Dynamics of "Classical" Bosons, Fermions, and beyond

TL;DR

This work builds a geometry-based classical mechanics formalism that retains memory of quantum statistics by encoding exchange statistics in the symplectic form of a projected quantum-state manifold. By deriving a classical Lagrangian from a manifold of two-particle quantum states and employing statistics-dependent Berry connections and Kähler structure, the authors reveal how bosons and fermions exhibit distinct phase-space dynamics in both 1D inverted-oscillator scattering and 2D lowest-Landau-level vortex motion under quadratic potentials. The results demonstrate classical manifestations of exclusion and bunching, show statistics-driven differences in elliptical and saddle potentials, and suggest possible chaotic regimes for three or more particles, while clarifying the relationship to quantum dynamics and the geometric interpretation of quantum mechanics. The framework provides a tractable route to study many-body quantum-statistics effects in complex landscapes and offers a path toward extensions to anyons and classical-field limits relevant for quantum Hall physics. The work thus bridges quantum statistics, classical phase-space geometry, and potential experimental platforms for interferometry and beam-splitter-like setups.

Abstract

We study the classical mechanics and dynamics of particles that retains some memory of quantum statistics. Our work builds on earlier work on the statistical mechanics and thermodynamics of such particles. Starting from the effective classical manifold associated with two-particle bosonic and fermionic coherent states, we show how their exchange statistics is reflected in the symplectic form of the manifold. We demonstrate the classical analogues of exclusion or bunching behavior expected in such states by studying their trajectories in various quadratic potentials. Our examples are two-particle coherent states in one dimension and two-particle vortex motion in the lowest Landau level. We finally compare and contrast our results with previous simulations of the full quantum system, and with existing results on the geometric interpretations of quantum mechanics.

Figures (13)

• Figure 1: The symplectic factor $f_{\bar{z}z}=f(r^2)$ defined in Equation \ref{['stwoform']} for anyons of exchange phase $\nu\pi$ as a function of the relative coordinates, in units of $\sim \ell_B^{-2}$, which we have scaled to be dimensionless. This factor carries the crucial quantum statistics information going into the classical equations of motion. For distinguishable particles, the symplectic factor is always 1, which is also the asymptotic value for all identical particles. The difference between fermions and bosons is most pronounced for distances $\sim 0.5-2$ units.
• Figure 2: Scattering against a 1D inverted harmonic potential with $m\omega=1$: (a) A pictorial representation of two-particle scattering. The maximum of the potential is at $(x,p)=(0,0)$. For all trajectories, the center of mass is assumed to be located at $(X_{CM},P_{CM})=(0,0)$; all dynamics is in the relative coordinate. (b) Energy of each particle type for a fixed $x(0)$ and varying $\dot{x}(0)$ as described by Equation \ref{['ene']}. It is seen that for a given initial position, the energies of the particles are ordered by type as $E_{bos}>E_{dist}>E_{fer}$, as expected from the magnitude of their symplectic forms in this regime.
• Figure 3: Scattering trajectories for two particles in the presence of an inverted harmonic potential with initial condition $x(0)=0.7$ and varying initial velocities. Time is measured in dimensionless units, and the the scale of the x-axis is in 0.01 of this unit. The figures show the evolution of relative distance $x_1-x_2$ and relative velocity $\dot{x}_1-\dot{x}_2$ between the two particles respectively for initial relative velocity (a), (b) $\dot{x}(0)=-0.3$, (c), (d) $\dot{x}(0)=-0.8$ and (e),(f) $\dot{x}(0)=-1.3$. Insets: lin-log plots for large times. As initial velocities are increased, all particle species transition from reflecting against the potential to passing through each other. The critical initial velocity for this transition depends on the type of particle species under consideration.
• Figure 4: The function $R(x_L)$ defined as the number of time steps for which the trajectory $x(t)<x_L$ for the initial conditions: $x(0)=0.7$ and $\dot{x}(0)=-0.3$, $x(0)=0.7$ and $\dot{x}(0)=-0.8$ and $x(0)=0.7$ and $\dot{x}(0)=-1.3$, for the left center and right panels respectively. Note that the fermions spend less time in any given interval than the other particle types.
• Figure 5: Trajectories for harmonic elliptic and saddle potentials. In each case, the initial conditions for the two particles are shown as an enlarged green dot. Left Panel: All particles - single, distinguishable and identical - behave in a similar way when subjected to an elliptic harmonic trap with $v/u=0.4$. one particle is initially at the origin and the other at a distance $\sim$ 1.4 units away. Right Panel: A symmetric saddle potential with $u=-v=1$, again with one particle starting at the origin. The grey lines $y=\pm x$ are the asymptotes of the applied saddle potential.