Maxwell-Boltzmann in the Mosh Pit

Original paper: Collective Motion of Humans in Mosh and Circle Pits at Heavy Metal Concerts


You must have observed a flock of birds or a school of fish form wonderful patterns. The entire group behaves like one big organism. Have you ever wondered if humans behave similarly when many of them get together? Are there similarities between violent mobs or cheering crowds and a herd of sheep or a flock of birds? Today’s paper studies human behaviour in one such form of gathering – A mosh pit!

Before jumping into a mosh pit, it is worthwhile to discuss how systems made up of a collection of moving objects form complex patterns. One type of system where complex patterns arise occurs when each moving unit in the system interacts with its neighbours while maintaining the same absolute velocity. Systems of this sort are very common in biology, from schools of fish to flocks of starlings and swarms of locusts. In all these examples, each moving unit rarely communicates with the entire group to coordinate their motion. However, each individual in these groups is trying to move in the general direction of their nearest neighbours, and with the same speed as them. Tiny errors may occur while trying to follow the neighbourhood. Under the right circumstances, these tiny errors can give rise to complex patterns. This phenomenon, where patterns emerge because self-propelling particles try to align with their neighbours, can be modelled by a set of very simple equations introduced by Tamas Vicsek in 1995. This is one of the simplest models physicists use to study how patterns emerge in flocks.

1
Figure 1: Left: Starlings in flight form patterns. Flocking of birds is an example of collective motion where patterns emerge because each bird in the flock tries to follow in the general direction of their neighbours. Right: A bird labeled k with position X_k and velocity w_k tries to align with the mean velocity of its neighbours within the disk of radius R

Human beings are capable of intelligent decision making on their own. Yet a crowd’s behaviour may not have any trace of the intelligence of an individual. Spontaneous formation of lanes of pedestrians, jamming during a panic-induced motion of a crowd and Mexican waves (also known as ‘the wave’ in the US) of cheering fans during a football match are all collective behaviours in response to stimuli from the surroundings. It is worth asking if the equations that describe the motion of flocks of birds can be applied to a collection of humans with reasonable accuracy.

In today’s paper, Silverberg and his colleagues study the dynamics of the crowd in heavy metal concerts. They study YouTube videos to calculate the speed distribution in mosh pits. While they refer to previous studies of crowd dynamics, they start this paper with a few surprising observations. Even though the density of people in a mosh pit is much higher than gaseous systems, their behaviour resembles gaseous particles. They report that the speed distribution in mosh pits fits well with the 2D Maxwell-Boltzmann distribution.

mbdist
Figure 2: An example of the Maxwell-Boltzmann distribution of speed of particles in a gas.

The Maxwell-Boltzmann distribution is used to describe the probability that a particle in a volume of gas moves with a certain speed. Such a distribution occurs when a volume of gas is in thermal equilibrium, meaning that the gas has the same temperature as its surroundings and its temperature does not change with time. The shape of that distribution suggests that there is a range of speeds, in the middle of the distribution, that includes the majority of the particles (denoted by the light blue bars in Figure 2). A much smaller fraction of particles is likely to move very fast or slow. Similarly, a small fraction of the participants in a mosh pit moves very fast, while the majority moves at a much slower pace.

A mosh pit resembles gaseous systems in equilibrium although it has all the characteristics of a non-equilibrium system –  it continuously changes with time as participants join or leave the pit, it changes shape on the suggestion of the performers on stage. Although it may be expected that each participant in a mosh pit moves with similar velocity as their neighbour, the authors show that there is no such dependence beyond one shoulder length. Thus it is not necessary that your neighbours have the same velocity as you, in a mosh pit. The authors go on to ask and answer the question: “why does an inherently non-equilibrium system exhibit equilibrium characteristics?”

3
Figure 3: The forces that can be used to describe the motion of crowds in a mosh pit. The repulsion force occurs due to collision and does not exist beyond two human shoulder length. The propulsion force describes the individual velocity and the force needed to move against or with the crowd. The flocking force describes the tendency to follow in the average direction and speed of the crowd. Noise term is used to quantify any other random behaviour.

They simplify the complex dynamics exhibited in a mosh pit by breaking it down to a few phenomena that can be observed intuitively. They observe that a mix of forces that describe repulsion due to collision, self-propulsion, flocking interaction and some random noise, can model the crowd.

Using these equations and suitable parameters, they simulate the dynamics of the crowd (Figure 4). It is interesting to note that the equations lead to three major phenomena that may dominate at various time scales. They are flocking, noise and collision. Noise and collision tend to disrupt any patterns, whereas the tendency to form flocks and follow the neighbours creates patterns. If it takes a long time to form flocks, the disruption from collisions and noise gives rise to random motion. Random motion has the nice property of making its velocity distribution look like a bell curve, which gives rise to the Maxwell-Boltzmann distribution[1].

 

 

circlepit
Figure 4: This is an example of a simulation of the crowd using the equations in figure 3. Red dots represent active participants who move around and thus experience flocking interactions. Black dots represent participants who prefer to remain stationary and are not subject to flocking interactions or random forces. This simulation can be accessed at: http://mattbierbaum.github.io/moshpits.js/

 

Therefore, a majority of random movers have a speed around the most probable one, whereas a tiny fraction moves fast or slow. This is similar to gases in equilibrium and answers the question why mosh pits that seem to be out of equilibrium behave like particles in an undisturbed gas. If, however, people form flocks faster than they collide with each other they tend to separate themselves out from non-participants. In this case, the flocking dominates for the active participants leading to a different distribution. The active participants are confined but they can move with a large angular momentum.

While studying human behaviour is a fun exercise, the authors conclude that such studies may help design better crowd management strategies and architectural safety protocols. Heavy metal concerts put a large number of humans in extreme conditions thus creating an opportunity to study how humans behave as a group.

[1] Vicsek, T. & Zafeiris, A. Collective motion. Phys. Rep. 517, 71–140 (2012).

[2] Simulation can be found at: http://mattbierbaum.github.io/moshpits.js/


1. The Gaussian distribution, or bell curve, describes the velocity of particles in a gas. The Maxwell-Boltzmann distribution for the speed (= absolute value of the velocity vector) can be derived from the Gaussian distribution of the velocities. (Link to: Distribution for the velocity vector)

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