Ensemble probability distributions for the position

The rules of particles' motion in the proposed model have been described as a random walk in this post, while in this post we have evaluated the probability mass function of position for one single particle emission.

Each emission is characterized by one specific value of the source momentum v₀ that, in the absence of quantum or external forces, coincides with the momentum propensity V. However, for an ensemble of similarly-prepared particles, the source momentum is a random variable with uniform distribution in the range from -1 to 1 (see this post). We are now interested in evaluating the position probability mass function (pmf) for the ensemble of particles.

To do so, we just need to multiply the pmf w(x, V; t, x₀) by the source momentum probability density and integrate over all possible values,

The probability density function (pdf) of the source momentum is simply 1/2 (uniform distribution). Thus the ensemble pmf can be rewritten as a constant multiplying a Beta function (with the support variable played by the quantity (1+V)/2 and the two coefficients given by the quantities t+x-x₀+1, resp., t-x-x₀+1). From the relation between the Beta function with the Gamma function (factorials), we obtain that

a result that can be verified by inspection for small times. Not surprisingly, the pdf of position is uniform over the admissible light cone that, in 1D, contains exactly 2t+1 lattice nodes (from -t to t w.r.t. the source).

This result can be verified with the computer code described in this post. Simulations with a large but finite number of emissions N_p give the results shown in the figure below. Clearly, the frequency of arrivals at position x after t iterations tends to a uniform theoretical distribution as the number of particle emissions increases.

Frequency of arrivals (# of arrivals/ # of emissions) as a function of the position after 500 iterations for Np = 500 (black), 5000 (blue), and 50000 (red) emissions. The frequency tends to the probability density 10^-3.

We obtain a similar result if we use the large-time approximation of the binomial pmf w(x) derived in a previous post. On the other hand, consider that ⟨x⟩=x₀+v₀t, where the brackets denote the average or expected value of the position at a certain time. The relation between the source momentum and the expected position is monotonic (actually, it is linear). Therefore, for the properties of probability density functions under changes of variable, we have that

which is a very good approximation of the position pmf for large times. This approximation will be used whenever it won't be possible to evaluate explicitly the "true" pmf. Note that, while x is an integer, ⟨x⟩ can take real values (actually, rational values, as it will be explained in a future post).

Are these results in agreement with Quantum Mechanics? Consider the propagator for a free particle and assume, as done with the proposed model so far, a perfectly localized source at x₀. The probability amplitude is evaluated (in lattice units) as

which yields a probability density (through the Born rule) equal to 1/(2t), i.e., the same approximated result above. Standard QM would say that this wave function is non-normalizable, since it extends with a non-zero value along the whole 1D position space. However, we must rather intend it as bounded within the light cone of the source, that is, between x₀-t and x₀₊t. The integral of the QM probability density within these bounds correctly gives the value of one, as it should be.

In other terms, we have found a first correspondance between the proposed model and QM in terms of probability density predictions. Other, and less trivial correspondance, are discussed in this paper and will be treated in future posts.