We are going to review the physics of the walker problem in the context of computational models. This will deal with the in-silico approaches to the problem and shed some light on what we call 'surfers vs walkers'.

It is also meant as a 'checkpoint journal entry' for the participants of the facebook group studying the walkers.

**The walker problem**

**The object of study is the association of a particle and the wave it creates in a media. We focus on the silicon walkers as observed by Couder et al and modeled by Bush et al. The physics is rather straightforward in the Newton (force/acceleration) view. The forces are as such:**

**Slope force.**Aka "field force". Each time the particle bounces it creates a Bessel standing wave. These waves sum (interfere) and create a wavefield. The gradient of the wavefield at the point of impact gives an impulse to the particle when it bounces off the surface (like a ball bouncing off an inclined surface). The important points here are a/ the memory of the field (loosely defined in the literature as the number of waves that contribute to the wavefield). It is the main control parameter in our studies. As the memory increases so does the height of the total wave (you sum many little waves). For example in our simulation, the basic wave is of height 0.02 faraday length and the sum is about 0.1. b/ this is a**DISCRETE**set of waves.**Viscous force**. Every time the particle bounces on the surface it is slowed down. This force is proportional to the speed.**Elastic force**. This force was introduced recently by the Couder team in a technical tour de force. They injected the bouncing particle with ferromagnetic material and submitted the particle to a centropotential EM field. While it is described as a EM force, it is akin to an elastic potential and the force is modeled as -kx with the exact form of a elastic force. The author prefers the 'elastic force' description for various reasons not having to do directly with the QM interpretation. It is in this elastic potential that the walker system exhibits intermittence and provides the most intriguing insights into coherence/decoherence and the superposition principle.

These 3 forces are the ones identified in the problem so far. It should be stressed that the 'historical' walker study focused on 1 and 2. For example the initial Couder/Fort paper balanced the field force with the viscous force to calculate the speed of walkers. The Bush papers dealt with 1 and 2 as well. The elastic force is a recent addition in experiments.

**The Bush approximation: analytical solutions**

The mathematical description of the hydrodynamics at play in the walker system has been done by Bush et al. Essentially they make analytical way by approximating the discrete sum of Bessels by a integral. Long story short, instead of a discrete sum, we hypothesize the intermediate steps and calculate with a step that is shorter than the actual bounce. This is not physical, in the sense that these intermediate waves do not exist in reality but they are expedient from a mathematical standpoint. We are able to compute the trajectory with this 'smooth' approximation by inverting matrices.

**Surfer vs Walker**

In the case of the Bush approximation the particle is always informed by the wave. The mental picture is 'the surfer'. The waves are constantly generated and driving the particle, the particle surfs on its waves.

The silicon droplet system is a 'walker' system. The particle just 'bounces' off the surface and only picks up an acceleration at the time of impact.

The important part here is that surfers assume a continuous set of waves generated along the path, while the walkers deal with a discrete set of waves. This really impacts the 'field force' only.

**Integration routines**

The main advantage of the 'surfer' approximation is that following Bush et al, we can compute an exact solution to the problem by inverting matrices. This results in a smooth flow and has been implemented by the Dotwaves crowd in Python (by Burak Budanur) and Matlab (by Samuel Bernadet). The disadvantage of this routine from a computational standpoint is that it is heavy and therefore slow. "Slow" is a relative term in computing and a lot of information is coming from these simulations, they also recover many of the features of the 'real' walkers. The important part from a model standpoint is that all forces (including field force) are continuously integrated and smoothly changing.

The discrete routine has been implemented in java (by Marc Fleury) and it's main advantage is that it is very fast (by an order of 10000) so allows for quick visual inspection of the behaviors. The disadvantage is that the integration routine, if done naively, introduces a lot of noise. To address this noise, and after a lot of prodding, the author has implemented a hybrid walker integration. The integration routine has been refined as such:

- Field force is still discrete (there is a discrete set of waves in the real problem) and the integration done in one step. In the future we may move the force to a 'Runge Kutta" like average but this is not straightforward and will be arbitrary.
- EM force is continuous. The integration routine has been smoothed out to account for the continuous nature of the EM force (it applies when the particle is in flight) and gives us much a much smoother integration.
- Viscous force is continuous. This is debatable as the viscous force in the real experiments only apply when the particle is in contact with the liquid. It does however increase the smoothness of the integration routine.

**Physical accuracy**

In a nut-shell the main features of walkers integration in java is a discrete treatment of waves field with a continuous EM and viscous. The surfer integration is a continuous treatment of all forces including the field force. The discrete nature is closer to the physical picture (as the walkers only create discrete waves, as simple as that). The speed is a secondary point, as we can run both types of simulations over long periods of time.

**Noise: discrete vs continuous**

The a posteriori justification for the surfer approximation, besides the fact that it is mathematically expedient, is that it seems to replicate the observed facts about walkers at least in the straight walker regime (from the published literature) and gives us good results with the elastic force in the dotwaves efforts.

However we know that discrete sets of waves create a different wavefield than continuous sets of waves, specifically at short distance. Tesselations are important in field theory, continuous and discrete sets giving different results. It should be pointed out that the walkers orbit at about 1 faraday length, meaning are subjected to the short term structure of the field. The presence of a 'lattice' (discrete points) as opposed to a continuous source of waves gives very different wavefields in general.

In other words, the surfer approximation introduces noise of it's own in the computational simulations (mathematical in nature) that is not present in the physical system or the walker implementation.

**Surfers vs Walkers: the 3D SURFER case and QM analog**

The interest in these systems comes from the fact that they offer a compelling mental picture with which to understand the (obscure) formalism of QM. As detailed in the blog referenced above it is the emergence of intermittences, characteristic of chaotic systems, that gives us the transition probabilities between states and a clear image of what 'superposition' means.

However this walker system is inherently 2D. And because we are 2D we have a notion of "steps" and the walker image only can exist in 2D, we bounce in the vertical direction and create discrete waves on the 2D surface. In 3D, the particle would ALWAYS be in contact with the media and would be generating waves continuously. In 3D we cannot escape to another dimension to only create discrete waves.

The 3D case can only be a SURFER case.