Random Systems
Linear stochastic equation
Random walks
How it works
A two-dimensional random walk is a concatenation of random movements, where each movement depends on its previous step. In this example, the initial position is at Point (0,0). With each new step, another random point within range is marked, starting from the last point. If a point is stepped on several times, it will be displayed a little thicker in this example. If there are no more untouched steps in any direction, the walk ends here.
Most examples of random walks have a fixed number of steps as the only end condition.
Where it is used
Random walks are used, for example, in financial mathematics to model stock prices, and in physics to simulate particle movement. They are mainly used for simple illustration of complex seemingly random motions.
What you can do
In this example, you can display individual steps or let it run
on its own with the "Go" button. "Reset" resets the graphic.
Check out a simplified python implementation
here.
Simulation of 2D Diffusion
What is diffusion?
Diffusion describes the process by which a substance or particle spreads out from a concentrated area into the surrounding area and thus decreases in concentration. This can be observed especially in liquids and gases, as these particles seem to move randomly in all directions.
Why is this so important for nature?
In nature, diffusion is everywhere. Animals use their sense of smell to detect tiny particles, some of which move and spread randomly through space, and can thus locate their prey or pack. The process of diffusion is also found in the human body, as respiration releases excess carbon dioxide, which concentrates in the cells and diffuses outward through the cell membrane into the blood. The process of osmosis in plants is also due to the diffusion of water.
Diffusion in everyday life
We humans also experience diffusion processes every day. The smell of coffee in the morning, which spreads in the apartment, the smell of freshly mowed lawn or even dead creatures. All these smells and particles spread in the space by the process of diffusion.