Z. Phys. B 96, 409-416 (1995)

ZEITSCHRIFT FORPHYSIKB 9 Springer-Verlag 1995

Relaxation in self similar hierarchies C. Uhlig 1, K.H. Hoffmann

2 p.

Sibani 3

1Fachbereich Physik, Universitfit-GH-Essen, D-45117 Essen, Germany (e-mail: [email protected]) 2 Physik, TU Chemnitz-Zwickau, D-09009 Chemnitz, Germany (e-mail: [email protected]) 3Fysisk Institut, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark (e-mail: [email protected]) Received: 27 June 1994

Abstract. We investigate in some detail the relaxation process in self similar hierarchies. We find that the process can be divided in four different time regimes. After an initial phase in which the connectivity of the hierarchy determines the relaxation, the system enters a kind of stationary state, which can be accurately described by a simple analytical sink-picture. At longer times the behavior of the process is correctly described by the idea of quasiequilibrium. In this regime, propagators decay with power-laws. Finally, the global equilibrium state is reached, and the evolution stops. PACS: 05.40; 05.70L; 02.50

1. Introduction 'Complex' systems, like spin glasses, glasses, proteins, and biological or artificial systems capable of learning, have received steadily increasing attention during recent years. These systems usually possess a highly dimensional statespace, where a strongly non-convex energy or cost function can be defined. The coupling to an external bath leads to a stochastic relaxational dynamics. Typical behavior includes non-exponential relaxation, aging [1, 2], and strong sensitivity to temperature fluctuations, as best demonstrated in spin-glass experiments. Although there might not be a unique mechanism behind all these phenomena, it is useful to consider theoretical models which yield to mathematical analysis, while at the same time embodying at least some of the observed features of complex relaxation, as for instance discussed by Palmer [3] long ago. Hierarchical models belong to this class, and have therefore extensively been studied [4, 5, 6]. The applications range from turbulence [7] to several aspects of spin-glass dynamics [4, 5, 10]. Recent numerical investigations demonstrate that the relaxational dynamics of other standard problems as the Travelling Salesman Problem [8] and the spin-glass prob-

lem indeed show emergent hierarchical features, thus supporting the relevance of hierarchical modelling. The simplest and best known class of models consist of a set of states arranged hierarchically in levels to form a regular upside rooted tree. For each level one defines an energy, which is assumed to increase with the level. The stochastic dynamics is then defined by a thermally activated random walk on the tree. Although simply defined, random walks on regular trees already have rich mathematical properties. At sufficiently low temperature, their propagators have a characteristic algebraic decay. The exponents depend on the parameters of the walk, i.e. the temperature for the thermal case [5]. In some cases, a transition to exponential relaxation at higher temperature is also found. It is natural to investigate the generality of these properties by enlarging the class of models under consideration. A first step in this direction was taken by two of the authors [6], who introduced and studied the so called LS-tree. This model contains two energy scales rather than one, whereby the degeneracy of the local minima is lifted. It was found that the introduction of local minima changes the dynamics in important respects. One finds a family of temperature dependent exponents for the algebraic relaxation, rather than a single one as in the degenerate case. This result was obtained by an approximate analytical calculation, where the relaxation is described as a succession of quasiequilibrium states in a sequence of subtrees containing the initial node. In this paper we extend the previous short paper in several ways. We first numerically demonstrate the correctness of the quasiequilibrium assumption by an exact numerical calculation. We then provide a more complete description of the relaxation behavior on trees by introducing a quantity, the 'lateral variance', which quantifies the degree of equilibration. We show that the relaxation behavior on 'intermediate' time scales, which was not mentioned in the short paper, can be understood analytically. Finally, we examine the effect of spatially inhomogeneous rates, which can play a role in modelling the relaxation behavior under temperature variations.

410

2. The model

I=A 1 or A 2"

2.1. The tree

klu eeq (x) = k I Peq (Y)"

Following [6], the state space of our model is constructed as shown in Fig. 1. The building block is a 'mother' node connected to two 'daughters' of lower energy, the energy differences being A t and A 2 respectively. The tree is constructed iteratively out of one initial block by i) duplicating the already existing structure, and ii) identifying the top nodes of the two resulting twins with the daughters of the building block. For Am =A2 the structure is regular and coincides with that studied e.g. in [7]. A level in the tree is the set of nodes connected by the same number of bonds to the root. Each node has thus a unique level index l, with the local energy minima being at level 0, their mothers being at level 1 and so forth, up to the root node being at level lm~x, which is the height of the tree. Each node is contained in a unique subtree of height m, dubbed the m'th subtree containg that node. The whole tree Z contains N = 2 ~m=+~- 1 nodes. For each node x we introduce a degeneracy g (x), which can account for the number of microstates lumped into the node in a coarse graining procedure. We assume a level dependent degeneracy g (x) = g (l) = x / for all nodes x at level/. The parameter tc is called entropic factor.

It is well known [9] that the Master equation can be symmetrized by introducing a new dependent variable

2.2. The dynamics The model is completed by a probabilistic dynamical rule. We assume a thermal activated marcovian process described by the master equation:

(3)

1

u(x,t)=- )'r-q-x~/P ( P(x,t)

(4)

whereby the solution can be expanded in a complete set of eigenvectors to the symmetrized evolution operator: N

P(x,t)=~Peq(X)

~, ~t(x,i)e~itgt(k,i) i,k=l

P0 (k) x V e~/-eq(k) "

(5)

Here Po is the initial distribution, and 6(k, i) is the kth component of the ith normalized eigenvector of the symmetric matrix. The largest eigenvalue of the transition matrix is 21 = 0 corresponding to the equilibrium solution a (k, 1) = ~FPeq (/s

We are particularly interested in the solution of (1) for the sharp initial condition P(y, O[xo)= Oyx , which can be conceived as being prepared by a quench~ i.e. the conditional probability:

P(Y'tIx~

V -~'q~O) X a(Y,i)a(Xo, i)eZ~t"

(6)

i=l

N

O,P(x,t)= ~, F ( x , x ' ) P ( x ' , t ) ,

(1)

x,=l

3. The concept of quasiequBibrium

where the transition matrix F fulfills the stochasticity condition ~ F ( x ' , x ) = O and the matrix elements (or x"

rates) must obey detailed balance:

1"(x', x) Beq(x) = 1"(x, x') Peq (x').

(2)

The off-diagonal elements are nonzero only if x and x" are connected by a bond on the tree, i.e. if x and x' are neighboring nodes. Equation 2 translates than into a constraint on the ratio between the 'upwards' rate k[ and 'downwards' rate k~ along a bond with energy difference

Some analytical insight in the dynamical behavior of tree structures [6] can be obtained by an approximate method of analysis based on the idea that relaxation proceeds as a succession of quasiequilibria. The approximation has two main ingredients, namely that at any given moment, most of the probability mass is to be found within a subtree of a certain characteristic size, and that inside that subtree the distribution is, in some appropriate sense, Boltzmann-like. We recall for convenience that the Boltzmann distribution is given by:

Peq(x) -

g(x) e-~E(":) Zz/ with (7)

Z X = ~, g (x) e - ~(x). In quasiequilibrium we expect that the difference

P(y, tlXo) Peg(Y) p(x, tlx0 ) Peq (x)

Fig. 1. The tree construction scheme

(8)

should be uniformly small. To quantify this expectation, we define, for any subtree of height I containing the starting point x o of the random walk, the 'lateral variance':

411

v (t, 0

l~.1 ( = ~o

)2 p(i(k),tIXo)---Peq(i(k))p(j(k,l),tlXo) eeq (J (k, l))

.

The indices i (k) and j (k, l) point to the leftmost respectively rightmost node on level k, k < l:

i ( k ) = 2 lm''-Ic,

j(k,l)=2-k(21m"x+2l)--l.

(9)

metric downward rates imply that the initial condition for the thermal relaxation process following the quench can be concentrated into a small region. In our simple model, an arbitrarily sharp initial condition can be obtained by choosing s I sufficiently large. More generally we expect the value of s t to play a role when temperature variations are important. The branching ratio z of the tree is assumed to be z = 2. We performed most calculations on a tree of height

The lateral variance gauges the unbalance of the probabilities at the the leftmost and rightmost node, relative to their equilibrium values. It vanishes in equilibrium and should be very small in quasiequilibrium. Due to the self similar structure of the tree, this overall behavior would not change if the contribution from intermediate nodes were included in the sum. A second measure of the degree of quasiequilibrium follows from detailed balance. We expect the ratio of probabilities at neighbor nodes to be cIose to

R1=~=xe -~, I=A,,A 2.

-12

I

~ i

\

(10)

We therefore monitor this ratio along the leftmost branch of a sequence of nested subtrees containing the initial node. Finally, we looked at the time development of the total probability mass in a subtree containing the initial node. This quantity is expected to be close to one, for times up to the characteristic escape time of the subtree. For larger times, it should slowly decay towards its equilibrium value.

X N

N \, ".,. N\ 1"-,,x 2\\3"\ 4", 5',,,, 7\

10-6

\

\

\ \ ' " \ . ",. ,\ '~

.....

\ \

"

0,\~

"

"~

\ \\~;\/

"

C

~o-~

4. Results We performed several numerical experiments on three types of trees: 1. regular tree with A~ = A 2. LS-tree with A 1 > A 2 3. LS-tree with A1 < A=, which is the mirrored version of case 2. We note that the symmetry between case 2 and 3 is broken by the initial form of the probability distribution, which is always concentrated at the leftmost node.

4.1. Model parameters

442

vA', \NX.\ "....'

q4 ~ 1

:"

10 -~

_

.,,

_

',,,, ~

-

\

1(512

~.\' N \ \ \,"..~

\

-=',

\'.:~. xxd

\ \"- ~.

.\ \,]..~ "~ "X".'. ........\

We choose transition rates k~, k~ of the following form:

I k~ = sZlce - ~

(11)

q~1~

, io-~

k~ = s~.

(12)

While the parameter s ~ does not change the (Boltzmann) equilibrium distribution on the tree, it modulates the attempt frequen W along the branch of length I = d 1, A 2. In order to see its effect, let us consider the typical situation in which a complex system is quenched into a low energy state. I f the low energy part of the phase space can meaningfully be modelled by a tree, highly asym-

, io~

L

1 ~o ~

I\,\

N\",~ ~o ~

~o~

t ~

Fig. 2. The parameters used are: a A~=A~=2, s a ' = s ~ = l , bA~=2.5, A2=l , SA~=sA:=I, eA~=l, A2=2.5 , s a ' = s a : = l , d A 1= 1, A2=2.5 , s ~' = 100, s a== 1. The lateral variance ofa subtree of height l (label refer to height l) rapidly decays towards zero for times t ~ r (O In contrast to the regular tree a the lateral variance of the LS-tree decays as a power law b-d. The cusps in e and d result from a change of sign of the difference between the conditioned probability of the nodes, i.e. the right node is more populated than the left nodes.

412 /max = 7, leading to a 255 • 255 transition matrix F. Finite

size effects for the temperature range of interest were found to be quite negligible for this system size. The entropic factor is chosen to be x = 2. As shown in [6] the eigenvalues of F mainly depend on the value of p = k~' k A2 A 1 A2 + ~. The exponents ~ - and ~ - of the upward rates kf 1 of the three different models are chosen such that they roughly yield the same value of p. We choose 1. A~ T

A2

2,

SAI=s

in the energy level of their root node. We can still identify for each subtree of height l a 'mean' eigenvalue )t (~ or 1

'mean' relaxation time ~eq ~q) 3t(l) (see Table 1). This 'mean' eigenvalue 2(o plays an important role in the re0.3 0.25

11 U d 41 U d 7/

/ !//..i//

0.2

A~= 1

T 0.15

2. 3.

=2.5 A1 ~-=1 a) b)

and

~-=1,

s~=s a~=l ,

0.1 0.2

and

s~ ' = 1 0 0 ,

! s A== 1.

4.2. Eigenvalues

We numerically diagonalize the symmetrized transition matrix of the problem, and find that the gross features of the spectra are the same in the three cases considered: the eigenvalues are distributed almost uniformly on a logarithmic scale. However, for the case of accelerated kinetics along the short branch we find a deviation from this behavior for the 'fast' eigenvalues, which are enhanced by a factor of hundred relative to the case with s ~' = 1. We have checked numerically that, as expected, the relaxation exponents at any fixed temperature are not effected by the value of s I. A detailed discussion of the spectrum for the regular tree can be found for the purpose of comparison in [7]. The eigenvalue distribution of the regular tree has an expected degeneration: for each level index l = 1..... /max there are nx = 2 x="x-t eigenvalues which are equal, corresponding to the permutation symmetry within the level. The symmetry is partially lifted in the LS-tree. There are still n~ equivalent subtrees of height l, differing at most

Table 1. Relaxation times: 2 nd column: mean relaxation time of a regular subtree of height l, 3 rd column: relaxation time of a irregular subtree of height l with s A~ = s ~2= 1, 4 m column: relaxation time of a finite irregular tree of height l with s a' = s a2= 1, 5 tu column: relaxation time of a irregular subtree of height l, where the time constant on the short branch is s A' = 100

1/ 2/ ?' q / / / i

,, ,.. ! 7 s../ 6./ 7/"

/ i /

i'/i"ii/

I/

o.~ ..Q 12 .13 0 L_ 13.

,

f ~f~,>..----f--

f'7,

0

In the third case we investigated two choices of the parameter s A L For s ~ = 100 the transitions along the short branch are accelerated by a factor 100. The random walk always starts in the leftmost node on level 0: % = 2 ~. . . . 128. This corresponds to the lowest and highest lying local minimum in case 2 and 3 respectively.

,

b

A2=2"5"T

s A ~ = s A2= 1

,

I/~,~.:_~.i

,

,

,

,

0.8 -c

~>"~J>'7-"2%"

Jill!i~ 1/ 21 3/ 4# 5i

0.6

6/' 7,

.."

.

o, /X.,'2:4> 0.2.~

/,//ff

:

I

,

,

i

4

5

6

L

I

I

i

0.75"d

1

2

3

7

o

0.71 |11

I

10-~

I

101

I

103 10 5 t ~,--

I

107

Fig. 3. The parameters used are: a A ~ = A 2 = 2 , Sm=sA2=I, b A]=2.5, A 2 = l , S ' ~ = s A ~ = I , e A I = I , A2=2.5, S ~ = s A 2 = I , d A~ = 1, A2=2.5, s m = 100, s ~2= 1. The figure shows the ratios of probabilities belonging to neighbouring nodes along the leftmost branch of the tree. The label 1 and 7 refer to the lowest and the highest lying bond respectively. These ratios have a plateau in the intermediate time phase, which is explained in the paper by a simple sink-picture. The final equilibration is reached in each case at times of the order of the relaxation time, as expected

l

~(~) ~eq~ A ~1 = A~2

~.(l) -eq, A I ~ A 2

~'eq' AIz/zZI2

~eq,'r(l) ~1"4
1 2 3 4 5 6 7

3.6945 34.2901 266.4862 1,989.1141 14,724.9603 108,838.1986 804,253.5785

2.3481 23.0318 186.0267 1,410.0652 10,542.3978 78,796.4127 603,682.6600

2.4397 23.8180 191.9426 1,454.1824 t0,868.5477 81,019.6080 603,682.6600

1.5114 12.6180 104.7521 801.8793 6,045.6904 46,289.5264 386,017.1787

sAI=100

413

1

laxation of the subtree. The numeric value is near to the real equilibration time of a finite tree of height l (see Table 1). A subtree of height I should enter a quasiequilibrium state at a time of order _(o ~eq"

10-3 ~ . . . ~ . ~ _

4.3. The time development of the relaxation process

10 i / . 1042i/,

The time scales entering the relaxation process can be interpreted as characteristic relaxation times for a sequence of subtrees. Figure 3 a - d show that, within each subtree, the relaxation proceeds in four stages:

1

1 10-3 i~

I

a

I

,

i

I . ~l-l..

1

b

.il~

10-s

............

lo -7

1. t < 10: probability development at short times,

I

.

.

.

.

.

.

2. 10 < t < Zeq-(1)'.intermediate phase,

3. r(J)q< t < Zeq-(7)"quasiequilibrium and development towards equilibrium,

lo-3

4. t > r ~ )" the state of equilibrium.

10-6

:.\ x <-.:

4.3.1. Short. times, For times smaller than 101 the fast modes have great influence on the behavior of the system. Their properties for the case of regular tree have been discussed in details in [7]. In general, the time development of the probability in each node is given by different power laws. Expanding the master equation for short times and using the hierarchical structure of the tree one obtains that the exponent is given by the number of bonds along the shortest path to the initial node. Thus the probability of the nodes on the leftmost branch can be written as P ~ d where I is the level index.

1o-3 10-7 I

i 0 -11

10-1

4.3.2. The intermediate state. In this relaxation stage, the flux out of a subtree is nearly constant over many decades of time (Fig. 4a-c). In addition the probability ratio shows a plateau at times 10 < t < %q (1--1) (Fig. 3a-c). These features can be described by the following ansatz for the probability along the leftmost branch: P (r (n, t l Xo) = f (t) a'

(13)

where r (~ is the root node of the subtree of height 1 containing the initial node. The time dependent function f (t) is constant for a regular tree, an increasing power law in the case of the LS-tree with A ~ > A2 and a decreasing power law in the case of the LS-tree with A ~ < A 2 (Fig. 4a-d). The weak dependence of the plateau value of the probability ratios, a is neglected in the approximation. For times smaller than I'eq ~(l) all the subtrees not containing the initial node and connected to the node r u), i > l are treated as sinks. These absorb probability at an effective downward rate kffr which takes into account their internal dynamics. We thus write an approximate Master equation for the node r ~ i > I as

= -- (k At + k~' + kffr) e (r (`), t IXo)

Inserting the ansatz (13) into (14) we find

I

101

,

' I I ~ { t ~S% i 10 3 10 5 107 t--I,,--

4. The parameters used are: a A~=A2=2, S A I = s A 2 = I , b A1=2.5, A2=l, s~=sa2=l, c Al=l, A2=2.5 , s~'=s~2=l, d A~= 1, A2=2.5, s~ = 100, sa~= 1. The probability flux out of a subtree of height l (label refer to height l) is nonvanishing and only weakly time dependent in the intermediate time region. For times greater than the corresponding relaxation time the flux decays algebraically with an exponent/T Fig.

a2_

1 ( f ' (t) +ka,+k~,+kw ~u' \ f (t)

a

k A1

05)

=o with solutions

1 { ' f f ( t ) 4_zA,~_z.A,.z.~rf" ~ a=2kg~ \ f (t) --'~" --'~a --'~a j

~_(

t

jfl2 zll (16)

d, P (r% t IXo)

+k,zl i P(r ( i - l),tlxo)+kg'P(r(e+~),tlXo).

s

"

(14)

The ' + ' solution is discarded, since it leads to a probability ratio value a greater than one. In the case of the f ' (t) regular tree the term ~ appearing on the rhs is nearly zero, while for the LS-trees it decreases as t - t. It is there-

414 fore neglected for long times, whereby the ansatz becomes self-consistent. In order to find the effective downward rate kfff, we let c~ be the root of the subtree of height i - 1, i > l, which is connected to r (~ but does not contain the initial node. In the course of the stationary evolution, this subtree gets filled up with probability. Our numerical investigation shows that the ratios of probabilities along the rightmost branch of all subtrees have a plateau value, which we call b. We therefore have (17)

P ( a , t l Xo)= b P (r (i), t i Xo) . The Master equation for the node ~ is

4 p ( a , tlxo) = - ( k , 42 + k aA, + k aA2)P(o~,t]Xo)

+ kJ ~P(r% t[ Xo)

(18)

where in a first approximation the two subtrees connected to ~ are considered as sinks9 Inserting (13) and (17) into (18), we find

kg 2

b=

(19)

f ' (t) . ,.A~_c 1 . 4 , . Z~A2 f (t) ''~u --'~d --'~d Inserting (17) into the Master equation for r (~

dt e (r (i), t l Xo) = -- (k.~ + k~ ~+ kff ~) P (r (0, t[ Xo)

+ kA. 'P(r (i-1), tlxo) + k g ' P ( r (*+1), t[ Xo) +

kuA2 P(~,tlXo)

(20)

we can finally identify the effective downward rate k f of (14) as

kf=g (1

).

f ' (t) f(t)

f ' (t)

Again considering the ratio ~

(21)

to be zero and using

the numerical values for the transition rates we can calculate the value of a. We find a = 0 . 1 3 4 for the A1 A~ regular tree. For the LS-tree with ~ - = 2 . 5 , ~ = = 1, s 4' = s ~2= 1 we find a = 0.091, while for the LS-tree with A I - - 1 , --A2=2.5, s4'--sa2--1--- we get a=0.314. These T T values are in excellent agreement with the numerical results (Fig. 3 a-c). The plateau value for the highest bond can now be calculated by discarding the terms _ ,~.~4~p_,-r'("), t lx0) and k if' P (r (~+ ~), t [x0) in (14). Simple manipulations yield kf~ kf'

a - - kdA1 +

(22)

with a = 0.144 for the regular tree, and a = 0.095 for the LS-tree with A1 = 2 5, A~ = 1, s 4~= s 42= 1 Finally, for T A" T A 9 the LS-tree wlth ~ -1= 1 , ~ -2= 2 . 5 , s 4 2 = s A 2 = 1 we get a = 0.382. Again this is in excellent agreement with the numerical findings (Fig. 3 a-c). In case 3 with s 4' = 100, s 4~ = 1 the simple sink picture is not valid anymore. Due to the fast relaxation along the short bonds, boundary effects resulting from the finite height of the tree have greater influence on the relaxation process. We find that the plateau in the probability ratios is not independent of the level index, as in the previous cases. Ignoring this difference anyway, the previous calculation supplies a plateau value a = 0.711, which agrees with the numerical value within 1.25%. The calculation for the top bond yields a = 09 which also fits the correct value for the upper bond very well. The numerical plateau values lie in between the above results (Fig. 3 d).

4.3.3. Quasiequilibrium. At times t=r(Jq) a subtree of height l enters a quasiequilibrium state: for the regular tree the lateral variance rapidly decreases towards zero, as shown in Fig. 2a, while in the LS-trees it does so as a power law, which is shown in Fig. 2b-d. The difference can be understood as follows: in the LS-tree, nodes within one level have different equilibrium probabilities. Since the downward rates are the same throughout the level, the probability flux out of a subtree of height l is supplied to its sibling subtree, via the common parent node, in such a way that the lateral variance of the latter does not vanish before an iternal rearrangement. During the regime of quasiequilibrium, i.e. --eq ~:(o < t < Zeq-(7),the probability ratios R x, I = A 1 , A 2 inside the subtree of height l (Fig. 3a-d) have reached a value very close to their equilibrium values, which is obtained from detailed balance considerations and is therefore independent of the constant s ~, I = A 1, A 2. A t the onset of the quasiequilibrium regime t = rq) --eq the outgoing flux from the subtree of height l containing the initial node becomes strongly time dependent (Fig. 4a-d). This is due to the fact that the sibling of the subtree starts providing a back flow of probability9 This internal equilibration of the twin subtrees does not however affect the outflux at level I + 1. For the three cases considered i = 1, 2, 3 with s ~= 1, the flux out o f a subtree decays as a power law t - a' during the whole quasiequilibrium regime (Fig. 4a-c). The exponent p~ depends on i, but not on the subtree level. The total probability mass in each subtree is a time integral of the flux, and decays therefore with the exponent Yi = fli - 1 (Fig. 5a-c). Not surprisingly the comparison to the exponents given analytically in [5, 6] is in excellent agreement with our numerically calculated values: In the case of the regular tree we obtain a value of Yl = 0.349, while the theoretical value is In z

Yl -- A / T + lnz-- In ~: -- 0.347. For the LS-tree with

415

~

- ~ -

-'---~--'---"

']

, " " . \ ""'-...~ ' ~ . \\

\

Finally, for the LS-tree with

"-

6.~

A1-1, T

\ ' \ \ ""'"'-" ' - - . . . . .5. . . . . . . .

A2-2.5, T

the numerically calculated value is ?3 = 0.848, while the analytical value is

10-I

In (1 + exp [ ( A 2 - A ~ ) / T ] ) ?3 - A 1 / T + in (1 + exp [(A z - A 1)/T]) - In x = 0.847. ' -4'.. >..._~.~_1 ~.~_<-<~<-..--~-~ ........... '~

0

o.6

"~'~"'"-." ................... 5......... I

~"~

~_2

,

110-1:

,

,

j_

" ~---~"-<'t~i<.i~....:i.i;.:i.

E c~ CJ

4.3.4. Equilibrium. In this regime all the quantities have reached their equilibrium values. The lateral variance is zero, while the probability ratios have reached their Boltzmann-values.

n0-a

10-5 .

I

I

1

I

...............- .... -

~o

\

\

-.

10-4

5. Summary and conclusion

s|

-----.-~--4

-

;%-

_ I

L

\<< \

\

\\

\~ ...... - \ 9

10 <

",<<-,<, \\

"....,

9

.

.......

10-s 10-6

I 101

Id I

, 10 3

, 10 5

,"5-107

Fig. 5. The parameters used are: a At=A2=2, sAJ=s~2=l, b ,01=2.5, ,02=1, Saa=SA2=I, e At=l, `02=2.5, S~=sA2=I, d Aj=I, Az=2.5, sZJ=100, s~Z=l, e lm~=9, ,0~=1, A2=2.5 , s A~= 100, s ~2= 1. For times smaller than the corresponding relaxation time, the total probability mass is concentrated in the subtree of height l (label refer to height l) containing the initial node. Later, in the quasiequilibrium regime, the probability mass decays algebraically with an exponent )/= B ~- 1. This power law is however not observable in d due to boundary effects

A1

T

The accelerating factor s ~ ' : 100 modifying the rates along the short bonds of the tree drops of the analytical expressions. Hence it cannot affect the theoretically predicted exponents within the approximation used in [6]. Numerically, the seven level tree shows finite size effects in this case, and the probability mass has no power law decay (Fig. 5 d). We therefore computed the probability mass in a larger tree with /max= 9 (Fig. 5e) and found a power law with Y3 = 0.786. The discrepancy relative to the analytical expression is 7%.

-2.5,

As

T

- 1,

we obtain ~12= 0.099, while the theoretical value is in (1 + exp [(A 2 - A 1)/T]) )'~--A~/T+ln(1 +exp[(A2--AO/r])-lnK

---0.100.

In this paper we presented a detailed investigation of the relaxation behavior in a class of hierarchical structures. The aim was to check the quasiequilibrium assumption, which states that at any given time the probability is concentrated in a certain subsystem, and that within this subsystem it is Boltzmann-like distributed. This assumption was often used in literature [5, 6]. We also provided a better understanding of the probability development in the subtree on time scales shorter than the corresponding quasiequilibrium time. The relaxation process inside a subtree is characterized by four different time regimes. The first one describes the short time behavior of the probability and is mainly influenced by the decay of the fast modes. After this first period the relaxation process enters a kind of stationary state, the intermediate phase. During this period a subtree of certain size supplied probability to all nodes outside, while its internal distribution only changes slowly. The probability ratios and the outgoing flux do not change much over many time decades. These features are explained analytically with a very simple sink-picture. At times corresponding to the equilibration of a finite tree of given height l, the subtree of height l enters a quasiequilibrium state, characterized by a rapid decay of the lateral variance and by probability ratios near the equilibrium Boltzmann-value. For these reasons, the probability distribution inside a subtree has already equilibrium features, but is nonetheless time dependent. Due to the different quasiequilibrium times for subtrees of different height, the picture of successive quasiequilibria can describe the relaxation process. The last phase of the relaxation is the equilibrium state, where the probability distribution is really stationary.

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