**4. Statistical Theory of the Transition to the Final State**

We are now ready to investigate the statistical consequences of the *μA*-interaction modelled in the previous section. We introduce the total variance of the parameters in Equation (5), Ξ = *Nχ*2, and define an aggregate variable *Y* = *Y*(*α*) of *A*, suitably normalized,

$$Y(\boldsymbol{\alpha}) = \frac{1}{\boldsymbol{\Sigma}} \sum\_{n=1}^{N} \kappa\_n(\boldsymbol{\alpha}) \tag{6}$$

to represent the overall degree of enhancement/suppression (so that *Y* > 0 for net enhancement of + and *Y* < 0 for net enhancement of −). It follows that *Y* is characterized by mean and variance *Y* <sup>=</sup> 0 and *Y*2 <sup>=</sup> <sup>Ξ</sup>−1. Then, for sufficiently small *<sup>κ</sup>n*, we can rewrite (5) as

$$|b\_{\dot{\jmath}}(a)|^2 = e^{\sum\_{a} \log(1+j\kappa\_a)} = e^{\Xi \dot{\jmath} \varGamma(a) - \frac{1}{2} \sum\_{a} \kappa\_a^2} = e^{\Xi \big(j \varUpsilon(a) - \frac{1}{2}\Big)},\tag{7}$$

with *j* = + or −. Here, in the exponent, we have done the calculation to second order in *κ<sup>n</sup>* and we have replaced ∑*<sup>n</sup> κ*<sup>2</sup> *<sup>n</sup>* by *Nχ*2.

Since all factors in the product (5) are independent, the distribution *q*(*Y*) over the aggregate variable *Y* = *Y*(*α*), defined by Equation (6), in the ensemble of initial states of *A*, is well described by the Gaussian distribution,

$$q(\boldsymbol{\chi}) = \sqrt{\frac{\boldsymbol{\Xi}}{2\pi}} \ e^{-\frac{1}{2}\boldsymbol{\Xi}\boldsymbol{\chi}^2} \,. \tag{8}$$

centered around *Y* = 0 with variance Ξ−1.

Initial states differ in their efficiency in leading to a transition to a final state, since the total transition rate may depend strongly on *α*. The transition to the final state (3) with |*bj*| <sup>2</sup> given by (7) has the rate (2*π*)−1*w*(*Y*) where

$$w(\boldsymbol{Y}) = |\psi\_+|^2 e^{\Xi(\boldsymbol{Y} - \frac{1}{2})} + |\psi\_-|^2 e^{\Xi(-\boldsymbol{Y} - \frac{1}{2})} \, , \tag{9}$$

with *w*(*Y*) = 1. The terms in (9) are the partial transition rates for the + and − channels.

The total transition rate (9) depends strongly on *Y*. We shall now go into the statistics of the final states which is strongly influenced by *w*(*Y*). To get the distribution *Q*(*Y*) over *Y* for the *final states*, corresponding to *q*(*Y*) for the initial states, we must multiply *q*(*Y*) by the transition rate (9) which is normalized in the sense that its mean value is 1. This is the standard approach in scattering theory, see, e.g., Ref. [17]. Here, it can be interpreted as a selection process, as previously discussed, that favors initial states which are efficient in leading to a transition, with a selective fitness being proportional to the transition rate (9). Thus, the distribution over final states can be written (see Figure 2)

$$Q(Y) = q(Y)w(Y) = |\psi\_+|^2 Q\_+(Y) + |\psi\_-|^2 Q\_-(Y) \, ,$$

$$Q\_\pm(Y) = \sqrt{\frac{\Xi}{2\pi}} \, e^{-\frac{1}{2}\Xi(Y \mp 1)^2} \,. \tag{10}$$

The normalized partial distributions, *<sup>Q</sup>*+(*Y*) and *<sup>Q</sup>*−(*Y*), also with variance <sup>Ξ</sup>−1, are centered around *Y* = 1 and *Y* = −1 and refer to *μ* ending up in the state |+*<sup>μ</sup>* and |−*μ*, respectively. The coefficients of *Q*+(*Y*) and *Q*−(*Y*) in *Q*(*Y*) are |*ψ*+| <sup>2</sup> and <sup>|</sup>*ψ*−| 2, expressing Born's rule explicitly.

**Figure 2.** The distribution *Q*(*Y*) over *Y* of transitions taking place in *μA*-interaction for increasing size of *A* corresponding to Ξ = 1 (broken line), Ξ = 5 (thin line), and Ξ = 60 (thick line). *Q*(*Y*) is composed of two distributions *Q*+(*Y*) and *Q*−(*Y*) with weights |*ψ*+| <sup>2</sup> and <sup>|</sup>*ψ*−| 2, respectively. These distributions become separated as Ξ increases. Each initial state of *A*, |0, *αA*, is represented by a certain *Y* = *Y*(*α*). As the size of *A* increases and Ξ becomes larger, states that are efficient in leading to a transition are found around *Y* = −1 and *Y* = +1, respectively. These initial states then lead to *μ* ending up in either |−*<sup>μ</sup>* or |+*μ*, respectively, with probabilities confirming the Born rule.

It is instructive to follow the distribution *Q*(*Y*) with growing Ξ. For small Ξ (= *Nχ*2) it is broad and unimodal; it then turns broad and bimodal with narrowing peaks. For large Ξ, it is split into two well separated distributions with sharp peaks, weighted by the squared moduli of the state components of *μ*, |*ψ*+| <sup>2</sup>*Q*+(*Y*) and <sup>|</sup>*ψ*−| <sup>2</sup>*Q*−(*Y*), at *<sup>Y</sup>* <sup>=</sup> 1 and *<sup>Y</sup>* <sup>=</sup> <sup>−</sup>1, respectively. They represent two different sub-ensembles of final states (see Equation (2)). Other values of *Y* correspond to non-competitive processes. The aggregate variable *Y* is "hidden" in the fine unknown details of *A* that can influence the *μA*-interaction.

From Equation (7) we see that the dominance of either channel is characterized by |*b*+(*α*)| 2/|*b*−(*α*)<sup>|</sup> <sup>2</sup> = *e*2Ξ*<sup>Y</sup>* being either very large or very small. For a small system, i.e., small *N* and hence small Ξ, this ratio does not deviate much from unity which means that we have an entangled superposition of the two final states. As Ξ increases, the ratio deviates more strongly from unity and one of the channels starts to dominate. When Ξ is of the order of 10, the sub-distributions, *Q*+(*Y*) and *Q*−(*Y*), are essentially non-overlapping, *Q*(*Y*) is concentrated around ±1, and the ratio of dominance between the channels is of the order of *<sup>e</sup>*<sup>20</sup> <sup>≈</sup> <sup>10</sup>10.

The initial state for *μ* in (1) is a superposition, a '*both-and* state', and it ends up in (2) which is again a product state, with *μ* in *either* |+*<sup>μ</sup> or* |−*μ*. The initial states of *A* vary widely in their efficiency to lead to a final state. When one transition-rate term in (9) is large, the other one is small. The selection of a large transition rate therefore also leads to a bifurcation with one of the terms in (3) totally dominating the final state.
