# On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory

## Abstract

**:**

## 1. Introduction

## 2. Pilot-Wave Theory and the Born Rule

## 3. The Dynamical Relaxation Justification

“The exact (fine-grained) density is given by [$P=\rho /g$]. Now, starting from an arbitrary [$g(q,0)$], the initial values of [g] are carried along the system trajectories in configuration space. If the system is sufficiently complicated, the chaotic wandering of the trajectories [$Q\left(t\right)$] will distribute the [g] values in an effectively random manner over the accessible region of configuration space. On a coarse-grained level, P will then be indistinguishable from [$\rho ={|\Psi |}^{2}$]. Another (equivalent) picture sees the increase of subquantum entropy as associated with the effectively random mixing of two ‘fluids’, with densities P and [$\rho $], each of which obeys the same continuity equation, and is ‘stirred’ by the same velocity field [so that] the two ‘fluid’ densities... will be thoroughly mixed, making them indistinguishable on a coarse-grained level”.[31]

## 4. Objections to Dynamical Relaxation

“Even if one proves that the universe as a whole is in quantum equilibrium, we really want to prove that patterns inherent in subsystems of the universe are in quantum equilibrium. Whether any [of Valentini’s results] survive the move to the proper understanding of [$p=\rho $] is not clear”.[27]

“you might well imagine that it follows that any variable of interest, e.g., X, has the ‘right’ distribution. However, even if this were so (and it is), it would be devoid of physical significance! As Einstein has emphasized, ‘Nature as a whole can only be viewed as an individual system, existing only once, and not as a collection of systems’.”[36]

“[t]he very concept of a smooth distribution [p] is limited, being strictly valid only in the purely theoretical limit of an infinite ensemble ($n\to \infty $). This implies for example that in a laboratory consisting of a finite number of atoms, the actual distribution (say of electron positions) has the discrete form [of our (38), just above] so that one necessarily has some disequilibrium [$p\ne {\left|\psi \right|}^{2}$] on a fine-grained level”.[30]

“It is this assumption which introduces a distinction between past and future: Essentially, it is assumed that there is no special ‘conspiracy’ in the initial conditions, which would lead to ‘unlikely’ entropy-decreasing behaviour”.[31]

## 5. The Argument from Typicality

“However, this question arises: what is the good of ... giving distributions over a hypothetical ensemble (of worlds!) when we have only one world. The answer [... is that...] a single configuration of the world will show statistical distributions over its different parts. Suppose, for example, this world contains an actual ensemble of similar experimental set-ups. [...I]t follows from the theory that the ‘typical’ world will approximately realize quantum mechanical distributions over such approximately independent components. The role of the hypothetical ensemble is precisely to permit definition of the word ‘typical’.”[37]

“... by far the largest number of possible velocity distributions have the characteristic properties of the Maxwell distribution, and compared to these there are only a relatively small number of possible distributions that deviate significantly from Maxwell’s”.[39]

## 6. Objections to the Typicality Argument

“may be illustrated by the case of a universe consisting of an ensemble of n independent subsystems (which could be complicated many-body systems, or perhaps just single particles), each with wavefunction ${\psi}_{0}\left(x\right)$. Writing ${\Psi}_{0}^{univ}={\psi}_{0}\left({x}_{1}\right){\psi}_{0}\left({x}_{2}\right)\cdots {\psi}_{0}\left({x}_{n}\right)$ and ${X}_{0}^{univ}=({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n})$, a choice of ${X}_{0}^{univ}$ determines – for large n – a distribution ${\rho}_{0}\left(x\right)$ which may or may not equal $|{\psi}_{0}{\left(x\right)|}^{2}$ .“Now it is true that, with respect to the measure $|{\Psi}_{0}^{univ}{|}^{2}$, as $n\to \infty $ almost all configurations ${X}_{0}^{univ}$ yield equilibrium ${\rho}_{0}={\left|{\psi}_{0}\right|}^{2}$ for the subsystems. It might then be argued that, as $n\to \infty $, disequilibrium configurations occupy a vanishingly small volume of configuration space and are therefore intrinsically unlikely. However, for the above case, with respect to the measure $|{\Psi}_{0}^{univ}{|}^{4}$ almost all configurations ${X}_{0}^{univ}$ correspond to the disequilibrium distribution ${\rho}_{0}={\left|{\psi}_{0}\right|}^{4}$. This has led to charges of circularity: that an equilibrium probability density $|{\Psi}_{0}^{univ}{|}^{2}$ is in effect being assumed for ${X}_{0}^{univ}$; that the approach amounts to inserting quantum noise into the initial conditions themselves...”

“because of the law of large numbers, the set of typical points [for which the statistical distribution of positions in the ensemble approximately matches ${\left|\varphi \right|}^{2}$)] will have a ${|\Psi |}^{2}$-measure close to 1, for N large. So, if one picks a microscopic configuration of the universe Q that is typical relative to ${|\Psi |}^{2}$, it will give rise... to an empirical distribution satisfying Born’s statistical law...

“[Dürr, Goldstein, and Zanghí] claim that quantum equilibrium and therefore Born’s law is actually very natural. However, for that last claim to be right, one needs to argue that the measure with respect to which the configurations under discussion are typical is itself ‘natural’ (every configuration is typical with respect to at least one measure—the delta measure concentrated on itself)”.[44]

“In fact the only ‘explanation’ of the fact that we obtain a ${\left|\varphi \right|}^{2}$ distribution rather than a uniform distribution is probably that God likes quantum equilibrium and Born’s law and so put it there at the beginning of times.

“The upshot of this discussion is that quantum equilibrium, in Bohmian mechanics, should, in my view, be presented as a postulate, independent of the other ones, rather than as somehow being the only natural or reasonable choice. It is not a particularly unnatural choice and it is true that quantum equilibrium is still far less mysterious than classical non equilibrium at the origin of the universe... But one should not present it as more natural than it is”.[44]

## 7. Discussion

“provided the sense [$\mu $] of typicality were given, not by ${|\Psi |}^{4}$ (which is not equivariant), but by the density to which $|{\Psi}_{t}{|}^{4}$ would backwards evolve as the time decreases from t to THE INITIAL TIME 0. This distribution, this sense of typicality, would presumably be extravagantly complicated and exceedingly artificial.

“More important, it would depend upon the time t under consideration, while equivariance provides a notion of typicality that works for all t”.[38]

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Density plot of $\rho \left(0\right)={|\Psi \left(0\right)|}^{2}$. Please note that $\rho $ evolves periodically with period T. Note also that, for the particular non-equilibrium initial distribution $P\left(0\right)=\mathrm{constant}$, $g\left(0\right)=\rho \left(0\right)/P\left(0\right)=\rho \left(0\right)$. So this same figure can be taken also as illustrating $g\left(0\right)$ for this particular non-equilibrium distribution. The corresponding $g\left(t\right)$ for $t=T$, $t=2T$, and $t=-T$ are shown in subsequent figures.

**Figure 3.**The left panel is the same as panel (d) of the previous figure, but for an ensemble of 50,000 initially-uniformly-distributed particles, allowing one to see some of the fine-grained structure in the distribution. The right panel zooms in on the portion of the box highlighted by the gray box on the left, for an ensemble of 200,000 particles, allowing even more fine-grained structure to be visible.

**Figure 4.**The time-evolution of $g=\rho /P$, for an initially-uniform P. The evolution through each period is like a “kneading” operation which results in the non-uniformities in g being systematically mixed down to smaller and smaller length scales. Further time-evolution would eventually result in a map whose $\overline{g}$ was uniform.

**Figure 6.**This is the distribution for $g=\rho /P$ that is required at $t=-T$ in order to make $g=\rho $, i.e., to make $P=$ constant, at $t=0$. A very specific fine-grained micro-structure in P is required, at the earlier time, to generate a smooth, easily-describable non-equilibrium distribution at a later time.

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Norsen, T.
On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory. *Entropy* **2018**, *20*, 422.
https://doi.org/10.3390/e20060422

**AMA Style**

Norsen T.
On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory. *Entropy*. 2018; 20(6):422.
https://doi.org/10.3390/e20060422

**Chicago/Turabian Style**

Norsen, Travis.
2018. "On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory" *Entropy* 20, no. 6: 422.
https://doi.org/10.3390/e20060422