Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer.
ABSTRACT Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.
arXiv:cond-mat/0601679v6 [cond-mat.soft] 20 Feb 2008
Near mean-field behavior in the generalized Burridge-Knopoff
earthquake model with variable range stress transfer
Junchao Xia∗and Harvey Gould
Department of Physics, Clark University, Worcester, MA 01610
Department of Physics and Center for Computational Science,
Boston University, Boston, MA 02215
J. B. Rundle
Department of Physics and Center for Computational Science and Engineering,
University of California, Davis, CA 95616
Simple models of earthquake faults are important for understanding the mecha-
nisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Be-
cause of the importance of long-range interactions in an elastic medium, we general-
ize the Burridge-Knopoff slider-block model to include variable range stress transfer.
We find that the Burridge-Knopoff model with long-range stress transfer exhibits
qualitatively different behavior than the corresponding long-range cellular automata
models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer,
depending on how quickly the friction force weakens with increasing velocity. Exten-
sive simulations of quasiperiodic characteristic events, mode-switching phenomena,
ergodicity, and waiting-time distributions are also discussed. Our results are consis-
tent with the existence of a mean-field critical point and have important implications
for our understanding of earthquakes and other driven dissipative systems.
∗Present address: Department of Chemistry, The University of Iowa, Iowa City, IA 52240
Earthquake faults are important examples of driven dissipative systems . Models of
fault systems are important for understanding Gutenberg-Richter (power law) behavior, the
occurrence of characteristic events, and the relation between small and large earthquakes [1,
2, 3, 4, 5]. Understanding driven dissipative systems is important, for example, in the
context of avalanches , neural networks , depinning transitions in charge density waves
and superconductors , magnetized domains in ferromagnets , domain rearrangements
in flowing foams , and granular materials under shear stress .
A relatively simple dynamical model that contains much of the essential physics of earth-
quake faults is the spring-block model proposed by Burridge and Knopoff . This model
consists of blocks connected by linear springs to their nearest neighbors with spring constant
kc. The blocks are also connected to a loader plate by linear springs with spring constant
kL, and rest on a surface with a nonlinear velocity-weakening stick-slip friction force which
depends on a parameter α that controls how quickly the friction force decreases as the ve-
locity is increased. The model was studied numerically in one dimension in Ref.  and
more recently in Refs. [13, 14, 15, 16, 17, 18, 19, 20, 21].
An earthquake event is defined as a cluster of blocks that move (slip) due to the initial
slip of a single block. In addition to the amount of energy released in an earthquake event,
a quantity of interest is the moment M, which is defined as?
displacement of block j during an event and the sum is over all the blocks in the event.
j∆xj, where ∆xjis the net
Carlson and Langer simulated the one-dimensional Burridge-Knopoff model for N = 100
and N = 1000 blocks. The main result of their simulations [13, 14, 15, 16, 17, 19] is that for
∼2 the moment probability distribution P(M) scales as M−xfor small localized events
with an exponent x ≈ 2 . There also is a peak in P(M) for large events indicating a
significant presence of characteristic (non-power law) events.
Because simulations of the Burridge-Knopoff model require solving Newton’s equations
of motion and are time consuming, several cellular automata (CA) models have been pro-
posed that neglect the inertia of the blocks and simplify the effect of the friction force by
assuming that the motion is overdamped. These cellular automata include those due to
Rundle, Jackson, and Brown  and Olami, Feder, and Christensen . In these models
P(s), the distribution of the number of blocks in an event, does not exhibit power law scal-
ing  for nearest-neighbor stress transfer if periodic boundary conditions are used . A
generalization of these CA models to include more realistic long-range stress transfer 
yields considerable differences with the nearest-neighbor CA models  and with the orig-
inal Burridge-Knopoff model. In particular, for long-range stress transfer P(s) exhibits
Gutenberg-Richter scaling consistent with the system being near a mean-field (spinodal)
critical point [29, 30, 31, 32]. In addition, the long-range CA models can be described by
a Langevin equation [29, 31, 33] and small and medium size events can be interpreted as
fluctuations about a free energy minimum [29, 30, 34]. Large events drive the system out of
equilibrium from which the system decays back to an equilibrium state .
The CA and Burridge-Knopoff models lack several elements that would make them more
realistic. In particular, the long-range CA models do not include inertia and more realistic
friction laws, and the Burridge-Knopoff model does not include long-range stress transfer.
Both types of models do not include elastic (seismic) wave radiation (phonons) because there
is no medium in which seismic waves can propagate. However, the lack of seismic waves is
a reasonable approximation, because seismic waves carry little energy in real faults .
In this paper we discuss our extensive simulations of a generalized Burridge-Knopoff
model with long-range stress transfer between the blocks . For various values of the
dynamic friction parameter α and the range of stress transfer R, we observed phenomena
similar to real fault network systems, including Gutenberg-Richter scaling, quasiperiodic
characteristic events, and mode-switching. Our primary results are that the behavior of
the long-range Burridge-Knopoff model differs significantly from the short-range Burridge-
Knopoff model, the behavior of the long-range Burridge-Knopoff and CA models is similar
only for small α, and the nature of the friction force is important and strongly affects the
behavior of the Burridge-Knopoff model. In particular, we find numerical evidence for two
types of scaling behavior: a mean-field spinodal critical point similar to that found in the
long-range CA models [29, 30, 31, 32] for R ≫ 1 and α<
found in Refs. [14, 15, 19] for α>
∼2 and all values of R studied in the range 1 ≤ R ≤ 500.
∼1  and the scaling behavior
The original Burridge-Knopoff model in one dimension is governed by the equation of
motion [12, 14, 15]
dt2= kc(xj+1− 2xj+ xj−1) − kLxj− F(v + ˙ xj),(1)
where xjis the displacement of block j from its equilibrium position, v is the speed of the
substrate, which moves to the left, F(˙ x) = F0φ(˙ x/˜ v) is a velocity-dependent friction force,
˜ v is a characteristic velocity, and m is the mass of a block. The loader plate is fixed.
As in Ref.  we introduce the scaled variables τ = ωpt, ω2
p= kL/m, and uj= (kL/F0)xj,
and rewrite Eq. (1) in dimensionless form as
¨ uj= ℓ2(uj+1− 2uj+ uj−1) − uj− φ(2αν + 2α˙ uj), (2)
with 2α = ωpF0/kL˜ v, ℓ2= kc/kL, and ν = vkL/(ωpF0); a dot denotes differentiation with
respect to τ. The form of the friction force is plotted in Fig. 1 and is given by 
1 − σ
y = 0
1 − σ
, y > 0.
Note that φ(y) decays monotonically to zero from φ(0+) = 1 − σ and prohibits slip in the
same direction as the motion of the substrate .
The four dimensionless parameters, ℓ, α, ν, and σ govern the behavior of the system.
The parameter α appears in the argument of φ in Eq. (2) and determines how quickly the
dynamic friction force decreases with increasing velocity; α = 0 means that the dynamic
friction force is equal to the constant 1−σ. Larger α means that the friction force decreases
more rapidly with velocity and the motion is less damped; α → ∞ implies that the friction
force drops to zero immediately for positive velocities.
We generalize the Burridge-Knopoff model by assuming that a block is connected to R
neighbors (in each direction) with the rescaled spring constant kc/R ; R = 1 corresponds
to the usual Burridge-Knopoff model. We used the second- and fourth-order Runge-Kutta
algorithms [39, 40] with the time step ∆t = 0.001 to solve Eq. (2) generalized to arbitrary
R. Both algorithms and other fourth-order algorithms  give similar results.
1 − σ
1 − σ
FIG. 1: The form of the velocity-weakening friction force φ(y). The friction force decays monoton-
ically to zero from the initial value φ(0+) = 1 − σ with the initial slope −1.
III. IMPLEMENTATION OF THE BURRIDGE-KNOPOFF MODEL
Because the velocity of a block is a continuous variable, we need to introduce a criterion
for when a “stuck” block begins to move and when a moving block becomes stuck so that we
can define the beginning and end of an event. We define a block to be stuck if its velocity
is less than a parameter v0. In addition, the stress on the block, defined to be the force due
to all the springs coupled to it including the loader plate spring, must be smaller than the
maximum static friction force F0(taken to be unity in dimensionless units). If a block is
stuck, we choose the value of the static friction force to be such that it cancels the stress.
At the next time step, a stuck block will remain stuck if the stress on it is still smaller than
F0. A moving block will become stuck at the next time step if its speed is less than v0and
decreasing and if the stress on it is less than F0. In our simulations we take v0= 10−5, which
yields reasonable results.
An earthquake event begins with the slip of a block and ends when all blocks become
stuck. A block is said to “fail” when it begins to move after being stuck. A moving block
can become stuck and then move (slip) again during an event.
We initially set ˙ uj= 0 for all j and assign small random displacements to all the blocks;
hence all blocks are initially stuck. We compute the force on all the blocks and update
˙ ujand uj for all j using the generalization of Eq. (2) for arbitrary R. We continue these
updates until all blocks become stuck again. We then move the substrate (the loader plate
is fixed) until the stress on one block exceeds F0. This stress loading mechanism is known