Absence of hidden analytic conserved quantities in harmonically confined rods

Abstract

Systems of hard rods of equal length in a one-dimensional harmonic trap have been observed to exhibit peculiar non-ergodic behavior that might suggest the existence of a novel hidden conserved quantity beyond the two well known ones, i.e., the total energy and the center-of-mass energy. In this work, we investigate this possibility by systematically constraining the forms of the conserved quantities, and we rigorously rule out the existence of any extra hidden conserved quantity that is analytic in the positions and momenta of the rods involved. We do so by showing two key results: conservation during free motion demands the $U(1)$ invariance of these quantities under rotations of the position and momenta of each rod, and conservation during collisions demand an $S_N$ invariance under the permutation of the momenta of the rods as long as one of the rods have non-zero length. We then show that these conditions imply that any conserved quantity is functionally dependent on the two known conserved quantities. In addition, we show that in the special case where all rods have zero length (i.e., when they are point particles), conservation under collisions only requires invariance under a smaller $S_N$ group of permutations of the labels of the rods, which leads to a much larger set of analytic conserved quantities that we explicitly write down. In all, this rigorously clarifies the structure of conserved quantities in the hard rod problem, and motivates the application of such systematic methods to other classical systems.