Position-Momenta Uncertainties in Classical Systems

TL;DR

The paper shows that classical particles in thermal baths with either strictly conserved or mean-biased angular momentum exhibit quantum-like position-momentum uncertainty bounds. By constructing radial and generalized Gibbs baths with temperature $T$ and chemical potential $\mu$, the authors derive steady states and compute variances, establishing lower bounds on $\Delta x \Delta p_x$ (and $\Delta y \Delta p_y$) proportional to the mean angular momentum, with a universal factor $c\le 1/2$ for central potentials and a $T\to0$ equality. The framework extends to noncentral and higher-dimensional potentials, where bounds depend on $\langle p_\phi\rangle$ and include non-conjugate pairs with coefficients $c_{ij}$, offering a classical mechanism for uncertainty-like constraints tied to angular-momentum symmetry. These results bridge classical stochastic thermodynamics and quantum-like uncertainty, suggesting experimental tests via active/chiral particles or Brownian gyrators and enriching the understanding of how conservation laws shape fluctuations. The work highlights the role of generalized ensembles in producing fundamental bounds and points to broader implications for the quantum-classical boundary in nonequilibrium statistical physics.

Abstract

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.

Figures (4)

• Figure 1: Sample paths for $V(r)=r^\alpha/\alpha$ at $T=10^{-3}$ with $\ev{p_\phi}=1$ and $\gamma=2.$ (a) $\alpha=2$ (two sample paths) (b) $\alpha=2.1.$ The insets show the spiraling nature of trajectories.
• Figure 2: (a) Four sample paths, (b) steady state distribution $P(x,y)$, for $V(r)=r^2/2$ with conserved $\ell=1$ ($\gamma=2$, $T=0.5$). Dashed line in (a): limiting trajectory as $T\to0.$
• Figure 3: Time evolution of $\Delta x$, $\Delta p_x$ and their product $\Delta x\Delta p_x$ for $V(r)=kr^\alpha/\alpha$ with $\gamma=4$, $k=2$ and conserved $\ell=1$ at $T=10^{-1}$. (a) and (b) show this for $\alpha=2$. (c) and (d) show this for $\alpha=2.1$. The lower bound, $\abs{\ell}/2$ of $\Delta x\Delta p_x$, is shown in a black solid line in (b) and (d). Data is averaged over 5000 samples. The stochastic differential equations \ref{['eq:2dLangevin']} were integrated using the second order technique discussed in Mannella1989.
• Figure 4: Time evolution of $\Delta x$, $\Delta p_x$ and their product $\Delta x\Delta p_x$ for $V(r)=kr^\alpha/\alpha$ with $\gamma=4$, $k=2$ and fluctuating $p_\phi$ with $\ev{p_\phi}=1$ at $T=10^{-1}$. (a) and (b) show this for $\alpha=2$. (c) and (d) show this for $\alpha=2.1$. The lower bound, $\abs{\ev{p_\phi}}/2$ of $\Delta x\Delta p_x$, is shown in a black solid line in (b) and (d). Data is averaged over 5000 samples. The stochastic differential equations \ref{['eq:Lang_2d_noncons']} were integrated using the second order technique discussed in Mannella1989.