Stochastic quantum hydrodynamic model from the dark matter of vacuum fluctuations: The Langevin-Schrödinger equation and the large-scale classical limit
11 Stochastic quantum hydrodynamic model from the dark matter ofvacuum fluctuations: The Langevin-Schrödinger equation and thelarge-scale classical limit
Simone Chiarelli1 and Piero Chiarelli2,3 (1) Scuola Normale Superiore, Consoli del Mare, 1 , 56126, Pisa, ItalyEmail: [email protected]
Phone: +39-050-509686Fax: +39-050.563513 (2) National Council of Research of Italy, San Cataldo, Moruzzi 1, 56124, Pisa, Italy
Email: [email protected]: +39-050-315-2359Fax: +39-050-315-2166 (3) Interdepartmental Center “E.Piaggio”, Faculty of Engineering, University of Pisa, Diotisalvi 2, 56122,Pisa, Italy.
Abstract: The work derives the quantum evolution in a fluctuating vacuum by introducing therelated (dark) mass density noise into the Madelung quantum hydrodynamic model. The papershows that the classical dynamics can spontaneously emerge on the cosmological scale allowingthe realization of the classical system-environment super system. The work shows that the darkmatter-induced noise is not spatially white and owns a well defined correlation function with theintrinsic vacuum physical length given by the De Broglie one. The resulting model, in the case ofmicroscopic systems, whose dimension is much smaller than the De Broglie length, leads to theLangevin-Schrodinger equation whose friction coefficient is not constant. The derivation puts inevidence the range of application of the Langevin-Schrodinger equation and the approximationsinherent to its foundation.The work shows that the classical physics can be achieved in a description whose length scale ismuch bigger both than the De Broglie length and the quantum potential range of interaction. Themodel shows that the quantum-to-classical transition is not possible in linear systems, and definesthe long-distance characteristics as well as the range of interaction of the non-local quantumpotential in order to have a coarse-grained large-scale classical phase. The theory also shows thatthe process of measurement (by a large-scale classical observer) satisfies the minimum uncertaintyconditions if interactions and information do not travel faster than the light speed, reconciling thequantum entanglement with the relativistic macroscopic locality.
PACS : 03.65.Ud, 03.67.Mn, 03.65.Ta, 03.75.Gg
Keywords: stochastic quantum hydrodynamics, quantum decoherence induced by dark matter,quantum to classical transition, quantum dissipation, qbits, mesoscale dynamics, wave functioncollapse
1. Introduction
The conflict between the quantum mechanics and the classical ones attracts the interest of manyresearchers of the noways physics [1-3].This lack of knowledge has lead to many logical paradoxes that contrast with our sense of reality[1-3]. A quantitative tentative to investigate the problem was given by Bell [3] in response to the socalled EPR paradox [2] a critical analysis of the quantum non-locality respect to the notion of themacroscopic classical freedom and local relativistic causality .The Copenhagen interpretation of quantum mechanics [3–5] treats the wave function asrepresenting ‘the probability’ of finding a particle at some location’ [12]. However, such atreatment leads to the non-intuitive conclusion that the physical state is just a probability waveuntil observed. The absence of an analytical link with the pre-measure physical state fights againstthe common sense of reality and the existence of a real world independent by the observer and themeasure process [5].If the Copenhagen probabilistic connection with the pre-measure world is strictly assumed, theconclusion that the real state is not physically defined before the measure is unavoidable.Actually, the completeness and self-consistency of this logical result cannot be achieved since theprocess of the observation is out the Hamiltonian description of quantum mechanics. The need ofhaving a classical environment in order both to perform the measure and to define the quantumeigenstates, indeed, leads to a great theoretical loophole: Is the classical world necessary to thequantum mechanics or the quantum evolution is the fundamental law?Besides, the unavailability of the theoretical connection between the quantum and the classicalmechanics, that would explain how the laws of physics pass from the deterministic quantumbehavior to the classical one (even irreversible), leaves open many questions about how conceptsof the classical experience such as, measure, principle of causality, locality, physical state of theexternal reality, wave and particle behaviors, can be compatible or related to the quantummechanics.The connection between the quantum state and the statistical (classical) process of measure isdefined by a postulate that, is a matter of fact, makes the quantum mechanics a semi-empiricaltheory without a self-consistent theoretical framework.On the other hand, if the wave function is something physically real, then, there must exist adefined mechanism (e.g., the so-called wave function collapse, out of the canonical law of quantummechanics) expressing the interaction with the observer embedded into a classical universe.In this case, there also exists the problem about how the Schrödinger equation can be generalized[13, 14] or derived in the frame of such a more general quantum theory [15].In order to fill this theoretical lack, there exist various interpretations of quantum mechanics likethe many-worlds interpretation [17], the Bohmian mechanics [18, 19], the modal interpretation [20],the relational interpretation [21], the consistent histories [22], the transactional interpretation [23,24], the QBism [25], the Madelung quantum hydrodynamics [26-28] and the decoherence approach[29] .The Madelung approach (that is a particular case of the Bohmian mechanics [30]) owns theimportant peculiarities to be both mathematically equivalent to the Schrödinger one [31] and totreat the wave function evolution Si | | e in the classical-like representation as the motion ofthe mass density | | owing the impulse i i Sp q . In this way it introduces the concept oftrajectories of motion and naturally hosts the notion of physical reality before the measure.The Madelung description has the advantage to disembogues into the classical mechanics as soonas and the so-called quantum pseudo-potential are set to zero.Nevertheless, if we wipe out (by hand) the quantum potential from the quantum hydrodynamicequations in order to obtain the classical mechanics, we also cancel the stationary quantum eigenstates where the total force exerted by the Hamiltonian potential and the quantum one (on themass density distribution | | ) is null. Doing that, we change the nature of the equation ofmotion. Thence, a more correct and analytic mechanism is needed to pass from the quantum non-local description to the classical one in the frame of the hydrodynamic approach.Others characteristics of the quantum to classical transition are captured by the decoherenceapproach that investigates the possibility of obtaining the classical state through the lost ofquantum coherence generated by the presence of the environment. The decoherence is shown to beproduced into the system by treating it as a sub-part of the overall system, comprehending theenvironment whose interaction is semi-empirically defined by non unitary interaction [29].However, this approach is not able to explain how, by having a quantum overall system, theobserver can perform the irreversible processes of the statistical measure (and to be quantum de-coupled with the measured system). To overcome this problem, the relational quantum mechanicsintroduces the super-observer that is not entangled with the overall system [21]. Actually, this “adhoc” postulate, is unsatisfactory and brings logical contradictions.From the experimental and numerical simulation point of view, there exist the important evidencethat the decoherence and the localization of quantum states come from the interaction with thestochastic fluctuations of the environment [32-35] and/or dark matter.In this work the authors generalize the Madelung quantum hydrodynamic approach to itsstochastic version, where the noise, due to the quantum-mechanical properties of a fluctuatingvacuum (in term of curvature associated to dark matter), owns a non-white spectrum showing theemergence of the intrinsic De Broglie physical length into the vacuum.The work also shows that the stochastic Langevin-Schrodinger equation is derived from the theoryfor systems whose physical length is much smaller than the De Broglie length.In the final section the authors analyze how the classical mechanics can be achieved, underappropriate conditions, on a large scale description. The uncertainty principle in the measureprocess is investigated in the frame of the stochastic quantum hydrodynamic model (SQHM). Thepaper analyzes how the measure in a classical large-scale system can satisfy both the uncertaintyprinciple and the finite velocity of transmission of light and information.
2. The quantum hydrodynamic equation in presence of vacuum darkmass density fluctuations
In the present work we go beyond the flat static solution
81 02
R Rg c TG as forclassical matter vacuum, and assume that there is still energy and momentum within the space-time due to the possible presence of gravitational waves that can give a contribution T 'Gc . Solution to such equations has been introduced by de Sitter that illustrates thatmatter may not be the only source of gravity and thus wrinkles in the space-time may not be due tomatter only.By considering the vacuum as a fluctuating background, we define the stochastic generalization ofthe quantum-hydrodynamic equations [26-28,31] that for the wave function iS | | e are givenby the conservation equation for the mass density n | | ii n ( nq )t q . (2.1) and by the motion equations ( q,t )ii i Spq m m q , (2.2) qucli i ( H V )p q , (2.3)where H is the classical Hamiltonian of the system and where // qu i i nV m q qn , (2.4)The ripples of the vacuum curvature are assumed to manifest themselves by an additionalfluctuating mass density distribution (MDD) vac n tot vac n n n (2.5)where nvac lim n n , that, through the quantum potential // totqu( n ) tottot i i nV nm q q , (2.6)leads to the fluctuating force qu( n )toti V q . (2.7)Being the mass density vac n defined positive, the vacuum fluctuations (as a mean vac n )give rise to an additional mass that, owning just the gravitational interaction, is dark matter.For the purpose of this work, we assume that the vacuum dark matter (DM) does not interact with thephysical system (the gravity interaction is disregarded for its weak constant and it is not included in H ).As far as concerning the dark matter evolution, it is defined by additional (gravitational) motion equationdescending by the cosmological dynamics. Nevertheless, we disregard the DM cosmological evolutionand assume, for our laboratory macroscopic systems, that the dark mass vac n is locally uniformlydistributed with a constant amplitude of fluctuations ( q,t ) n such as ( q,t )vac vac n n n (2.8) In deriving the characteristics of the quantum potential fluctuations (and hence of its force), we use thecondition that the vacuum dark matter, described by the wave function vac , does not interact with thephysical system (this due to the weak gravity constant). In this case the wave function tot of the overall,system reads tot vac (2.1.1)leading to the overall quantum potential
22 22 2 vacqu vac( n )tot i ivac vacvac vaci i i i i i | || |V = | | | |m q q| | | || | | || | | | | | | |m q q q q q q .(2.1.2)Moreover, given the energy fluctuations qu tot( q,t ) qu( q,t )V E n V dq , (2.1.3)due to the vacuum dark mass density noise of wave-length vac( ) vac( ) n | | cos q (2.1.4)(associated to the dark matter wave-function fluctuation vac cos q ) (2.1.5)where
22 2 12 22 vac vacqu( q,t ) vac vaci i i iii | | | | | |V | | | | | |m q q q q| | | | cos q sin qm q| | | | tanm q q , (2.1.6)at small wave length ( / V ), reads
222 22 2 qu( ) tot( q,t )tot iV tot( q,t ) tot( q,t )tot iV V | |E n | | tan q dqn V m q | | n dq n | | tan q dqn V m q m (2.1.7)(D.12)where it has been used the normalization condition tot( q,t ) totV n dq n V and where, on largevolume, it has been used the approximation tot( q,t ) toti | |lim n | | tan q dq n Vq . (2.1.8)The result (2.1.7) shows that the energy, due to the mass density fluctuations, increases as theinverse squared of . Being so, the associated quantum potential fluctuations, on very short distance(i.e., ), can lead to unlimited large energy fluctuations even in the case of vanishing noiseamplitude (i.e., T ).In order to warrant the convergence of equations (2.2-3, 2.6) to the deterministic limit (2.2-4) of quantummechanics for T , this behavior imposes the need of a supplemental condition on the spatialcorrelation function of the noise (we name it G( ) ).The derivation of conditions on the noise correlation function shape G( ) , brings a quite heavystochastic calculations [36]. A more simple and straight derivation of G( ) can be obtained byconsidering the spectrum of the fluctuations.Since each component of spatial frequency k brings the quantum potential energycontribution (2.1.6), the probability of happening kTEp exp , (2.1.9)by (2.1.7) reads
22 2 qu c Ep exp kTmexp expkT (2.1.10)where c / ( mkT ) (2.1.11)is the De Broglie length.From (2.1.10) it comes out that the spatial frequency spectrum S( k ) c c kS( k ) p( ) exp exp (2.1.12)is not white and the components with wave-length smaller than c go quickly to zero.Thence, given the mass density noise correlation function, that for the sufficiently general case, to beof practical interest, can be assumed Gaussian with null correlation time, isotropic into the spaceand independent among different co-ordinates such as ( q ,t ) ( q ,t ) ( q ) ( q ) ( T ) n , n n , n G( ) ( ) , (2.1.13) the spatial shape G( ) reads
221 2 c( ) ( k )/c c G exp[ ik ] S dk exp[ ik ] exp k dk exp . (2.1.14)The expression (2.1.14) shows that uncorrelated MDD fluctuations on shorter and shorter distance areprogressively suppressed by the quantum potential allowing the deterministic quantum mechanics torealize itself for systems whose physical length is much smaller than the De Broglie one. The characteristics of the stochastic force noise induced by the fluctuations of the quantumpotential (due to the vacuum mass density fluctuations) can be derived by assume the quantumpotential as composed by a regular part qu( n ) V (to be defined) plus the zero mean fluctuatingpart st V such as // totqu qu st( n ) tottot nV n V Vm q q . (2.2.1)Moreover, given the force noise st( q,t ,T ) i Vm q , (2.2.2)it is possible to show (see appendix A) that it owns the correlation function ( q ,t ) ( q ,t ) ( q ) ( q ) ( T ) , , F( ) ( ) (2.2.3)with the condition lim ( q ) ( q ) ( T )T , (2.2.4)where the spatial shape F( ) is connected to G( ) of the dark matter and where T is thefluctuation amplitude parameter (of DM). Thence, the motion equation acquires the form i pq m , (2.2.5) ( q ) qu( n )toti ( q,t ,T )i (V V )p mq , (2.2.6) As shown in Appendix A, the correlation function of the quantum potential fluctuations, at thesmallest order in c , reads ( q ,t ) ( q ,t ) ( q ,t ) ( q ,t )c ( q ) ( q )c( q ) ( q ) ( T )c cc a, n , nm a n , n G( ) ( )m n , na exp ( )m (2.3.1)where a is the boson-boson s-wave scattering length for Lennard-Jones interacting particles (see(3.6) in section 3).By using the variance (2.3.1), for systems, whose physical length L is much smaller than the De Broglielength (i.e., c L ), it follows that
24 24 24 24 2 ( q ,t ) ( q ,t ) ( T ) ( q ,t ) ( q ,t )c( q ) ( q ) ( T )c cc c ( q ) ( q )cc , lim x , x n , na lim exp ( )m n , na ( ) D ( )m (2.3.2)where ( q ) ( q )cc n , naD = m (2.3.3.)where ( q ) ( q ) n , n l t m s Besides, about the regular regular part qu V , for microscopic systems, without loss ofgenerality, we can pose
22 2 2 1 21 2
12 2 // /totqu st qu st/tot ( n ) n nV
42 2 /c( q ) ( q ) ( T ) ( q ) ( q ) ( T ) mkT mkT D= n , n a a n , n (2.3.14) /( q ) ( q ) ( T )/ a n , n kTm , (2.3.15)That, by posing T ( q ) ( q ) ( T ) lim n , n kT n (with ) (2.3.16)leads to / /T mlim D kTa n (2.3.17) T / lim kTm n . (2.3.18)Moreover, by by assuming in the limit of small fluctuations amplitude T lim kT , , (2.3.19)it follows that / / mD kT n (2.3.20) / kTm n , (2.3.21)so that (2.3.11-12) are satisfied by . (2.3.22)For instance, for / and / it follows that / mD kT n (2.3.23) // kTm n ,. (2.3.24)
22 205 2 / D D kTm n ; (2.3.25)For and / , we obtain / mD kT n (2.3.26) // kTm n , (2.3.27) // D D kTm n (2.3.28)Furthermore, by posing pD c D m L (2.3.29)and pcD kT L (2.3.30)where D is the dimensionless constant
32 2 1 2 2 202 20 p / p/D p p m kT L n , (2.3.31)that for p reads
21 2 2 1 202 20 / /D m kT L n (2.3.32)and for / (2.3.23-4, 2.23.6-7)) gives
21 2 02 20 /D m L n . (2.3.33) The quantum-hydrodynamic equation (2.3.7) for the complex field /( q,t )q,t q,t q,t q,t | | exp n exp
S S (2.4.1)where S qm q (2.4.2)reads
22 2 1 2 1 2 1 2 1 21 2 / / / /( q ) ( t )/ d S S S S S S Sq m dt q t q m q q q q t m q q| |V S q q mDm q q| |m q (2.4.3)leading to the partial stochastic differential equation / / / /( q ) ( t ) ( t )/ S S S | |m V S q q mD Ct q q m q q| | (2.4.4)(C.2.4)
The presence of unnoticeable small dark matter fluctuations, even negligible on the ordinary scalesystems, is suffice to lead to a finite De Broglie length that is much smaller than the cosmologicalscale allowing the quantum decoherence and the emergence of the classical behavior into theuniverse.The possibility of dividing the universe in classical sub-parts, allows to correctly introduce theexistence of the environment. Besides, given that the action of force noise of the environment onthe MDD derivatives generates an increase of the energy of the quantum potential in the same wayas the dark matter, it follows that the spatial shape of the correlation function of the force noise ofthe environment owns the same form of that one of the dark matter.Thence, in presence of the physical environment it is possible to assume the stochastic interaction / / /ext ext ext ( t )st
V S q q D (2.4.1.1)from which it follows that / / /( q ) ext ( t ) ( t )st / / /( q ) ( t ) ( t ) V V S q q mD CV S q q mD C (2.4.1.2)where, by assuming both ext D D .and ext , it follows that ext ext D D D D (2.4.1.3)and ext ext , (2.4.1.4)and that all preceding formulas can be retained with the substitution ext
D D (2.4.1.5) ext (2.4.1.6) It must be noted that, in absence of dark matter fluctuations, the quantum-decoupling of theenvironment cannot be assumed and that extst V cannot be formulated in the form of (2.4.1.1) .Thence, (2.4.1.5-6) can be assumed only in a physical vacuum that, following the general relativity,is constituted by a fluctuating geometrical background. Once the dark matter makes possible to have the classical environment, a system of microscopicphysical length L (i.e., c L ) n obeys to the conservation equation (see (C.3.9, C.3.11) inappendix C) ( q,t )t ( q,t ) diss( q,t )ic ( q,t )t ( q,t ) diss( q,t )ic ( n q )lim n Qq( n q )lim n Qq L L (2.4.1.7)where (see ( C.3.11) in appendix C)
01 12 k ( k ) ,t ) h,t ) ........diss( q,t ) h ( k terms )
CDQ d p...p n! p ... p (q,p(q,p NN (2.4.1.8)and where it has been used the identity c lim q q L . Thence, by (2.4.1.7) it follows that diss( q,t ) Q| | | | S S| |t m q q m q q | | (2.4.1.9) leading to the generalized Langevin-Schrodinger equation (LSE) diss( q,t )/ / /( q ) ( t ) ( t ) Qi V S q q D i Ct m q q | | (2.4.1.10)(C.2.7)
Moreover, since for a quantum system that owns and hence D , able to stronglymaintain its quantum coherence (we name it “robust” quantum systems), it holds diss( q,t ) lim Q , (2.4.2.1)and, for
22 1 20
T < k (2.4.2.2)(e.g., T < k for / and / ), also that finite lim , (2.4.2.3)for sufficiently small temperature, the term diss( q,t ) Qi | | can be disregarded in (2.4.1.10) and the LSEreads / / /,T ( q ) ( t ) ( t ) lim i V S q q D Ct m q q .(2.4.2.4) (LSE)As already observed, the sensibility of the system to fluctuations is related to the Lyapunovexponents of its classical trajectories of motion. This aspect goes beyond the purpose of this workand is not analyzed here. It is suffice to say that for linear systems (non classically chaotic) and the (LSE) can be applied to them.Generally speaking, for the case of classically chaotic systems, the LSE (2.4.1.10) has to beconsidered.Moreover, it must be noted that the SLE description is made possible by the integrability of thevelocity field Sq m q that can be warranted in small scale (slightly perturbed) quantumsystem but it may fall in macroscopic large-scale system whose velocity field can be non-integrable. For slow kinetics with the characteristic time ch satisfying the condition / Dch m mkT n L , (2.5.1) equation (2.3.7) (for T ) reduces to p p/ /( q ) qu /D ( t )( t ) Dc c V Vq mkT q m L L . (2.5.2)that for p reads
16 2 2 ( q ) qu /D ( t )( t ) D/ ( q ) qu / ( t )
V V kTq qV Vm kT mq mkT m L L nn (2.5.3)For a quantum system with Kg m and m L , equation (2.5.2) can be applied tokinetics with characteristic time down to D Dch ( T ) s mL (2.5.4)that being /D m L n n (2.5.5)gives /ch m skT kT n n (2.5.6)It is worth mentioning that equation (2.5.3) leads to a simplified Smolukowski equation that is onlya function of the space variables [37].
3. Emerging of the classical behavior on coarse-grained large scale
Is matter of fact that, if the quantum potential is canceled by hand in the quantum hydrodynamicequations of motion (2.1-3), the classical mechanical equation of motion emerges [28]. Even if this istrue, this operation is not mathematically correct since it changes the characteristics of the QHAequations. Doing so, the stationary configurations (i.e., eigenstates) are wiped out because wecancel the balancing of the quantum potential force against the Hamiltonian force [37] thatgenerates the stationary condition. Thence, an even small quantum potential cannot be neglectedinto the deterministic QHA model.On the contrary, in the SQHM it is possible to correctly neglect the quantum potential (at least inclassically chaotic systems) when its force is much smaller than the noise such as. qu( n ) ( q,t ,T )i V| | | |m q . (3.1)When the non-local force generated by the quantum potential is quite small (respect to thefluctuations amplitude) so that / /qu( n ) D Di c V mkT| |m q m m
L L , (3.2)its effect can be disregarded in (2.3.7) .Besides, even if the noise ( q,t ,T ) has zero mean, the mean of the quantum potential fluctuations st( n,S ) V S is not zero, and the stochastic sequence of inputs of noise alters the coherentreconstruction of the quantum superposition of state by the dissipative force ( t ) q in (2-3-7).Moreover, by observing that /D ( t )c m L (3.3)grows with the scale of the system (i,e., c L for macroscopic systems), condition (3.2) issatisfied if qu( n )( q )q ic Vlim | | lim itedm q (3.4)and the classical behavior can emerge in systems of sufficiently large physical length. Actually, inorder to have a large-scale description, completely free from quantum effects, we can more strictlyrequire qu( n ) qu( n ) qu( n )( q ) ( q ) ( q )q i i ic V V Vlim | |m q m q q . (3.5)By observing that for linear systems q qu( q ) lim V q , (3.6)from (105) the SQHM shows that they do never have a macroscopic classical phase. Generallyspeaking, stronger the Hamiltonian potential higher the wave function localization and larger thequantum potential behavior at infinity [38]. Given the MDD k( q ) | | exp P (3.7)where k( q ) P is a polynomial of order k , in order to have a finite quantum potential range ofinteraction it must be k (it results k for uni-dimensional linear interaction). Actually,since the linear interaction is not maintained up to infinity (for energetic reason, a finite boundenergy requires a weaker than linear interaction such as q ( q ) lim V ), there exists a large-scale classical description when the physical length of the system is much larger than the range oflinear interaction. A physical example comes from solids owning a quantum lattice. If we look atthe intermolecular features where the interaction is linear, the behavior is quantum (such as the x-ray diffraction shows), but if we look at their macroscopic properties (e.g., low-frequency acousticwave propagation) the classical behavior is shown.For instance, systems that interact by the Lennard-Jones potential for which the long distance wavefunction reads [39] /r lim | | a r (3.8)that leads to the quantum potential r qu( n ) q | |lim V lim a | |m | | r r mr (3.9)and to the quantum force
22 2 2 2 3
11 122 2 qu( n )r q
V | | rlim lim rr m r | | r r m r r r m r , (3.10)the large scale classical behavior can appear [38, 40] in a sufficiently rarefied phase (see section 4.4).It is interesting to note that in (3.6) the quantum potential acquires the form of the hard spherepotential of the pseudo potential Hamiltonian model of the Gross-Pitaevskii equation [15, 41]where a is the boson-boson s-wave scattering length.By observing that, in order to fulfill the condition (3.5) a sufficient condition reads qu( n )( q ) ( r , , )i Vr | | dr lim ited ,m q , (3.11)it is possible to define the quantum potential range of interaction [38] qu( n )( q ) ( r , , )iqu c qu( n )( q ) ( r , , )ci Vr | | drqV| |q (3.12)that gives a measure of the physical length of the quantum non-local effects.The convergence of the integral (3.11) for r is warranted for L-J type potentials since, near theequilibrium point ( r ), the L-J interaction is linear and being
20 0 r qu( n )( q ) lim V r it followsthat qu( n )( q )r ( r , , )i Vlim r | | cons tan tq . (3.13) By discretizing the phase space conservation equation given by the current equation (2.2.5-6, 2.4,2.3.4, 2.3.8) for the system of N coupled particles [42], it is possible to obtain the quantumhydrodynamic master equation for a macroscopic system of a huge number of molecules.In order to obtain the macroscopic description, we may procede by discretization of the stochasticquantum hydrodynamic equations i n ( nq )t q i pq m , (20) ( q ) qu( n )i j ( q,t ,T )( t ) j V Vp m q mq , (D.5) (3.1.1)
For the purpose of this work, we can exemplify the to the simpler case kinetics with a characteristictime ch larger than ch kT , (20) (3.1.2) leading (for T ) to i n ( nq )t q ( q ) qu( n )j ( q,t ,T )( t ) j V Vq m q Given the conserved equation in the local j -th cell of side l with the current j( q,t ) j( t ) J nq thatreads ( q,t ) q ( q ) qu ( q,t ,T )( t ) q ( q ) qu ( q,t ) ( t ) J nq n V V n V V mL mL , (3.1.3)by posing j tot( q ,t )j x l n , (3.1.4)the discrete spatial generalization of the SDE (3.1.3)reads j jm m mk k jk k k k( t )qukm,k kj dx x V V dt x dW mLD' D D'' (3.1.5)where k ( q )k V V , ( n )qu ( q ) quk k V V , k ( q ,t )k and where the terms jk D , jk D' and jk D'' arematrices of coefficients that corresponding to the discrete approximation of the derivaties q andwhere l j k ( q ) ( q ) ( T ) ( l( k j ))j j lim l , , G . (3.1.6)Generally speaking, the form of the overall interaction (classicl plus quantum) ( q ) quk k V V steming by the k -th cell, depends by the physical system and its evolution.For instance, by assuming c qu l , L (where L is the available volume per molecule), for asystem of sufficiently rarefied phase of L-J interacting particles with a asymptotically vanishingquantum potential (3.9-10) , the quantum potential interaction between adjacent cells is null and,hence, the classical master equation is obtained.Here, generally speaking we observe that, given the range of interaction of the quantum potential qu , the De Broglie length c , and the system size L ( L the mean available volume permolecule), we can generally distinguish the cases:i. qu c , L ii. qu c , L iii. qu c > L > iv. c > L Typically, for the L-J potential the quantum potential range of interaction qu extends itself a littlebit further than the linear zone around the equilibrium position r (let’ say up to r ) .By using this approximation for the L-J interaction, so that for r r qu( n ) V rq , (3.1.7)and for r r [39] qu( n ) V q m r , (3.1.8) qu reads
11 3 r r cqu c c c drdr r r a r (3.1.9)that, since for T k and microscopic mass Kg m , mc r , we obtain qu r (3.1.10)Thence, the rarefied phases owing qu c r L , for particles interacting by a Lennard-Jones potential, is fully classic since the mean molecular distance L is much larger both than theDe Broglie length and the quantum potential length of interaction qu .The second case qu c r L refers to dense phases (e.g., fluid phase) that still own aclassical behavior since, as a mean, the particle are distant each-other more than the range ofinteraction of the quantum potential. The inter-particle distance mostly lies in the non-linear rangeof L-J interaction [38].The case “iii” qu c r L applies when the neighboring molecules lie in the linearrange of the intermolecular potential at a distance smaller than the non-local quantum potentialinteraction qu .The observables on such physical scale show quantum behavior (e.g., the Bragg’s diffraction of theatomic lattice).In the case “iv” c > L , when the condensed fluid phase (i.e., r L ) persists down to a verylow temperature so that the De Broglie length becomes larger than the mean intermoleculardistance ( T k for typical intermolecular distance of order of m ), the fluid shows anextreme decrease of molecular viscosity [38]. The super-fluidity is induced by the quantumpotential interaction between the molecules [40]By changing the temperature and, accordingly, both c and the mean inter-molecular distance L ,we can have quantum-to-classic phase transition in the case iii and iv, respectively: i. qu + L (with c qu < ) solid-fluid transition with melting of crystalline lattice(e.g., ice -water transition [38])ii. c L (with qu c < ) superfluid-fluid transition (e.g., He4 lambda point [38,40]) The SQHA model shows that the measure is not necessarily a decoherent process by itself: Thesensing part of the measuring apparatus (the pointer) and the measured system may have acanonical quantum interaction that, after the measurement when the measuring apparatus isbrought to the infinity (at a distance much beyond c ), ends. Then the reading and the treatment ofthe “pointer” state is done by the measurement apparatus: This process is practically a classicirreversible process (with a defined arrow of time) leading to the macroscopic output of themeasure.On the other hand, the decoherence is necessary for the measurement process in order to have,both before the initial time and after the final one, quantum-decoupling between the measurementapparatus and the system in order to collect a statistical ensemble of data from repeated measures. If for physical length much smaller than c any system approaches the quantum deterministicbehavior and behave as a wave so that its sub-parts are not independent each-other, it follows thatin order to perform the measurement (with independence between the measuring apparatus andthe measured system) it is necessary that they are far apart (at least) more than c and hence, forthe finite speed of propagation of interactions and information, the measure process must lastlonger than the time c / c ( mc kT ) . (3.3.1)For qu c > the measurement time can be even bigger than (3.3.1) but not less. Moreover, since higher the amplitude of the noise T lower the value of c and higher thefluctuations of the energy measurements ( T ) E , it follows that the minimum duration of themeasurement c c multiplied by the precision of the energy measurement ( T ) E has a lowerbond.Given the Gaussian property of the noise (2.3.2), we have that the mean value of the energyfluctuation is ( T ) E kT . Thence, for the non-relativistic case ( mc kT ) a particle of mass m owns an energy variance E / / /( T ) ( T ) E ( ( mc E ) ( mc ) ) ( mc E ) ( mc kT ) (3.3.2) from which it follows that / c ( mc kT ) hE t E c , (3.3.3)It is worth noting that the product E is constant since the growing of the energy variancewith the square root of T is exactly compensated the decrease of the minimum time ofmeasurementThe same result is achieved if we derive the experimental uncertainty between the position andmomentum of a particle of mass m in the quantum fluctuating hydrodynamic model.If we measure both the spatial position of a particle with a precision c L (so that we are ableto not perturb the quantum configuration of the measured system) and the variance p of themodulus of its relativistic momentum mc)pp( / due to the fluctuations that reads ( T ) / // /( T ) Ep ( ( mc ) ( mc ) ) ( ( mc ) m E ( mc ) )c ( m E ) ( mkT ) (3.3.4)we obtain the experimental uncertainty /c L p ( mkT ) h (3.3.5)If we measure the spatial position with a precision c L , we have to perturb the quantumstate. Due to the increase of the spatial confinement of the wave function (by increasing theenvironmental temperature or by an external potential), the increase of both the quantum potentialenergy and its fluctuations are generated so that the final particle momentum gets a variance p higher than (3.3.5).It is worth mentioning that the SQHM leads to the minimum measurements uncertainty as aconsequence of the relativistic postulate of finite speed of light and information.Even if the quantum deterministic behavior ( c ) in the low velocity limit ( c ) leads tothe undetermined inequalities c c (3.3.6) / c cE ( mc kT ) (3.3.7)their product E (3.3.8)remains defined and constitutes the minimum uncertainty of the quantum deterministic limit.Beside, (3.3.6) in the relativistic limit shows that the duration of the measurement process in thedeterministic limit becomes infinite. Being it endless, it is not possible to perform it in the canonicalquantum mechanical universe.Moreover, since non-locality is confined in domains of physical length smaller than c andinformation cannot be transferred faster than the light speed (otherwise also the uncertaintyprinciple will be violated) the local realism is obtained in the coarse-grained large scale physics andthe paradox of a “spooky action at a distance [43]” is limited on a distance of order of c or of qu . The above result holds for particles with rest mass different from zero, while for determining thelength of non local interaction (entanglement) of the photon ( c ?) the relativisticgeneralization of the SQHM is required. The theory that describes how the quantum entanglement is maintained up to a certain distanceand how it can be maximized, can lead to important improvements in the development ofmaterials for high-temperature superconductors and Q bits systems.Moreover, the theory owing a self-defined quantum correlation distance can be also veryimportant in defining different regimes of chemical kinetics in complex reactions and phasetransitions.Besides, the SQHM can furnish an analytical self-consistent theoretical model for mesoscalephenomena and quantum irreversibility.
4. Conclusions
The SQHM describes how the quantum dynamics realizes itself in a vacuum whose metricfluctuates. In this scenario the canonical quantum mechanics is the limiting description achieved ina flat static vacuum.Figuratively, in a 3-dimensional space time, where the space can be represented by the surface of asee with very small ripples (instead by a flat static plane) the non local interaction of quantummechanics breaks down on large scale and, in huge systems of weakly bounded particles, theclassical mechanics emerges.The SQHM, shows that in the physical fluctuating vacuum, the spatial spectrum of the noise is notwhite and it owns the De Broglie characteristic length. Due to this fact, the quantum entanglementis effective in systems whose physical length is much smaller than such a length. The model showsthat the non-local quantum interactions may extend themselves up to a finite distance in the case ofnon-linear weakly bonded systems.The SQHM shows that the Schroedinger equation can be derived by taking into account, at the firstorder of approximation, the effect of fluctuations on microscopic systems.The derivation of the LSE from the general SQHM allows to define its basic assumptions and itsrange of applicability limited to microscopic systems whose physical length is much smaller thanthe De Broglie one.The SQHM shows that the minimum uncertainty condition is satisfied during the process ofmeasurement in a fluctuating environment and that it can have a finite duration.The theory shows that the minimum uncertainty in the measurement process is satisfied if, andonly if, interactions and information do not travel faster than the speed of light, making compatiblethe relativistic postulate (at the base of the large scale locality) with the non-local quantuminteractions at the micro-scale.The SQHM makes compatible the hydrodynamic description of quantum mechanics with thedecoherence approach showing that the quantum potential is not able to maintain the quantumcoherence in presence of fluctuations, generating a frictional force leading to a relaxation process(decoherence). The theory shows that the superposition of states does not physically exist inmacroscopic systems made up of molecules and atoms interacting by long-range weak potentialssuch as the Lennard-Jones one. References
1. T. Young, Phil. Trans. R. Soc. Lond. 94, 1 (1804).2. R. P. Feynmann, The Feynman Lectures on Physics, Volume 3, (AddisonWesley, 1963).3. G. Auletta, Foundations and Interpretation of Quantum Mechanics, (World Scientific, 2001).4. G. Greenstein and A. G. Zajonc, The Quantum Challenge, (Jones and Bartlett Publishers, Boston,2005), 2nd ed.5. P. Shadbolt, J. C. F. Mathews, A. Laing and J. L. OBrien, Nature Physics 10, 278 (2014).6. C. Josson, Am. J. Phys, 42, 4 (1974).7. A. Zeilinger, R. Gahler, C.G. Shull, W. Treimer and W. Mampe, Rev. Mod. Phys. 60, 1067 (1988).8. O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).9. W. Sch¨ollkopf and J.P. Toennies, Science 266, 1345 (1994).10. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw and A. Zeilinger, Nature 401,680 (1999).11. O. Nairz, M. Arndt and A. Zeilinger, Am. J. Phys. 71, 319 (2003).12. M. Born, The statistical interpretation of quantum mechanics - Nobel Lecture, December 11,1954.13. G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D. 34, 470 (1986).14. G.C. Ghirardi, Collapse Theories, Edward N. Zalta (ed.), The Stanford Encyclopedia ofPhilosophy (Fall 2018 Edition).15. P.P.Pitaevskii,”Vortex lines in an Imperfect Bose Gas”, Soviet Physics JETP. 1961;13(2):451–454.16. H. Everette, Rev. Mod. Phys. 29, 454 (1957).17. L. Vaidman, Many-Worlds Interpretation of Quantum Mechanics, Edward N. Zalta (ed.), TheStanford Encyclopedia of Philosophy (Fall 2018 Edition).18. D. Bohm, Phys. Rev. 85, 166 (1952).19. S. Goldstein, Bohmian Mechanics, Edward N. Zalta (ed.), The Stanford Encyclopedia ofPhilosophy (Summer 2017 Edition).20. O. Lombardi and D. Dieks, Modal Interpretations of Quantum Mechanics, Edward N. Zalta(ed.), The Stanford Encyclopedia of Philosophy (Spring 2017 Edition).21. F. Laudisa and C. Rovelli, Relational Quantum Mechanics, Edward N. Zalta (ed.), The StanfordEncyclopedia of Philosophy (Summer 2013 Edition).22. R.B. Griffiths, Consistent Quantum Theory, Cambridge University Press (2003).23. J.G. Cramer, Phys. Rev. D 22, 362 (1980).24. J.G. Cramer, The Quantum Handshake: Entanglement, Non-locality and Transaction, SpringerVerlag (2016).25. H. C. von Baeyer, QBism: The Future of Quantum Physics, Cambridge, Harvard UniversityPress, (2016)26. Madelung, E.:. Z. Phys. 40, 322-6 (1926).27. Jánossy, L.: Zum hydrodynamischen Modell der Quantenmechanik. Z. Phys. 169, 79 (1962).28. Weiner, J.H.,
Statistical Mechanics of Elasticity (John Wiley & Sons, New York, 1983), p. 315-7.29. Lidar, D. A.; Chuang, I. L.; Whaley, K. B. (1998). "Decoherence-Free Subspaces for QuantumComputation". Physical Review Letters. (12): 2594–2597.30. Tsekov, R., Bohmian mechanics versus Madelung quantum hydrodynamics, arXiv:0904.0723v8[quantum-phys] (2011).31. I. Bialyniki-Birula, M., Cieplak, J., Kaminski, “Theory of Quanta”, Oxford University press, Ny,(1992) 87-111.32. Cerruti, N.R., Lakshminarayan, A., Lefebvre, T.H., Tomsovic, S.: Exploring phase spacelocalization of chaotic eigenstates via parametric variation. Phys. Rev. E 63, 016208 (2000).33. E. Calzetta and B. L. Hu, Quantum Fluctuations, Decoherence of the Mean Field, and StructureFormation in the Early Universe, Phys.Rev. D, , 6770-6788, (1995).
34. C., Wang, P., Bonifacio, R., Bingham, J., T., Mendonca, Detection of quantum decoherence dueto spacetime fluctuations, 37 th COSPAR Scientific Assembly. Held 13-20 July 2008, in Montréal,Canada., p.3390.35. F., C., Lombardo , P. I. Villar, Decoherence induced by zero-point fluctuations in quantumBrownian motion, Physics Letters A 336 (2005) 16–24.36. P., Chiarelli, Can fluctuating quantum states acquire the classical behavior on large scale? J.Adv. Phys. 2013; , 139-163.37. P., Chiarelli, Stability of quantum eigenstates and kinetics of wave function collapse in afluctuating vacuum , in progress.38. P., Chiarelli, Quantum to Classical Transition in the Stochastic Hydrodynamic Analogy: TheExplanation of the Lindemann Relation and the Analogies Between the Maximum of Density atHe Lambda Point and that One at Water-Ice Phase Transition, Physical Review & ResearchInternational, 3(4): 348-366, 2013.39. Bressanini D. “An accurate and compact wave function for the 4He dimer”, EPL. 2011;96.40. P., Chiarelli, The quantum potential: the missing interaction in the density maximum of He atthe lambda point?, Am. J. Phys. Chem.. (6) (2014) 122-131.41. Gross EP. Structure of a quantized vortex in boson systems. Il Nuovo Cimento,1961;20(3):454–457. doi:10.1007/BF02731494.42. C.W. Gardiner, Handbook of Stochastic Method , 2 nd Edition, Springer, (1985) ISBN 3-540-61634.9,pp. 331-41 .43. A. Einstein, B. Podolsky and N. Rosen, Can Quantum-Mechanical Description of PhysicalReality Be Considered Complete? Phys, Rev., 47, 777-80 (1935).44.
Weiner, J. H. and Forman, R.: Rate theory for solids. V. Quantum Brownian-motion model. Phys. Rev. B10, 325 (1974). Appendix A
In orderto derive the correlation function of the quantum potential fluctuations, we assume thatthe the dark matter density (MD) fluctuations ( q,t ) n own an amplitude that is very muchsmaller than the MD of the physical system n and hence it follows that //totqu / /tot // / // / n nnV m q q m q qn n nnn nnm q qn n n n n nnq q q n qnm nn n nn
22 2 1 2
12 22 1 2 / q qn nn ln nn nnm q q q q q qn (A.1) where T lim n n . (A.2) Therefore, / / /qu qu/ nm n n n ln nnV ,V , ,q q q q q q q qn n nn n ln n ln nn n n, ,q q n q q q q q qn nln n n nn n,q q n q q n nnn n,n q q n q q (A.3) that since the mean value n is not random, leads to
240 20 2 2 qu( q ) qu( q' q )c c mlim V ,Vn n n n' 'ln n ' ln n ln n nn n n nlim , ,q q q' q' q q n ' q ' qn n'n nn n,n q q n ' q ' qli
20 2 2 c n n n n' 'ln n ' ln n ln n nn n n nm , ,q q' q q' q q n ' q ' qn n'n nn n,n q q n ' q ' q (A.4) that at first order in reads
240 20 2 20 qu( q ) qu( q' q )c ( q') ( q')( q') ( q')c ( q')( q')c mlim V ,V n nln n ' ln n ' n ln n ' nlim , ,q q' q q' n n q q ' q ' q n nn' n ,q q ' q ' q n nln nlim q
11 21 242 2 2 2 2 ( q')( q') ( q')( q') ( q') ( q')( q') ( q') ' n' ln n ' ln n ' ln n nn, n n, nq' n n q q' q q' q q'n nln n ' n ' n, ,q q q' q' n n q q q' q' n n (A.5)Given that the terms with first derivatives q and ' q give terms proportional to q q' ,in the limit of they are null and thence it follows that
240 23 30 2 223 30 qu( q ) qu( q' q )c ( q')( q')c ( q')( q') ( q')( q')c mlim V ,V n ' nlim n, nq q'n n ' n, nn n q q q' q'n ' nlim n, nq q'n n ( q')( q') c n, nn n (A.6)and, given that c kT , for very low temperature, it follows that
240 6 2 3 2 3 20 ( q')T qu( q ) qu( q' q ) ( q')/ /( q')c nnlim lim V ,V n, nq q'm n n (A.7) As far as it concernes the quantm potetntial force fluctuations, the zero order term can be generallyassumed of the form
T qu( q ) qu( q' q ) ( q,q') ( q')c lim lim V , V f n, nq q' (A.8)
Since n n lim n n , for Lennard-Jones potential we have that I. r / r r / r r / r lim n lim n lim | | a r (A.9) where a is the boson-boson s-wave scattering length. (see (3.6) in section 3), and hence that II. ( r')/ /( r') nn a r r r' r' ar r'n n (A.10) for r r .Moreover, by assuming in the linear range of interaction for r r , the Gaussian localization
20 2 rn n exp r (A.11) where r r r , it follows that the diffusion coefficient owns a parabolic behavior
22 226 2 3 2 2 3 2 2 2 2 2 202 2 22 222 2 2 2 2 2 2 2 2 20 0 ( r')/ /( r') r r' r r' r r'n n expr n r' n n r r' r r'r r' r r' r r' r r'n r r' r r' n r r' (A.12) that tends to zero for r r' or .Moreover, given that r and that for r r L (i.e., ( ) G ) the ratio nn reaches the lowest value (since about all the mass is localized there), the wave function is poorlyperturbed by MDD fluctuations (and is well described by the deterministic quantum limit), we canssume (A.10) over all the space to obtain
420 020 024 2 04 04 24
T qu( q ) qu( q' q ) T ( q ) ( q')c c( q ) ( q ) Tc ccc lim lim V ,V a lim lim n , nmn , na lim lim exp ( )m n, na ( )m . (A.13) As far as it concerns the force correlation function, in this case we obtain ( q ,t ) ( q ,t ) T qu( q ) qu( q' q )cT qu( q ) qu( q' q )cT T cc cc c , lim lim V , Vq q'lim lim V ,Vq q'a n, nlim lim lim lim exp ( )m
24 2 c c a n, n ( ) D ( )m . (A.14) Appendix B
The irreversible force induced by fluctuations in small scale systems
In order to obtain the explicit expression of the term qu( n )tot V q (B.1)let’s start by equation (2.2.6 ) ( q ) qu( n )tot V Vq m q (B.2)that we can rearrange as
22 2 1 21 2 / tot( q ) qu /( n ) tot nnV V m q q n q qnq m q (B.3)where the term
22 1 21 2 / tot/ tot nnq q q n q qn (B.4)in (B.3) generates an additional acceleration respect to the deterministic case leading to a change ofthe velocity field q of the mass density. It is noteworthy that, in the deterministic case (B.4)becomes null and D tot lim n n (actually, in thre limit of small fluctuations (i.e., small sizesystems), n is close to the value of the deterministic limit of the eigenstates).Moreover, by observing that in the stationary states (i.e., q , the analogouses of the eigenstatesof the deterministic limit [37] (let’s name them quasi-eigenstates), the mean MDD tot n does notchanges with time and both tttot t tot ( q ) vac ( q )( q ) ( q, )tt n lim n d n n n constt (B.5) and tt( q ,t ) t( t ) tt q lim qdt , (B.6)approaching the stationary state (i.e., q ) it follows that / /tot tot/ /tot tot n nn nq q q n q q q q q q n q qn n ,(B.7)and thence, generally speaking, for small q , sufficiently close to the stationary quasi-eigenstates,(B.4) can be developed in the series approximation
22 1 2 0 11 2 / ntot n/ tot nn A A q .... A qq q q n q qn (B.8) where A is a stochastic noise whose mean A is defined by the stationary state condition / /tot tot/ /tot totn( q,t ) ( q,t ) n( q,t ) n nn nq q q n q q q q q q n q qn nA q .... q A .(B.9)Thence, at first order in q , close to the deterministic limit of quantum mechanics (i.e., c L ) ,leads to
22 2 1 2 1 2 11 2 / /tot ( t )/ tot nn D* A qm q q q n q qn , (B.10)to ( q ) qu / ( t ) V VAq q Dm m q (B.11)where // D*D m (B.12) The first order approximation (B.10) allows the Marcovian process to become self-consistent (independent by the dark matter evolution) reducing toand where q,t ) q,p,t ) n d p ( ( N is defined by the Smolukowski equation (C.3) in appendix C,of the Marcovian process (2.3.7).Moreover, for system with irrotational velocity field (that admits the action function S ) suchas i S qq , equation (2.3.7) can read ( q ) qu( n ) /i ( t )i j V VSp A mDq q (B.13) that by posing ( q,t ) S SAq q , (B.14)leads to ( q ) qu / ( t ) V V Sq Dm q , (B.15)Besides, by comparing (B.15) with (2.3.4), it follows that // totst / /tot nnV Sm q q q qn n ( B.16)and that / / /st ( t ) V q q D (B.17)Finally it interesting to note that, for m const the quantum hydrodynamic equation ofmotion leads to the quantum Brownian particle given by [44] ( q ) qu / ( t ) V Vq q Dm q . (B.18)is recovered.Unfortunately, the validity of (B.18) is not general since is not constant.This agree with the results given in ref. [45-46] that show that only in the case of linear harmonicoscillator, in contact with a classical heat bath, the friction can be a constant. Besides, since inorder to have the quantum decoupling with the environment (i.e., a classical super-system), thenon-linear interaction is needed (see identity (3.5-6) of section 3), actually, the case constant isnever rigorously possible except for the deterministic limit of the canonical quantum mechanicswith .It can only approximaltely accepted for locally linear oscillators (non-linearly coupled to theenvironment) for which we can assume . Appendix C
The environmental Marcovian noise in presence of the quatum potential
Since tot n is postulated by the approximation (2.8) the determination of as well as of all themodel is not complete.Nevertheless, o nce infinitesimal dark matter fluctuations have broken the quantumcoherence on the cosmological scale (i.e., c m that for barionic particles withmass
30 27 Kg m , it is enough
410 10
T K Kmk ) and the resultingclassical universe can be divided in subparts (the Newtonian limit of gravity issufficiently weak force for satisfying condition (3.5)), we can define the super-systemmade up of the system and the environment.At this stage, we can disergard the vacuum fluctuations associated to the dark matter (i.e., n ) and consider the Markovian process (2.3.7) i pq m , (C.1) ( q ) qu( n ) /i ( t )j V Vp m Dq , (C.2) In presence of the quantum potential the evolution of the MDD ( q,t ) n tot n lim n dueto the stochastic motion equation (2.3.7)) depends by the exact sequence of the forceinputs of the Marcovian noise.On the other hand,, the probabilistic mass density (PMD) q,p,t ) ( N of the Smoluchowskiequation P( q, p,q , p | t' t ,t ) P( q, p,q', p' | ,t')P( q', p',q , p | t' t ,t )d q' d p' (C.3)for the Marcovian process (2.3.7) (where the PTF P x, z | ,t ) ( represents the probabilitythat an amount of the PMD) q,p,t ) ( N at time t , in a temporal interval , in apoint z ( q , p ) , transfers itself to the point x ( q, p ) [47]) is somehow indefinitesince the quantum potential depends by the exact sequence of the inputs of the forcenoise.Even if the connection between ( q,t ) n and n ,t ) (q cannot be generally warranted, theapproximation (B.10) that reads / / ( t )/ n n mD A qm q q q n q qn , (C.4)introduces the linkage between ( q,t ) n and q,t ) q,p,t ) n d p ( ( N (C.5)leading the motion equation ( q ) qu( n ) /i j ( t )( t ) j V Vp m q m Dq (C.6)It is worth mentioning that the appplicability of (C.6) is not general but it is stronglysubjected to the condition of being applied to small scale systems with c L that admitstationary states ( q ) whose MDD is sufficently close to that of the deterministiceigenstates (i.e., small force noise amplitude) for which it is possible to assume that thecollection of all MDD ( q,p,t ) n configurations will reproduce the PMD n ,t ) (q such as tt ( q )( q ) t tt n lim nd nt (C. 7)This assumption is at the basis of the relation (C,4) that expresses the connection betwenthe PMD n and the MDD n (i.e., the information about n can be obtained by knowing n and q ).Besides, if the system is sufficiently close to the deteministic limit of the quantummechanics (for which c L (i.e., very small force noise amplitude) it is a sufficient condition) and it owns (irrotational [31]) stationary states (i.e., quasi-eigenstates) so that itis still quantum and. the action function S (as integral of the momentum field) exists, wehave that Sq m q and n can be defined by knowing n and S . C.1. The conservation equation of the Smolukowski equation inpresence of the quantum potential
By using the method due to Pontryagin [47] the Smolukowski equation leads to thedifferential conservation equation for the PTF
P q, z | ,t ) ( x,z|t , ) x,z|t , ) ii P Pt x ( ( V , (C.1.1)where the current i x,z|t , ) i J P ( V is given by the series of cumulants
000 0 2 2 k ( k ) x,z|t , )im x,z|t , ) im........lx,z|t , ) i x,z|t , ) ii hm m l( k terms )
C PD PP P x ...x n! x ... x ((( ( V (C.1.2)where ( k ) hi i m m l l ,x| ,t )im........l ( k terms ) C lim y x y x ... y x P d y (y (C.1.3)and where ( q,p,t ) ( q ) quj qq V Vx p q , (C.1.4)being /( q ) ( q ) ( T )( q ) qu ( t )j c qq ,V Vdx d dtp m dWq . (C.1.5)Moreover, for one particle problem or many decoupled particle system (e.g., linearoscillators)) it is possile the diagonal description ( q ) ( q ) ( T )im p c ,p| ,t ) ,D D mlim p p p p P d p (p (C.1.6) ( k ) ( k )im....l i ...k indexes k k ,p| ,t )( k terms ) C Clim p p p p ... p p P d p ....d p (p (C.1.7) C.2.1 The non-Gaussian PTF generated by the quantum potential q,z|t , ) q,z|t , ) ii P Pt q ( ( V , (25) (C.2.1.1) where the current i x,z|t , ) i J P ( V is given by the series of cumulants [47]
000 0 2 1 n ( n ) q,z|t , )im ........mim q,z|t , ) nq,z|t , ) i q,z|t , ) i nm m mn
C PD PP P q ...q n! q .... q ((( ( V (C.2.1.2)
00 00 0 02 1 ( q ) qu( P q,q |t ,t )) im q,z|t , )q,z|t , ) i q,z|t , ) i mn ( n ) q,z|t , )im ........mnn m mn
V V D PP P q qC P.. n! q .... q ( (( ( ( V (C.2.1.3) where
30 1 1 ( n ) hi i m m m m ,q| ,t )n nim ........mn C lim y q y q ... y q P d y (y (C.2.1.4)that owns an infinite number of terms due to the presence of the quantum potential.If on one hand, the continuity of the Hamiltonian potential warrants that velocities i i y q arefinite and ( n )im ........mn C on very short time increment, on the other hand, since the quantum potentialdepends by the derivatives of ( q,t ) n , it can lead to very high values of force also in the limit of veryshort time increment so that very far away points i i y q can contribute to the probability transitionfunction P y q | ,t ) ( , and the cumulants higher than two cannot be disregarded in (C.2.1.3).Thence, being the cumulants higher than two non-vanishing, the PTF P q, z | ,t ) ( is not Gaussianand. and equation (C.1.1) does not reduce to the FPE. C.3. The motion equation for the spatial densities By integrating over the momenta, the conservation equation ii ( ) n ( n ),t ) ,t )im im........lm m ln,t ) ,t ) ii i( ) n,t )im im.......m ,t ),t )t x C C...x n! x .... xxt x xC C...x n!pqt q p (q,p (q,p(q,p (q,p (q,p N(q, p VN(q, p N NN N NNNN ( n ) ,t ).lm ln n termsi x .... xx (q,p N (C.3.1)
00 20 0 2 k ( k ) x,z|t , )im x,z|t , ) im........li i hx,z|t , ) m x,z|t , ) m l( k terms )
C PD Px ...P x n! P x ... x ((( ( QV P (C.3.2) i ( q ) qu( t ) qqx V Vp mq q (C.3.3) i ( q ) qu /( t ) p ( t ) p / mqx V Vp mq Dq (C.3.4)where h( q,t ) n ,t )d p N(q, p , (C.3.5) /qu / nV m q qn (C.3.6)we obtain h h h( q,t )( ) n ( n ),t ) ,t )im im........lm m ln n terms hi d p q d p p d pt q pC C...x n! x .... x d px (q,p (q,p N N NN N (C.3.7) that with the condition p ,t ) lim (q,p N and by posing hh q d pq d p NN (C.3.8) leads to ( ) n ( n ),t ) ,t )im im........lm m ln n terms hi C C...x n! x .... xn qn d pt q x (q,p (q,p N N (C.3.9)where ( )im C p D and where ( ) n ( n ),t ) ,t )im im........lm m ln n terms hi k ( k ) ,t ),t ) ........h ( k terms ) C C...x n! x .... x d px CD ...p n! p ... pp (q,p (q,p (q,p(q,p N N NN h d p (C.3.10)
01 12 k ( k ) ,t ) h,t ) ........diss( q,t ) h ( k terms )
CDQ d p...p n! p ... p (q,p(q,p NN (C.3.11)gives the compressibility of the mass density distribution that is linked to the generation of entropyand quantum dissipation. Thence, equation (C.3.9) can read diss( q,t ) n n q n q Qt q q