Modular Schr"{o}dinger equation and dynamical duality

Mo dular Sc hr¨ odinger equation and dynamic al dualit y Piot r Garbaczew ski ∗ Institut e of Ph ysics, Universit y of Op ole, 45-0 52 Op ole, Poland August 2 2, 2021 Abstract W e discuss quite surprising prop ertie s of the one- p arameter family of mo dular (Aub erson and Sabatier (1994)) nonlinear Sc hr ¨ odinger equations. W e develo p a unified theoretical fr amew ork for this family . Sp ecial atten tion is paid to the emergen t dual ti me ev olution scenarios which, alb eit run ning in the r e al time parameter of the p ertinen t n onlinea r equation, in eac h considered case, ma y b e mapp ed among eac h other by means of a suitable analytic con tin u at ion in time pro cedure. Th is dynamical dualit y is c haracteristic for non-d issipativ e quan tum motions and their dissip at ive (diffusion-t yp e pro cesses) p artners, and naturally extends to classical motions in confin ing and scattering p oten tials. P A CS num bers: 02.5 0.Ey , 05.20 .-y , 05.40.Jc 1 Motiv ation An inspiration f or the presen t pap er comes from the recen t publication [1] discussing effects of v arious scale tra ns fo r ma t ions upo n the free Sc hr¨ odinger picture dynamics. In particular, it has b een noticed that an appropriate definition of the scale cov ariance induces Hamiltonia ns whic h mix, with a p ertinen t scale exp onen t as a h yp erb olic ro- tation angle, a n orig ina l free quan tum dynamics with its free dissipativ e coun terpart (effectiv ely , a suitable v ersion o f the free Br ownian motion), [1] c.f. also [2]. The t wo disparately differen t time ev olution pa t t erns do run with resp ect to t he same real time lab el. Ho w ev er, the ultimate ”mixing” effect of the a bov e men tioned scale transformations ta k es t he form o f the t he Lorentz-lik e transformation Eq. (35) where one Hamiltonian tak es the role of a regular ” t ime ” while another of the ”space” dimension lab els. This ob vious affinit y with the E uclidean ”space-time” notion and the in v olv ed complex ana lysis metho ds (e. g. Wic k rotation, imaginary time transforma- tion, analytic con tin uation in time) sets b oth conceptual and p ossibly phenomenological obstacles/prospects p ertaining to an existence of suc h dynamical patterns in nature. ∗ Electronic address : pg ar@uni.op ole.pl 1 Most puzzling for us is the apparently dual notion of the time lab el whic h while a priori referring to the r eal time ev olution, may as we ll b e interpreted as a Euclidean (imaginary time) ev olution. It is our aim to address an issue in its full generality , b y resorting to a o ne-parameter family o f mo dular Schr¨ odinger equations, where external conserv ativ e p oten tials admitted. The adopted p erspectiv e relies on standard approa c hes to nonlinear dynamical sys- tems where a sensitiv e dep endenc e on a control ( here, coupling strength) parameter ma y arise, p ossibly inducing global changes of prop erties of solutions to the equations of motion. That is exactly t he case in the presen t analysis. W e wish to demonstrate that a dualit y prop ert y (r ealized by an imaginary time transformation) do es relate solutions of the mo dular nonlinear Sc hr¨ odinger equation for v arious coupling constan t regimes. In Section 2 w e set a general Lag rangian and Hamiltonia n framew ork for the sub- sequen t discussion and indicate that effectiv ely the dynamics can b e reduced to three sp ecifi c coupling v alue c hoices, [3 , 4], eac h of them b eing sep ar ately discussed in the lit- erature. W e aim at a stationary a ction principle form ulation of the mo dular Sc hr¨ odinger dynamics and the relat ed hydrodynamics (the sp ecial cases are: the fa miliar Bohm- t yp e quantum h ydro dynamics [5] and the h ydro dynamical picture o f diffusion-t yp e pro cesse s [2]). Basic principles of the a ction principle w orkings a re patterned af ter the classical h ydro dynamics treatises, [7, 6 ]. In Section 3, in Hamiltonia n dynamics terms, w e g ive a new deriv at io n and further generalize the origina l arg umen ts of [1]. W e emplo y scaling prop erties of the Shannon en trop y of a contin uous probability densit y ρ . = ψ ∗ ψ . The scale co v ar ia n t patterns of ev olution, non- diss ipative with an admixture of a dissipativ e comp onen t, are estab- lished in external p oten tial fields. Section 4 is dev oted to a detailed analysis of the emergen t time duality not ion for the three sp ecial v alues 0 , 1 , 2 of the coupling parameter. There , w e pay more atten- tion to a sp ecific intert wine b et wee n confining and scattering dynamical systems. A prop erly addressed sign issue for the p oten tial function app ears to b e vital fo r a math- ematical consistency of the f o rmalism (links with the theory of dynamical semigroups that underlies the dissipativ e dynamics scenarios). As a b ypro duct of a discussion we establish the Ly apuno v f unctionals fo r the considered dynamical patterns of b eha vior and the (Shannnon) entrop y pro duction time rate is singled out. Section 5 addresses mo r e sp ecific problems t hat allow to gra sp the dualit y concept from a p ersp ectiv e o f the theory of classical conserv ative systems and diffusion-t yp e (sp ecific ally - Smoluc hows ki) sto c hastic pro cesse s. A n um b er of illustrative examples is w ork ed out in detail, with an emphasis on the time duality notio n in the classically inspired (standard Hamilton-Jacobi equations) and dissipativ e patterns o f ev olution 2 (mo dified Hamilton-Jacobi: e.g. Smoluc how ski diffusion pro cess es vs quan tum dynam- ics). 2 Mo dular S c hr¨ odinger equatio ns Let us consider a sub class o f so-called mo dular Sc hr¨ odinger equations [3, 4] whic h are lo cal a nd homogeneous nonline ar generalizations of the standard Sc hr¨ odinger equation: i ~ ∂ t ψ =  − ~ 2 2 m ∆ + V  ψ +  κ ~ 2 2 m ∆ | ψ | | ψ |  ψ , (1) where ψ ( x, t ) is a complex function, | ψ | . = ( ψ ∗ ψ ) 1 / 2 , V ( x ) is a real function a nd a coupling parameter κ is no n- negativ e, κ ≥ 0 . If κ > 0, the p ertinen t nonlinear dynamics is kno wn t o pr eserv e the L 2 ( R n ) norm of an y initially giv en ψ , but not the Hilb ert space scalar pro duct ( ψ , φ ) of t w o differen t, initially g iv en ψ and φ . The induced dynamics is non-unitary in L 2 ( R n ); unitarity is restored if κ = 0. The p ertinen t set of solutions is a subset of L 2 ( R n ), but not a linear subspace (no sup erp osition principle for κ 6 = 0). F or all κ ≥ 0 a usual contin uity equation holds true with the familiar quan tum mec hanical definition of a (probabilit y) densit y current: ∂ t ρ = −∇ · j , where ρ = ψ ∗ ψ and j = ( ~ / 2 mi )( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ). W e consider normalized solutions only , whic h sets a standard form of j . = ρ · v , where v = ( ~ / 2 mi )[( ∇ ψ/ ψ ) − ( ∇ ψ ∗ /ψ ∗ )] . = (1 /m ) ∇ s is regarded as a gradient velocity field and ρ ( x, t ) = | ψ | 2 ( x, t ) is a probabilit y densit y on R n . 2.1 Lagrangian formalism The one-pa r ameter fa mily of mo dular equations (1), together with its complex con- jugate v ersion, deriv es from the lo cal Lagra ngian densit y L by means of the sta- tionary action principle [8]-[5]. Let us consider a functional of ψ -functions, their space and time deriv ativ es, including complex conjuga tes : I [ ψ , ψ ∗ ] = R t 2 t 1 L ( t ) dt where L ( t ) = R L ( x, t ) dx , (w e leav e unsp ecifie d, p ossibly infinite, inte gra tion volume ). W e imp ose the stationary a ctio n condition δ I [ ψ , ψ ∗ ] = 0 for indep enden t v ariat io ns δ ψ , δ ψ ∗ whic h are b ound to v anish at inte gra tion v olume b oundaries. By in v oking elemen ts o f t he p edestrian functional calculus [5], like e. g. δ L /δ ψ ≡ ∂ L /∂ ψ − P i ∇ i [ ∂ L /∂ ( ∇ i ψ )], o ne ends up with the Euler-Lag range equations: ∂ t [ ∂ L /∂ ( ∂ t ψ ∗ )] = δ L /δ ψ ∗ and ∂ t [ ∂ L /∂ ( ∂ t ψ )] = δ L /δ ψ . If w e prop erly sp ecify the Lag rangian densit y L . = L κ : L κ ( x, t ) = i ~ 2 [ ψ ∗ ( ∂ t ψ ) − ψ ( ∂ t ψ ∗ )] − ~ 2 2 m ∇ ψ · ∇ ψ ∗ − V ( x ) ψ ψ ∗ + (2) 3 κ ~ 2 8 m  ∇ ψ ∗ ψ ∗ + ∇ ψ ψ  2 ψ ψ ∗ . the stationary action principle yields a pair of adjoin t mo dular equations whic h com- prise Eq. (1) in conjunction with its complex conjugat e: − i ~ ∂ t ψ ∗ =  − ~ 2 2 m ∆ + V  ψ ∗ +  κ ~ 2 2 m ∆ | ψ | | ψ |  ψ ∗ . (3) W e ha ve previously introduced a curren t ve lo cit y field v ( x, t ) b y means of j . = ρ · v . Our g r a dien t assumption v = (1 /m ) ∇ s follows f rom an implicit reference to t he f amiliar Madelung substitution: ψ = | ψ | exp( is/ ~ ), where | ψ | 2 = ρ . Its consequen t exploitation allo ws us to rewrite the Lagrangian densit y (2) as follo ws (w e lea ve intact an orig ina l order of resp ectiv e en tries): L κ ( x, t ) = − ρ h ∂ t s + m 2 ( u 2 + v 2 ) + V ( x ) − κ m 2 u 2 i (4) where u ( x, t ) . = ( ~ / 2 m ) ∇ ρ/ρ is another ve lo cit y field (named an osmotic velocity ). Here, δ I [ ρ, s ] = 0 giv es rise to a con tin uity equation ∂ t ρ = −∇ ( ρ · v ) and yields: ∂ t s + 1 2 m ( ∇ s ) 2 + V + (1 − κ ) Q = 0 , (5) where, in view o f | ψ | = ρ 1 / 2 , the fa miliar notion of the de Broglie-Bohm quan tum p oten tial, [5, 11], na t urally app ears: Q = Q ( x, t ) . = − ~ 2 2 m ∆ ρ 1 / 2 ρ 1 / 2 = − ~ 2 4 m " ∆ ρ ρ − 1 2  ∇ ρ ρ  2 # . (6) The mo dular Schr¨ odinger equation (1) t a k es the form: i ~ ∂ t ψ = [ − ( ~ 2 / 2 m )∆ + V ] ψ − κQ ψ . (7) 2.2 Hamiltonian formalism A symplectic structure can b e asso ciated with the dynamical system (1)- (3) b y in- tro ducing fields π ψ and π ψ ∗ that a re conjugate to ψ and ψ ∗ resp ec tively : π ψ = ∂ L /∂ ( ∂ t ψ ) = ( i ~ / 2) ψ ∗ and π ψ ∗ = ∂ L /∂ ( ∂ t ψ ∗ ) = − ( i ~ / 2) ψ . The subsequen t L egendre- t yp e t r ansformation defines the Hamiltonian densit y: H κ = π ψ · ∂ t ψ + π ψ ∗ · ∂ t ψ ∗ − L κ = ~ 2 2 m ∇ ψ · ∇ ψ ∗ + " V − κ ~ 2 8 m  ∇ ψ ∗ ψ ∗ + ∇ ψ ψ  2 # ψ ψ ∗ = (8) ρ h m 2 v 2 + V + (1 − κ ) m 2 u 2 i = π s ∂ t s − L κ where, this time with resp ect to the p olar fields ρ ( x, t ) and s ( x, t ), w e hav e: π ρ = ∂ L /∂ ( ∂ t ρ ) = 0 a nd π s = ∂ L /∂ ( ∂ t s ) = − ρ . 4 The Hamiltonian reads H κ ( t ) = R H κ ( x, t ) dx . F or the v ariationa l calculus it is not L but L = R L dx that really matters. Therefore, it is useful to note that L ( t ) = −h ∂ t s i − H κ ( t ) , (9) where, in view of R ρ dx = 1, w e can in tro duce the mean v alue h ∂ t s i = R ρ ∂ t s dx . Let us ev aluate the mean v alue of the g ene ralized Hamilton-Jacobi equation (5). By assuming a pro per b eha vior of ρ at the inte gra tion v olume b oundaries [11], w e readily get h Q i . = R Q ρ dx = +( m/ 2) h u 2 i > 0. Thence, on dynamically admitted fields ρ ( t ) and s ( x, t ), L ( t ) ≡ 0, i. e. h ∂ t s i = − H κ . Let us consider tw o function(al)s A = R A ( x, t ) dx and B = R B ( x, t ) dx , whic h may explicitly dep end on time t . W e define their Poiss on brac ke t { A, B } = − i ~ Z dx  δ A δ ψ δ B δ ψ ∗ − δ A δ ψ ∗ δ B δ ψ .  (10) In particular, while iden tifying A ≡ ψ ( x, t ) and B ≡ H κ ( t ), w e get the mo dular Sc hr¨ odinger equation (1) in t he form ∂ t ψ = { ψ , H κ } (11) while, b y setting A ≡ ψ ∗ , an adjoint equation arises ∂ t ψ ∗ = { ψ ∗ , H κ } . (12) W e recall e. g. that ˙ π ψ = − δ H κ /δ ψ while ˙ ψ = δ H κ /δ ψ . Since the time dep endence of H κ ( t ) is realized only throug h the canonical fields, the Hamiltonian surely is a constant of motion. Thence h ∂ t s i as w ell. The p olar decomp osition ψ = ρ 1 / 2 exp( is/ ~ ), ψ ∗ = ρ 1 / 2 exp( − is/ ~ ) preserv es a symplectic structure. In the self-explanatory notation there holds { A, B } . = { A, B } ψ, ψ ∗ = { A, B } ρ,s (13) and thence: ∂ t ρ = { ρ, H κ } = δ H κ δ s = − 1 m ∇ ( ρ ∇ s ) (14) while ∂ t s = { s, H κ } = − δ H κ δ ρ = − 1 2 m ( ∇ s ) 2 − V − (1 − κ ) Q . (15) The result is v alid for all κ ≥ 0. Let G ( t ) = R dx G ( x, t ). Clearly , c.f. [6 ], if the time-dep endenc e of G is realized only through canonically conjugate fields ρ ( x, t ) and s ( x, t ) (and t heir deriv ativ es), t hen d G dt = { G, H κ } . (16) One should realize that a particular time-dep endence pattern of G ( t ) critically relies on t he c hosen pa r a mete r range fo r κ ∈ R + . 5 2.3 Reduction to effectiv e κ = 0 , 1 and 2 self-coupling regimes In the notation o f Eq. (4), a distinguished r o le of κ = 0 , 1 or 2 coupling pa r ameter v alues is particularly conspicuous. The relev a nce of only these three κ = 0, 1, 2 para meter v alues ma y b e fur t her enhanced. Namely , let us recall [3] imp ortant differences b et w een prop erties of solutions of Eqs. (1) and (3), dep ending on whether 0 < κ < 1, κ = 1 or κ > 1. (i) In the par a mete r range 0 < κ < 1 (in fact 0 ≤ κ < 1) , if ψ ( x, t ) = | ψ | exp ( is/ ~ ) actually is a solution of (1), then ψ ′ ( x ′ , t ′ ) = | ψ ′ | exp( is ′ / ~ ), with x ′ = x , t ′ = (1 − κ ) 1 / 2 t , | ψ ′ | ( x ′ , t ′ ) = | ψ | ( x, (1 − κ ) − 1 / 2 t ′ ) and s ′ ( x ′ , t ′ ) = (1 − κ ) − 1 / 2 s ( x, t ), a uto matically solv es the linear Sc hr¨ odinger equation: i ~ ∂ t ′ ψ ′ =  − ~ 2 2 m ∆ + 1 1 − κ V  ψ ′ . (17) The scaling transformation replaces a nonlinear problem b y the linear one, alb eit with a r e-sc aled p oten tial. (ii) F or a sp ecifi c v alue κ = 1 w e encoun ter the formalism that deriv es (though actually generalizing) f rom the w av e picture of classical Newtonian mec hanics, [5]. (iii) In case of κ > 1, a rep etition of previous scaling steps, provide d w e replace (1 − κ ) 1 / 2 b y ( κ − 1) 1 / 2 in the p ertinen t form ulas, results in the follo wing outcome: ψ ′ ( x ′ , t ′ ) = | ψ ′ | exp( is ′ / ~ ) is a solution of the nonlinear Sc hr¨ odinger equation i ~ ∂ ′ t ψ ′ =  − ~ 2 2 m ∆ + V κ − 1  ψ ′ + 2  ~ 2 2 m ∆ | ψ ′ | | ψ ′ |  ψ ′ (18) Effectiv ely , in the whole coupling parameter ra nge κ ∈ R + , only the cases of κ = 0, κ = 1 and κ = 2 form a m utually exclusiv e family , b oth on mathematical and ph ysical grounds. Since for κ = 0 and κ = 2 , the ab o ve scaling transformatio ns trivialize, while b eing irrelev a n t for the distinctiv ely ”b orderline” case o f κ = 1, w e can safely restore the notation of Eqs. (1) and (3). In connection with the c hoice of κ = 2, one more observ ation is of utmost imp or- tance, [3], see also [12]- [16]. Namely , if a complex function ψ ( x, t ) = | ψ | exp( is/ ~ ) (19) is a solution of (1) with κ = 2, then t he r e al function θ ∗ ( x, t ) = | ψ | exp( − s/ ~ ) (20) is a solution of the generalized (forw ard) heat equation ~ ∂ t θ ∗ =  ~ 2 2 m ∆ + V  θ ∗ (21) 6 with a diffusion co efficien t D = ~ / 2 m and an external p oten tial V = V / 2 mD . By setting V = 0 w e w ould arrive at the standard heat equation ∂ t θ ∗ = D ∆ θ ∗ . Another real function θ ( x, t ) = | ψ | exp(+ s / ~ ) is a solution of the time-adjoint (bac kw ards) v ersion of Eq. (2 1 ): − ~ ∂ t θ =  ~ 2 2 m ∆ + V  θ . (22) Note that if w e would ha ve started fro m Eqs. (21) and (22) with the purp ose to arriv e at t he mo dular equation (1) with κ = 2, the ill-p osed Cauch y problem w ould p ossibly b ecome a serious obstacle. That in view of the o ccurrence of the backw ards parab olic equation. This ill-p osedness migh t b e healed b y in vokin g the theory o f strongly contin uous dynamical semigroups, [12, 13, 17]. T o this end w e need to choose V ( x ) to b e a con tin uous function that is b ounded from ab ov e, so that V ′ = − V b ecomes b ounded from b elo w. Then Eq. (21) w ould acquire a ” cano nical” form of the forw ard diffusion- t yp e equation related to the contractiv e semigroup op erator exp( − ˆ H t/ ~ ): ~ ∂ t θ ∗ = − ˆ H θ ∗ =  ~ 2 2 m ∆ − V ′  θ ∗ . (23) Eq. ( 2 3), to gether with its time adj o in t ~ ∂ t θ = ˆ H θ =  − ~ 2 2 m ∆ + V ′  θ (24) stand for principal dynamical equations of so- called Euclidean quantum mec hanics [13], while falling in to a bro ader framew ork of the Schr¨ odinger b oundary data and sto c hastic in terp olation problem, [12 , 13] see e.g. also [17]. Note, that if one assumes that the external p oten tial V ′ ( x ) is a con tin uous func- tion that is b ounded from b elo w, then the op erator ˆ H = − ( ~ 2 / 2 m )∆ + V ′ is (essen- tially) self-adjoint in L 2 ( R n ). So, we ha v e consisten tly defined unitary transformations exp( − i ˆ H t/ ~ ) in L 2 ( R n ) and i ~ ∂ t ψ = ˆ H ψ , together with plus − i ~ ∂ t ψ ∗ = ˆ H ψ ∗ , ( κ = 0 case), as their lo cal manifestations. A careful analysis rev eals [12, 13] that the standar d Schr¨ odinger picture dynamics can b e mapp ed into the con tractive semigroup (generalized diffusion equation) dynam- ics (23) and (24), b y means of an analytic con tin uatio n in time. This mapping w e repro duce in the b oo k-k eeping (Wic k rota tion) form; ( it ) → t . W e emphasize tha t one should not mys tify the emergen t ”imaginary time” trans- formation. The time lab el t p ertains to f airly standard (unequiv o cally real) dynamics scenarios. Ho w ev er the detailed patterns of temp oral b eha vior for tw o motion scenarios are v ery different. 7 Nonetheless, v arious (real) functionals of dynamical v ar ia bles , ev olving in a cc or- dance with the c hosen motio n rule (say , Sc hr¨ odinger’s ψ ( t ) and ψ ∗ ( t ), may b e consis- ten tly transformed into the affiliated functionals that ev olv e a cc ording to another, we call it dual, (dissipativ e θ ∗ ( t ) and θ ( t )) motion rule. The rev erse transformation w orks as w ell. W e shall address t his issue in b elo w, f rom v aried p ersp ectiv es. 3 Scale co v arian t patterns of e v ol u tion Let [∆ l ] stand f or an arbitrary ( a lbeit fixed for the presen t purp ose) unit o f length. F or a con tin uous probability densit y ρ on R 1 w e can in tro duce its (dimensionless) Shannon en trop y functional, also named differential en tropy : [2] S ( ρ ) . = − Z d  x [∆ l ]  ([∆ l ] ρ ) l n ([∆ l ] ρ ) = − Z dx ρ ln([∆ l ] ρ ) . (25) If ρ ( x, t ) dep ends on time, then S ( ρ ) = S ( t ) may ev olv e in time as w ell. The time rate equation for S ( t ) is dev oid of any [∆ l ] input a nd for all κ ≥ 0 has the v ery same functional form: D ˙ S = D { S, H κ } = −h u v i (26) Ob viously , h· i stands fo r the mean v a lue with r esp ect to a probability densit y ρ ( x, t ). The ab o ve time rate fo rm ula do es not dep end on a sp ecific unit of length, that is presen t in the definition of the dimensionless Shannon en tropy . How ev er, the en trop y functional itself is sensitiv e to scaling tra nsfor ma t ions. Setting x ′ = x/β , where β > 0 is the scale para meter, w e ha v e 1 = R ρ ( x ) dx = R β ρ ( β x ′ ) dx ′ . Accordingly , the scale transformation x → x ′ = x/β induces a tra nsfor - mation of the probability densit y in question: ρ ( x ) → β ρ ( β x ′ ) . = ρ ′ ( x ′ ) . (27) Giv en the Shannon en tropy of the densit y ρ ( x ), w e can compare the outcome with that for the densit y ρ ′ ( x ′ ): S ′ ( ρ ′ ) . = − Z dx ′ ρ ′ ( x ′ ) ln ([∆ l ] ρ ′ ( x ′ )) = − Z dx ρ ( x ) ln ( β [∆ l ] ρ ( x )) = S ( ρ ) − ln β . (28) Consequen tly , the x → x/β scaling is equiv alent to the sole c hange of the length unit [∆ l ] → β [∆ l ] in the Sha nno n en trop y definition. Since β is a fixed scaling parameter, the time rate formula (26) is scale indep enden t. Note that b y setting β . = exp α , w e get S ′ ( ρ ′ ) = S ( ρ ) − α . 8 W e o bserv e that u ′ ( x ′ ) = β u ( x ) and the scale in v ar ia nce of D ˙ S = −h u v i tells us that v ′ ( x ′ ) = (1 /β ) v ( x ). Since w e assume v ( x, t ) to b e the g r adien t field, v = (1 / m ) ∇ s , w e readily ar riv e at the corresp onding scaling prop ert y of s ( x, t ): s ′ ( x ′ ) = 1 β 2 s ( x ) − → v ′ ( x ′ ) = 1 β v ( x ) . (29) W e demand furthermore tha t the ( κ = 1) equations ∂ t ρ = −∇ ( ρ · v ) and ∂ t s + (1 / 2 m )( ∇ s ) 2 + V = 0 are scale in v arian t, e.g. they retain their form a fter in tro ducing x ′ , ρ ′ ( x ′ , t ) and s ′ ( x ′ , t ) instead x , ρ ( x, t ) and s ( x, t ) resp ectiv ely . This implies an induc e d scaling prop ert y of V ( x ) → V ′ ( x ′ ): V ′ ( x ′ ) = 1 β 2 V ( x ) (30) and th us H ′ 1 = (1 /β 2 ) H 1 . In view of Q ′ ( x ′ , t ) = β 2 Q ( x, t ) (31) the general ev olutio n equations for κ 6 = 1 are n ot scale inv ariant. Let us consider a cum ulativ e effect of the ab o v e scaling rules up on the Hamiltonian H κ = Z dxρ [( mv 2 / 2) + V + (1 − κ )( mu 2 / 2)] (32) according to: H κ . = H κ ( t ) → H ′ κ ( t ) = Z dx ′ H ( ρ ′ , s ′ )( x ′ , t ) . (3 3) F ollow ing the guess of [1 ], originally analyzed in the case of V ≡ 0, w e take β = exp( α/ 2); there fo llo ws H ′ κ ( α ) = exp( − α ) Z dx ρ [ m 2 v 2 + V ] + exp(+ α ) Z dx ρ (1 − κ ) m 2 u 2 (34) whic h can b e recast as the h yp erbolic transformation with a h yp erbo lic a ng le α H ′ κ ( α ) = cosh α H κ − sinh α K κ . (35) Here, in addition to H α , we encounte r a new Hamiltonian generator K κ K κ . = Z dx ρ h m 2 v 2 + V − (1 − κ ) m 2 u 2 i , (36) The difference b et w een H κ and K κ is enco ded in the sole sign inv ersion of the last (1 − κ )( mu 2 / 2) en try . W e can as we ll analyze an effect of the scaling transformation up on K κ , [1]: K ′ κ ( α ) = − sinh αH κ + cosh α K κ . (37) 9 Hyp erbolic rotations that are explicit in equations (35) and (37) form a conspicuous Loren tz-ty p e t r a nsformation (hy p erb olic rotation of co ordinates in Mink o wski space) of the direct analogue ( H κ , K κ , 0 , 0) of the familiar Mink ow ski space v ectors ( p 0 , p, 0 , 0) and/or ( ct, x, 0 , 0). W e recall that the time lab el t is left untouc hed b y hitherto considered scale trans- formations. Conseq uently , like H κ , the generator K κ and the induced α -family of generators H ′ κ ( α ), K ′ κ ( α ) ar e legitimate Ha miltonian generators of diverse time ev o- lution scenarios, all running with resp ect to the same time v ariable t . The p ertinen t motions arise through common fo r a ll Hamiltonians, a priori prescrib ed, symplectic structure Eq. (13) and t he generic time ev olution rule (16). Remark 1: By re-t r acing bac k the passage from the mo dular equation to the corresp onding Lagrangia n and Hamiltonia n, w e realize that K κ can b e asso ciated with another ( κ − 2)-family ( κ ≥ 0) of mo dular Sc hr¨ odinger equations: i ~ ∂ τ ψ = [ − ( ~ 2 / 2 m )∆ + V ] ψ + ( κ − 2) Q ψ . (38) to b e compared with the originally in tro duced κ - family i ~ ∂ t ψ = [ − ( ~ 2 / 2 m )∆ + V ] ψ − κQ ψ . (39) W e shall not b e elab orate on the general κ > 2 parameter regime and, in view of the previously established reduction pro cedure (Section 1.3), w e confine our further discussion to sp ecific κ = 0, 1 or 2 cases. Remark 2: Let us notice that K 0 ≡ H 2 , H 1 ≡ K 1 and K 2 ≡ H 0 . W e hav e H ′ 1 = exp ( − α ) H ′ 1 and the follow ing h yp erb o lic transformation prop erties hold t rue for H 0 and H 2 : H ′ 0 = cosh αH 0 − sinh αH 2 (40) and H ′ 2 = − sinh αH 0 + cosh α H 2 . (41) These t r a nsformation rules are a g ene ralizatio n of those presen ted for the V ≡ 0 case b y L. Brenig, [1]. Remark 3: Hamilto nians of the form H 0 = ( m/ 2) h v 2 + u 2 i + h V i (42) and H 2 = ( m/ 2) h v 2 − u 2 i + h V i (43) are kno wn in t he literature, [18, 14], and ar e in terpreted t o set quan tum-mec hanical and dissipativ e-dynamical frameworks resp ectiv ely . Some elemen tary hints in this connec- tion can also b e found in [1, 15, 16, 2]. Coming back to our observ ation that a r es caling 10 of the dimensional unit [∆ l ] → exp α [∆ l ] induces a transformation S ′ ( ρ ′ ) = S ( ρ ) − α of the Shannon en trop y S ( ρ ) for a con tin uous probability densit y ρ , w e realize that the c hoice o f α ≤ 0 implies a sharp ening o f the r esolution unit, hence an effectiv e growth of the Shannon entrop y with the low ering of α = −| α | . Indeed, w e hav e S ′ ( ρ ′ ) = S ( ρ )+ | α | . In this situation, effectiv e (rescaled form) Hamiltonians H ′ 0 and H ′ 2 , alw ays ha v e an admixture of b oth non-dissipativ e and dissipativ e comp onen ts, H 0 and H 2 resp ec tively . 4 Dynamical duality Remem b ering that fo r general functionals G ( t ) of ρ and s , w e ha v e ˙ G = { G, H κ } , let us consider a pro duct F ( x, t ) . = − ρ ( x, t ) s ( x, t ) of conjugate fields s and π s = − ρ . The time ev olution of F ( t ) = Z dx F ( x, t ) . = −h s i (44) lo oks quite interes ting: dF dt = { F , H κ } = − Z dx  s ( x, t ) δ H κ δ s − ρ ( x, t ) δ H κ δ ρ  = (45) − Z dx ρ h m 2 v 2 − V − (1 − κ ) m 2 u 2 i . A new Hamiltonian-type functional has emerged o n the rig h t- hand-side of the dynam- ical iden tity (45 ) . Let us denote H ± κ = Z dx ρ h m 2 v 2 ± V ± (1 − κ ) m 2 u 2 i . (46) W e p oin t out that in con trast to previous scaling-induced Hamiltonians, the negat iv e sign has b een generated b oth with resp ect to t erms ( m/ 2) h u 2 i and h V i . The + V → − V mapping w as pa inf ully lac king in the previous discussion to hav e prop erly implemen ted, the implicit, analytic contin ua tion in time pro cedure. The previous motion rule rewrites as dF dt = { F , H + κ } = − H − κ ( t ) , (47) where H + κ = H is t he time evolution generator Eq. (8). No te tha t H + κ is here a constan t of motion, while H − κ ( t ) is not. After accoun ting for the Poisson brac k et (13), w e encoun ter a complemen tary rela- tionship for the time ev o lution that is generated b y the induc e d Hamiltonian H − κ : dF dt = { F , H − κ } = − H + κ ( t ) . (48) Presen tly , H + κ is a constan t of mot ion, while H − κ ( t ) no longer is. 11 F or e ach v alue of κ ∈ R + , w e thus arrive at dual time-evolution scenarios, g enerated b y Hamiltonians H + κ and H − κ , resp ectiv ely . The duality notion stay s in confo rmit y with our previous discussion (Section 1.3) of the ”imaginary time” transformation. These dual motio ns, ev en if started from the v ery same initial t = 0 data a nd allo w ed to contin ue indefinitely , do result in disparately different patterns of b eha vior: non-dissipativ e and dissipativ e, resp ectiv ely . Nonetheless, if w e admit that the a priori c hosen generator of motio n is H + κ , then w e are intere sted as w ell in: d 2 F dt 2 = −{ H − κ ( t ) , H + κ } = − dH − κ dt = +2 Z ρ v ∇ [ V + (1 − κ ) Q ] dx (49) while, presuming that H − κ actually is the evolution g ene rato r, in the dual ev olutio n form ula d 2 F dt 2 = −{ H + κ ( t ) , H − κ } = − dH + κ dτ = − 2 Z ρ v ∇ [ V + (1 − κ ) Q ] dx . (50) The right-hand-sides of Eqs. (49), (50) asso ciate the mean p o w er t ransfer ra tes (gain, loss or none) to the dual Hamilto nia n ev olutions of F ( t ), generated b y H + κ and H − κ resp ec tively . This remains div orced from the fact that H + κ ( t ) and H − κ ( τ ) actually are constan ts of the p ertinen t dual motions. F or clarit y of our discussion we shall confine further attention to L 2 ( R n ) solutions of (1) and (3). W e assume that the external p otential V ( x ) is a con tinuous function tha t is b ounded from b elo w. If the energy op erator ˆ H = − ( ~ 2 / 2 m )∆ + V is self-adjoint, then w e hav e consisten tly defined unitary tr ansformations exp( − i ˆ H t/ ~ ) in L 2 ( R ) so that i ~ ∂ t ψ = ˆ H ψ holds true ( κ = 0 ). The case of κ = 1 we asso ciate with tw o classes of external p oten tials ± V ( x ), with + V ( x ) b ounded from b elo w. This will allo w us to discriminate b et wee n the confining and scattering regimes. The ” b orderline” meaning of κ = 1 can b e read out from Eq. ( 4 ), where mu 2 / 2 con tributions cancel a wa y . In conformity with our previous discussion of generalized heat equations, related to solutions of Eq. (1) with κ = 2, giv en + V ( x ) a nd ˆ H , w e pass to a pair of time-adjoin t parab olic equations: ~ ∂ t θ ∗ = − ˆ H θ ∗ and ~ ∂ t θ = ˆ H θ . Here, θ ∗ ( x, t ) = [exp( − ˆ H t/ ~ ) θ ∗ ]( x, 0) represen ts a forw ard dynamical semigroup ev olution, while θ ( x, T − t ) = exp(+ ˆ H t/ ~ ) θ ( x, T ) stands for a backw ard one. Both are unam biguously defined in a finite time interv al [0 , T ], provided one has prescrib ed suitable end-p oin t dat a [12]. The corresp onding mo dular Schr¨ odinger equations (plus their complex conjugate v ersions) read: (i) κ = 0 = ⇒ i ~ ∂ t ψ = [ − ( ~ 2 / 2 m )∆ + V ] ψ 12 (ii) κ = 1 = ⇒ i ~ ∂ t ψ = [ − ( ~ 2 / 2 m )∆ ± V − Q ] ψ (iii) κ = 2 = ⇒ i ~ ∂ t ψ = [ − ( ~ 2 / 2 m )∆ − V − 2 Q ] ψ . The asso ciated Lagra ngian densities (4) in the ( ρ, s )-represen tation and the induced dynamical rules are w orth listing a s well. In addition to the con tin uity equation ∂ t ρ = −∇ ( ρ · v ) w e hav e v alid the Hamilton- Jacobi t yp e equations: (i) κ = 0 ; L = − ρ [ ∂ t s + ( m/ 2)( v 2 + u 2 ) + V ] = ⇒ ∂ t s + (1 / 2 m )( ∇ s ) 2 + ( V + Q ) = 0 (ii) κ = 1 ; L = − ρ [ ∂ t s + ( m/ 2) v 2 ± V ] = ⇒ ∂ t s + (1 / 2 m )( ∇ s ) 2 ± V = 0 (iii) κ = 2; L = − ρ [ ∂ t s + ( m/ 2)( v 2 − u 2 ) − V ] = ⇒ ∂ t s + (1 / 2 m )( ∇ s ) 2 − ( V + Q ) = 0. On dynamically admitted fields ρ ( t ) and s ( x, t ), L ( t ) ≡ 0 , i. e. h ∂ t s i = − H . The resp ec tive Hamiltonians (8) do fo llo w: (i) H + . = R dx ρ [( m/ 2) v 2 + V + ( m/ 2) u 2 ] (ii) H ± cl . = R dx ρ [( m/ 2) v 2 ± V ] (iii) H − . = R dx ρ [( m/ 2) v 2 − V − ( m/ 2) u 2 ] W e emphasize that, from the start, V ( x ) is chosen to b e a con tin uous a nd b ounded from b elo w function. In the definition of the ab o v e Hamiltonians there is no κ la- b el an ymore and a subscript ”cl” refers to t he classically motiv ated (Hamilton-Jacobi theory) w av e formalism. The ev olution equations for F = −h s i , c.f. [1] and [1 6 , 2], clearly define d ual pairs: ˙ F = { F , H + } = − Z dx ρ h m 2 v 2 − V − m 2 u 2 i = − H − ( t ) , (51) ˙ F = { F , H − } = − Z dx ρ h m 2 v 2 + V + m 2 u 2 i = − H + ( t ) (52) and ˙ F = { F , H + cl } = − Z dx ρ h m 2 v 2 − V i = − H − cl ( t ) (53) ˙ F = { F , H − cl } = − Z dx ρ h m 2 v 2 + V i = − H + cl ( t ) . (54) It is instructiv e to notice that the functional F ( t ) in Eqs. (52) and (54) ma y consis- ten tly play the role of a Lyapuno v functional, indicating the preferred sense of time (”time arrow”) in the course of the ev olutio n pro cess. Namely , if we tak e h V i > 0, the righ t-hand-side express ion is negativ e definite. Hence F ( t ) is a monotonically de- ca ying function of time whic h is a standar d signature of a dissipation pro cess, c.f. the Helmholtz free energy and the relativ e entrop y discussion for diffusion-t yp e pro cesses, [16, 2, 19]. In the nota t ion of section 1, c.f. Eq. (8 ) , we ha v e H ± = H ± 0 and H ± cl = H ± 1 . The corresp onding (dual) time rate formulas (49), (50 ) do follow . The motion rules for ˙ F ( t ) can b e given more transparent fo r m by rein tro ducing constan ts H ± of the resp ectiv e motions. Then ˙ F ( t ) = − m h v 2 i ( t ) + H ± (55) 13 and ˙ F ( t ) = − m h v 2 i ( t ) + H ± cl . (56) Here, the non-negativ e term m h v 2 i ( t ) should receiv e due atten tion, b ecause of its ut- most imp ortance in the study of the Shanno n en tropy dynamics, [16, 2, 19], where it represen ts an entr opy pr o duction time rate. The latter is generated solely by the dynamical pro cesse s whic h are in trinsic to t he system and do es not in v olve a n y en- ergy/heat flow, in or out of the p oten tially dissipativ e system (that w ould need the notion of an external to the system, thermal reserv oir). Since H + and H − are constants of resp ectiv e motions, F ( t ) − t H ± are monot o nically decreasing in time quan tities. This prop ert y extends to the H ± cl generated dynamics as w ell. The sp eed (slowing down, or acceleration), with whic h the ab o v e deca y pro cess ma y o ccur, relies on the sp ecific dynamical pa ttern of b eha vior of h v 2 i . That is quan tified b y − md h v 2 i /dt , hence: ¨ F ( t ) = { ˙ F ( t ) , H ± } = ± 2 Z ρv ∇ ( V + Q ) dx . (57) F or H ± cl generated motions, w e ha v e: ¨ F ( t ) = { ˙ F ( t ) , H ± cl } = ± 2 Z ρv ∇ V dx . (58) W e p oint out that a ma jor distinction b et w een the dual dynamical rules is enco ded in the righ t-hand- side s of the ab ov e equations: the conspicuous sign in v ersion is worth con templation. The related in tegrals hav e a clear meaning of the p o we r transfer (re- lease, absorption or p ossibly none), in the mean, that is induced b y time ev olution of the p ertinen t dynamical system, [16, 2]. Let us stress that the ”imaginary time” transformation, ev en if not quite explicit in our discussion, hereb y has b een extended to dynamical mo dels of purely classical pro v enance. The p ertinen t Wic k r o tation connects confined and scattering motions, admitted to o ccur in a con tin uous and b ounded from b elo w (confining) p oten tial + V ( x ) and in its in v erted (scattering) coun terpart − V ( x ), resp ectiv ely . An obv ious example of a n inv erted har mo nic oscillator [20] is w orth men tioning at this p oin t. P arab olic p oten tial barriers [21] and repulsiv e 1 /r 2 p oten tials [22] b elong to the same catego r y . In the ab ov e discussion, the sign inv ersion issue is manifested in second time deriv a- tiv es of v ar io us functionals. In view of ±∇ V presence, w e can identify this b eha vior as a remnant o f the standard classical (Newtonian) reasoning, [12]: if the sign lo oks wrong with resp ect to the classical Newton equation (e. g.w e hav e + ∇ V ), w e can correct this ”defect” by in terpreting time t as an ” imaginary time” it (or iτ to av o id a not ational confusion). 14 5 Ph ysics-rel ate d implemen tation s of the d u al dy- namics p att erns: a (dis) illusion of an ”imaginary time” 5.1 Generalities With the not a tional conv entions D = ~ / 2 m , b ( x, t ) . = v ( x, t ) + u ( x, t ), while imp osing suitable b oundary conditions (e.g. ρ, bρ, v ρ v anishing at in tegration b oundaries or infinities), we can write the Shannon entrop y time rate of c hange in a n umber of equiv alen t w a ys, [2]. D ˙ S = D { S, H κ } = −h u v i = h v 2 i − h b v i (59) and D ˙ S = D h∇ v i = h∇ b i + D h u 2 i . (60) The non-negative entry (1 /D ) h v 2 i is inte rpreted as the entr opy pr o duction rate in the considered dynamical system [16, 2 , 19]. W e note that D h u 2 i = − D h∇ u i = 2 m h Q i (61) so that the mean div ergence of an osmotic v elo cit y is alwa ys negative. Here h Q i > 0 holds true for all finite times, [2]. The v alue 0 can b e ac hiev ed, if at all, o nly in the asymptotic regime t → ∞ . T o giv e a fla vor of the time duality (and sp ecifically the ”imag inary time transfor- mation”) connection, let us men tion that the free Brownian motion can b e embedded in the ab o v e sc heme b y setting b ≡ 0 and regarding D as the diffusion constan t. W e ha v e v = − u = − D ∇ ln ρ and Shannon en tropy t ime rate tak es the form of the de Bruijn iden tity D ˙ S = h v 2 i = h u 2 i , [2]. The Shannon en tropy S ( t ) of the Brow nian motion g r ows monotonically in time, solely due to the entrop y pro duction (1 /D ) h v 2 i ( t ). The la tter, in turn, is kno wn to b e a decreasing function of time (at least in the larg e time asymptotic) and ultimately is b ound to v anish at t → ∞ . In case of the diffusion pro cess in a conserv a tiv e p oten tial it is not Shannon entrop y , but the Helmholtz f ree energy that ta k es the role of the Ly apunov functional and sets the ”time arrow”, c.f. Eqs. (55 and see [2 , 19]. Note that F ( t ) − tH − is a monotonically deca ying in time quantit y . F or comparison, the Schr¨ odinger picture quan tum dynamics t ypically in v olve s b 6 = 0 . In the sp ecial case o f the free motion ˙ S > 0, hence S ( t ) grows indefinitely , [16]. In the large time asymptotic ˙ S → 0, while the en tropy pro duction (1 /D ) h v 2 i remains un tamed a nd nev er v anishes while approaching a finite p ositiv e v alue. 15 Ho w ev er, the would-be natural prop ert y ˙ S ( t ) > 0 is not generic for the quan tum motion in external p oten tials, [23]. Nonetheless F ( t ) − tH + do es monotonically decrease with time t → ∞ indicating t he Ly apunov functional-induced ”arr o w of time”. Ev en, though this dynamics is manifestly non-dissipativ e. 5.2 Harmonic oscillator and its inv erted partner Let us consider a standard classical harmonic oscillator problem, where H . = p 2 2 m + 1 2 mω 2 q 2 (62) is an obv ious constant of motion for the Newtonian system ˙ p = m ¨ q = − mω 2 q , q ( t ) = q 0 cos ω t + p 0 mω sin ω t (63 ) p ( t ) = p 0 cos ω t − mω q 0 sin ω t . Clearly H = p 2 0 / 2 m + ( mω 2 / 2) q 2 0 is a p ositiv e constan t. Let us p erform an analytic contin uation in time, b y considering the Wic k rot a tion t → − it , paralleled by the transformatio n of initial momen tum da t a p 0 → − ip 0 . Once inserted to the ab o ve harmonic o sc illato r expressions, w e arriv e at H − ip 0 = − p 2 0 / 2 m + ( mω 2 / 2) q 2 0 . = − H (64) and q − ip 0 ( − it ) . = q ( t ) = q 0 cosh ω t − p 0 mω sinh ω t (65) together with p − ip 0 ( − it ) . = + i p ( t ) = − ip 0 cosh ω t + imω q 0 sinh ω t , (66) whic h simply rewrites as p ( t ) = − p 0 cosh ω t + mω q 0 sinh ω t . (67) W e observ e that q ( − t ) = q 0 cosh ω t + p 0 mω sinh ω t (68) and − p ( − t ) = p 0 cosh ω t + mω q 0 sinh ω t (69) are the familiar inv erted oscillator solutions, g enerated by H , [20 ]. Indeed, equations of motion for q ( t ) a nd p ( t ) directly deriv e from the Hamiltonia n H − ip 0 = − H with H = p 2 2 m − 1 2 mω 2 q 2 (70) 16 They giv e rise to the Newton equation ˙ p = m ¨ q = + mω 2 q . Ho we ve r, the dynamics generated b y H is related to that g enerated b y − H by the time reflection: the latter dynamics runs back wards, if the former runs fo rw ard. Remark: At the first glance, the harmonic oscillator example may b e regarded as a unique sp ecial case (linear dynamical system) to whic h our imaginary time tra ns - formation argumen ts ma y b e applied. F ortunately , we can giv e a num b er o f nonlinear mo dels whose (Euclidean) in v ersion can b e consisten t ly implemen ted, [24]. The Eu- clidean connection g oes b ey ond the confining vs scattering p oten tial idea o f ours and extends to p erio dic p oten tials a s we ll. Examples from the instan ton ph ysics: static lo calized (kink) solutions o f the φ 4 nonlinear field theory in one space dimension ma y b e in terpreted as Euclidean time solutions of the double w ell p oten tial pro blem; the sine-Gordon kink ma y b e interpreted as a Euclidean time solution of a pla ne p endulum problem. 5.3 Time dualit y via analytic cont in uation in time The ab o v e pro cedure giv es clear hints on how to connect the dual classical wa ve theory ev olutions, asso ciated with the previously discussed Hamiltonians H ± cl . W e recall that in addition to the contin uity equation, w e infer the dual Hamilton-Jacobi equations ∂ t s + (1 / 2 m )( ∇ s ) 2 ± V = 0 and that there holds ∂ t ρ = −∇ · ( ρ v ) with v ( x, t ) = (1 /m ) ∇ s ( x, t ). In the adopted notational conv ention, w e define the initia l da t a s 0 ( x ) = − s 0 ( x ) and in tro duce an ”imaginary time” transformation : ψ ( x, t ) = ρ 1 / 2 exp( is/ 2 mD ) − → ψ ( x, t ) . = ψ − is 0 ( x, − it ) = (71) ρ 1 / 2 − is 0 ( x, − it ) exp[ is − is 0 ( x, − it ) / 2 mD ] . = ρ 1 / 2 ( x, t ) exp[ − s ( x, t ) / 2 mD ] . W e note that lim t ↓ 0 is − is 0 ( x, − it ) = i ( − is 0 )( x, 0) = s 0 ( x ). An analogous pro cedure for a n analytic con tin uation in time has b een w orked out in the general context of the Euclidean quan tum mechanic s in Ref. [13 ], ho w ev er with no mention of its extension t o the classically inspired Hamilton-Jacobi equation. Let us denote v = (1 /m ) ∇ s . Accordingly , the transformations implemen ted b y (71) replace H + cl = R dx ρ [( m/ 2) v 2 + V ] by − H − cl = R dx ρ [ − ( m/ 2) v 2 + V ], with the very same f unction V ( x ) in b oth expressions. Clearly: ∂ t ρ = −∇ · ( ρv ) − → ∂ t ρ = + ∇ · ( ρ v ) (72) whic h is an obv ious indicativ e of the time reflected (bac kw ards) ev olution. Analog ously ∂ t s + (1 / 2 m )( ∇ s ) 2 ± V = 0 − → ∂ t s − (1 / 2 m )( ∇ s ) 2 + V = 0 (73) 17 where, the time reflection t → − t induces an exp ected form of the dual Hamilton-Jacobi equation: ∂ t s + (1 / 2 m )( ∇ s ) 2 − V = 0 . (74) The ab o v e discuss ed analytic con tin uation in time directly extends to the general pair H ± of dual Hamiltonians, c.f. Section IV of Ref. [13]. The description b e- comes ev en more straigh tfo rw ard, b ecause in this case w e connect pairs of linear par- tial differen tial equations. If ψ ( x, t ) actually is a solution of the Sc hr”odinger equation i (2 mD ) ∂ t = ˆ H ψ , then ψ − is 0 ( x, − it ) = ρ 1 / 2 ( x, t ) exp[ − s ( x, t ) / 2 mD ] . = θ ∗ ( x, t ) (75) solv es a backw ards diffusion-type equation − (2 mD ) ∂ t θ ∗ = ˆ H θ ∗ (76) while θ ( x, t ) = ρ 1 / 2 ( x, t ) exp[+ s ( x, t ) / 2 mD ] (77) solv es the forward equation (2 mD ) ∂ t θ = ˆ H θ . (78) In the ab o v e one ma y ob viously iden tify D = ~ / 2 m . The whole pro cedure can inv erted and w e can tr a ce bac k a non-dissipativ e quan tum dynamics pattern whic h sta ys in affinit y (duality) with a given dissipativ e dynamics, c.f. [13]. 5.4 Diffusion-t yp e pro cesses 5.4.1 Smoluc ho wski pro cess The Hamilto nia n appropriate for the description of dissipativ e pro cesse s (strictly sp eak- ing, diffusion-ty p e sto c hastic pro cesse s) has the form H − . = Z dx ρ  ( m/ 2) v 2 − V − ( m/ 2) u 2  (79) with the a priori c hosen, contin uous and b ounded from b elo w p oten tial V ( x ). It is the functional form of V ( x ) whic h determines lo cal c haracteristics of the diffusion pro cess, [2]. Once the F o kk er-Planc k equation is inferred ∂ t ρ = D ∆ ρ − ∇ ( b · ρ ) , (80) where ρ 0 ( x ) stands for the initial condition, w e adopt the forw ard drift b = f /mγ of the pr o cess in the standard Smoluc hows ki f o rm, characteristic for the Brownian motion 18 in an external fo rce field f ( x ) = −∇V . Here, γ is a friction (damping) parameter and, instead of D = ~ / 2 m , w e prefer to think in terms of D = k B T /mγ where T stands for an ( equilibrium) temp erature of the reserv oir. An admissible for m o f V → f = −∇V m ust b e compatible with the Riccatti-type equation, pro vided the p otential function V ( x ) has b een a priori chos en: V ( x ) = m " 1 2  f mγ  2 + D ∇ ·  f mγ  # . (81) The F okk er-Planck equation can rewritten as a con tin uity equation ∂ t ρ = −∇ · j with the diffusion current j in the form: j = ρv = ρ mγ [ f − k B T ∇ ln ρ ] . = ρ m ∇ s . (82) W e recall the general definition of the curren t velocity v = (1 /m ) ∇ s . Since the time-indep enden t s = s ( x ) is here admissible, w e hav e actually determined s = − 1 γ ( V + k B T ln ρ ) (83) whose negativ e mean v alue F = −h s i determines for the Helmholtz free energy of the random motion, as follows: Ψ . = γ F = U − T S , (84) where S . = k B S stands f or the Gibbs-Shannon en tro py of the contin uo us pro babilit y distribution, while an internal energy reads U = hV i . Since we assume ρ and ρV v to v a nish at the in tegrat io n v olume b oundaries, w e get ˙ Ψ = − ( mγ )  v 2  = − k B T ( ˙ S ) int ≤ 0 . (8 5 ) Clearly , the Helmholtz free energy Ψ decreases as a function of time, or remains con- stan t. The Shannon entrop y S ( t ) = −h ln ρ i t ypically is not a conserv ed quan tit y . W e im- p ose b oundary restrictions that ρ, v ρ, bρ v anish at spatial infinities or other in tegrat ion in terv al b orders and consider: D ˙ S =  v 2  − h b · v i . (86) whic h rewrites a s follo ws ˙ S = ( ˙ S ) int + ( ˙ S ) ext (87) where k B T ( ˙ S ) int . = mγ  v 2  ≥ 0 (88) 19 stands for the en trop y pro duction rate, while k B T ( ˙ S ) ext = − Z f · j d x = − mγ h b · v i (89) (as long as negativ e whic h is not a m ust) may b e in terpreted as the heat dissipation rate: − R f · j d x . Let us consider the stationary regime ˙ S = 0 asso ciated with an (a priori assumed to exist) in v arian t density ρ ∗ . Then, b = u = D ∇ ln ρ ∗ and − (1 /k B T ) ∇V = ∇ ln ρ ∗ = ⇒ ρ ∗ = 1 Z exp[ −V /k B T ] . (90) Hence − γ s ∗ = V + k B T ln ρ ∗ = ⇒ Ψ ∗ = − k B T ln Z . = γ F ∗ (91) with Z = R exp( −V /k B T ) dx . Ψ ∗ stands for a minim um of the time-dep enden t Helmholtz free energy Ψ. Because of Z = exp ( − Ψ ∗ /k B T ) (92) w e hav e ρ ∗ = exp[(Ψ ∗ − V ) /k B T ] . (93) Therefore, the conditional Kullbac k-Leibler en tropy H c , of the density ρ relat ive to an equilibrium (stationa r y) densit y ρ ∗ acquires the form k B T H c . = − k B T Z ρ ln( ρ ρ ∗ ) dx = Ψ ∗ − Ψ . (94) In view of the conca vity prop ert y of the function f ( w ) = − w ln w , H c tak es only negativ e v alues, with a maxim um at 0. W e ha ve Ψ ∗ ≤ Ψ a nd k B T ˙ H c = − ˙ Ψ ≥ 0. H c ( t ) is b ound t o grow monotonically tow ards 0, while Ψ( t ) drops down to its minim um Ψ ∗ whic h is reache d up on ρ ∗ . The Helmholtz free energy minim um remains divorced from an y extremal pro p- ert y of the Gibbs-Shannon en tropy . Only t he Kullback -Leibler en trop y sho ws up an exp ected (grow th) asymptotic b eha vior. See e.g. also [2]. Note that prop erties of the fr ee Bro wnian motion can b e easily inferred b y setting b ≡ 0 in the ab o v e discussion. Then, the diffusiv e dynamics is sw eeping and there is no asymptotic in v aria n t density , nor a finite minim um for Ψ( t ) whic h decreases indefinitely . 20 5.4.2 Rein tro ducing dualit y T o set a connection with the pr evious time duality (”imaginary time” transformation) framew ork, we need only to observ e some classic prop erties of Smolucho wski diffusion pro cesse s. Once we set b = − 2 D ∇ Φ with Φ = Φ( x ), a substitution: ρ ( x, t ) . = θ ∗ ( x, t ) exp[ − Φ( x )] (95) with θ ∗ and Φ b eing real functions, conv erts the F okke r- Planck equation into a gener- alized diffusion equation fo r θ ∗ : ∂ t θ ∗ = D ∆ θ ∗ − V ( x ) 2 mD θ ∗ (96) and its (here trivialized in view of the time-indep endence of Φ) time adjo int ∂ t θ = − D ∆ θ + V ( x ) 2 mD θ (97) for a real function θ ( x, t ) = exp[ − Φ( x )], where V ( x ) 2 mD = 1 2 ( b 2 2 D + ∇ · b ) = D [( ∇ Φ) 2 − ∆Φ] . (9 8) Let us note an o b vious factorization prop ert y for the F okk er-Planck probabilit y densit y: ρ ( x, t ) = θ ( x, t ) · θ ∗ ( x, t ) (99) whic h stays in affinit y with a quan tum mec ha nical factorizatio n formula ρ = ψ ∗ ψ , alb eit presen tly realized in terms of t w o real functions θ and θ ∗ , instead of a complex conjugate pair. In view of (at this p oin t we restore an original notation of Section 4): ρ 1 / 2 ( x, t ) exp[ − s ( x, t ) / 2 mD ] . = θ ∗ ( x, t ) (100) w e immediately r ec ov er s = (2 mD )[Φ − (1 / 2) ln ρ ] (101) in conformit y with the previous definition Eq. (83). If there are no external forces, Φ disapp ears and w e are left with the free Brownian motion asso ciated with s = − mD ln ρ . F or the record it is useful to men tion explicit tr a nsformations b et w een Green’s func- tions appro pr ia te for quantum mot ion and transition probabilit y densities of standard diffusion t yp e pro cesse s, [25]. Explicit examples of the free dual dynamics and those in the harmonic p oten tial, ha v e b een w orke d out there. 21 6 Conclus ions An analytic con tin uatio n in time ( o r an ”imaginary time” transformatio n) stands for a ma pping b et w een tw o differen t t yp es of dynamics, b oth running with resp ect t o t he equally ”r eal” (istic) time. In our case, the mo dular Sc hr¨ odinger equation has b een a unifying departure p oin t for an analysis of, sometimes not quite expected, affinities b et w een dynamical pat t erns of b eha vior generated by the same primary nonlinear field, but typic ally considered disjoin tly - as researc h problems on their ow n, f or prop erly selected coupling parameter v alues. One may p ossibly take the view that κ = 0 , 1 , 2 cases ar e just formally analo- gous ma t hem atical descriptions of differen t phy sical systems. Our standp oin t is that a primary dynam i c al system is a mo dular Sc hr¨ odinger equation with an arbitrarily adjustable coupling parameter. Differen t coupling regimes refer to physic al ly differ ent patterns of b eha vior, but there is a deep intert wine b et we en them, to b e further ex- plored. The global c hanges of prop erties of solutions of nonlinear dynamical equations as the control parameter is v aried are a ro utine wisdom for nonlinear system exp erts. This prop ert y is shared b y the mo dular Sc hr¨ odinger equation as w ell. If one tak es seriously the dynamical duality (or time dualit y) concept, the mo dels considered here should not b e view ed indep enden tly an ymore. O ne can trace the dynamical patterns of one mo del in terms of t hose for another, and in reve rse. Ev en, if at the first glance, the p ertinen t dynamics patterns ma y seem to hav e nothing in common. It is the Hamiltonian analysis of the nonlinear field (e.q. Sc hr¨ odinger equation) whic h has led to a v ariety of emergen t Hamiltonian motion scenarios. Tw o classes of them, w e call them confining or scattering, sta y in close affinity o n quan tum, classical and sto c hastic (dissipativ e) dynamics lev els of description. Euclidean metho ds are o f ten used in v arious (esp ec ially quantum) branc hes of the statistical phys ics researc h of equilibrium and near- equilib rium phenomena. Our dis- cussion w as basically concen t rated on the real time flow not io n which, ev en after a Euclidean (imaginary time) transformation, still remains a real(istic) time flo w, alb eit with a new ph ysical meaning. It is w orth men tioning that an an indep enden t , quan tum theory motiv ated approac h [26] (deformation quan tization with an ”imagina r y transformed” deformation constant ~ ) has obvious links with the Euclidean map viewp oin t tow ards diffusion-ty p e pro cesses, explored in the presen t pap er. 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