1/f Noise and Self-Organized Criticality

• Dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point
• Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state
• This gives insight into the origin of fractal objects

Classic Problem: "1/f" Noise

• 1/f noise has been detected for transport in diverse systems like resistors, hourglasses, the flow of the Nile, and the luminosity of stars
• The low-frequency power spectra of these systems show a power-law behavior fP over different time scales
• Despite effort, there isn't a general theory that explains the widespread occurrence of 1/f noise

Spatial Observation

• Spatially extended objects, including cosmic strings, mountain landscapes, and coastal lines, appear to be self-similar fractal structures
• Turbulence exhibits self-similarity in both time and space
• The commonality: power-law temporal or spatial correlations extend over decades, where physics might vary dramatically

Argument

• Dynamical systems with extended spatial degrees of freedom naturally evolve into self-organized critical structures/states that are barely stable
• Self-organized criticality is the underlying mechanism behind the phenomena
• Combination of dynamical minimal stability and spatial scaling results in a power law for temporal fluctuations
• Noise propagates through scaling clusters through a "domino" effect, disrupting minimally stable states
• Long-wavelength perturbations cause a cascade of energy dissipation on all length scales, characteristic of turbulence

Theory

• Criticality here differs from the critical point at phase transitions in equilibrium statistical mechanics
• Equilibrium is reached only by tuning a parameter like temperature
• Critical point in dynamical systems studied is an attractor reached by starting far from equilibrium
• Scaling properties of the attractor are parameter-insensitive, implying no fine-tuning is needed to generate 1/f noise and fractal structures

One-Dimensional Array

• There is a one-dimensional array of damped pendula, with coordinates u„, connected by torsion springs
• The springs are weak compared with gravity
• An infinity of metastable or stationary states exist
• Pendula point almost down, u„ = 2πN, where N is an integer and the spring winding numbers N differ
• Initial conditions: forces C„=u„+1 — 2u„+u„—1 are large, so all pendula are unstable
• Pendula rotate until spring forces on all pendula assume a value + K, just able to balance gravity and keep the configuration stable
• If all forces start positive, then the final forces will all be K
• The array is also stable in any further relaxed configuration
• Dynamics stop upon reaching first, maximally sensitive state considered locally minimally stable

Minimally Stable Structure

• A small "kick" on one pendulum in the forward direction relaxes the force
• The force on a nearest-neighbor pendulum exceeds the critical value, and the perturbation propagates by a domino effect until it hits the end of the array
• At the end of this process, forces return to their original values, and all pendula have rotated one period
• The system is stable with respect to small perturbations in one dimension, and the dynamics are trivial

Differing Dimensions

• Relaxation dynamics will take the system to a configuration where all pendula are in minimally stable states
• If one pendulum is relaxed slightly, this will render the surrounding pendula unstable, and the noise will spread to the neighbors in a chain reaction, ever amplifying
• The pendula are generally connected with more than two minimally stable pendula, and the perturbation eventually propagates throughout the entire lattice
• This configuration is unstable with respect to small fluctuations and cannot represent an attracting fixed point for the dynamics
• More and more more-than-minimally stable states will be generated, and these states will impede the motion of the noise

Stability

• The system will become stable precisely at the point when the network of minimally stable states has been broken down
• Noise signal cannot be communicated through infinite distances
• At this point there will be no length scale in the problem
• One might expect the formation of a scale-invariant structure of minimally stable states
• The system might approach, through a self-organized process, a critical point with power-law correlation functions for noise and other physically observable quantities
• "Clusters" of minimally stable states must be defined dynamically as the spatial regions over which a small local perturbation will propagate

Sense

• Dynamically selected configuration is similar to the critical point at a percolation transition where the structure stops carrying current over infinite distances, or at a second-order phase transition
• Magnetization clusters stop communicating
• The arguments are general: do not depend on details of the physical system, including details of local dynamics and presence of impurities
• Self-similar fractal structures are widespread in nature: "physics of fractals" could be that they are the minimally stable states originating from dynamical processes which stop precisely at the critical point

Scaling

• Naturally gives rise to a power-law frequency dependence of the noise spectrum
• At the critical point, there is a distribution of clusters of all sizes
• Local perturbations will propagate over all length scales, leading to fluctuation lifetimes over all time scales
• A perturbation can lead to anything from a shift of a single pendulum to an avalanche, depending on where the perturbation is applied
• The lack of a characteristic length leads directly to a lack of a characteristic time for the resulting fluctuations

Sand

• A distribution of lifetimes D(t) leads to a frequency spectrum
• Visualize through a pile of sand
• If the slope is too large, the pile is far from equilibrium, and the pile will collapse until the average slope reaches a critical value where the system is barely stable with respect to small perturbations
• The "1/f" noise is the dynamical response of the sandpile to small random perturbations

Dimensions

• Numerical simulations are performed in one, two, and three dimensions on several models
• One model is a cellular automaton, describing the interactions of an integer variable z with its nearest neighbors
• In two dimensions z is updated synchronously: z(x,y) → z(x,y) - 4, z(x ± 1,y) → z(x ± 1,y)+1, z(x,y ± 1) → z(x,y ± 1)+1, if z exceeds a critical value K

Aspects

• There are no parameters here since a shift in K simply shifts z
• Fixed boundary conditions are used, i.e., z=0 on boundaries
• Cellular variable may be thought of as the force on an individual pendulum, or the local slope of the sand pile (the "hour glass") in some direction
• If the force is too large, the pendulum rotates (or the sand slides), relieving the force but increasing the force on the neighbors
• The system is set up with random initial conditions z ≫ K, and then simply evolves until it stops, i.e., all z's are less than K
• The dynamics is then probed by measurement of the response of the resulting state to small local random perturbations
• Response are found on all length scales limited only by the size of the system

Depiction

• Shows a structure obtained for a two-dimensional array of size 100×100
• The dark areas indicate clusters that can be reached through the domino process originated by the tripping of only a single site
• Clusters are defined operationally: in a real physical system one should perturb the system locally in order to measure the size of a cluster
• Shows a log-log plot of the distribution D(s) of cluster sizes for a two-dimensional system determined simply by counting the number of affected sites generated from a seed at one site and averaging over many arrays
• The curve is consistent with a straight line, indicating a power law D(s)~s¹, τ≈ 0.98
• The fact that the curve is linear over two decades indicates that the system is at a critical point with a scaling distribution of clusters
• Shows a similar plot for a three-dimensional array, with an exponent of τ≈ 1.35
• At small sizes the curve deviates from the straight line because discreteness effects of the lattice come into play
• The falloff at the largest cluster sizes is due to finite-size effects, as checked by comparing simulations for different array sizes
• A distribution of cluster sizes leads to a distribution of fluctuation lifetimes

Perturbation

• If the perturbation grows with an exponent y within the clusters, the lifetime t of a cluster is related to its size s by t¹+γ=s
• The distribution of lifetimes, weighted by the average response s/t, can be calculated from the distribution of cluster sizes: D(t)=(s()/t)ds/dt = t(-(γ+1)τ+2γ)/γ = t-α
• Shows the distribution of lifetimes corresponding to Fig
• namely how long the noise propagates after perturbation at a single site, weighted by the temporal average of the response
• This leads to another line indicating a distribution of lifetimes of the form with α≈ 0.42 in two dimensions α≈ 0.90 in three dimensions
• These curves are less impressive than the corresponding cluster-size curves, in particular in three dimensions
• The lifetime of a cluster is much smaller than its size, reducing the range over which we have reliable data
• The resulting power-law spectrum is

Summary

• Power distribution of cluster sizes and time scales as expected from general arguments about dynamical systems with spatial degrees of freedom
• More numerical work is needed to improve accuracy, and to determine the extent to which the systems are "universal," how the exponents depend on the physical details
• The dynamics of a self-organized critical state of minimally stable clusters of all length scales generates fluctuations on all time scales
• The noise at a given frequency f is spatially correlated over a distance L(f) which increases as f decreases