Time Reversibility of Quantum Diffusion in Small-world Networks

0.29 MB·English

by  Sung-Guk Han

Time Reversibility of Quantum Diffusion in Small-world Networks
∗
Sung-Guk Han and Beom Jun Kim
Department of Physics and BK21 Physics Research Division,
Sungkyunkwan University, Suwon 440-746, Korea
(Dated: January 4, 2012)
We study the time-reversal dynamics of a tight-binding electron in the Watts-Strogatz (WS)
small-worldnetworks. Thelocalized initialwavepacketattimet=0diffusesastimeproceedsuntil
the time-reversal operation, together with the momentum perturbation of the strength η, is made
2
at the reversal time T. The time irreversibility is measured by I ≡|Π(t=2T)−Π(t=0)|, where
1
Π is the participation ratio gauging the extendedness of the wavefunction and for convenience, t is
0
measured forward even after the time reversal . When η=0, thetime evolution after T makes the
2
wavefunctionatt=2T identicaltotheoneatt=0,andwefindI =0,implyinganullirreversibility

n oracompletereversibility. Ontheotherhand,asηisincreasedfromzero,thereversibilitybecomes
a weaker, and we observe enhancement of the irreversibility. We find that I linearly increases with
J increasingη intheweakly-perturbedregion,andthattheirreversibilityismuchstrongerintheWS

2 network than in thelocal regular network.

] PACSnumbers: 89.75.Hc,73.20.Jc,05.70.Ln
h
Keywords: Small-worldnetwork,Quantum diffusion,Time-reversaldynamics
c
e
m
Time-irreversiblephenomenainnature,suchasspread quantum system shows τ ∼ logN in the former while
- ofaninkblobinwaterandagingoflivingorganisms,are τ ∼N for the latter structure. An abrupt change in the
t
a ubiquitous in our macroscopic world [1]. We always see scaling behavior of τ has been shown to occur as soon
st the inkblobspreadinateacupandpeople getolder,not as the rewiring probability becomes nonzero. In other
. the other way around. These time-irreversible behaviors words, a transition from the slow world (τ ∼ N) to the
t
a lookpuzzlingbecauseallthesedynamicsarebasedonmi- fast world (τ ∼ logN) occurs at null rewiring probabil-
m
croscopicequationsofmotion,quantumorclassical,that ity simultaneously with a structural transition from the
- containtime-reversalsymmetry. Inthelate19thcentury, largeworld(l ∼N)tothesmallworld(l ∼logN),withl
d
this seemingly paradoxical observation of time-reversal being the averagepath length connecting two arbitrarily
n
microscopic dynamics and time-irreversible macroscopic chosenvertices. Consequently,achangeinthescalingbe-
o
c thermodynamicsperplexedmanyscientistsuntilabreak- haviorofτ reflectsadynamicalaspectofthesmall-world
[ through was made by Boltzmann. It is now well-known transitioninWSnetworks. Inthiswork,weexaminethe

that the directionalityof the arrow of timein the second effect of shortcuts on the time reversal dynamics of the
1
law of thermodynamics can be understood by the huge tight-binding electron system at different perturbation
v
1 difference in the numbers of allowed microscopic states strengths.
7 between the initial and the final macroscopic states. The time evolution of the tight-binding electron in
4 Recently, the quantum and the classical time-reversal a network structure is governed by the following time-
0 dynamics of a diffusion system and a spin system have dependent Schr¨odinger equation:
.
1 been studied to answer the long-standing question of
∂|Ψi
0 where does the irreversibilitycome from[2–5]. In Ref. 3, i~ =H|Ψi, (1)
2 quantumdiffusionsystemswereshowntodisplaytheuni- ∂t
1 versal behavior of time irreversibility in terms of per- where ~ is the Planck constant, and the quantum me-
:
v turbation strength. Because the complex networks have chanical ket |Ψi and the Hamiltonian H in position rep-
i drawn much interest, there have been studies of quan- resentation are written as Ψn ≡hn|Ψi and
X
tum systems in Watts-Strogatz (WS) small-world net- ′
ar works [6]. Localization phenomena of quantum systems Hnn′ =Hn′n =(cid:26)∆0 ofotrhenrw∈isΛe,n (2)
in small-world networks have been studied [7, 8], and
quantum diffusion problems have been investigated in respectively. Here, n is the vertex index, Λ is the set of
n
various ways in WS networks [9–12]. directly connected vertices of n, and the on-site energy
In the present work, we aim to study the time re- hasbeenassumedtobeuniformandsetequaltozero. As
versibility of the quantum diffusion problem in WS net- afurthersimplification,wealsoassumethatthehopping
′
works. The diffusion of a wave packet has been shown energy∆doesnotdependonnorn. Henceforth,weuse
to occur much faster in WS networks than in a regular the dimensionless units: the time is measuredin units of
network [10, 11]: A suitably defined diffusion time in a ~/∆, the position in units of the lattice spacing a of the
one-dimensional (1D) regular lattice without shortcut,
and the momentum in units of ~/a.
Once the WS network for a given rewiring probability
∗ Correspondingauthor: [e-mail] p is constructed following the standard procedure (see
2
Ref.6),weperformthenumericalintegrationofthetime-
80
dependent Schr¨odingerequationstarting fromthe initial (a) p=0.0
localized wave packet at time t = 0 (Ψn = δn,0) by us- 60
ingthefourth-orderRunge-Kuttaalgorithmwiththedis-
crete time step dt = 0.01. Except for p = 0, all results Π 40
are obtained from the average of 1000 different network
realizations. We use the time-reversal test similarly to 20
Ref. 3: At the reversal time T, a momentum perturba-
0
tion of the strength η is made by applying the operator 0 0.5 1 1.5 2 2.5 3 3.5 4
exp(iηxˆ) to |Ψ(T)i with the position operator xˆ. For
t
convenience, we measure t in the forward direction even
after the reversal time T so that at t = 2T, the system 80
(b) p=0.2
goes back to the initial state in the absence of the per-
60 η=0.1
turbation (i.e., when η = 0). During the numerical time
integration, we compute the participation ratio
Π 40
Π(t)= Nn=1|Ψn(t)|2, (3) 20
PNn=1|Ψn(t)|4 0 η=0.0
P 0 0.5 1 1.5 2 2.5 3 3.5 4
which has been frequently used in the study of localiza-
tion phenomena. When the wavefunction is completely t
localized, Π takesthe value ofunity. Onthe other hand, 103
as the wavefunction spreads in space and the quantum (c)
system becomes extended, Π is known to be O(N). The
101 p=0.0
time irreversibility I is then defined as 0.1
I 0.2
I ≡|Π(2T)−Π(0)|. (4) 10-1 0.4
0.6
0.8
It is to be noted that Π(t < T) is independent of η 1.0
whereas Π(2T) and I are functions of η. When η =0, I 10-3
0.001 0.01 0.1
takes the value zero, meaning the null irreversibility (or
η
the complete reversibility), because Π(2T) = Π(0). As
η is increased from zero, the perturbed momentum at T
FIG. 1. (Color online) The participation ratio Π(t) as a
makes it difficult for the system to go back to the initial
function of time t in (a) the local regular network and (b)
quantum state; thus, I is expected to increase.
the WS network at rewiring probability p = 0.2. The sys-
Figure 1 displays Π(t) in Eq. (3) (a) for the local 1D
tem size N = 1600 and the reversal time T = 2 are used.
regularnetworkcorrespondingtothe WSnetworkatthe
Although not clearly discernible, (a) and (b) contain 101
rewiring probability p = 0 and (b) for the WS network different curves for the momentum perturbation strength
at p = 0.2. Although not clearly discernible, each of η =0.0,0.001,0.002,··· ,0.1 (from bottom to top). The per-
Fig. 1(a) and (b) has 101 different curves corresponding turbed and time-reversed quantum state in the WS network
to η = 0.0,0.001,0.002,···,0.1 (from bottom to top). in(b)showsamuchstrongerdeviationfrom theinitiallocal-
We also show the time irreversibility I at various values ized quantumstate. (c) Theirreversibility I=|Π(2T)−Π(0)|
of p in Fig. 1(c). The system size N = 1600 and the versus η at various rewiring probabilities p. When p = 0.0
reversaltimeT =2areused. WefirstobservethatΠ(T) for the local regular network, I increases with increasing η.
On the contrary, when p 6= 0.0, I first increases, but soon
is about four times larger for p = 0.2 than for p = 0.0,
saturates. Theoscillatory behaviorobservedforp=0.0orig-
which is explained by the fast diffusion in the WS net-
inatesfromthediscretenessoftheunderlyingperiodiclattice
workstructure,asdiscussedinRef.10. Atboth(a)p=0
structure.
and (b) p=0.2, time reversaldynamics is shown to give
risetoalargerreflectionasymmetryofΠ(t)aroundt=T
as η is increased. This is not surprising because a larger
momentum perturbation will surely make the time re- from the unperturbed one. The enhancement of the ir-
versed quantum state at t = 2T more different from the reversibility in the WS network is more clearly seen in
initial state at t = 0. A more interesting observation Fig. 1 (c) for I versus η at various rewiring probabil-
one can make from the comparison of Fig. 1 (a) and (b) ities p. Figure 1(c) shows that the irreversibility is a
is that the deviation of Π(2T) from Π(0) is much big- monotonically increasing function of the rewiring prob-
ger for the WS network (p = 0.2) than for the regular ability p. In other words, the enhanced structural ran-
network (p =0.0). This clearly indicates that the struc- domness caused by more shortcuts drives the quantum
tural irregularity in the WS network plays an important diffusion dynamics in the WS network to become more
role in making the perturbed dynamics deviate severely sensitive to the momentum perturbation. Interestingly,
3
104 of the average distance ξ between the two endpoints of
N=1600 shortcuts, i.e., ξ ∼ N/pN = 1/p. If the tail part of the
103 3200 (a) T=1 wavefunctionof the tight-binding electron does not have
6400
102 12800 enough time to arrive at the closest shortcut endpoint,
I 101 the system should behave just like a 1D regular lattice.
Also,quantumdiffusioninaregular1Dsystemisknown
100 tohaveadiffusiontimeproportionaltothesystemsizeto
10-1 becomefullyextended(τ ∼N),whichindicatesthatthe
10-3 10-2 10-1 100 101 wavepacketin the 1D systemspreads ona distance scale
pT ℓ∼T before reversal time. Accordingly, if T ∼ℓ.1/p,
104 the system behaves just like a 1D regular lattice while
N=1600 (b) T=10 it changes its behavior as T ∼ 1/p is crossed. Let us
3200
103 6400 consider several cases: (i) For pT . 1, the system be-
12800 haves like a 1D system because T is short enough that
I thewavepacketdoesnothavetimetoarriveatashortcut
102 endpoint. Furthermore, in this case of short-time diffu-
sion, all observed results should be independent of N as
101 long as T ≪ N; i.e., T is too short for the particle to
10-2 10-1 100 101 102 feel the finiteness of the system. (ii) For pT & 1, the
pT tight-binding particle now begins to meet shortcut end-
100 points and the diffusion behavior changes. As soon as
the WS networkbegins to have a finite fractionofshort-
10-1 (c) cuts,thequantumdiffusionbehaviorisknowntorapidly
N= 3200, T= 5 change so that the diffusion time τ increases only log-
N 10-2 N= 3200, T=10
I/ N= 6400, T= 5 arithmically with increasing N. This implies that the
10-3 N= 6400, T=10 wavefunction spreads in a distance that increases expo-
N=12800, T= 5 nentially with time. We, thus, conclude that this very
N=12800, T=10
10-4 fast diffusion occurs when pT & 1 and that the parti-
10-2 10-1 100 101 102 cle can arrive at the other side of the system, giving rise
pT to an N-dependence in the irreversibility. Consequently,
the size-independence of I in the region of pT . 1 is
FIG. 2. (Color online) Thetime irreversibility I in Eq.(4) is changed as we enter the intermediate region of pT & 1.
measured for various values of the rewiring probability p at
(iii) Forthe limiting caseofpT ≫1,the wavefunctionof
a fixed value of the reversal time T to plot I versus pT for
the particle has already completely spread; thus, the ir-
(a) T = 1 and (b) T = 10. For pT . 1, the system is in
reversibility should not depend on pT. In this long-time
the 1D region and I does not depend on the system size N.
limit, the wavefunction becomes fully extended, and the
As pT is increased to become pT &1, the system enters the
participation ratio is Π ∼ N. Consequently, for a big
small-worldregionanddifferentsizesbegintoshowdeviations
from each other. As pT is increased further, I saturates to enoughsystem,I(pT)/N saturatesaspT becomeslarger,
some finite value: I(pT → ∞)/N ∼ const., independent of approaching a value that does not depend on N.
T,as shown in (c). Thestrength of momentumperturbation
The results for the irreversibility I versus pT curve at
is η = 0.1, and all results are averages over 1000 different
η =0.1showninFig.2clearlyfittheabovehand-waving
network realizations.
scalingarguments well. In Figs.2(a) and (b), I is shown
to be independent of N for pT . 1 whereas we see a
clear size dependence for pT & 1. As pT is increased
theWSnetworkatasufficientlylargerrewiringprobabil-
even further, I saturates to some values that depend on
itybehavesverydifferentlyfromthelocalregularlattice:
N, as clearly seen in Fig. 2(b). Figure 2(c) shows that
For p > 0, I increases linearly with increasing η for the
the saturated value of I/N in the limit of pT ≫ 1 is
weakly-perturbedregionandthen soonsaturates to a fi-
independent of N.
nite value. In contrast, for p = 0, I increases linearly
with increasing η without showing saturation. However, In summary, we studied the time reversaldynamics of
these findings need to be carefully examined in view of the tight-binding electron system in the WS small-world
thefinitescalesinthesystem,i.e.,finitenessofthelength networks. Initially, the localized quantum mechanical
scale (N) and the time scale (T). stateofthe electronevolvesintime untilthe momentum
It should be noted that in WS networks there is an perturbation of the strength η is made at T, at which
additionallengthscaleotherthanthemicroscopiclength time reversal operation is also made. The 1D regular
scale of the lattice spacing a and the macroscopiclength lattice andthe WS small-worldnetworkexhibit different
scale of the system size N. When a nonzero rewiring behaviors: the irreversibility,measured by the difference
probability p is given, the number of shortcuts is pro- betweentheparticipationratioofthefinalandtheinitial
portionalto pN, whichdetermines the thirdlengthscale states, is found to be bigger in the latter network due to
4
the structural randomness caused by the random short- ACKNOWLEDGMENTS
cuts. It is also found and argued that the irreversibility
does not depend on the system size for pT .1 and that ThisworkwassupportedbytheKoreaResearchFoun-
it saturates to a value as pT becomes larger. dation Grant KRF-2009-013-C00021.
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