Phenomenological Universalities: Coherence, Supersymmetry and Growth

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by  Marcin Molski

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 (clas-
sical)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 Schr¨odingerequationforvibratingharmonicandan- 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 Schr¨odinger 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
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