Dislike of general opinion makes for tight elections.

In modern democracies, the outcome of elections and referendums is often remarkably tight. The repetition of these divisive events are the hallmark of a split society; to the physicist, however, it is an astonishing feat for such large collections of diverse individuals. Many sociophysics models reproduce the emergence of collective human behavior with interacting agents, which respond to their environment according to simple rules, modulated by random fluctuations. A paragon of this class is the Ising model which, when interactions are strong, predicts that order can emerge from a chaotic initial state. In contrast with many elections, however, this model favors a strong majority. Here we introduce a new element to this classical theory, which accounts for the influence of opinion polls on the electorate. This brings about a new phase in which two groups divide the opinion equally. These political camps are spatially segregated, and the sharp boundary that separates them makes the system size dependent, even in the limit of a large electorate. Election data show that, since the early 1990s, countries with more than about a million voters often found themselves in this state, whereas elections in smaller countries yielded more consensual results. We suggest that this transition hinges on the electorate's awareness of the general opinion.

Laboratory rivers adjust their shape to sediment transport.

An alluvial river builds its own bed with the sediment it transports; its shape thus depends not only on its water discharge but also on the sediment supply. Here we investigate the influence of the latter in laboratory experiments. We find that, as their natural counterpart, laboratory rivers widen to accommodate an increase of sediment supply. By tracking individual particles as they travel downstream, we show that, at equilibrium, the river shapes its channel so that the intensity of sediment transport follows a Boltzmann distribution. This mechanism selects a well-defined width over which the river transports sediment, while the sediment remains virtually idle on its banks. For lack of a comprehensive theory, we represent this behavior with a single-parameter empirical model which accords with our observations.

Boltzmann Distribution of Sediment Transport.

The coupling of sediment transport with the flow that drives it allows rivers to shape their own bed. Cross-stream fluxes of sediment play a crucial, yet poorly understood, role in this process. Here, we track particles in a laboratory flume to relate their statistical behavior to the self-organization of the granular bed they make up. As they travel downstream, the transported grains wander randomly across the bed's surface, thus inducing cross-stream diffusion. The balance of diffusion and gravity results in a peculiar Boltzmann distribution, in which the bed's roughness plays the role of thermal fluctuations, while its surface forms the potential well that confines the sediment flux.

Growth and shape of a laboratory alluvial fan.

Using laboratory experiments, we investigate the influence of water and sediment discharges on the morphology of an alluvial fan. In our flume, a single-thread laminar river deposits corundum sand into a conical fan. We record the fan progradation with top-view images and measure its shape using the deformation of a Moiré pattern. The fan remains virtually self-affine as it grows, with a nearly constant slope. We find that, when the sediment discharge is small, the longitudinal slope of the fan remains close to that of a river at the threshold for sediment transport. Consequently the slope depends on the water discharge only. A higher sediment discharge causes the fan's slope to depart from the threshold value. Due to the downstream decrease of the sediment load, this slope gets shallower towards the fan's toe. This mechanism generates a concave fan profile. This suggests that we could infer the sediment flux that feeds a fan based on its proximal slope.

Laplacian networks: Growth, local symmetry, and shape optimization.

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

Laboratory alluvial fans in one dimension.

When they reach a flat plain, rivers often deposit their sediment load into a cone-shaped structure called alluvial fan. We present a simplified experimental setup that reproduces, in one dimension, basic features of alluvial fans. A mixture of water and glycerol transports and deposits glass beads between two transparent panels separated by a narrow gap. As the beads, which mimic natural sediments, get deposited in this gap, they form an almost one-dimensional fan. At a moderate sediment discharge, the fan grows quasistatically and maintains its slope just above the threshold for sediment transport. The water discharge determines this critical slope. At leading order, the sediment discharge only controls the velocity at which the fan grows. A more detailed analysis reveals a slight curvature of the fan profile, which relates directly to the rate at which sediments are transported.

Width of laminar laboratory rivers.

A viscous fluid flowing over plastic grains spontaneously generates single-thread channels. With time, these laminar analogues of alluvial rivers reach a reproducible steady state, showing a well-defined width and cross section. In the absence of sediment transport, their shape conforms with the threshold hypothesis which states that, at equilibrium, the combined effects of gravity and flow-induced stress maintain the bed surface at the threshold of motion. This theory explains how the channel selects its size and slope for a given discharge. In this light, laboratory rivers illustrate the similarity between the avalanche angle of granular materials and Shields's criterion for sediment transport.