Position-Momenta Uncertainties in Classical Systems

We demonstrate that classical particles coupled to thermal baths that conserve angular momentum, or allow it to fluctuate about a nonzero mean, obey a position–momentum uncertainty relation formally analogous to the Heisenberg bound. For motion in an arbitrary central potential, this relation universally reduces to  $\Delta x  \Delta p_x   >  L /2 $   where  $L$  is the mean angular momentum (or the conserved initial value). We establish the physical realizability of such baths by constructing Langevin dynamics that preserve a Boltzmann energy distribution in steady state for both conserved and non-conserved angular momentum ensembles. We also outline experimental routes for observing this emergent classical uncertainty bound.

Ref: Dipesh K. Singh, P. K. Mohanty, Phys. Rev. E 112, 054129 (2025)