Coarse-graining complex dynamics: Continuous Time Random Walks vs. Record Dynamics

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by  Paolo Sibani

epl draft
Coarse-graining complex dynamics: Continuous Time Random
Walks vs. Record Dynamics.
Paolo Sibani1
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11 FKF, University of Southern Denmark, Campusvej 55, DK5230, Odense M.
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2

n
a PACS 02.50.Ey–Stochastic processes
J PACS 89.75.Da–Systemsobeying scaling laws
2 PACS nn.mm.xx–Fluctuation phenomena, random processes, and Brownian motion
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Abstract – Continuous Time Random Walks (CTRW) are widely used to coarse-grain the evo-

lution of systems jumping from a metastable sub-set of their configuration space, or trap, to
]
h another via rare intermittent events. The multi-scaled behavior typical of complex dynamics is
c provided by a fat-tailed distribution of the waiting time between consecutive jumps. We first
e
argue that CTRW are inadequate to describe macroscopic relaxation processes for threereasons:
m
macroscopic variablesarenotself-averaging, memory effectsrequireanall-knowingobserver,and
- different mechanisms whereby the jumps affect macroscopic variables all produce identical long
t
a time relaxation behaviors. Hence, CTRW shed no light on the link between microscopic and
st macroscopic dynamics. We then highlight how a more recent approach, Record Dynamics (RD)
. provides a viable alternative, based on a very different set of physical ideas: while CTRW make
t
a use of a renewal process involving identical traps of infinite size, RD embodies a dynamical en-
m trenchment into a hierarchy of traps which are finite in size and possess different degrees of
- meta-stability. WeshowinparticularhowRDproducesthestretchedexponential,power-lawand
d logarithmic relaxation behaviorsubiquitousincomplex dynamics,togetherwiththesub-diffusive
n
time dependence of the Mean Square Displacement characteristic of single particles moving in a
o
complex environment.
c
[

1
v
9
9 Introduction. – Statistical physics is largely about counted for. Nevertheless, a stationary renewal process is
1
coarse-grainingmicroscopic descriptions into macroscopic not a natural choice to describe the macroscopic changes
5
ones more closely related to experiments. Thermal relax- occurring in e.g. non-stationary relaxation processes.
.
1ation of ‘glassy’ systems is a case in point: Due to their
0large number of microscopic configurations from which a As emphasized in the much touted weak ergodicity
3
deterministic(zerotemperature)trajectoryneverescapes, breaking scenario [4,5] time and ensemble averages dif-
1
:configuration space can be partitioned into catchments fer for renewal processes involving fat-tailed waiting time
v
basins which, atfinite temperature, retaintrajectoriesfor distributions. This property is closely related to a well-
i
Xalapseoftime offinite andrandomduration. We refer to known mathematical result of Sparre-Andersen [6,7] by
rthese basins as traps, to the time spent in them as wait- the fact that the number of jumps in the interval[0,t) re-
aing time and to the transitions between traps as jumps. mains a distributed quantity in the limit t → ∞. Hence,
Describing relaxationin terms of traps and jumps greatly inaCTRWdescriptionmacroscopicquantitieshavebroad
reduces the number of variables and constitutes the first distributions even in the thermodynamic limit. A second,
step of coarse-graining. Based on Continuous Time Ran- related, issue is related to the system size dependence of
dom Walks (CTRW) [1–3], a well established approach the average and variance of macroscopic observables. As
furtherassumesthateachjumpbringsthe systembackto we argue, both quantities must scale linearly with system
the same situation, i.e. that the sequence of jumps con- size, but fail to do so in CTRW. Thirdly, the memory
stitutes a renewal process. Using a fat-tailed distribution mechanismimpliedbyCTRWrequiresanall-knowingob-
for the waiting time, the multi-scaled relaxation behavior server and, lastly, the long-time tail of the waiting time
characteristicofcomplexsystemscaninmanycasesbeac- distribution can hardly be justified in many applications.
Insummary,eventhoughCTRWappearflexibleandemi-
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Paolo Sibani1
nentlyapplicable,their useto modelcomplex dynamicsis variables are invariably delta-distributed, including cases
adubious endeavor. We arguebelowthat Record Dynam- where broken ergodicity stems from a broken symmetry.
ics(RD)isaviablealternativewhichreliesonacompletely To the best of the author’s knowledge, no experimental
different physical picture and which avoids the problems evidence has ever contradicted this result.
affecting CTRW, technically because the jumps are there Since a single CTRW process cannot consistently de-
a Poissonprocess. scribe macroscopic relaxation, let us instead try N inde-
A record in a time series is an entry larger (or smaller) pendent and simultaneous jumping processes, each sup-
than all the entries that precede it. Records have always portedinoneofN domains,asituationtypicalofspatially
beenapopulartopic,butarecentsurgeofinterestintheir extendedsystems withshort-rangedinteractions. The av-
statistical properties [8] seems motivated by the ongoing erageandthevarianceofthenumberofjumpsthroughout
debateonclimatechange,whichisaccompaniedbyanum- the system are in this case both proportional to N, and
ber ofrecordbreakingevents. Thatthermalnoiserecords σ (t)/µ (t) ∝ N−1/2 hence vanishes for large N, taking
j j
haveanimpactincomplexdynamicswasproposed[9]ina weakly broken ergodicity along. This sounds reassuring,
modelstudyofChargeDensityWaves. Overtheyearsthe but, as we shall see, the memory behavior implied by the
sameidea,whichwenowrefertoasRecordDynamics,has description requires an all-knowing observer.
found applications in condensed matter physics [10–13], For any choice of W(t), renewalequations for the jump
evolutionary biology [14] and the dynamics of ant soci- probability P(n,t),
j
eties[15]. Theterm‘record’inRDsignalsthatovercoming
t
arecord-sizeddynamicalbarrierelicitsajump–henceforth P(n,t) = P(n−1,t′)W(t−t′)dt′ ; n=1,2.(.3.)
in this connection termed quake— which brings the sys- j j
Z0
tem from one trap to a new and previously unexplored t
trap. RD hence describes a process of entrenchment into Pj(0,t) = 1− W(t′)dt′, (4)
a hierarchy of traps indexed by dynamical barriers of in- Z0
creasing size [16,17]. Focusing on the temporal statistics are solved in the s domain by
and the macroscopic effects of the quakes, RD provides a
coarse-graineddescription of glassy dynamics. n 1−W˜(s)
P˜(n,s)= W˜(s) . (5)
j
s
Critique of CTRW. – The probability P(n,t) of n
j (cid:16) (cid:17)
jumps in the time interval[0,t) and its first two moments Theaveragenumberofjumps, µ (t)= ∞ kP(k,t)and
paroeseddistciludsesetdodbeenloowte,thuesinLgaptlhaceelveattreiarbsleaannddtahesuLpaeprliamce- hthaeveatuhxeinliatrryanqsufoarnmtisty µj2−j(t) j= ∞kP=0k(=k02 −jk)Pj(k,t)
transform of a function, respectively. Central to the de- P
2
scription is the waiting time probability density (PDF) W˜(s) 2 W˜(s)
W(t). Wheneveritsaverageisfinite,theexponentialform µ˜j(s)= s(1−W˜(s)) and µ˜.j2−j(s)= s 1−W˜(s)! ,
W(t) = exp(−t/t )/t is a natural choice and, we stress,
0 0 (6)
a choice to which our critique does not apply. To model
respectively.
complex relaxation a ‘fat-tailed’ PDF lacking a finite av-
To derive Eq. (2), insert the small s expansion
erage
α t −α−1 W˜(s)=1−(t0s)α+O(sα+1) (7)
W(t)= , 0<α<1, t≥t , (1)
0
t t
0 (cid:18) 0(cid:19) oftheLaplacetransformofEq.(1)intoEq.(5). Inverting
the outcome yields
is utilized. Through mathematical steps detailed further
below, the average and variance of the number of jumps α
1 t
occurringin(0,t)areshown,asymptoticallyforlarget,to µ(t)≈ (8)
j
αΓ(α) t
be connected by the equation (cid:18) 0(cid:19)
and
2α
σj2(t)≈µj(t)+(cid:18)tt0(cid:19) (cid:18)αΓ(12α) − Γ2(α1+1)(cid:19), (2) µj2−j(t)≈ αΓ(12α)(cid:18)tt0(cid:19)2α. (9)
where Γ is the gamma function. For α=1, σj2(t)=µj(t). The result follows from σj2(t)=µj(t)+µj2−j(t)−µ2j(t).
Otherwise, in the large t limit, σ2(t) ∝ µ2(t) and since Ifthejumpsconstitutethetrueclockofthedynamics,it
j j
σ (t)/µ (t) then approaches a constant, the number of isnaturaltodescribetheireffectonrelaxationasaMarkov
j j
jumps retains a broad distribution in the same limit. As chain. Thequestionisthenhowthepropertiesofthelatter
the same is true for time averages of quantities subordi- affect the relaxation in the time domain. In general, the
nated to the jumps but not for the corresponding ensem- propagator of a Markov chain is a linear superposition of
ble averages, ergodicity is ‘weakly’ broken. In contrast, exponentially decayingmodes, eachof the form bn, where
textbookstatisticalmechanicsteachesusthatmacroscopic b< 1. The same is true for averages calculated using the
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Coarse-grainingcomplex dynamics: Continuous Time Random Walks vs. Record Dynamics.
propagator. Without loss of generality, we now consider suffice to elicit a quake, simply because there might be
thetimedependenceg(b,t)correspondingtoasinglemode no barrier to cross. Marginal stability [18] further posits
bn, which is obtained by averaging n over the probability thatthebarrierssuccessivelycrosseddifferbyaminuscule
P(n,t) that n jumps occur. In the Laplace domain this amount. Inthislimiteveryrecord-sizedenergyfluctuation
j
amounts to leads to the crossingofa barrierandrecord-sizedthermal
∞ 1−W˜(s) fluctuations triggerquakes. The temporalstatistics ofthe
g˜(b,s)= P˜j(n,s)bn = s(1−bW˜(s)). (10) qPuoaiskseosnopcrcoucrersins,gwbheotsweeaevnetrwagaenµd(tt>,tt)w∝is(ilnn(tth)i−sllnim(tit)a)
n=0 q w w
X
is independent of the temperature [9,14].
IfW(t) hasafinite averaget ,expandingEq.(10)tolow-
0 To generalize the above results to cases where energy
est order, we find that the mode decays exponentially in
barriersdifferbyafiniteamount,considerthat,ingeneral,
time, with a time scale t /(1−b) diverging as expected
0
forb→1. Wealsonotethatsinceg(b,t)actuallydepends t
′ ′
µ (t ,t)= r(t )dt, (12)
on b, the eigenvalue spectrum of the Markov chain mat- q w
Ztw
ters in the time domain. This hinges on the s term in the
denominator and the t s term in the nominator canceling where r(t) is the quaking rate and where the form r(t) =
0
out. The situation radically differs if W˜(s) = 1−(t s)α a/t correspondsto the logarithmic behavior associatedto
0
with 0 < α < 1. To leading order, Eq. (10) gives a term marginal stability. The generalized form
proportionaltosα−1,whichinthetimedomaintranslates
r(t)=atx−1, 0≤x≤1, (13)
into a power-law decay whose exponent, −α, is indepen-
dent of b. In other words, the value of the exponent α is
i)reducestoa/tforx=0,ii)producestime-homogeneous
unrelated to the dynamical effects of the jumps.
behavior for x = 1 and, iii) integrated with respect to
Consider now the simple scaling description known as
time, yields
pure orfull aging behavior,whichapproximatelycaptures
someaspectsofmemorybehavioringlassydynamics. Ac- µ (t ,t)=atx−txw d=ef f(t)−f(t ). (14)
q w w
cording to pure aging, certain macroscopic variables de- x
pendontheratio t ,e.g. inthe thermoremnantmagneti-
tw Since, as later argued, a Poissondistribution still applies,
zationofspin-glasses[12],t>t isthetimecountedfrom
w a particle happening to reside in a trap at time t leaves
w
the initialthermalquenchandt isthe time atwhichthe
w it at time t>t with probability
w
external magnetic field is switched off.
Knowingthatthesystemhasremainedinthesametrap P (t ,t)=exp(−f(t)+f(t )). (15)
0 w w
up to time t at which observationscommence, the prob-
w
ability density for exiting the trap at time t>tw is In terms of the lag time ∆ = t − tw (0 ≤ ∆ < ∞)
commonly used in lieu of t, the residence time R spent in
W(t) α t −α−1 a trap has PDF
WR(tw,t)= t∞w W(t′)dt′ = tw (cid:18)tw(cid:19) , (11) W (t ,∆) = −dP0(tw,tw+∆) (16)
R w
which is identicalRto the RD expression (17) obtained be- d∆
low by a different route. Since all traps are equivalent in = aexp[−a(t +∆)x−tx)](t +∆)−1+x.
x w w w
CTRW,the memorybehaviorimplied by Eq.(11)rests on
the observer knowing when a trap is entered. This might We note in passing that ∆ is often denoted by τ or by t
be experimentally achievable if a single trapdescribes the in the literature, both usages unfortunately clashing with
whole system, a possibility however already discarded as our present notation. For x≪1, one obtains
unphysical. If, however, N independent processes unfold
−a−1
atthesametime,theobservermusttrackwheneverytrap W (t ,∆)≈ a 1+ ∆ (17)
R w
is accessed, a task hardly feasible in experiments. tw (cid:18) tw(cid:19)
Dynamical hierarchies, records and marginal which is equivalent to the CTRW expression given by
stability. – Upward rooted binary trees [16, 19–21] Eq. (1). Importantly, the time scale parameter which is
whose nodes and height respectively represent traps and fixed inCTRW is simply the system agein RD. Secondly,
theirenergyprovideaconvenientcoarse-grainedrepresen- Eq. (17) contains a stretched exponential, and its simi-
tation of energy landscapes with multiple minima. In low larity to W(t) is restricted to the limit x → 0. Thirdly,
temperature thermalization, the ‘bottom’ states of low- the RDparameteraispositivebutnota priori limitedto
est energy are those mainly occupied, and gaining access the unit interval. For a single hopping process and in the
to nodes not previously visited entails crossing an energy limit x→0,ifeachbarrierrecordtriggersa quake,a=1,
barrierof recordmagnitude. Hence, diffusion on a hierar- otherwise 0 < a < 1. In extended systems, where several
chical structure can be described in terms of RD. In the independent hopping processes occur simultaneously, a is
general case, a record-sized energy fluctuation does not proportional to the size of the system, as we later argue.
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Paolo Sibani1
AccordingtoEq.(17),theaverageorcharacteristictime ageisexperimentallyobservedincolloidalsystems[22,24].
spent in a trap occupied (but not necessarily entered) at The age dependence of the life-time of the exponential
time t is approximation given in Eq. (21) is not usually discussed,
w
but follows nevertheless by the simple Taylor expansion
x 1−1 a 1 a
t (t )=a−1 x exp( tx)Γ( , tx), (18) given above. In the limit x → 0, Eq. (20) reduces to the
0 w a x w x x w
power-law
where Γ(s,z) = ∞(cid:16)ex(cid:17)p(−y)ys−1dy is the incomplete up-
z
pergammafunction. Asacheckwenotethatt0(tw)=a−1 t −a(1−b) a(1−b)
for x = 1 and thRat in the limit x → 0 t (t ) → t , for g(tw,t)=c(tw) ≈c(tw)exp(− ∆),
0 w w t t
a > 1. In the same limit and for a ≤ 1, the average is (cid:18) w(cid:19) w (22)
infinite, but tw still provides the characteristic time scale where the exponential approximation holds for ∆ ≪ tw.
for the power-law decay implied by Eq. (17). Anticipatingalaterobservation,wenowletµ bepropor-
q
Assume now that an applicationspecific function f has tional to the system size N of a macroscopic system via
been found such that the probability density for the oc- a = Na˜, where a˜ is a new constant. Secondly, we treat
currence of a quake is uniform in the stretched observa- bn =exp(˜bn) as one mode of a relaxation process param-
tion interval f(t)−f(tw). Partitioning the interval into eterized by n in lieu of time. Of the N eigenvalues in the
M subintervals, let p be the probability that a quake falls spectrum most will approach zero as N → ∞. A glance
intoanyoftheseandnotethattheprobabilityfornquakes at Eq. (20) shows that only those for which ˜b = O(1/N)
occurringisthebinomialB(p,n,M). Intherelevantlimit produce a macroscopic decay independent of N. If the
p → 0, M → ∞ and pM → µq, the binomial tends to a decay of ˜b with N is faster respectively slower than 1/N,
Poissondistribution, as claimed. the corresponding mode in the time domain either has a
Using a binary tree to coarse-grain an energy land- ‘frozen’ constant value or immediately decays to zero in
scape [16], we just argued that RD dynamics arises in the large N limit. Note that the stretching exponent x is
two ways: in the limit x→0, successive barriers increase independent of system size, while the exponent −a(1−b)
marginally, records in the impinging noise induce barrier in Eq. (22) is only N independent if, as just discussed,
crossings and, on average, the typical number n ≈ µq of ˜b=O(1/N).
barriers crossed at time t is proportional to ln(t). The
In summary, simple and general RD arguments lead
Arrhenius relation ln(t)∝b(n)/T where T is the temper-
to dynamical behaviors common in complex systems:
′
ature and b(n) is the height of the nth barrier then im-
stretchedexponentialrelaxationandpowerlawswithpure
plies b(n) ∝ Tn. If marginal stability is relinquished, i.e
aging scaling. The sub-diffusive behavior of a single par-
for x>0, we find ln(µ )≈ln(n(t))∝xln(t) for t>>t ,
q w ticle moving in a complex environment is discussed next.
′
fromwhichweinferthatthesizeofthenthbarriercrossed
is b(n)∝(T/x)ln(n). Subdiffusion. – Irreversible single particle jumps in
The time dependence of a macroscopic quantity, say g, complex environments, e.g. binary Lennard-Jones mix-
is calculatedinRD by averagingits dependence g˜(n)over tures in their glassy phase [25] are indicative of collective
the probability of n quakes occurring in (t ,t), i.e. configurationalre-arrangements. The same is, we believe,
w
∞ true for single particle diffusion in a living cell, a problem
(µ (t ,t))n
g(t ,t)=e−µq(tw,t) g˜(n) q w , (19) which has recently been modeled using CTRW [26]. It is
w
n! difficult to imagine howa living cell cancontainthe traps
n=0
X
of infinite, or at least very large,spatial extension needed
where µ (t ,t) is given in Eq. (14). As a first example,
q w to produce a waiting time distribution with a long-time
assume g˜(n) = c(n = 0)bn, where c expresses the initial
tail, especially considering that the diffusing particle and
condition and where b < 1. The stretched exponential
its enclosure have similar length scales.
behavior ubiquitous in glassy dynamics [22,23]
Experimental data for dense colloidal system [27] re-
g(tw,t)=c(tw)e−µq(tw,t)(1−b) =c(tw)e−(tx−txw)a(1x−b), analyzed in [13] show that single particle Mean Square
Displacement(MSD) growlogarithmically,a propertyex-
(20)
plained in Ref. [13] using RD. This result, which corre-
immediately follows. If g is a one-point average, c(t ) =
w
c(t0)e−txwa(1x−b) and there is in reality only one time argu- gspleonpdasrttiocletsheinlimgeitnexra→l p0roinbeEtqh.e(2lo4)c,alsurgeg-aersrtasntgheamtesnints-
ment t. In contrast, a two-pointcorrelationfunction with
of their aging environment. This leads to sub-diffusion
c(t =1)trulydependsontwoarguments,aswellknown.
w formulasrathersimilartotheirCTRWcounterparts. Dis-
Again using the lag time ∆=t−t Eq. (20) is recast,
w tinguishing between the two approaches can therefore be
for ∆/t ≪1 into
w experimentally challenging, as it e.g. requires an analy-
g(tw,∆)=c(tw)e−τ(∆tw), (21) sis of higher moments and/or an explicit investigation of
age dependencies via ensemble averages. To avoid convo-
an exponential decay with a characteristic time constant lutedtypographythesamesymbolisusedforthemoments
τ(tw)= ax(t11w−−bx). A relaxation time increasing with system of the particle position, irrespective of the method used.
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Coarse-grainingcomplex dynamics: Continuous Time Random Walks vs. Record Dynamics.
Note however that CTRW formulas have one time argu- To better discriminate between CTRW and RD, con-
ment, while RD formulas mostly have two. sider the ratio B between the fourth and the squaredsec-
After performing n independent jumps, eachassociated ond moment of X. Given n jumps, the fourth moment
toarandomadditivepositionchange∆x ,apointparticle is
i
is located at µ (n)=ne +(n2−n)e2. (30)
n X4 4 2
X(n)= ∆xi. (23) Correspondingly in the time domain
i=1
X
Assumeforsimplicitythattheidenticallydistributed∆xi µ (t)≈µ (t)e + t 2α e22 (31)
havevanishing oddmoments anddenote their secondand X4 j 4 t αΓ(2α)
(cid:18) 0(cid:19)
fourth central moments by e and e , respectively. The
2 4
form of these moments will depend on e.g. whether the for CTRW and
particlesmoveinapotentialwell,buttheargumentsbelow
µ (t ,t)=µ (t ,t)e +(µ (t ,t))2e2 (32)
do not. X4 w q w 4 q w 2
After n jumps, the variance of the particle position or,
for RD. For CTRW, the ratio
equivalently its MSD, is σ2 (n)=ne . Hence,
X 2
µ (t) αΓ2(α) e 1
σX2 (t)=µj(t)e2 and σX2 (tw,t)=µq(tw,t)e2 (24) B(t)= (σ2X4(t))2 ≈ Γ(2α) + e42µ (t) (33)
X 2 n
for CTRW and for RD, respectively. Explicitly, using
Eq. (14), we find the sub-diffusive behavior approaches αΓ2(α) as t → ∞. In the same limit, the RD
Γ(2α)
expression
σX2 (tw,t)=atx−xtxwe2. (25) B(tw,t)=1+ ee42µ (t1 ,t) (34)
2 q w
Note that if the first jump moment e1 differs from zero a approaches unity, independently of the exponent x. This
formula of the same type holds for the average position. difference offers an opportunity to discriminate between
Writing for conveniencetw =y andt=y+∆, where ∆ is the twodescriptions. Assuming thata salienteventdefin-
the lag time, and expanding Eq. (14) to first order in ∆, ing the age of the system can be identified, a second pos-
we find sibility is to investigate whether the particle MSD has an
a d(yx)
µ (y,y+∆)≈ ∆. (26) aging dependence by performing ensemble averages. This
q x2 dy dependence is present in RD but not in CTRW.
Experimentally, the variance is estimated using the time
Discussion. – The eminent applicability of CTRW
integral
conceals a number of theoretical issues. Firstly, fat-tailed
1 tmax−∆ waiting time PDFs for spatially confined processes, such
σ2 (t)= [X(y+∆)−X(y)]2dy, as diffusion in cellular environments are in general cur-
X t −∆
max Z0 tailed by finite size effects. Secondly, macroscopic vari-
(27)
ables modeled with a single CTRW feature an unphysical
where t is the largest observation time. This corre-
max lack of self-averaging. Finally, since memory in CTRW
sponds to averaging µ (y,y +∆) with respect to y over
q cannot be rooted in the unchanging physical properties
thesametimespan. Tofirstorderin∆,thetimeaveraged
of the traps visited, it must be rooted in the observer’s
particle MSD is then
awareness of the time at which a trap is entered. This
a knowledge is only available (in principle) if traps pertain
σ2 (∆,t )≈ tx−1∆ for ∆<t (28)
X max x2 max max totheentiresystem,thepossibilityalreadyinvalidatedby
the lack of self-averaging.
If, on the other hand, ∆ ≈ t , time averaging is of
max Broadly speaking, RD has the same range of applica-
dubious value, and Eq. (24) directly implies
tions as CTRW, but shares none of their problems: Res-
idence times have, with a single exception, a finite aver-
a
σX2 (∆)≈ x∆x. (29) age which increases systematically with system age. This
means that, in contrast to CTRW, macroscopic configu-
Taken together, Eqs. (28) and (29) describe a cross-over rations contain information on the system’ s age, a fact
oftheMSDfromalineartoasub-lineartimedependence, which naturally explains memory behavior in RD. Since
abehaviorobservedbyJeonetal.[26]intheirexperiments quakes are a Poisson process, albeit of an unusual kind,
on lipid granules in an intracellular environment. These subordinatedphysicalquantitiesarealwaysself-averaging.
authors claim that their findings ‘unanimously’ point to Using averages over the number of quakes, RD produces
CTRW as the mechanism behind sub-diffusive behavior, a wide ranging analytical description of glassy relaxation
butaswejustarguedRDoffersanalternativeexplanation. and of single particle diffusion in complex environments.
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Paolo Sibani1
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seemstofindits wayinto macroeconomics[29], wasadvo- diffusion in the logarithm of time. Journal of Physics-
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