A dimensionless particle in a square heat reservoir:
|convergence to the centre of the room using the radius of gyration as a proxy measure|
When simulating the toy problem of a dimensionless particle in a square heat reservoir, I made the following assumptions:
- The room is a 10x10 square and the walls are inelastic.
- Given that state is represented by the particle's position and the room is convex, the euclidean distance is a good metric for measuring the difference between states.
- Assuming that the Causal Path Entropy varies continuously over states, we have a second argument for discretisation and may use the max operator rather than the nabla operator to discover local maxima.
- Assuming that the Causal Path Entropy is proportional to a propensity for mixing, we may approximate variations in Causal Path Entropy with Euclidean proxy measures for diffusion such as average nearest neighbours and the radius of gyration.
- The particle isn't quite dimensionless though it's relatively small with respect to the room which allows us to approximate the Causal Path Entropy with the Boltzmann Entropy.
The details of the experiment are contained in the following notebook.
- Causal Entropic Forces (A. D. Wissner-Gross & C.E. Freer. 2013. Physical Review Letters.)
- Causal Entropic Forces: Intelligent Behaviour, Dynamics and Pattern Formation (Hannes Hornischer. 2015. Masters Thesis.)