It is amazing how different the world appears on different scales. Let us take as an example fluid flow, say the river Isar moving at slow pace through the city of München. Its flow patterns are governed by the Navier-Stokes equations. These are deterministic field equations (partial differential equations). If we could increase the degree of spatial resolution by a factor 107, then we would see H2O (and other) molecules moving fairly rapidly in a random like fashion. To strengthen our imagination of an unfamiliar world, computers are of great help. It is a striking experience to watch a movie of a molecular dynamics simulation of a large number of interacting particles governed by Newton’s equation of motion. Particles collide incessantly and move erratically without any particular aim. How do these particles then manage to organize themselves in such a way as to form a flow pattern on a large scale? Very crudely the reason is that the local conservation laws of mass, momentum and energy impose constraints not immediately visible on the microscopic scale. In what will follow our goal is to elucidate how from a random motion on the small scale a deterministic motion on the large scale emerges.
KeywordsBoltzmann Equation Random Motion Brownian Particle Hydrodynamic Limit Deterministic Chaos
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