# Particle transport at arbitrary timescales with Poisson-distributed collisions.

## Summary (1 min read)

### Introduction

- The collisions are instantaneous velocity changes where a new value of velocity is generated from a model probability function.
- The authors study three different cases of velocity transition functions and compute the transport properties from the evolution of the variance.
- Many results on random walks rely on the assumption of long timescales.
- The authors study the time evolution of the variance of the walker position and show that the character of transport, understood here as the scaling of the variance as a function of time, changes over time.

### II. THEORETICAL FRAMEWORK

- The walker moves with constant velocity except at collisions, when the velocity transitions instantaneously to a new value (see Fig. 1). (3) We use the transform variable κ , instead of x, to distinguish the transformed function from the original one.the authors.the authors.
- The transformed function is written as the original one but uses the transform variable s. Equation (4) means that the initial location and velocity are assumed to be statistically independent [28].
- For what follows, the authors only consider the case where these initial PDFs have finite first and second moments and can be Fourier-transformed.
- This form of the equation is very useful once one realizes that the Fourier transform of a PDF is its characteristic function [28].

### VI. CONCLUSIONS

- Borrowing some ideas from PRW [9,10,15] and biological dispersion [7], the authors have developed a technique to find exact analytic expressions for the mean, variance and transport exponent of a 1D random walker subject to Poisson-distributed collisions (with constant rate) which moves at constant velocity between collisions.
- Transport properties, as established with the value of the transport exponent λ [Eq. (13)], have in general been shown to be time-varying and dependent on the possibly random initial conditions of location and velocity.
- Three different types of velocity transitions are studied.
- In particular, the authors follow some basic steps in the approach described in Refs. [7,9,10,15].
- In that case, the walker only has two possible motion states (towards increasing x or decreasing x, respectively).

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### "Particle transport at arbitrary tim..." refers background in this paper

...We state clearly the definition of the transform to avoid confusion with references that may use a different convention [29]:...

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2,660 citations

### "Particle transport at arbitrary tim..." refers methods in this paper

...The case σ f0 = σ fR models a situation similar [14] to PRW and indeed shows the same smooth transition from ballistic to diffusive around γ t = 1....

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...The number of collisions is then distributed Binomial ( t/dt, dt/τ ) which, for short dt , converges to Poisson(t/τ ) [28,33]....

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...APPENDIX A: TRANSPORT EQUATION To start, we borrow some basic ideas from PRW and biological dispersal....

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...Many other methods have been developed to investigate similar situations in PRW [13,14,32,33] but are not pursued here....

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...Borrowing some ideas from PRW [9,10,15] and biological dispersion [7], we have developed a technique to find exact analytic expressions for the mean, variance and transport exponent of a 1D random walker subject to Poisson-distributed collisions (with constant rate) which moves at constant velocity between collisions....

[...]

1,543 citations

### "Particle transport at arbitrary tim..." refers result in this paper

...This constitutes an interesting analogy to a similar result observed in the formally different problem of relative separation of particles [24,25]....

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##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "Particle transport at arbitrary timescales with poisson-distributed collisions" ?

Although the authors focus on first and second-order moments ( relevant to transport studies ), the procedures presented here can straightforwardly be extended to higher-order statistics of the random walker location ( as a function of time ), for example, the skewness and kurtosis. These will be the subject of future studies. Indeed, obtaining t/τ collisions by time t can be seen as successes in t/dt Bernoulli trials [ 33 ] with probability of success ( t/τ ) / ( t/dt ) = dt/τ. The authors can then rewrite Eq. ( A2 ) as ( ∂ ∂t + v ∂ ∂x + γ ) p ( x, v, t ) = γ ∫ f ( u, v ) p ( x, u, t ) du.