Can a system really be random?

How random is dice?

How much chance is there in a die roll? Can you predict the outcome of a litter, maybe even influence it? Jan Nagler from the Max Planck Institute for Dynamics and Self-Organization in Göttingen works, among other things, in the field of chaos research and develops mathematical models to get to the bottom of this question.

When one speaks of chaos in everyday life, one usually means a state of great disorder. In physics, a chaotic system is characterized by the fact that it reacts extremely sensitively to given initial conditions. This dependency means that a small change in the start-up requirements leads to large changes in the output. This makes long-term predictions as good as impossible, the dynamics of the system appear "chaotic".

"Real randomness does not play a major role in chaotic systems - in contrast to the sensitivity of the initial conditions, they are not necessary for chaotic behavior," says Jan Nagler, physicist at the Max Planck Institute for Dynamics and Self-Organization and at the University's Institute for Nonlinear Dynamics Goettingen. He finds chaotic systems fascinating because they are still largely not understood today.

A model of chance

This is because the behavior of chaotic systems can usually not be predicted, although the systems under consideration are deterministic, that is, the future development is completely determined by the initial conditions. Since no elements of chance appear in the equations that describe the movement of the system, one often speaks of so-called deterministic chaos.

Physicists are faced with the apparently paradoxical situation that a system that is inherently predictable, despite everything, hardly allows any predictions about the future. “I think that's charming,” emphasizes Nagler. The attraction of chaos theory is that in chaotic systems the knowledge of the future state is limited by the precision with which scientists can measure the initial conditions.

A dumbbell as a model of chance

A cube has always been considered the perfect generator for random numbers. “The cube is an interesting system, especially because not many have dealt scientifically with the process of rolling the dice.” Nagler investigated the question of how one can describe the dynamics of a dice and to what extent chance plays a role in the process of throwing the dice .

In order to investigate the randomness of throwing the dice, the physicist developed a model that at first glance has little to do with the classic six-sided dice: throwing a dumbbell. Because the chance of throwing the dice comes to the fore when it stands vertically on an edge and can either fall to one side or the other. This situation is well described by the mechanism of the dumbbell throw. “This is the simplest cube model that describes chaotic behavior,” says Nagler.

Coincidence in the simulation

Video: dumbbell throw in simulation

In the model, two point masses are connected by a massless rod. The dumbbell constructed in this way can move in one plane, i.e. two spatial dimensions, and falls down due to gravity. If the dumbbell hits the floor in the simulations, it jumps back, falls down and jumps up a little, maybe still turning and at some point comes to a standstill - just like a cube hops over the table several times before it hits one Number shows.

The movement of the dumbbell is determined by various initial conditions: the height of the throw, the orientation of the dumbbell to the floor, the relationship between the two masses and of course the friction of the floor. The frictional forces extract energy from the dumbbell and ultimately brake its fall to a standstill. “The free fall is clear. Then the dumbbell comes on the table top, but the effects of friction in general are not yet fully understood, ”says Nagler, explaining the problem.

A symbolic code decodes the dynamics

In order to describe the dynamics of a dumbbell throw under different initial conditions, Nagler came up with a symbolic code with which he can characterize chaotic behavior. Each time the dumbbell hits, the physicist looks at the orientation of the two masses compared to the starting position. If the orientation has not changed, he assigns a 0 to the impact, if the position of the masses is reversed, he marks it with a 1. In addition, he distinguishes whether the dumbbell rotates counterclockwise or clockwise during a rollover and describes this with the Letters L or R.

The movement of a dumbbell

Nagler used simulations to test many different initial conditions to find out how they determine the position of the dumbbell after a throw. For example, he determined the friction, the throwing height and the mass ratio and then systematically tested the possibilities of how the dumbbell could be thrown. Such a throw is then only determined by the orientation of the barbell to the ground and by the force with which it is thrown.

The results of the simulations can be illustrated in a so-called orbit flip diagram. In this diagram, the force (angular momentum) is plotted against the orientation (angle of rotation) and each point corresponds to a unique set of initial values. If the orientation of the masses has not changed after the throw, the point is colored yellow. The yellow color corresponds to the number 0 of the symbolic code. A change in orientation is marked in red. The brightness of the color also reveals how often the dumbbell hopped on the floor before it came to a standstill. The darker the color, the more jumps the barbell made.

Dice are unfair

Orbit flip charts

The orbit flip diagrams clearly show that the initial conditions determine the predictability or unpredictability of rolling the dice. Because large, single-colored areas appear in the diagrams, in which the dynamics of the throw can be well described. There are no random elements here that could influence the result of the throw. This is different in the mixed areas, where you can see chaotic, apparently random behavior: the red and yellow dots alternate again and again. In these areas there is the sensitive dependency of the initial conditions that make up chaotic behavior.

The graphs show that the predictable areas shrink the higher the barbell is thrown and the smaller the friction when bouncing. Both of these increase the number of jumps the dumbbell makes on the floor before it comes to a rest, and thus more often leads to the random situations in which the dumbbell is almost on the tip.

With the help of his analyzes, Nagler found that the model dumbbell makes an average of five such jumps with realistic friction levels and thus the throwing result is relatively easy to predict. The physicist therefore ascribes the fact that a dice is used as a generator for random numbers to the “external randomness” of the hand: A normal dice player cannot sufficiently reproduce the starting conditions with which he could repeatedly roll similar results. However, Nagler adds: “I do believe that cheating can be trained well. Dice are always unfair because they don't forget how they started. "