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Quantum Equilibrium in Stochastic de Broglie–Bohm–Bell Quantum Mechanics

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Abstract

This paper investigates dynamical relaxation to quantum equilibrium in the stochastic de Broglie–Bohm–Bell formulation of quantum mechanics. The time-dependent probability distributions are computed as in a Markov process with slowly varying transition matrices. Numerical simulations, supported by exact results for the large-time behavior of sequences of (slowly varying) transition matrices, confirm previous findings that indicate that de Broglie–Bohm–Bell dynamics allows an arbitrary initial probability distribution to relax to quantum equilibrium; i.e., there is no need to make the ad-hoc assumption that the initial distribution of particle locations has to be identical to the initial probability distribution prescribed by the system’s initial wave function. The results presented in this paper moreover suggest that the intrinsically stochastic nature of Bell’s formulation, which is arguable most naturally formulated on an underlying discrete space-time, is sufficient to ensure dynamical relaxation to quantum equilibrium for a large class of quantum systems without the need to introduce coarse-graining or any other modification in the formulation.

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Notes

  1. As was briefly discussed in Ref. [9] the time step size \(\epsilon\) could be time (index) dependent, with a magnitude that is self-consistently determined by the system’s dynamics. Here, such a time dependence will be ignored, as it does not materially impact the results presented below.

  2. See Ref. [17] for the formal definition of weak (and strong) ergodicity and below in Sect. 2.3.1.

  3. More precisely (dropping the argument (t)): Suppose \(\tilde{T}\pi\) is invariant under an infinitesimal shift of \(\pi\) of the form, \(\pi \rightarrow \pi + \delta ^j - \delta ^{j'}\) with \(\delta ^j\) a vector which is zero everywhere, except on location j where it has value \(\epsilon\). Then the infinitesimal change of \(\tilde{T}\) must be zero for all pairs \(j,j'\), i.e., \(\delta (\sum _m\tilde{T}_{nm}\pi _m) = \sum _m\tilde{T}_{nm}(\delta ^j_m - \delta ^{j'}_m) = \epsilon \sum _s \lambda ^s v^s_n(\overline{v}^s_j - \overline{v}^s_{j'}) = \epsilon (\tilde{T}_{nj} - \tilde{T}_{nj'}) = 0\), which shows that \(\tilde{T}\) must have identical columns, as in Eq. (23).

  4. Technically, the matrices must furthermore be bounded and have non-zero elements well-separated from zero.

  5. Units are such that \(\hbar = c = 1\) and the scale is set using the linear box dimension L.

  6. To test the impact of the time step size, a few computations were repeated with \(\epsilon /L=0.01/N\), which produced essentially identical results.

  7. As can be seen in Fig. 2, the exponential suppression of the eigenvalues of \(\tilde{T}(t)\) is not strictly linear in the eigenvalue index s: the slope is not quite constant and there are clusters of near degenerate eigenvalues. This introduces additional uncertainty in the fitted results for c(t) at different values of 1/N, which is not included in the size of the error bars.

  8. Cf. [19,20,21] for detailed work on the nature and role of chaos in the causal de Broglie–Bohm theory.

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Acknowledgements

I would like to thank Marc Artzrouni for helpful comments and discussion.

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    Jeroen C. Vink

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Vink, J.C. Quantum Equilibrium in Stochastic de Broglie–Bohm–Bell Quantum Mechanics. Found Phys 53, 27 (2023). https://doi.org/10.1007/s10701-022-00668-5

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