ISEGORIA / MATH ENCYCLOPEDIA
Complex systems: order from many simple parts
Many identical units, each following a rule that mentions only its neighbours, and a whole that does something none of them does. Oscillators that synchronise past a critical coupling, a sandpile that tunes itself to the edge of an avalanche, a random lattice that suddenly connects, and self-propelled particles that flock. In each case an order parameter, a threshold, and a mechanism.
Before you begin: Probability, dynamical systems, and statistical mechanics
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Synchronisation: the Kuramoto model
N oscillators run round a circle, each at its own natural frequency ωᵢ, and each is pulled towards every other by a coupling K. The dots are the oscillators, coloured by natural frequency; the arrow is the mean field r e^{iψ}, whose length r is the order parameter: 0 for phases spread uniformly, 1 for all in step. Kuramoto’s discovery is that for frequencies drawn from a Lorentzian of half-width γ the transition is exact: below K꜀ = 2γ the mean field dies away and r = 0; above it, r = √(1 − K꜀/K), and the oscillators with |ωᵢ| ≤ K r lock to the mean field while the rest drift. Raise K slowly and watch the arrow grow from nothing.
Worked example. With \(\gamma=1\) and \(K=3\), \(K_c=2\) and the locked-state order parameter is \(r=\sqrt{1-2/3}=0.577\). An oscillator locks when it has a fixed point, \(\omega_i=K r\sin(\theta_i-\psi)\) with a solution, that is \(|\omega_i|\le Kr=1.73\); for the Lorentzian that is the fraction \(\tfrac2\pi\arctan(1.73)=0.67\) of the population. At \(K=6\), \(r=0.816\) and 87% lock.
Watch out. The formula is for \(N\to\infty\). With the \(N=120\) oscillators here, \(r\) fluctuates by about \(1/\sqrt N\approx0.09\) even below \(K_c\), so the measured curve on the right rounds off the sharp corner at \(K_c\) and never quite reaches zero. The frequencies are the quantiles of the Lorentzian rather than a random sample, which removes sampling noise but puts two very fast oscillators at the tails; they never lock and are the dots that keep spinning. The Lorentzian is special in giving a closed form; for a Gaussian the threshold is \(K_c=2/(\pi g(0))\) as well, but \(r(K)\) has no elementary expression.
Why is the threshold K꜀ = 2/(π g(0)) set by the density of oscillators at the centre frequency alone?
Just above threshold \(r\) is tiny, so only oscillators with \(|\omega|\le Kr\), a sliver around the centre, can lock; the drifting ones contribute nothing to \(r\) on average by symmetry. The locked ones sit at \(\sin(\theta-\psi)=\omega/Kr\), and their contribution to \(r\) is \(\int_{-Kr}^{Kr}\cos\theta\,g(\omega)\,d\omega=Kr\int_{-\pi/2}^{\pi/2}\cos^2\theta\,g(Kr\sin\theta)\,d\theta\). Self-consistency \(r=\) this, with \(g(Kr\sin\theta)\to g(0)\) as \(r\to0\), gives \(1=K g(0)\pi/2\). Only \(g(0)\) survives the limit.
2. Self-organised criticality: the sandpile
Drop grains one at a time onto a grid. A cell holding four grains topples, sending one grain to each neighbour; grains that fall off the edge are lost. Neighbours that reach four topple in turn, and the chain reaction is an avalanche whose size is the number of topplings. Nobody tunes a parameter, yet the pile drives itself to a state in which avalanches of every size occur with a power-law distribution: that is self-organised criticality. Drop at the centre and the pile grows into the famous deterministic pattern; drop at random sites and read the avalanche histogram on the right. The order of additions never matters: the final pile is the same, which is why the model is called abelian.
Worked example. Four grains on one cell of an empty grid: one toppling, the four neighbours hold one grain each, the centre is empty, and no grain is lost. Add four more at the centre: the centre topples again, the neighbours reach two. After the third four, the neighbours hold three; the fourth four topples the centre, then all four neighbours, and then the centre once more as each neighbour hands a grain back: an avalanche of size 6, after which the diamond’s corners hold their first grains.
Watch out. A power law needs many decades to be seen, and a grid of 101 by 101 gives about three: the histogram bends down at the largest sizes, where avalanches reach the boundary and lose grains. The exponent \(\tau\) of the two-dimensional sandpile is not known exactly; numerical estimates cluster near 1.3 with logarithmic corrections, and the fitted slope here depends on the range of sizes used. The centre-drop pile is not critical in the statistical sense at all: it is a single deterministic object whose pattern has a scaling limit described by a system of partial differential equations (Pegden and Smart, 2013), which is why it looks like a fractal rather than like noise.
Why does the pile organise itself to the critical state instead of overshooting into a permanent avalanche or settling flat?
Two opposite drives balance. Adding grains raises the average height, and a higher pile topples more readily, so avalanches grow larger. Larger avalanches reach the boundary more often and lose more grains, which lowers the pile. The only stationary state is the one where the mean loss per grain added is exactly one, and that forces avalanches to span every scale up to the system size: if they were all small, grains could not reach the edge and the pile would keep rising; if they were all system-wide, the pile would be draining faster than it fills. Criticality is the fixed point of this feedback, and it is reached from any start.
3. Percolation: the lattice that suddenly connects
Each site of a square lattice is open with probability p, independently, and open neighbours join into clusters. For small p the clusters are tiny; for large p one giant cluster covers the lattice. The change is not gradual: in the infinite lattice there is a sharp threshold p꜀ ≈ 0.5927 below which no infinite cluster exists and above which exactly one does. Drag the gold point to change p; the same random numbers are reused, so raising p only ever adds sites and you can watch clusters merge. The spanning cluster, the one that touches both top and bottom, is drawn in gold. The graph on the right is the probability that a spanning cluster exists, for two lattice sizes: the larger the lattice, the sharper the step.
Worked example. At \(p=0.5\) on the 48 by 48 lattice the largest cluster holds about a tenth of the open sites and nothing spans; at \(p=0.65\) the largest cluster holds over 80% of them and spans in almost every sample. Between them the spanning probability rises from near 0 to near 1 over a width that shrinks like \(L^{-3/4}\) as the lattice grows, since the correlation-length exponent is \(\nu=4/3\).
Watch out. Site percolation on the square lattice has no exact threshold: 0.592746 is numerical, while bond percolation on the same lattice has \(p_c=1/2\) exactly (Kesten, 1980) and site percolation on the triangular lattice also has \(p_c=1/2\). The exponents such as \(5/36\) and \(4/3\) are universal, shared by every two-dimensional lattice, and proved for the triangular lattice through Smirnov’s conformal invariance. On a finite lattice “spanning” is a substitute for “infinite”: the spanning probability at \(p_c\) is neither 0 nor 1 but a definite number that depends on the shape of the box and not on the lattice.
Why must there be a sharp threshold at all, rather than a spanning probability that rises smoothly with p?
Whether an infinite cluster exists does not depend on any finite set of sites: changing finitely many sites can shrink or extend a cluster but cannot create or destroy an infinite one. Kolmogorov’s zero–one law then says the event has probability 0 or 1 for each \(p\). Monotonicity, more open sites can only help, makes the set of \(p\) with probability 1 an interval \((p_c,1]\), and that is the threshold. The smooth curves on the right are finite-size effects: for a lattice of side \(L\) the step has width about \(L^{-1/\nu}\).
4. Flocking: the Vicsek model
Particles move at a fixed speed and, at every step, each one turns to the average heading of its neighbours within a radius, plus a random error of size η. There is no leader, no goal and no force, only imitation with noise. At low noise the whole population ends up moving together: the polarisation φ, the length of the mean velocity, is near 1. At high noise the headings decorrelate and φ falls to the 1/√N of a random crowd. Between them lies a phase transition, an ordered phase of self-propelled particles that no equilibrium system of the same dimension could have, since the Mermin–Wagner theorem forbids it. Change η and watch the flock form or dissolve; drag ρ to see that density helps.
Worked example. With \(N=300\), \(\rho=2\) and \(\eta=0.5\), the polarisation climbs from about 0.06 at the random start to above 0.9 within a few hundred steps and stays there. At \(\eta=2\pi\) the new heading is uniformly random whatever the neighbours do, so \(\varphi\) fluctuates around \(1/\sqrt{300}=0.058\). The measured curve on the right crosses \(\varphi=0.5\) between \(\eta=2.5\) and \(3\) at this density.
Watch out. The box is periodic, which is what lets a flock persist: on a wall-bounded field it would scatter at the boundary. With \(N=300\) the transition is smeared over a broad range of \(\eta\), and whether it is continuous, as Vicsek reported, or discontinuous with travelling density bands, as later large simulations found (Grégoire and Chaté, 2004), only shows up at sizes far beyond what runs in a browser. The averages on the right are over 150 steps after a transient of 150 at each \(\eta\), so the curve is a snapshot of a slow relaxation, not an equilibrium.
Why can two-dimensional flocks be ordered when two-dimensional magnets with continuous spins cannot?
In an equilibrium XY magnet a long-wavelength twist of the spins costs energy proportional to the square of its wavenumber, and summing the thermal fluctuations of all such twists in two dimensions gives a logarithmically divergent integral: the spins cannot agree over infinite distances. Flocking particles are not in equilibrium: they move, so a particle carries its heading into new neighbourhoods and the information about the mean direction is convected across the system rather than diffusing. Toner and Tu showed that this convective term makes the fluctuations converge, so true long-range order survives in two dimensions.