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arXiv:2511.06217 · Physics Letters A

Finite-sample deviations and convergence in the statistics of Bohmian trajectory ensembles

Bingyu Cui, Yanting Cao

quant-phSubmitted Nov 9, 2025• Announced Nov 11, 2025• 0 citations
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Abstract

We analyze finite-sample statistics of Bohmian trajectories for single spinless and spin-1/2 particles. Equivariance ensures agreement with $|ψ|^2$ in the quantum equilibrium limit, yet experiments and simulations necessarily use finite ensembles. We show that in regular flows (e.g., wavepackets or low-mode superpositions of eigenstates of harmonic oscillators) sample means and/or variances over modest $N$ are consistent with Born-rule moments. In contrast, degenerate superpositions of 3D oscillators with nodal barriers and chaotic Bohmian dynamics exhibit sensitive dependence on initial conditions and complex flow partitioning, which can yield noticeable finite-sample deviations in the mean and variance. For the spin-1/2 particle, both convective and Pauli currents conserve $|ψ|^2$, but they are associated with different velocity fields and thus might yield different finite-sample trajectory statistics. These findings calibrate the interpretation of trajectory-based uncertainty and provide practical guidance for numerical Bohmian simulations of spin and transport, without challenging the equivalence to orthodox quantum mechanics in the quantum equilibrium ensemble.

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We analyze finite-sample statistics of Bohmian trajectories for single spinless and spin-1/2 particles. Equivariance ensures agreement with ∣ψ∣2|ψ|^2∣ψ∣2 in the quantum equilibrium limit, yet experiments and simulations necessarily use finite ensembles. We show that in regular flows (e.g., wavepackets or low-mode superpositions of eigenstates of harmonic oscillators) sample means and/or variances over modest NNN are consistent with Born-rule moments. In contrast, degenerate superpositions of 3D oscillators with nodal barriers and chaotic Bohmian dynamics exhibit sensitive dependence on initial conditions and complex flow partitioning, which can yield noticeable finite-sample deviations in the mean and variance. For the spin-1/2 particle, both convective and Pauli currents conserve ∣ψ∣2|ψ|^2∣ψ∣2, but they are associated with different velocity fields and thus might yield different finite-sample trajectory statistics. These findings calibrate the interpretation of trajectory-based uncertainty and provide practical guidance for numerical Bohmian simulations of spin and transport, without challenging the equivalence to orthodox quantum mechanics in the quantum equilibrium ensemble.
We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule P=|Ψ|2 by initial distributions of Bohmian particles with P0≠|Ψ0|2.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.
The arrival time statistics of spin-1/2 particles governed by Pauli's equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrival time experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.
The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.
We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of the moving nodal points, in cases with 1, 2 and multiple nodal points. Then, we find the contributions of the ordered and chaotic trajectories in the Born distribution, and when the latter is accessible by an initial particle distribution which does not satisfy Born's rule.
Abstract We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the ‘nodal point-X-point complex’ mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is manifested after a short time. We then study the mixing properties of ensembles of Bohmian trajectories with initial conditions satisfying Born’s rule. The trajectory points are initially distributed in two sets S 1 and S 2 with disjoint supports but they exhibit, over the course of time, abrupt mixing whenever they encounter the nodal points of the wavefunction. Then a substantial fraction of trajectory points is exchanged between S 1 and S 2 , without violating Born’s rule. Finally, we provide strong numerical indications that, in this system, the main effect of the entanglement is the establishment of ergodicity in the individual Bohmian trajectories as t → ∞ : different initial conditions result to the same limiting distribution of trajectory points.