydrodynamics and quantum me- During t = (1+3k)π/3ω1 and t = (2+3k)π/3ω1 chanics. First quantum theory in a hydrody- (k =1, 2, ) SP travels through the positions 1 namic representation was formulated by Erwin ··· and 2 both in the forward and in backward direc- Madelung [32]. Remarkably that Madelung’s tions, respectively. In these points the velocities equations exhibit a close relation of the Bohmian ~v+ and ~v− give the resulting velocity ~v0 directed mechanics [26, 39, 54] and hydrodynamics. It along the lying in the plane (x, y). gives the reason to hypothesize that quantum By adding the vortex rings with other phases medium behaves like a fluid with irregular fluc- φ1 and φ2 ranging from 0 to 2π we fill the ball tuations [4]. everywhere densely by these rings. The resulting One more impressive example comes from velocities ~v 0 in some points of the ball will lie on the fluid mechanics. It is a demonstration of 6 quantum-like behavior of droplets bouncing on We accept the following assumptions: (a) the fluid surface [8, 9, 13, 42]. After the articles velocity consists of two components - irrotational appeared in the press, attempts at an explana- and solenoidal [28] that relate to vortex-free and tion of this experiment were undertaken with the vortex motions, respectively. The basis for the point of view of [5, 7, 12, 22– latter is the Helmholtz theorem; (b) we admit 25, 27, 36, 40, 49]. Since we know that pres- that the density is a function of the pressure, i.e., ence of the quantum potential [2, 3] is a sign the fluid is barotropic. The Madelung’s nonlocal of the nonlocal interaction this summary gives a quantum pressure [32] has a deep relation to the perspective to find an analogue of the quantum osmotic pressure in vacuum, as follows from Nel- potential for the case of the classical fluid me- son’s [38]. Then the Navier-Stokes equa- chanical system. One may suppose from quan- tion can be reduced to the Schr¨odinger equa- tum mechanical point of view that behavior of tion if we assume that the pressure P and the an incompressible can be described by a Bohmian quantum potential [2] are compatible; Schr¨odinger-like equation. However we need to (c) in order to neutralize the viscosity we shall replace in this case the Planck constant by other shake a bath with the fluid vertically with some parameter [7, 49] exceeding the Planck constant frequency and amplitude, so that the Faraday at many orders. oscillations will stay subcritical [13]. Two equations, describing flow of an incom- (a) The irrotational and solenoidal compo- pressible fluid [29], are the NavierStokes equa- nents submit to the following equations tion (6) and the mass conservation equation; it is Eq. (3) equated to zero. The mass density 1 ρM we replace further by a probability density ρ ~v = ~v + ~v = S + ~v , S R m∇ R according to the following formula  (15)  M mN ( ~vR)=0, [ ~vR]= ~ω. ρ = = = mρ. (13) ∇· ∇ × M ∆V ∆V  Here the mass M is a product of an elementary Subscripts S and R hint to scalar and vector mass m by the of these masses, N, fill- (rotational) potentials underlying emergence of

ing the volume ∆V . Then the mass density ρM these two components of the velocity. The scalar is defined as a product of the elementary mass field is represented by the scalar function S - ac- m by the density of quasi-particles ρ = N/∆V . tion in . Both velocities are The quasi-particle is a droplet-like inhomogene- to each other. Now we may de- ity, which moves with the local stream veloc- fine the momentum and the of the ity of the equivalent fluid. We assume that quasi-particle each quasi-particle moves like a Brownian par- ticle with the diffusion coefficient inversely pro- ~p = m~v = S + m~v , (16) portional to m and subjected to the kinematic R 2 ∇ 2 v 1 2 vR viscosity ν = µ/ρM . So that, ρ is the probability m = ( S) + m . (17) density of finding the quasi-particle within the 2 2m ∇ 2 volume ∆V and obeys the conservation law ∂ ρ Now we may rewrite Navier-Stokes equation (6) +(~v )ρ =0. (14) in the following form ∂ t ·∇

∂ 1 1 ( S+m~v )+ ( S)2 + m2v 2 + ~ω ( S + m~v ) = P U +ν 2( S+m~v ). (18) ∂ t ∇ R 2m ∇ R × ∇ R −ρ∇ −∇ ∇ ∇ R   h i (a) | {z } 7

Note that the term embraced by the curly thermalized fluctuating force field [22, 23]. Here bracket (a) is the term m(~v )~v in Eq. (6). the fluctuating force is expressed via the gradient (b) The quantum-like behavior·∇ of droplets of the pressure divided by the density distribu- shown in [8] hints that the Navier-Stokes equa- tion of particles chaotically moving. tion under some modes can be reduced to the The term Q in Eq. (22) fits for further trans- Schr¨odinger equation [7, 49]. Consider in this formation of∇ the Navier-Stokes equation to the connection the first term from the right side of Schr¨odinger-like equation. But the second term Eq. (18) in detail. More definitely, consider the from the right side of this equation is superflu- pressure P which we shall represent consisting ous. For that reason we need to admit that the of two parts, P1 and P2. We begin from the fluid should be incompressible, ρ = 0. Fick′s law [22, 23]. The law says that the dif- The second term from right side∇ of Eq. (18) is fusion flux, J~, is proportional to the negative a gradient of the U, which rep- value of the density gradient J~ = D ρ, where resents negative value of the − ∇ D = ησ/(2m) is the diffusion coefficient [38]. acting to the quasi-particle. The parameter ησ is the parameter replacing (c) The last term from the right side of the Planck constant for the case of the fluid Eq. (18) describes kinetic losses in the fluid be- flows [49]. Since the term η J~ has of cause of the viscosity. To compensate decay of σ∇ the pressure, we define P1 as the pressure having the waves in the fluid it is proposed to use the diffusion nature gravitational force by shaking the fluid periodi- η η2 cally in vertical direction [8, 13]. The frequency P = σ J~ = σ 2ρ. (19) ω of the periodic shaking in this case should 1 2 ∇ −4m∇ F satisfy to the following condition Kinetic energy of the diffusion flux is 2 µ (m/2)(J/ρ) . It means, that there exists ησωF = νm( ~v)= ( ~v). (23) one more pressure as the average momentum ∇ ρ ∇ transfer per unit area per unit time Here µ = νmρ is the dynamical viscosity of the 2 2 2 fluid. The viscosity poses by itself as a resistance m J ησ ( ρ) P2 = ρ = ∇ . (20) to motion of the fluid. Observe that measure of 2 ρ ! 8m ρ the viscosity is determined by the force required

Observe that sum of the two , P1 +P2, to shear the fluid. The shaking of the container divided by ρ gives a term should be weak enough in order to initiate the Faraday waves slightly below the critical thresh- η2 2ρ ρ 2 Q = σ ∇ ∇ (21) old. Due to this trick, effect of the viscosity may −2m " 2ρ − 2ρ ! # be neutralized [13]. Further, for that reason we shall not take into account the term ν 2 S. having the same form as the quantum potential. ∇ ∇ E. Nelson has shown that this pressure has an Let us multiply Eq. (18) from the left by ~vS and take that (~v ~v ) = 0 and (~v ~ω) = 0. osmotic nature [38]. It can be interpreted as S · R S · follows: a semipermeable membrane where the Next, by integrating the equation over the vol- osmotic pressure manifests itself is an instant ume of the fluid we obtain the following modified which divides the past and the future (that is, by the quantum potential, Q, Hamilton-Jacobi the 3D brane of our being is the semipermeable equation membrane). Now we may rewrite the first term ∂ 1 m S + ( S)2 + v 2 + U from the right side in Eq. (18) as follows ∂ t 2m ∇ 2 R 2 P 1 P ρ η2 2ρ ρ ∇ = ρ = Q + ∇ Q. (22) σ ∇ ∇ = C. (24) ρ ρ∇ ρ ! ∇ ρ −2m " 2ρ − 2ρ ! # Gr¨ossing noticed that the term Q, the gradient In this equation C is an integration constant. of the quantum potential, describes∇ a completely Here we used Q instead of P/ρ as follows ∇ ∇ 8

(a) t = 0.02 (b) t = 0.14 by the Helmholtz theorem. By substituting into Eq. (25) the wave function Ψ represented in a 7.5 7.5 5 5 polar form 2.5 2.5 Ψ= √ρ exp iS/η (26) 0 0 · { σ} -2.5 -2.5 -5 -5 and separating on real and imaginary parts we -7.5 -7.5 obtain Eqs. (24) and (14).

-7.5 -5 -2.5 0 2.5 5 7.5 -7.5 -5 -2.5 0 2.5 5 7.5 A solution of the linear Schr¨odinger equa- (c) t = 0.42 (d) t = 0.98 tion with the potential simulating a barrier with

7.5 7.5 two slits discloses interference patterns from 5 5 them both forward and backward. Scattering 2.5 2.5 of a Gaussian soliton-like wave on the slits pro- 0 0 duces two waves - reflected and transmitted, -2.5 -2.5 Fig. 5. Both waves evolve against the back- -5 -5 ground of the subcritical Faraday waves. As for -7.5 -7.5 the Faraday waves, Eq. (25) under conditions -7.5 -5 -2.5 0 2.5 5 7.5 -7.5 -5 -2.5 0 2.5 5 7.5 stated in [34, 35] can be reduced to the Miles- FIG. 5: Scattering of a Gaussian soliton-like wavepacket on Henderson that describes their a barrier containing two slits [47, 49]. Time here is in seconds different modes. and scale of the area is given in centimeters. Velocity of the wave is about 150 mm/s. It corresponds to the phase velocity of the Faraday waves stated in [8]. A. The material dependent parameter ησ from Eq. (22), where Q is given in Eq. (21). We The vortex ball can be viewed as a droplet, see that Eq. (24) contains a term which repre- when moving it on a fluid surface. In this case sents energy of the vortex. Also we can see that instead of the viscosity an important parameter the vortex in Eq. (7) is replenished by the kinetic is the . The surface tension of a energy coming from the scalar field S, namely via liquid is the maximum restoring force that pro- the term (~ω )~v. Solutions of these two equa- vides spherical form to the droplet. Actually the tions, Eqs (7)·∇ and (24), representing the vortex best way to think of this physically is that it is and scalar fields, depend on each other. the energy cost of forming a liquid-air interface. Because of presence of the quantum potential The surface tension σ and the viscosity µ are in Eq. (24) this equation is called the quantum not directly related. The units of the surface Hamilton-Jacobi equation. Together with the tension are [N/m]. Whereas the units of the vis- cosity are [N s/m2]. However, Frenkel’s theory continuity equation (14) it can be extracted from · the following wave Schr¨odinger-like equation of sintering [20] takes into account relationship of these parameters through the following differ-

∂Ψ 1 2 ential dependence [43]: iησ = ( iησ + m~vR) Ψ+ UΨ CΨ. ∂ t 2m − ∇ − dr 3 σ (25) = . (27) The kinetic momentum operator ( iησ +m~vR) d t −4 µ contains the term m~v which describes− ∇ a con- R It describes change of the drop radius, r, with tribution of the rotation field conditioned by time. Extent of the sintering is defined by [17] the vortical motion. This term is analogous to the vector potential multiplied by the ratio of 3 σ r2 = r t. (28) the charge to the light which is repre- 2 µ sented in quantum electrodynamics [33]. Exis- tence of the vector rotation field is supported Here r2 is a contact area for a given time t. Let 9

vortex ball (a) -10 -8 -6 -4 -2 0 2 4 6 8 10 15 z, 30 E t = (cm)

10 20 subcritical Faraday oscillations 5 10 FIG. 6: Feynman-like representing the motion of (A) the vortex ball that rolls along the fluid surface. Exchange by 0 0 energy via a generated surface wave occurs at each bounce. -5 -4 -3 -2 -1 0 1 2 3 4 5 Here relation ∆E∆t = ησ simulates the well-known quantum- = 4.75 mm, b = 5 mm d = 6b x, (cm) mechanical relation ∆E∆t =h ¯. (b) -7 -5 -3 0 3 5 7 15 z, 20 us now substitute this result in Eq. (23) (cm) 10 15 3 σ 1 v v 10 ησωF = 2 r t( ~)= σS t( ~). (29) 2 r ρ ∇ 2 · ∇ 5 5 Here we took into account, that the volume of (A) the droplet is ∆V = (4/3)πr3 = 1/ρ and the 0 0 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 area S of the sphere is 4πr . Observe that t( ~v) = 6.95 mm, b = 5 mm d = 6b x, (cm) is a dimensionless parameter. It gives a contri-∇ bution only on stages of the periodic bouncing FIG. 7: Bohmian trajectories going out from two slits [47, 49] of the droplet [9]. The periodic shaking prevents are shown by black curves against the background of the prob- the process of the merging. Therefore we adopt ability density depicted by gray (light gray refers to low den- sity; dark gray refers to high density): (a) and (b) relate to that (t/2π)( ~v)S = ∆S is the contact area of ∇ the bouncing droplets having wavelengths λ = 4.75 mm and the droplet with the fluid through a thin air film. λ = 6.95 mm [8], respectively. Arrows (A) point to places As a result we get where Bohmian trajectories dramatically change direction be- cause of the existence of more than one slit.

2π T0 ησ = σ∆S = σ∆S . (30) 2ωF 2 Such a recoil of the fluid depends on preced- ing history of the droplet motion. It is the ef- Its value is about 10−11 J s [49] for the case of fect of the path memory stored in the Faraday the silicon oil [8]. Here T ·/2 is a half-period of 0 waves [13]. So, the droplet moves along an op- the Faraday oscillations. timal path, along the Bohmian trajectory, with the velocity B. The Bohmian trajectories and interference phenomena Ψ i ησ Ψ * − m ∇ + S

In the example of the vortex ball given in Fig. 3 ~v0 = Re = ∇ . (31) Ψ Ψ m the droplet will revolve on 120o after each bounc- h | i ing. As the droplet moves on the fluid surface, it Here i(ησ/m) is the velocity operator. induces Faraday waves on this surface, which, in It should− be∇ noted that the fluid medium turn, effect on motion of the droplet [9, 13]. Fol- should have a high susceptibility to any touching lowing to Feynman’s ideas [18, 19] the moving to its surface. It is achieved by generating the droplet induces in the fluid waves accompanying Faraday waves that are supported slightly be- the droplet, Fig. 6. This follows, in particular, low the supercritical threshold of the excitability. from the third Newton’s law - when an object The closer the threshold the higher the suscep- exerts force on the surface, the surface will push tibility. Due to such a super-high susceptibil- back that object with equal force in the opposite ity the Faraday waves are easily excitable long- direction. lived waves. Reflecting from the environment 10

boundaries they have a time for creating inter- 80 ference pattern around the droplet. The inter- eV ference guides the droplet along an optimal path 60 (p) that is the Bohmian trajectory, Fig. 7. It is in 40 good accordance with the de Broglie-Bohm the- ory [3, 10] where the wave function Ψ plays a 20 role of the pilot-wave guiding the particle along a most optimal trajectory. 0 0 2.1010 4.1010 6.1010 p⁄h-, m-1 IV. PHYSICAL VACUUM AS A SUPERFLUID FIG. 8: The dispersion relation ǫ vs. p. The dotted curve shows the non-relativistic square dispersion relation ǫ ∼ p2. The hump on the curve is a contribution of the roton compo- 10 − Let us define a material dependent parameter nent pRf(p − pR), pR/¯h ≈ 1.89 · 10 m 1, and σ = 0.5pR. Z h¯ = e2 0 α−1 1.0545 10−34 J s. (32) 4π ≈ × · mass of the pair, mp, is equal to doubled mass of 1/2 the electron, me. The charge of the pair is zero. Here Z0 = (µ0/ǫ0) is the wave impedance of The total spin of the pair is equal to 0. Whereas the free space (µ0 is the permeability of the free the , L, is nonzero. For the space, ǫ0 is its permittivity) and e is the electron charge. A dimensionless parameter α is the fun- first Bohr L =h ¯. The velocity of rotating about this orbit is L/(r m ) 2.192 106 m/s. damental physical constant (the fine-structure 1 · e ≈ · constant) characterizing the strength of the elec- So, it means that there is an elementary vortex. tromagnetic interaction. One can see that the Ensemble of such vortices forms a vortex line. parameterh ¯ is the reduced Planck constant. Now we may evaluate the dispersion relation By substituting the constanth ¯ into Eq. (25) in- between the frequency, ǫ(p) =hω ¯ , and wave number, p =hk ¯ , as it done in [30]. As follows stead of the parameter ησ we get the Schr¨odinger equation describing flow of a peculiar fluid - the from Eq. (25) we have: physical vacuum. The vacuum consists of pairs 1 2 of particle-antiparticles. The pair per se is the ǫ(p)= (p + pRf(p pR)) . (33) 2mp − Bose particle. These pairs stay at a temperature close to zero. It means, that the pairs make up Here pR = L/r1 = mpvR is the momentum of the Bose-condensate and, consequently, the vacuum rotation. The function f(p pR) is a formfactor − represents a superfluid medium [11, 50]. The su- relating to the electron-positron pairs rotating perfluid medium represents a ’fluidic’ nature of about the center of their masses. The formfactor space itself. Another name for such an ’ideal describes dispersion of the momentum p around fluid’ is the aether [33]. pR conditioned by fluctuations about the ground The physical vacuum (the aether), is a strongly state with the lowest energy. The formfactor is correlated system dominated by collective ef- similar to the Gaussian curve fects [44] with the viscosity equal to zero. Near- 2 (p pR) est analogue of such a medium is the superfluid f(p pR) = exp − . (34) − (− 2σ2 ) helium [53], which will serve us as an example for further consideration of this medium. The vac- Here σ is the variance of this dispersion. And uum is defined as a state with the lowest possible it is smaller or close to pR. The dispersion re- energy. We shall consider a simple vacuum con- lation (33) is shown in Fig. 8. The hump on sisting of electron-positron pairs. The pairs fluc- the curve is due to the contribution of the ro- tuate within the first Bohr orbit having energy tating electron-positron pair about the center of about (13.6 eV) 2 27 eV. Bohr radius of this masses. This rotating object is named the roton. · ≈ −11 orbit is r1 5.29 10 m. These fluctuations The rotons are ubiquitous in the vacuum space. occur about≈ the center· of their masses. The total Motion of the roton in a free space is described 11

by the following equation 2 B R ∂Ψ 1 2 ih¯ = ( ih¯ + mp~vR ) Ψ CΨ. (35) ∂ t 2mp − − Here CΨ determines an uncertain phase shift of the wave function, and most possible this phase relates to the chemical potential of a boson (the electron-positron pair) [30]. We did not take into account contribution of this term in the disper- sion diagram because of its smallness. As follows from the above computations Eq. (35) can be reduced to the Euler equation:

∂~vR P + [~ω ~vR]= ∇ , (36) A ∂ t × −mpρ describing a flow of the inviscid incompressible 1 fluid under the pressure field P . One can see FIG. 9: Rotation of the superfluid is not uniform but takes from here, that the Coriolis force appears as a place via a lattice of quantized vortices, whose cores (yellow) restoring force, forcing the displaced fluid par- are parallel to the axis of rotation [31]. Green arrows are the vorticity ω. Small black arrows indicate the circulation of ticles to move on circles. The Coriolis force the superfluid velocity uR around the cores. The vacuum is is the generating force of waves called inertial between two nonferromagnetic disks, A and B, fixed on center waves [30]. So the Euler equation admits a sta- shafts, 1 and 2, of electromotors [45]. Radiuses of the disks are R = 82.5 mm and distance between them can vary from tionary solution for uniform swirling flow under 1 to 3 mm and more. The vortex bundle rotates rigidly with the pressure gradient along z. the disk A. As soon as the vortex bundle reaches the top disk The twisted vortex states observed in super- B it begins rotation as well. fluid 3He-B [15] are closely related to the iner- tial waves in rotating classical fluids. The su- the ratio of the geometric mean of the velocities perfluid initially is at rest. The vortices are nu- v =¯h/(r m ) 2.192 106 m/s and V = RΩ cleated at a bottom disk platform rotating about R 1 e D to the · ≈ mean of· these velocities. Here axis z [31], see Fig. 9. The platform is in the nor- Ω is an angular rate of the disk A. So, we have mal state. The Coriolis forces take part in twist- ing of the vortices. The twisted vortices grow upward along the cylinder axis [31]. Samohvalov √vR VD VD 15 N = Nmax · = Nmax 6 10 . has shown through the experiment [45], that vR + VD s vR ≈ · the vortex bundle induced by rotating the bot- (37) tom nonferromagnetic disk A leads to rotation At the angular rate Ω = 160 1/s [45] the disk of the top nonferromagnetic motionless disk B, velocity VD = 13.2 m/s. Now we can evaluate Fig. 9. Both disks at room temperature have the kinetic energy of the vortex bundle induced been placed in the container with technical vac- by the rotating disk A. This kinetic energy is E = N m v 2 /2 0.026 J. It is sufficient for uum at 0.02 Torr. The utmost number of the · p R ≈ vortices that may be placed on the square of the transfer of the moment of force to the disk B. 2 2 18 Measured in the experiment [45] the torque is disk A is about Nmax = (2πR )/(2πr1)=2 10 , where R = 18.5 mm is the radius of the· disk about 0.01 N m. So, the disk B can be captured −11 · and r1 5.29 10 m is the radius of the by the twisted vortex bundle to be rotated. first Bohr≈ orbit.· Really, the number of vor- Light traveling along two paths through the tices situated on the square, N, is consider- space between the disks suffers a phase shift [33] ably smaller. It can be evaluated by multiply- as it is in the famous experiment of Aharonov & ing Nmax by a factor δ. This factor is equal to Bohm [1]. 12

V. CONCLUSION about a fluid-like quantum medium. This medium, called also the aether, has mass and is The Navier-Stokes equation contains terms de- populated by the particles of matter which exist scribing existence of vortex motions in the fluid in it and move through it [33, 41, 46, 51]. The media. Under some conditions the Navier-Stokes particle traveling through that medium perturbs equation can be reduced to the Schr¨odinger-like virtual particle-antiparticle pairs which, in turn, equation. This equation can contain terms de- create both constructive and destructive inter- scribing both the irrotational solutions and vor- ference ahead the particle. Thereby the virtual tex ones. The latter solutions are due to appear- pairs provide an optimal, Bohmian path for the ance of a term in the kinetic momentum opera- particle. It is interesting to notice that the vac- tor responsible for the vortex motion. This term uum undergoes fluctuations in the ground state looks as the vortex velocity multiplied by mass that have wave-like nature. These wave-like fluc- of the particle. The Helmholtz theorem gives a tuations are akin the subcritical Faraday oscilla- basis for such an inclusion of this term. The in- tions owing to which motion of the droplet along clusion is similar to that of the magnetic vector optimal, Bohmian path can be attainable. potential to the kinetic momentum operator [33]. Particle-antiparticle pairs composing vacuum Due to such an inclusion the Schr¨odinger equa- are on the lowest ground states and fluctuate tion admits emergence of solutions like vortex near the first Bohr orbit. This orbit has the balls moving along optimal paths called as the angular momentum L =h ¯. Sum of all angular Bohmian trajectories. Works of the French sci- momenta in the case of their alignment within a entific team [8, 9, 13, 42] throw light on dynam- container containing two disks (one is rotating) ics of such a motion. The vortex ball rolls, by is sufficient in order to initiate a torque in the bouncing from one plot to another, as it moves disk initially unmoved. Initiation of the torque along some path. Because of such bounces the is observed only in the vacuum, but not at ball induces weak long-lived waves on the fluid normal [45]. It occurs owing to self- surface. The waves form an interference pattern organization of the vortex lines into the vortex guiding the ball along the path due to creating bundles growing from the bottom rotating disk constructive and destructive interference ahead. up to the upper unmoved disk [15, 31]. This observation supports the de Broglie Bohms theory about the guiding wave function. Returning to the quantum realm we find that the Schr¨odinger equation describes mo- Acknowledgments tion of a special fluid - the physical vac- uum as a superfluid fluid presented by self- The author thanks prof. Robert Brady for organized Bose ensemble which consists of vir- valuable propositions. The author would like to tual particle-antiparticle pairs. It confirms in- thank Mike Cavedon for useful and valuable re- sights of Madelung [32] and Bohm with Vigier [4] marks and offers.

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