Position-momenta uncertainties in classical systems

We design a thermal bath that preserves the conservation of a system's angular momentum or allows it to fluctuate around a specified nonzero mean while maintaining a Boltzmann distribution of energy in the steady state. We demonstrate that classical particles immersed in such baths exhibit position-momentum uncertainties with a strictly positive lower bound proportional to the absolute value of the mean angular momentum. The proportionality constant, c, is dimensionless and does not depend explicitly on the system's parameters. Remarkably, while c is universally bounded by unity, it attains the exact value c=1/2 for particles in central potentials.