Random walk dispersion
Consider first a single particle with source at x0, with a source momentum that coincides with the momentum propensity in this case (no external forces, no quantum forces), v0 = V. The position pmf is given by the formula:
for any x comprised between x0-t and x0+t (the pmf is zero otherwise). Clearly, particles cannot be faster than light and thus cannot reach lattice nodes that are outside of the light cone from the source.
For example, if we take x0 = 0, at t = 1 we have w(1) = (1+V)2/4, w(0) = (1-V2)/2, and w(-1) = (1-V)2/4, as it should be given the rules of motion shown in this post. At t = 2, it is still rather easy to compute w(-2), w(-1), w(0), w(1), and w(2) by hand and verify that their sum is one. In the figure below, we plot the curve w(x) at t = 5 for a value V = 0.2.
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| pmf of the position (blue circles) and continuous-time approximation (red) for a free particle emitted at the source at zero position, with momentum 0.2 and after a time of 5 units. |
The pmf w(x) is a binomial distribution with number of trials equal to 2t, number of successes equal to t+x-x0 and probability of success equal to (1+V)/2. The mean of this pmf is thus ⟨k⟩ = (1+V)t and, consequently, the expected value of the position is ⟨x⟩ = x0+Vt as it should be. Check the figure in this post.
The figure above also shows an approximation of w(x) that is rather good for large times. From the De Moivre-Laplace theorem, w(x) can be approximated by a Normal distribution:
\approx&space;\frac{1}{\sqrt{2\pi&space;Dt}}&space;\exp{\left(-\frac{(x-x_0-Vt)^2)}{2Dt}&space;\right)} "w(x,V;t,x_0)\approx \frac{1}{\sqrt{2\pi Dt}} \exp{\left(-\frac{(x-x_0-Vt)^2)}{2Dt} \right)}")
where D = 1-e = (1-V2)/2. Note that the quantity Dt is the variance of both the original and the approximated distribution. Particles with higher energy propensity tend to be less dispersed around their mean position. This behavior is clearly visible in the figure below. In the limit when the energy propensity is 1, that is, V = ±1, the variance is zero and the particle trajectory is identical to x0+Vt.
The figure above also shows an approximation of w(x) that is rather good for large times. From the De Moivre-Laplace theorem, w(x) can be approximated by a Normal distribution:
where D = 1-e = (1-V2)/2. Note that the quantity Dt is the variance of both the original and the approximated distribution. Particles with higher energy propensity tend to be less dispersed around their mean position. This behavior is clearly visible in the figure below. In the limit when the energy propensity is 1, that is, V = ±1, the variance is zero and the particle trajectory is identical to x0+Vt.
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| Probability distributions of the position for a free particle emitted at a zero position source, with momentum 0.2 (blue) and 0.8 (orange) and after a time of 51 units. |
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