Coarse graining

The idea behind coarse graining is that the position of points in phase-space cannot be measured with arbitrary precision. As an example, we might only be able to measure positions and momenta to within some small error, .

This is essentially the same situation as if phase space was separated into cubes with size and we could only determine which cube the phase point was in.

This picture nicely demonstrate why the dynamics become non-invertible after coarse graining:


In the figure two coarse grained orbites, light gray squares, are formed from two trajectories, black. Although the trajectories do not intersect, the coarse grained orbits merge at the dark-grey square.

After coarse graining non-invertibility occurs both for and for . Coarse graining thus fails to generate an asymmetry between past and future.

A further problem with coarse graining is that the entropy depends on . If we take coarse graining as the origin of irreversibility, the status of the second law of thermodynamics will be more similar to a measurement problem than to a law of nature.

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Author Per Stoltze stoltze@fysik.dtu.dk