Peter Hnggi, Institut fr Physik, Universitt Augsburg V. Zaburdaev, S. - - PowerPoint PPT Presentation
Perturbation spreading in many particle systems: a random walk approach Peter Hnggi, Institut fr Physik, Universitt Augsburg V. Zaburdaev, S. Denisov and P. Hnggi Perturbation spreading in many particle systems: a random walk approach
Peter Hänggi, Institut für Physik, Universität Augsburg
Perturbation spreading in many particle systems: a random walk approach
Perturbation spreading in many particle systems: a random walk approach
Transport in Hamiltonian systems is anomalous [Zaslavsky
(2002)] Casual cone: general feature of many-body systems [Lieb & Robinson (1972), Marchioro et al. (1978)] What is inside the cone? What is on the front? What about correlations?
[V. Zaburdaev, S. Denisov, and P. Hanggi PRL 106 180601 (2011)]
N i¼1
i i=2, mi ¼ m or M,
i + 1
i ðtÞ HiðtÞ.i.e., N
i¼1 4 Eði; tÞ ¼ Ep.
∞
non-negative anomalous: casual cone
1000 2000
i
0.5 1 E(i,t=400) 1000 2000
i
0.25 0.5 1000 2000
i
0e+00 5e-03 1e-02
N = 1 N = 10
3
N = 10
6
FPU
PDF of flight times:
[Geisel (1985), Klafter (1982)]
γ > 0
ψ(τ) = γ τ0 1 (1 + τ/τ0)1+γ
∝ 1 √ te−x2/t
∝ t |x|1+γ
PDF of flight times:
50 100 150 x 10
10
10 P(x,t=100)
superdiffusion: scaling: ballistic peaks:
1 < γ < 2
P(x, t′) ≃ 1 Ku1/γ P
Ku1/γ , t
erence between
|x| = v0t
σ2(t) ∝ t3−γ
γ > 0
PDF of flight times:
50 100 150 x 10
10
10 P(x,t=100)
superdiffusion: scaling: ballistic peaks:
1 < γ < 2
P(x, t′) ≃ 1 Ku1/γ P
Ku1/γ , t
erence between
|x| = v0t
σ2(t) ∝ t3−γ
γ > 0
1000 2000
i
0.5 1 E(i,t=400) 1000 2000
i
0.25 0.5 1000 2000
i
0e+00 5e-03 1e-02
N = 1 N = 10
3
N = 10
6
FPU
position of the w equation ˙ x = v0 + ξ(t), Gaussian process of vanishing
Gaussian process of vanishing mean i.e., ξ(t)ξ(s) = Dvδ(t − s). wn biased Wiener process
Phump(x, t) = Φ(t) [p(x + v0t, t) + p(x − v0t, t)] /2
x(t + τ) = x(t) + v0τ + w(τ),
p(w, τ)
Gaussian PDF
50
x/t
1/2
10
10
p(x,t)t
100 200 300
x/t
10
10
p(x,t) t
the ballistic humps (4)
Phump(¯ x, t′) ≃ u−1/2Phump(¯ x/uγ−1/2, t),
and u = t′/t, erence between
width of the hump ∝
√ t
2000
i
10
10
10
ρ(i,t/ε
1/2)
200 10
10
1000
i/t
γ
10
10
10
ρ(i,t) t
γ
a) b)
ε = 4 ε = 2 ε = 1
times t ¼ 1000, 2000, 4000, and 6000 (the
is ¼ 5=3.
v0; D / ffiffiffi " p :
%"ðx; tÞ ¼ %"0ðx; t=s0Þ; ffiffiffiffiffiffiffiffiffiffi where s0 ¼ ffiffiffiffiffiffiffiffiffiffi "0=" p (see
200 400 600 800 1000
i
10
10
10
10
e(i,t)
1000 2000 3000
i
10
10
e(i,t)
10
10
e(i,t)
b)
ε = 4 ε = 3 ε = 1
c) a)
t = 1000 t = 2000
is ¼ 5=3.
that K / 11= v0,
P(x, t′) ≃ 1 Ku1/γ P
Ku1/γ , t
. we demonstrated that
Single particle PDF describes the energy correlation in many particle system
Levy-walk-like dynamics in ergodic many particle systems
New scalings - relation to phonon/mode coupling theories Velocity correlations can be calculated analytically Additional measurement/diagnostic tool
[V. Zaburdaev, S. Denisov, and P. Hanggi PRL 106 180601 (2011)]
Dynamical Heat Channels
School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel (Received 11 June 2003; published 4 November 2003) We consider heat conduction in a 1D dynamical channel. The channel consists of an ensemble of noninteracting particles, which move between two heat baths according to some dynamical process. We show that the essential thermodynamic properties of the heat channel can be obtained from the diffusion properties of the underlying particles. Emphasis is put on the conduction under anomalous diffusion conditions.
DOI: 10.1103/PhysRevLett.91.194301 PACS numbers: 44.15.+a, 05.45.Ac, 05.60.Cd
The link between thermodynamic phenomena and mi- croscopic dynamical chaos has been a subject of interest for a long time [1]. An example of such a relationship is deterministic diffusion [2], where local dynamical prop- erties, such as stability (or instability) of several fixed points can change the global diffusion from normal to anomalous [3]. Another question which has attracted a lot
ministic extended systems [4–8]. A large number of models have been proposed in order to understand the conditions under which a system obeys the Fourier heat conduction law [4]. Recently, a new class of 1D models, ‘‘billiard gas channels,’’ has been proposed [5–8]. These channels consist of two parallel walls with a series of scatterers, distributed along walls, and noninteracting particles that move inside. The two ends of the channel are in contact with heat baths. By changing the shapes and positions of scatterers, it is possible to change the conductivity of the channel [5–8]. In this Letter we show that such billiard gas channels belong to a wider class of models, which we call dynami- cal heat channels. The absence of interactions between particles and the independence of the particle dynamics
tween the thermodynamic aspect, which is governed by the properties of the thermostats, and the dynamics inside channel, which is governed by diffusion properties within the channel. All the essential information on heat con- ductivity of such a dynamical heat conductor can be
Dynamical heat channel.—To model dynamics within the channel, we consider N particles that move along direction X, following the equations of motion _ x fx; t; X 2 x; (1) where the function f can be either deterministic or random. To consider transport of heat, two heat baths with temperatures T and T are attached to the left and right ends of the channel. Each heat bath is characterized by a velocity probability density function (pdf), PTvTh, where vTh is the thermal velocity. After colliding with the heat bath, the particle is ejected back to the channel with a velocity vTh, which is chosen from PTvTh. As particles do not interact, the dynamics of the en- semble inside the channel can be described by a long trajectory of a single particle and the flux should be rescaled by the factor N. The trajectory of a particle is independent of the particle velocity vTh. The only differ- ence between ‘‘hot’’ and ‘‘cold’’ particles is that the hot
The velocity vTh does not change during the propagation through the channel and it can be interpreted as a tem- perature ‘‘label’’ of a particle. The dependence of the dynamics in Eq. (1) on vTh can be taken into account by introducing a scaling factor for the time t ! t=vTh. Because of the separation of the thermodynamic characteristics from the dynamical ones, the proposed approach is not limited to Hamiltonian systems only [5–8]. We assume only that the dynamics inside the channel has a diffusional character and can be charac- terized by the evolution of the mean square displace- ment (msd) hX2ti t: (2) This diffusion can be normal ( 1), subdiffusive ( < 1), or superdiffusive ( > 1) [9]. Following Ref. [5], heat transfer by a particle through the channel is Qt X
Mt j1
Ej X
Mt j1
qjEin
j Eout j ;
(3) where Ein
j and Eout j
are the energies before and after the j collision with the heat bath. qj is the direction factor: qj 1 if the j 1 collision is with the hot end, and qj 1 in the case of the cold end. Mt is the total number of collision events during time t. In the case of normal heat conductivity, Q grows linearly with t and the heat flux is defined by J lim
t!1
Qt t : (4) Let us start from a situation where the particle is initially located at the hot end. During diffusion it can P H Y S I C A L R E V I E W L E T T E R S
week ending 7 NOVEMBER 2003
VOLUME 91, NUMBER 19 194301-1 0031-9007=03=91(19)=194301(4)$20.00 2003 The American Physical Society 194301-1
return and collide with the hot bath again. According to
end, Q increases on average as R1
0 v2 Th=2PTvTh
that the process is reiterated starting from the cold end. Thus, the problem of heat transfer is reduced to the problem of diffusion in a finite interval under reflecting and absorbing boundary con-
the particle is located at the reflecting end. The average time needed to reach an absorbing boundary is the first moment of the pdf t of first arrival times. To take into account the effect of thermodynamic velocity on the time should be rescaled, as mentioned above, ! R1
0 1=vThPTvThdvTh . Because of the absence of
mass flux, the number of transitions from left to right and vise versa should be the same. Finally, we obtain the following equation for the one-particle heat flux through the channel of length L: JL 1 R1
0 v2 ThPTvTh PTvThdvTh
R1
1 vTh PTvTh PTvThdvTh
: (5) For an ensemble of particles the heat flux is written as Jens NJL. Here we take into account that in order to keep a density of particles in the channel fixed N should be proportional to L; namely, N / L. Equation (5) demonstrates a complete separation be- tween thermodynamic and dynamical aspects. Without loss of generality we consider the ‘‘delta’’
simple pdf, PTvTh vTh
p [8]. As a model for the dynamics in the channel, we con- sider continuous time random walks (CTRW) [10] in a discrete lattice. This allows one to cover the spectrum of diffusion regimes from subdiffusion to superdiffusion and to describe kinetics of deterministic Hamiltonian [11,12] and dissipative [3] systems. The CTRW model is a stochastic process which repre- sents an alternating sequence of waiting and jumping events [10]. A particle waits at each point for a time chosen from a waiting time pdf wt and makes a jump (flight) to the left or right with equal probabilities. The jumps are characterized by the pdf fx; t, the probability density to move a distance x at time t in a single flight event. We consider power laws for both pdfs [10] wt tw1; fx; t xf1: (6) Depending on the asymptotic properties of wt and fx; t, we have regimes of superdiffusion, normal dif- fusion, or subdiffusion [10]. Superdiffusion.—To achieve superdiffusion with a fi- nite msd hx2ti, there should be a correlation between the length and duration of the individual flights. Such corre- lation leads to the model of Le ´vy walks with the spatio- temporal pdf [10] fx; t ftjxj vt; ft tf1 (7) that corresponds to flights with a constant velocity v. Here we assume that w > 1, so there is a finite mean waiting time. Depending on f we distinguish among three regimes of diffusion [10], according to the exponent in Eq. (2), 8 < : 2; 0 < f < 1 3 f; 1 < f < 2 1; 2 < f : (8) In order to derive the dependence of the heat flux J on the length L of the channel, we start from a consideration
the finite interval of the length L bounded by reflecting and absorbing boundaries.We find that for Le ´vy walks the survival probability is t / est, where s / L1 for the ballistic motion ( 2), s / Lf for superdiffusion (1 < < 2), and s / L2 for normal diffusion ( 1) [see inset in Fig. 1(a)]. Then the pdf for first arrival times t is t _ t / sest [Fig. 1(a)] and has a finite first moment . As a result, in the asymptotic limit the heat Q grows linearly with time t, Qt E=t [line (3) in Fig. 2], where the mean arrival time is R1
diffusion, due to Le ´vy walks, we get a normal linear in time heat conductivity. Finally, for the one-particle heat flux, following
JL in terms of the msd exponent : JL / L; 8 < : 1; 2 f 3 ; 1 < < 2 2; 1 : (9) In Fig. 3(b) we show the numerical results for JL
The results are in good agreement with the scaling law suggested by Eq. (9) [straight lines in Fig. 3(b)].
1×10 10-4
1×10 10-3
pdf t for first arrival times for (a) superdiffusion with w 2:6, f 1:6 for L 100. Inset: Scaling of the exponent s with the channel length L, Lf. (b) Subdiffusion, w 0:6, f 2:6. Straight lines correspond to the asymp- totic behavior (see text).
P H Y S I C A L R E V I E W L E T T E R S
week ending 7 NOVEMBER 2003
VOLUME 91, NUMBER 19 194301-2 194301-2
Then taking Eq. (9) into consideration the thermal conductivity k JensN=rT shows the following be- havior for 1 < < 2: k / L2f / L1: (10) This diverges as one goes to the thermodynamic limit L ! 1. For the ballistic case 2, k / L, and for 1 we obtain k / const as expected. The Le ´vy walk model is an adequate approach for modeling of Hamiltonian kinetic in a mixed phase space [12]. Previous numerical results which have been ob- tained for Hamiltonian billiard channels show that 1:3 corresponds to the flux exponent 1:72 [7] and 1:8 to 1:178 [8], which are in good agreement with the relation in Eq. (9). In the case of Le ´vy flights [10], unlike Le ´vy walks, the velocity is not introduced explicitly. Here we assume that all flights have the same duration tf and use the following decoupled representation for flight pdf: fx; t fxt tf; fx xf1: (11) In this case the msd diverges and one is tempted to use hjxtji2 instead to characterize the dynamics. In the asymptotic regime this quantity scales as hjxtji2 t2=f [10]. Here the exponent f determines the dynamics in the channel and the dependence of flux JL on L. The scaling of the mean arrival time / Lf is consistent with the result obtained from a direct solution of a frac- tional Fokker-Planck equation [13]. We note that using hjxtji2 instead of the msd gives a different, yet unphysi- cal scaling, JL / L2=f3, with an exponent which is close to that in the scaling in Eq. (9) and coincides with it at the points f 2 (normal diffusion) and f 1 (bal- listic diffusion). Subdiffusion.—In the case of subdiffusion, w < 1 and f > 2 in Eq. (6), the mean waiting time diverges [10]. The motion is characterized by long localized events and the msd grows sublinearly, hx2ti tw. The pdf of the first arrival time t has a power-law asymptotics t / tw1 [Fig. 1(b)]. This finding is in line with a result for subdiffusion under a bias in a semi-infinite interval with an absorbing boundary [14]. The anomalous character of the pdf t leads to a divergence of . Thus, in the case
grows sublinearly with time [see curve (1) in Fig. 2], Qt tw: (12) Within a traditional definition of the flux, Eq. (4), the subdiffusive anomalous heat conductivity cannot be dis- tinguished from a heat insulator. We note that subdiffu- sion regimes cannot be achieved in the case of channels with Hamiltonian dynamics due to a finiteness of the recurrence time [9]. Coexistence of long localized events and flights.—In the case of competition between flights and localization events, the msd exponent is given by a relation [10] 2 w f; 0 < w < 1; 1 < f < 2: (13) The pdf for the arrival time t is governed, as in the case of simple subdiffusion, by the pdf for waiting times
anomalous heat conductivity, Eq. (12), even in the case of superdiffusion spreading, hx2ti t, > 1, Eq. (13). Figure 2 [line (4)] shows the dependence Qt, for the
1×10 10-3
1×10 10-5
1×10 10-7
(a) The mean first arrival time versus L; (b) one- particle flux JL, Eq. (5), versus L ( T 2 and T 1). Results correspond to different diffusion regimes with normal diffusion, w 1:6, f 2:4 (squares), superdiffusion, w 1:6, f 1:6 (stars), and Le ´vy flights, w 1:6, f 1:6 (triangles).
Evolution of Qt vs t for subdiffusion, w 0:6, f 2:6 [curve (1)], normal diffusion, w 1:6, f 2:4 [curve (2)], and superdiffusion, w 1:6, f 1:6 [curve (3)]. Curve (4) corresponds to a rescaled, for one particle, Qt for an ensemble N 100 in the case of competition between anomalous flights and localized events, w 0:4, f 1:2. Straight lines correspond to the asymptotics (see text). For all cases the length of channel is L 1000, T 2, and T 1.
P H Y S I C A L R E V I E W L E T T E R S
week ending 7 NOVEMBER 2003
VOLUME 91, NUMBER 19 194301-3 194301-3
ensemble of N 100 particles (w 0:4, f 1:2, 1:2), rescaled for one particle. One can go continuously among the various diffusional behaviors by tuning parameters in an iterated map. As an example of deterministic channel, we consider a com- bined map [3]. This one-dimensional map generates in- termittent chaotic motion with coexisting localized and ballistic motion events. The map is defined for one unit cell by the iterative rules, Xn1 fXn; fX ( X aXz 1; 0 < X < 1
4 ;
X ~ a1
2 X~ z; 1 4 < X < 1 2 :
(14) The parameters are a 4z~ z=z ~ z z and ~ a 4~
zz=
z ~
acterized by the pdfs in Eq. (6), with the exponents f z 11 and w ~ z 11. Varying the map parame- ters z and ~ z allows one to cover the diffusional regime in the range 0 < 2 . The thermodynamic velocity vTh can be included in the map description. We fix a time step dt and after one iteration of the map, Eq. (14), stretch the system time following t t dt=vTh. The length of channel is deter- mined by a number of unit cells, 0 > X > L.When the particle reaches the boundary cells, X 0 or X L, it is randomly placed into the interval 0 < X < 1=2 or L 1=2 < X < L, respectively. This corresponds to a diffusive reflection of particles back to the channel. The values of the flux exponents obtained for the map in Eq. (14) are in agreement with the prediction in
show the time evolution of Qt for the case of subdiffu- sion, w 0:5, f 1:5. The steps in the Qt depen- dence correspond to anomalously long waiting events when a particle is trapped near marginal stable fixed points x j 1
2 ; j 0; . . . ; L [3].
In summary, we have introduced a class of dynamical heat conductors. This class includes as particular cases recently proposed Hamiltonian billiard channels [5–8]. In the absence of interactions between the particles and the independence of the dynamics on particle energy, the proposed approach goes beyond Hamiltonian dynamics and allows one to express heat conductivity in terms of channel diffusion properties. The Hamiltonian character
tween particles are introduced (such as in the case of dynamical lattices [4]) or a dependence of dynamics on particle energy is included. Discussion with Professor B. Li and financial support from the Israel Science Foundation, the U.S. Israel BSF, and INTAS grants are gratefully acknowledged.
[1] N. S. Krylov, W
Physics (Princeton University Press, Princeton, NJ, 1979). [2] S. Grossmann and H. Fujisaka, Phys. Rev. A 26, 1779 (1982); T. Geisel and J. Nierwetberg, Phys. Rev. Lett. 48, 7 (1982). [3] G. Zumofen and J. Klafter, Phys. Rev. E 51, 1818 (1995). [4] S. Lepri, R. Livi, and A. Politi, Phys. Rep. 377, 1 (2003). [5] D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys.
[6] B. Li, L. Wang, and B. Hu, Phys. Rev. Lett. 88, 223901 (2002). [7] D. Alonso, A. Ruiz, and I. de Vega, Phys. Rev. E 66, 066131 (2002). [8] B. Li, G. Casati, and J. Wang, Phys. Rev. E 67, 021204 (2003). [9] M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature (London) 363, 31 (1993); J. Klafter, M. F. Shlesinger, and
[10] G. Zumofen, J. Klafter, and M. Shlesinger, in Anomalous Diffusion: from Basis to Applications, edited by R. Kut- ner, A. Pekalski, and K. Sznajd-Weron, Lecture Notes in Physics V
[11] J. Klafter and G. Zumofen, Phys. Rev. E 49, 4873 (1994). [12] S. Denisov, J. Klafter, and M. Urbakh, Phys. Rev. E 66, 046203 (2002). [13] M. Gitterman, Phys. Rev. E 62, 6065 (2000). [14] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
Evolution of Qt vs t for the map in Eq. (14) with w 0:5, f 1:5 (L1000, dt104, T 2, and T 1). Straight lines correspond to power-law dependence in Eq. (12). Inset: Numerically obtained relation between the scaling ex- ponent for one-particle flux and the exponent (w 1:6 for all cases). Straight line corresponds to 3.
P H Y S I C A L R E V I E W L E T T E R S
week ending 7 NOVEMBER 2003
VOLUME 91, NUMBER 19 194301-4 194301-4
From Anomalous Energy Diffusion to Levy Walks and Heat Conductivity in One-Dimensional Systems
1Istituto Nazionale di Ottica Applicata, No
¨thnitzer Strasse 38, D-01187 Dresden, Germany
2Max-Planck Institut fu
¨r Physik Komplexer Systeme, No ¨thnitzer Strasse 38, D-01187 Dresden, Germany (Received 26 February 2005; published 21 June 2005) The evolution of infinitesimal, localized perturbations is investigated in a one-dimensional diatomic gas
description, which was so far invoked to explain anomalous heat conductivity in the context of non- interacting particles is here shown to extend to the general case of truly many-body systems. Our approach does not only provide firm evidence that energy diffusion is anomalous in the HPG, but proves definitely superior to direct methods for estimating the divergence rate of heat conductivity which turns out to be 0:333 0:004, in perfect agreement with the dynamical renormalization-group prediction (1=3).
DOI: 10.1103/PhysRevLett.94.244301 PACS numbers: 44.10.+i, 05.45.Jn, 05.60.-k, 05.70.Ln
After the discovery of anomalous heat conductivity in classical one-dimensional lattice systems [1], in the past years a renewed attention has been devoted to the old problem of identifying the minimal ingredients required for the Fourier law to be ensured. As summarized in a recent review article [2], many different models have been numerically investigated to identify the physical conditions under which the thermal conductivity diverges with the system size L and, having assessed that L, to deter- mine the possibly different universality classes for the divergence rate . Simultaneously, several attempts have been made to estimate analytically the scaling behavior of : self-consistent mode-coupling theory [2] and the Boltzmann equation [3] suggest that 2=5, while the dynamical renormalization group indicates 1=3 [4]. Both predictions are compatible with numerical simula- tions which are, however, often affected by relatively strong finite-size corrections. The only system where con- vincing results have been obtained is the Fermi, Pasta Ulam model in the infinite temperature limit. Its behav- ior is consistent with 2=5 [5], but the symmetry of the potential casts doubt about the generality of this model [6]. A further simple system that can be effectively simulated
actions are provided by elastic collisions of pointlike par- ticles [7]. Unfortunately, the most detailed numerical simulations reported in the literature show a slow growth
that 1=3 [8], while others state that the conservative guess 0:25 is more realistic [9]. Settling this issue is not only conceptually important, but it is a necessary requisite to later quantify finite-size corrections, a crucial issue in applications to, e.g., carbon nanotubes, where one needs to know the prefactor as well. Although the problem involves intrinsically many de- grees of freedom, some researchers have tried to shed some light with reference to the simpler setup of noninteracting particles moving along a periodic array of convex scatter- ers (billiard gas channels) [10]. The absence of interactions simplifies the task of understanding heat conductivity and allows, in particular, tracing back heat conductivity prop- erties to the diffusion of single particles at equilibrium. Assuming that the mean square displacement hx2ti scales as t ( 1 corresponds to normal diffusion), it can be shown that 1 [11], under the assumption that each particle exchanges energy only at the channel borders, where thermal reservoirs operate. The limit of this ap- proach is, on the one hand, that the relationship between the diffusion exponent and the microscopic dynamics remains to be established [12] and, on the other hand, that particles do not mutually interact. Nevertheless, in this Letter we show that upon interpreting the energy density as a pseudoparticle density, the above reasoning can be fruitfully applied to the hard-point particles (HPG) to find convincing evidence that energy diffuses as in a Levy walk process [13], thus bridging two research lines that were so far basically disconnected from one another. As a further result, we are also able to establish that in the HPG, 1=3 with a 1% accuracy. The model consists of a chain of N pointlike particles with alternating masses mi lying on a segment of length L, 0 xi L (i 1; . . . ; N), where m2i m, with (i 1; . . . ; N=2), and m2i1 rm, with (i 0; . . . ; N 1=2), where square brackets indicate the integer part. Because of the absence of intrinsic energy and length scales, we can fix them at will. The only relevant parameter that cannot be scaled out is the mass ratio r. Without loss of generality, we set the number density : N=L 1, the energy per particle " hmiv2
i i=2 1 (which, in turn,
fixes the temperature, kBT 2) and one of the two particle masses equal to 1. The dynamics of this model is very simple, since the velocities change only in the collisions between adjacent particles while the updating rule is de- termined by the conservation laws of kinetic energy and PRL 94, 244301 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending 24 JUNE 2005
0031-9007=05=94(24)=244301(4)$23.00 244301-1 2005 The American Physical Society
linear momentum. By denoting with vi (v0
i) the velocity
before (after) a collision, the evolution equation amounts to v0
i vj 1 r
1 r vi vj; (1) where a plus (minus) sign is selected depending on whether i is even (odd) and j i 1. It is also interesting to notice that the positions xi contribute only indirectly to the evo- lution, by determining the collision times. Such properties, plus the conservation of the particle ordering along the chain, allow simulating the dynamics with an event driven algorithm [8] that exploits the heap structure of the future collision times. When r 1, collisions just exchange particle velocities that are, thereby, integrals of motion. Away from r 1, the system is no longer integrable, but remains nonchaotic. In fact, since the evolution Eq. (1) is linear, it describes the dynamics of an infinitesimal perturbation vit as well. Therefore, the weighted Euclidean norm Q X
i
2i; t X
i
mivit2 (2)
the same reason that energy is conserved in the phase- space dynamics, and a fortiori the maximum Lyapunov exponent cannot be larger than zero [8]. A more detailed characterization of the dynamics can be
exponents [14], which quantifies the space-time growth rate of a perturbation 2i; t 0 i;0 initially local- ized in i 0 (i;j is Kronecker function) u lim
t!1 1 2t log2i ut; t:
(3) u attains its maximum at u 0, where it coincides with the maximum Lyapunov exponent. In a standard chaotic model, upon increasing juj, u decreases until it crosses 0 at u us, a value that has been shown to coincide with the sound velocity [15]. It is then interesting to understand how the scenario modifies in a nonchaotic model such as the HPG. Previous studies revealed a slow convergence to the asymptotic regime both when r is close to 1 and for r 1 [8]. In order to avoid such problems we have chosen r 3, but we can stress that the same scenario has been confirmed by a few simulations run for r 2. The data reported in Fig. 1 show that converges to 0 for u < uc 1:02 [uc has been determined by fitting the long-time behavior of the secondary maximum of u], while it becomes strictly negative at larger velocities. This is analo- gous to harmonic chains, where the exponential growth rate of a perturbation is strictly zero as long as juj is smaller than the sound velocity [16]. We have verified that uc coincides, within the numerical error, with the sound ve- locity (determined from the position of the peaks in the structure function) at the very same temperature. The very observation that u 0 within a finite ve- locity range implies a subexponential scaling and thereby suggests looking for a more accurate scaling ansatz. The power-law hypothesis 2i; t t12i=t2; i < uct; (4) proves to be the correct choice. Because of the conserva- tion law (2), we expect that temporal and spatial rates coincide with one another, namely, 1 2 . The value of can be estimated from the behavior of 20; t which appears to follow a clean power law (see circles in Fig. 2). A best fit (solid line) yields 0:606 0:008, which strongly hints at 3=5. By adopting this value in Eq. (4), the rescaled profiles 2 are plotted in
equal to 1). The scaling ansatz (4) is finally validated by the extremely good overlap of the rescaled distributions in the interval delimited by the secondary peaks, observed in
1 2 u
Λ
uc
Spectrum of comoving Lyapunov expo- nents at different times for r 3; the chain length is N 8190. From bottom to top the curves have been obtained by comparing perturbations and times 100; 200, 200; 400, 400; 800, 800; 1200, and 800; 1600. The perturbation amplitude has been averaged over 104 different realizations.
10
210
3t 10
10
δ(2)(0,t) 10
410
510
610
7σ
2(t)Maximum height 20; t of the infini- tesimal perturbations profile (circles) and mean square displace- ment 2t (diamonds) versus time.
PRL 94, 244301 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending 24 JUNE 2005
244301-2
correspondence of the sound velocity us. It is interesting to notice that while the system size L 8190 allows obtain- ing a clean scaling behavior for the perturbation diffusion, direct simulations of the heat conductivity indicate that L 30 000 is not large enough to obtain a looser estimate
In view of the conservation law expressed by Eq. (2), it is tempting to interpret the perturbation profile 2i; t as a probability density function (PDF) and thereby compare
walks [13]. A simple schematization of a Levy walk con- sists of a particle moving ballistically between successive ‘‘collisions’’ whose time separation is distributed accord- ing to a power law, t / t1 ( > 0 [17]), while their velocity is chosen from some symmetric distribution which, in the simplest setup, reduces to Pu u 1 u 1=2. The propagator (the PDF to find in x at time t, a particle initially localized at x 0) of such a process is [18] Px; t / 8 > > > < > > > : t1= expax2
t2=
jxj & t1= tx1 t1= & jxj < t t1 jxj t jxj > t (5) It can easily be shown that, up to the length jxj t, the propagator Px; t scales as in Eq. (4) with the exponent 1=. The cutoff at jxj t is a consequence of the finite constant velocity and leads to sharp peaks at the
propagator for the Levy walk with 1= 5=3 is compared with the direct simulation of the HPG). In fact, these peaks correspond to single flights during the obser- vational time. The only relevant difference with the results
exhibited by 2, disappears as soon as the -Dirac func- tions in the definition of Pu are replaced by Gaussian distributions with a suitable width (see again the inset of
As for the secondary peaks, a best fit of their height gives a decay as t1:150:03. This is to be confronted with the theoretical prediction t11= [19], which, in this case, amounts to t19=15. Much of the deviation is to be attributed to the not yet established asymptotic decay in the tail. Let us now discuss the connection between dynamics in tangent space and energy diffusion. The latter process can be investigated from the evolution of an initially localized, finite perturbation vi0, Eit mi 2 vit vit2 mi 2 v2
i t
mivitvit mi 2 vit2; (6) and thereby perform an ensemble-average hEiti to get rid of statistical fluctuations. By directly computing hEiti, we found that even after averaging over 104 realizations, it was not possible to obtain meaningful re- sults on time scales shorter than those considered in the previous simulations. In fact, while an increasing accuracy is required to resolve tinier deviations, statistical fluctua- tions remain of order one at all times. Additionally, there are fluctuation amplifications due to vi sudden jumps [20]. However, in the limit of infinitesimal perturbations, the probability of such jumps vanishes and the last term on the right-hand side (rhs) of Eq. (6) can be identified with
perturbation is conserved, but, because of Eq. (2), the sums
hmivivii vanishes, so that the relevant properties of energy diffusion are captured by the pseudo-Euclidean norm of vit. Therefore, from the scaling behavior for 2 we can conclude that energy diffuses anomalously in the HPG. This can also be shown more directly from the behavior of the mean square displacement 2t ii22i; t, which is expected to scale as 2t / t. A best fit of our numerical data (see diamonds in Fig. 2) yields 1:35, a value in agreement with Ref. [8(b)]. Since in a Levy walk, the exponent is equal to 3 , we expect 4=3, which thus confirms the validity of the Levy walk interpretation. Energy superdiffusion can finally be related to the anomalous behavior of , by exploiting the link established by linear response theory between heat conductivity and the decay of spontaneous fluctuations of the energy density
10 20
x/t
γ
10
10
10
10
δ(2) t
γ
10 10
10
10
Rescaled perturbation profiles at t 40, 80, 160, 320, 640, 1280, 2560, and 3840 (the width increases with time), for 3=5. The profiles have been obtained by averaging over 104 realizations. In the inset, the profile at t 640 (solid line) is compared with the propagators of a Levy walk for an exponent 5
3 with a fixed velocity u 1 (dotted line)
and a Gaussian distribution with average equal to 1 and rms 0.036 (dashed line). In the last two cases, the propagators have been obtained by averaging over 108 different realizations.
PRL 94, 244301 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending 24 JUNE 2005
244301-3
[21]. By following Ref. [11], the divergence rate of heat conductivity can be connected to the anomalous diffusivity exponent through the simple relation 1 2 2 2=, which implies 1=3 in our case [22], a value in perfect agreement with the prediction of the dynamical renormalization group [4]. The clean results summarized in the three figures of this Letter are made possible by the peculiarity of HPG dy- namics, which allows for a strong reduction of statistical
whose presence would have obliged to averaging over an exponentially growing number of trajectories should not, however, be forgotten. Therefore, one might suspect that the Levy walk scenario is peculiar to the HPG, but this hypothesis contrasts the observation that the HPG scaling behavior coincides with the predictions based on very general hydrodynamic arguments. On the other hand, if Levy walks generally occur in 1D systems, evidence for this behavior should be found in, e.g., truly chaotic models: however, the above-mentioned numerical difficulties strongly suggest the need to define a completely new protocol to answer this question. Let us now briefly discuss the origin of the anomalous
mined by three elements: energy plus momentum conser- vation and the correlations between collisions. In Ref. [23], a 1D model was introduced where the energy of each particle is randomly exchanged (with one of the two neighbors) in uncorrelated collision events. In such a sim- plified context it was rigorously proved that Fourier law is
conductivity observed in the HPG is due to the additional conservation of momentum (which is absent in the model
set of completely uncorrelated events, we found a normal
aly is entirely contained in the long-range correlations between collisions.
work is part of the project PRIN2003 Order and chaos in nonlinear extended systems funded by MIUR Italy.
*On leave from Istituto dei Sistemi Complessi–Sezione di Frenze, CNR Largo E. Fermi 6, Firenze, I-50125 Italy. [1] S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 (1997). [2] S. Lepri, R. Livi, and A. Politi, Phys. Rep. 377, 1 (2003). [3] A. Pereverzev, Phys. Rev. E 68, 056124 (2003). [4] O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002). [5] S. Lepri, R. Livi, and A. Politi, Phys. Rev. E 68, 067102 (2003). [6] S. Lepri (private communication). [7] T. Hatano, Phys. Rev. E 59, R1 (1999). [8] (a) P. Grassberger, W. Nadler, and L. Yang, Phys. Rev.
and L. Yang, nlin.CD/0203019. [9] G. Casati and T. Prosen, Phys. Rev. E 67, 015203(R) (2003). [10] D. Alonso, R. Artuso, G. Casati, and I. Guarneri, Phys.
and I. de Vega, Phys. Rev. E 66, 066131 (2002); B. Li,
(2003). [11] S. Denisov, J. Klafter, and M. Urbakh, Phys. Rev. Lett. 91, 194301 (2003). [12] M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature (London) 363, 31 (1993). [13] A. Blumen, G. Zumofen, and J. Klafter, Phys. Rev. A 40, 3964 (1989). [14] R. Deissler and K. Kaneko, Phys. Lett. A 119, 397 (1987). [15] G. Giacomelli, R. Hegger, A. Politi, and M. Vassalli, Phys.
[16] M. Vassalli, Laurea thesis, Florence University, 1999. [17] Here, we are interested in the interval 1 < < 2, which corresponds to superdiffusive Le ´vy walks [13]. [18] J. Klafter and G. Zumofen, Physica (Amsterdam) 196A, 102 (1993). [19] By considering that the peak is positioned in z t1=1, the probability distribution decays as z1=1, and the remaining area is on the order of 1=. The result follows by taking into account the scaling of the y axis and assuming, as confirmed by numerics, that the peak does not broaden. [20] In the HPG, thermalization is induced by discontinuities in the evolution rule [P. Cipriani and A. Politi (unpublished)] which act very much as in ‘‘stable chaos,’’ where an irregular behavior arises and persists even in the presence
[21] B. J. Palmer, Phys. Rev. E 49, 2049 (1994). [22] In B. Li and J. Wang, Phys. Rev. Lett. 91, 044301 (2003), a different formula, 2 2=, is proposed that would yield 1=2. However, as shown in R. Metzler and I. M. Sokolov, Phys. Rev. Lett. 92, 089401 (2004), such a relation is based on an improper expression for the mean first passage time. [23] C. Kipnis, C. Marchioro, and E. Presutti, J. Stat. Phys. 27, 65 (1982).
PRL 94, 244301 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending 24 JUNE 2005
244301-4
makes sense only with a well defined velocity
Ψ(τ)e−ikvτvh(v)dv
ψ(τ)e−ikv0τv0h(v0)dv0
1 − [ hkτψ(τ)]s +
∞
Ψ(t)e−ikv0τv2
0h(v0)dv0
s
it seams no one calculated it for random walks We just described a perturbation in many particle system by a single particle PDF. Now we use the toolbox of many particle systems and apply it to the single particle process:
Levy Walk:
ψ(τ) ∝ (τ/τ0)−γ−1
diffusion
(γ > 2)
C(x, t) = v2 Dv γ − 1△P(x, t) = v2 1 γ − 1 ∂P(x, t) ∂t .
50
x 0.05 0.1 c(x,t=50)
50 100
10
superdiffusion(1 < γ < 2)
C(x, t) = v2 Kv γ − 1△γ/2P(x, t) = v2 1 γ − 1 ∂P(x, t) ∂t .
Correlation of a random walk model but with different exponent: ?
β = γ − 1/2