#002: Properly extend the classical uncertainty relationship to multiple d.o.f.

Category: Classical mechanics - Tags: Classical information, Hamiltonian mechanics, Entropy

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Find out how to generalize the uncertainty relationship we have found for classical mechanics (as it can be seen for example in this paper).

Mathematical problem. In a nutshell, the problem is equivalent to studying the evolution of the covariance matrix (a rank two tensor) over phase space (a symplectic manifold) under all possible linear Hamiltonian (symplectic) evolutions. That is, under the symplectic group.

As an example of the type of things we are looking for:

• If we start with a packet with no correlation across multiple degrees of freedom and equal uncertainty within each d.o.f. (a condition related to thermodynamic equilibrium), do we find that the uncertainty in each d.o.f. cannot become lower than the initial one?
• Still starting from the previous case, is it true that the uncertainty within a d.o.f. can only increase if we are adding a correlation to another d.o.f.? Does it mean that the “equipartition of entropy” over uncorrelated d.o.f. is the lowest bound?

Physical significance. The idea is to explore more in general the relationship between information theory, Hamiltonian mechanics and thermodynamics. The conjecture is that Hamiltonian dynamics preserves the entropic relationships that one would find at equilibrium. The classical uncertainty relationship is essentially an interplay between the total uncertainty (which is conserved) and the ones along the individual variables (which can be increased by introducing correlations between them). This work needs to understand the interplay between the total uncertainty and that of each individual degree of freedom.