Weak (non)conservation and stochastic dynamics of angular momentum

TL;DR

This work addresses how weak breaking of rotational symmetry induces the slow decay of total angular momentum in an isolated, multi-particle system. It develops a mesoscopic framework based on Langevin and Fokker-Planck dynamics, deriving a closed stochastic differential equation for the total angular momentum $L$ that captures both its drift and fluctuations as the system warms and reconfigures. By connecting microcanonical and canonical descriptions, the authors reveal how the conserved quantities evolve under weak anisotropy and validate the theory with molecular-dynamics simulations across low- and high-density regimes. The findings illuminate the fate of integrals of motion under near-symmetric conditions and suggest avenues for controlling rotational dissipation and energy conversion in mesoscopic systems.

Abstract

Angular momentum conservation influences equilibrium statistical mechanics, leading to a generalized microcanonical density for an isolated system and a generalized Gibbs density for a weakly coupled system. We study the stochastic decay of angular momentum due to weakly imperfect rotational symmetry of the external potential that confines the isolated many-particle system. We present a mesoscopic description of the system, deriving Langevin and Fokker-Planck equations, which are consistent with equilibrium statistical mechanics when rotational symmetry is maintained. When the symmetry is weakly violated, we formulate a coarse-grained stochastic differential equation governing the decay of total angular momentum over time. To validate our analytical predictions, we conduct numerical simulations of the microcanonical ensemble, an isolated system undergoing thermalization due to weak two-body interactions. Our coarse-grained Langevin equation accurately characterizes both the decay of the angular momentum and its fluctuations in a steady state. Furthermore, we estimate the parameters of our mesoscopic model directly from simulations, providing insights into the dissipative phenomenological coefficients, such as friction. More generally, this study contributes to a deeper understanding of the behavior of the integrals of motion when the corresponding symmetry is weakly violated.

Figures (9)

• Figure 1: Verification of the relationships between parameters of microcanonical and canonical distributions. The data points are computed from independent simulations, while the dashed lines correspond to functions defined in (\ref{['fdef']}). This analysis demonstrates that we have ensemble equivalence (\ref{['ensamble-equiv']}) in the case of low density.
• Figure 2: Trajectories of 523 particles over 5 units of time projected on $xy$ plane, corresponding to the system of Figure \ref{['fig-L']} at $L\simeq1500$. The gradient of the trajectory indicates the direction of time. The system is in the underdamped regime. The small red dot (at the bottom left corner) indicates the size of the Lennard-Jones particle.
• Figure 3: This figure illustrates the relaxation of total angular momentum $L$. The solid line is the ensemble average over 1200 simulations with the same initial $E$ and $L$. The shaded region surrounding the line corresponds to one standard deviation of the values within the ensemble. The inset shows the relaxation just after the violation of the rotational symmetry. For a different perspective, the same plot is presented on the right side with a logarithmic scale.
• Figure 4: Three methods of estimating $\gamma(t)$. The 1st and 2nd methods utilize the dissipation of $L$, while the 3rd method uses fluctuations of angular momentum in steady state to estimate $\gamma$.
• Figure 5: Temperature of the system during the relaxation, computed by maximum likelihood estimator. Again, the shaded region is the one standard deviation of estimated values. Note that temperature increases more than twice.