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Phenomenological Universalities: Coherence, Supersymmetry and Growth Marcin Molski1 1 Theoretical Chemistry Department, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznan´, Poland (Dated: February 2, 2017) Thephenomenological universalities(PU)areextendedtoincludetime-dependedquantumoscil- latory phenomena, coherence and supersymmetry. It will be proved that this approach generates minimumuncertaintycoherentstates oftime-dependentoscillators, which in thedissociation (classical)limitreducetothefunctionsdescribinggrowth(regression) ofthesystemsevolvingovertime. 7 The results obtained reveal existence of a new class of macroscopic quantum (or quasi-quantum) 1 phenomena, which may play a vital role in coherent formation of the specific growth patterns in 0 complex systems. 2 PACSnumbers: 89.75.-k,11.10.Lm,12.60.Jv,03.65.Fd,89.75.Fb n a J TheconceptofPUintroducedbyCastorina,Delsanto, generate the growth functions [1] 6 andGuiot(CDG)[1]concernsontologicallydifferentsys- 2 xdx tems, in which miscellaneous emerging patterns are de- ψ(q)=exp +C =exp x(q)dq+C − Φ(x) ] scribedbythe samemathematicalformalism. Universal- (cid:20) Zx (cid:21) (cid:20)Zq (cid:21) h (3) ity classes are useful for their applicative relevance and p for different powers n = 1,2,... of the truncated series facilitate the cross fertilization among various fields of - n research, including physics, chemistry, biology, engineer- (2). The integration constant C can be calculated from e a boundary condition for x(q) at q = 0. For example ing, economics and social sciences [1]-[9]. This strategy g for x(0) = 1, c = 0, c = 1, Φ(x) = x one gets the . isextremelyimportant,especiallyfortheexportofideas, 0 1 s Gompertz function, whereas for Φ(x) = x + c x the models and methods developed in one discipline to an- 2 2 c allometric WBE-type function can be derived [1] i other and vice versa. The PU approach is also a useful s y tool for investigation of the complex systems whose dy- ψ(q) = exp[(1 exp( q)], G h namics is governed by nonlinear processes. Hence, this − − p methodologycanbeemployed[1]toobtaindifferentfunc- ψ(q)W = exp[1+c2 c2exp( q)]1/c2. (4) − − [ tions of growthwidely applied inactuarialmathematics, Employing this approachthe PU can be classified as U1 1 biology and medicine. In this work the research area is (n=1),U2(n=2)etc. withrespecttothedifferentlev- v extended to include in the CDG scheme the quantum elsofnonlinearityutilizedbythecomplexsystemsduring 0 coherence and supersymmetry. In particular it will be 0 formationofthespecificgrowthpatterns. Unfortunately, provedthatCDGapproachcanbeusedtogeneratequan- 3 theCDGapproachinitsoriginalformdoesnottakeinto 0 tumcoherentstatesofthetime-dependentMorse[10]and accounta veryimportantphenomenonof regression(de- 0 Wei [11] oscillators, which in the dissociation (classical) cay) appearing in biological, medical, demographic and . limit reduce to the well-know Gompertz [12] and West- 2 economicsystems. Forexample,incancerbiologysucha 0 Brown-Enquist (WBE)-type [13] functions (e.g. logistic, situation appears under chemotherapeutic treatment of 7 exponential, Richards, von Bertalanffy) describing sig- tumors subjected to cycle specific (or nonspecific) drugs 1 moidal (S-shaped) growth of complex systems. causing regression of cancer [14]. To include this phe- : v AccordingtotheCDGconcept,variousdegreesofnon- nomenon in the CDG scheme, the first of Eqs.(1) should i linearity appearing in the systems under consideration X be modified to the form can be classified using the set of nonlinear equations [1] r dψ(q) a † + x(q)ψ(q )=0, dψ(q) dx(q) dq † x(q)ψ(q)=0, +Φ(x)=0. (1) dq − dq ψ(q)†G = exp[−(1−exp(−q)], Here,q =uttdenotesdimensionlesstemporalvariable,ut ψ(q)†W = exp[1+c2−c2exp(−q)]−1/c2, (5) is a scaling factor, whereas Φ(x) stands for a generating which for Φ(x)=x and Φ(x)=x+c x2 produces Gom- function, which expanded into a series of x-variable (it 2 pertzandWBE-typefunctionsofregression,whichdecay slightly differs from the original CDG formulae) [1] with time. Φ(x)=c (x+c /c )+c (x+c /c )2+... (2) Analysis of Eqs. (1) and (5) reveals that they can 1 0 1 2 0 1 be interpreted in the framework of temporal version produces different functions of growth ψ(q) for a variety [15] of the space-dependent quantum supersymmetry ofpatternsemergingincomplexsystems. Toobtaintheir (SUSYQM) [16], used among others to construct coher- explicit forms a combination of Eqs. (1) is integrated to ent states of oscillators and to obtain exact solutions of 2 the Schrödingerequationforvibratingharmonicandan- 1 exp √2x q (x 1)q ψ (q) = exp − − e exp e− (9) harmonic systems. In view of this, it is tempting to ap- 0 " 2(cid:0)xe (cid:1)# (cid:20) √2xe (cid:21) ply CDG methodology to generate the coherent states of time-dependent anharmonic oscillators and compare recently obtained by Molski [20]. Taking advantage of themwiththoseobtainedusingalgebraicprocedure[20]. theRiccatiequation(6)onemayderivethesecondorder ToprovethatmathematicalformalismofPUisahidden differential equation whose solution is function (9) formoftime-dependentsupersymmetry,letsdifferentiate growth equation (1) once with respect to q-coordinate 1 d2 1 2 + 1 exp( √2x q) P ψ (q)=0, and then rearrange the derived formulae to obtain the (cid:26)−2dq2 4xe − − e − 0(cid:27) 0 second order differential equation in a standard eigen- (cid:2) (cid:3) (10) value form whichincludeseigenvalueP0 =1/2 xe/4beingaground − d2ψ(q) dx(q) dψ(q) state(v =0)versionofageneralformulae[20](indimen- ψ(q) x(q) = sionless unit) P =v+1/2 x (v+1/2)2, v = 0,1,2.... dq2 − dq − dq v − e ItisinterestingtonotethatEq. (10)undersubstitutions 1 d2 +V(q) P ψ(q) = 0, (6) xe =¯hω/4De, ω =a 2De/mc2 and q =a(t t0)/√2xe −2dq2 − converts to the explicit form of the space-lik−e Feinberg- (cid:20) (cid:21) p in which Horodecki equation [19] 1 dx(q) V(q)−P = 2(cid:20)x(q)2+ dq (cid:21) (7) −2m¯h2c2ddt22 +De[1−exp[−a(t−t0)]]2−P0c ψ0(t)=0 (cid:26) (cid:27) represents(withaccuracytomultiplicativeconstant)the (11) time-dependent version of the Riccati equation, whose for the ground state of the time-dependent Morse oscil- spatial form is widely used in SUSYQM [16]. The quan- lator. Here P = (h¯ω/c)(1/2 x /4) stands for a mo- 0 e − tityx(q)appearinginEqs.(1)and(7)hasadualinterpre- mentum eigenvalue representing zero-point momentum tation: in algebraic methods +x(q) represents an anhar- ofvacuum[21,22], x isanharmonicityconstant,ω -fre- e monicvariable[17],whereasinSUSYQM, x(q)=W(q) quency, D - dissociation constant, m - mass, c - light e − stands for a superpotential[16], which permits construc- velocity. tionofthesupersymmetricquantalequationsstraightfor- Proceedingin the same mannerasfor the firsttermof wardtoanalyticalsolution. Thisquantityenablesalsoto Eq. (2)onecanderivex(q)andψ(q)forthesecondorder associate via Eq.(7) a potential energy V(q) and eigen- expansion of Φ(x) = c (x+c /c )+c (x+c /c )2 and 1 0 1 2 0 1 value P with all types of functions ψ(q) derived in CGD identical as before initial condition x(0)=(1 c )/c 0 1 − scheme. Ontheotherhand,ψ(q)inEq. (6)isinterpreted c exp( c q) c asthesolutionofthedifferentialequation(6),whoseform x(q) = 1 − 1 0, resemblesthenon-relativisticversionofthequantalFein- c21+c2−c2exp(−c1g) − c1 berg equation [18] derived by Horodecki [19], which is a (sc /c )exp[ c (q g )] c 1 2 1 0 0 = − − , space-like counterpart of the time-like Schrödinger for- 1 sexp[ c (q q )] − c 1 0 1 − − − mula. ψ(q) = 1 sexp[ c (q q )] 1/c2 To prove that the CDG approach produces not only { − − 1 − 0 } classical(macroscopic)growthfunctionsbutalsoquantal sexp[ c (q q )] c0/c21, 1 0 { − − } (microscopic) once, one may apply a linear expansion of 1 2c2 c2c +2c c the generating function Φ(x)= c0+c1x, which includes q0 = c ln (2c1−+c12)2(c2+0c 2) , a constant term c omitted in the CDG scheme and c 1 (cid:20) 0 1 1 2 (cid:21) 0 1 c (2c +c2) coefficient, which in the CDG approach was constrained s = 2 0 1 (12) 2c2 c2c +2c c to1[1]. Additionally,weassumethatx(0)=(1 c0)/c1, 1− 1 2 0 2 − which for c =0, c =1 gives the CDG initial condition 0 1 and then one may construct the quantal Feinberg- x(0)=1. EmployingEqs. (1)and(3)byintegrationone Horodecki equation gets (c ,c >0) 0 1 1 1 d2 1 exp[ c1(q q0)] 2 x(q) = [exp( c q) c ], +D − − − P ψ(q)=0, c1 − 1 − 0 (−2dq2 (cid:20)1−sexp[−c1(q−q0)](cid:21) − ) 1 c (13) 0 ψ(q) = exp [1 exp( c q)] exp q , (8) c2 − − 1 −c for the ground state of the time-dependent Wei oscilla- (cid:26) 1 (cid:27) (cid:18) 1 (cid:19) tor [11] in which q = t, D = (2c +c2)2/8c2(1 c ) = which by making use of correspondences c21 = 2xe, c0 = mc2D /¯h2 and P = D c2/2c20 = 1mc3P1/¯h2−. 2The 1 x can be converted to equations e − 0 1 0 − e solution (12) and parameters appearing in (13) can be exp( √2x q) 1+x rewrittentotheformappliedbyWei[11]byreplacements e e x(q) = − − , (cid:20) √2xe (cid:21) c1 =b, s=c, 1/c2 =ρ+1/2, c0/c21 =ρ0. 3 Continuing the search for further analogies, we find mayconstructthe coherentstates ofthe time-dependent that equations of growth (1) and regression (5) can be Morse oscillator, which for c =0 and c = 1 convert to 0 1 specified for α=α =0 in the forms the Gompertzian coherent states of growth (regression) ∗ first time derived by Molski and Konarski[24] Aˆα =αψ(q)exp[√2αq], αAˆ =α ψ(q) exp(√2α q), † ∗ † ∗ | i h | Aˆ= 1 d x(q) ,Aˆ = 1 d x(q) , (14) 1 d 1 [exp( c q) c ] α = √2 dq − † √2 −dq − √2 dq − c1 − 1 − 0 | i h i h i n o here [Aˆ,Aˆ†] = −dx(q)/dq = Φ(x), familiar in supersym- αexp c121 [1−exp(−c1q)] exp −cc01q exp(√2αq) mofestpriaccet-hdeeopreyndoefnmt ionsicmilluamtorusn[c1e6r]t.aIinntythicsohfoerrmenatlissmta,teAˆs nc0=−0→,c1=1 √12 ddq −o[exp(−hq)] i|αi= andAˆ† representannihilationandcreationoperators,re- αexp[1 exnp( q)]exp(√2αoq) α=0 spectively. Thecoherentstates,whichminimize the gen- − − −→ eralizedposition-momentum(localstates)ortime-energy d exp( q) exp[1 exp( q)]=0, (19) dq − − − − (nonlocal states) uncertainty relations are eigenstates of h i the annihilation operator. They not only minimize the Heisenberg relations, but also maintain those relations α 1 d 1 [exp( c q) c ] = h |√2 −dq − c1 − 1 − 0 in time (space) due to their temporal (spatial) stability, n o hence they are called intelligent coherent states [23]. To α exp 1 [1 exp( c q)] exp c0q exp(√2α q) ∗ −c21 − − 1 c1 ∗ prove that coherentstates (14) minimize the generalized time-energy uncertainty relation (¯h=1) nc0=0,c1=1 α 1 do [exhp( qi)] = −→ h |√2 −dq − − [∆x(q)]2(∆E)2 1 αΦ(x)α 2, Φ(x)= i x(q),Eˆ , α∗exp [1 expn( q)] exp(√2α∗qo)α∗=0 {− − − } −→ ≥ 4h | | i − h (1i5) d exp( q) exp [1 exp( q)] =0. (20) −dq − − {− − − } inwhichEˆ =id/dqisenergyoperatorwhereasx(q)plays h i the role of a temporal anharmonic variable associated In a similar manner, one may construct the coherent withagiventypeofpotential,thefollowingrelationships statesoftime-dependentWeioscillator,whichinthe dis- should be derived for normalized states α α =1 sociation (classical) limit convert to the coherent WBE- h || i type function of growth 1 1 αx(q)α = αAˆ+Aˆ α = (α+α ), † ∗ h | | i −√2h | | i −√2 1 d (sc1/c2)exp[−c1(q−q0)] + c0 α = hα|Eˆ|αi = i√12hα|Aˆ−Aˆ†|αi=i√12(α−α∗), α 1 √s2exnpd[q −c1(q1−sqe0x)p][−1c/1c(2q−sq0e)]xp[ cc11(oq| iq0)] c0/c21 { − − − } { − − } 2 αx(q)2 α = (α+α∗)2+ αΦ(x)α , exp(√2αq) h | | i h | | i H−av2ihnαg|Eˆd2e|rαivied=E(qαs.−(16α)∗)w2e−chaαn|Φp(xas)s|αtio. calculate(t1h6e) c0=−0→,c1=1 √12(cid:26)ddq − (s1′−/cs2′)exexpp[−[−(q(−q−q0q′0)′])](cid:27)|αi= squared standard deviations α{1−s′exp[−(q−q0′)]}1/c2exp(√2αq)−α=→0 ∆x(q)2 = hα|x(q)2|αi−hα|x(q)|αi2 = 12hα|Φ(x)|αi, (cid:26)ddq − (s1′−/cs2′)exexpp[−[−(q(−q−q0q′0)′])](cid:27){1−s′exp[−(q−q0′)]}1/c2 = ∆E2 = αEˆ2 α αEˆ α 2 = 1 αΦ(x)α , (17) ddq − 1+c2excp2(−exqp)( q) [1+c2−c2exp(−q)]1/c2 =0(.21) h | | i−h | | i 2h | | i − − h i which prove that Here q0′ = ln[(2−c2)/(1+c2)] and s′ = c2/(2 − c2). AnalogicallytheWBEstatesofregressioncanbederived 1 [∆x(q)]2(∆E)2 = αΦ(x)α 2. (18) fromquantalsolutionsofthecreationequation(14). The 4h | | i results obtained indicate that the concept of PU origi- Eq.(18) is satisfied both for α = 0 as well as α = 0 and nally applied only to macroscopic complex systems can 6 an arbitrary form of generating function Φ(x). Those be extended to include quantum phenomena such as co- facts indicate that ψ(q) in CDG approach can be inter- herence and supersymmetry playing a vital role on the preted as a minimum uncertainty coherent state of the microscopiclevel. Inconnectionpresentedamicro-macro time-dependent oscillators characterized by anharmonic conversion is accomplished by c 0, which transforms 0 → variable x(q). It is noteworthy that this interpretation quantum equations into classical ones. Only one excep- remains independent of the type of generating function tion is uncertainty relation (15), which is satisfied both Φ(x), hence it canbe applied both to micro- and macro- for micro- and macroscopic functions ψ(q) generated for scopic systems, characterized by c = 0 and c = 0, re- an arbitrary form of Φ(x). This fact has very impor- 0 0 6 spectively. In particular, using the CDG approach one tant interpretative implications. The time-like coherent 4 states, which minimize the position-momentum uncer- generate in the CDG scheme the coherent states of the tainty relation evolve coherently in time being localized space-dependent Morse and Wei oscillators, which min- on the classical space-trajectory [16]. On the contrary, imize the position-momentum uncertainty relation [26] the space-like coherent states which minimize the time- andin dissociationlimit c 0 or,equivalently E D, 0 → → energyuncertaintyrelationevolvealonglocalized(classi- reduce to the space-dependent sigmoidal Gompertz and cal) time-trajectory being coherent in all points of space WBE-like functions widely applied in a range of fields [20, 24]. Such states assumed to be coherent at an arbi- includinge.g. probabilitytheoryandstatisticswhereare trarypointofspaceremaincoherentinallpointsofspace. used to describe cumulative distribution of entities char- We conclude that the spatial coherence is an immanent acterized by different spatial sizes [28]. feature of all systems whose growth (decay) is described by functions derived in the CDG scheme independently oftheirquantalorclassicalnature. Althoughthenotions of coherence and supersymmetry are usually attributed [1] P. Castorina, P. P. Delsanto and C. Guiot, Phys. 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