> We introduce a theoretical approach that uses temporal coarse-graining akin to Einstein’s kinetic theory of Brownian motion (34). Our averaging scheme is valid in the case of nonjamming mixtures of hard particles, where the dynamics is dominated by pairwise interactions, which is a good approximation for typical pedestrian flows as well as dilute colloids. We recover and unify in a systematic manner the fundamental insights of Helbing and Vicsek (22) as well as Vissers et al. (11) and Klymko et al. (23) by showing that undulation-induced drift and diffusion can both contribute to lane nucleation. We also demonstrate that diffusive processes suppress the formation of very narrow lanes, thereby providing a dynamical selection mechanism that favors the nucleation of lanes of a particular width. We provide explicit formulas for the propensity of a given system to nucleate lanes, and we present a simple approximate rule that lanes emerge at a rate proportional to the product of agent speed, density, and an effective parameter related to the average magnitude of lateral displacement in agent-agent collisions.
So, there are 2 main mechanisms proposed in the literature on lane formation: "drift" and "diffusion". Drift arises from the tendency of people to have a preferred direction to turn to avoid a collision (i.e. right- or left-bias). When facing an opposing flow, repeated conflicts will cause you to slowly drift sideways in your preferred direction. Diffusion is the random Brownian Motion-like jostling that arises from conflicts with opposing pedestrians, which has no directional bias.
In both cases, you're getting jostled around more when there's an opposing flow in front of you than when you're in a lane of people going in the same direction, so there will be a tendency for lanes to form spontaneously ("nucleate"), even from a completely homogeneous initial configuration.
The point of the paper is that they derive, with minimal assumptions, a quantitative relationship that incorporates both mechanisms, and makes specific non-obvious predictions about the shapes and widths of lanes. It also predicts that the rate of lane formation has a simple form (density * speed * average jostle displacement from conflicts with other pedestrians). They do some real-world experiments to validate some of their predictions.
As someone who works on pedestrian simulation software, I don't see how to use this to write the simulation code, but it could be an interesting way to validate the software (i.e. verify that lane formation obeys the predicted relationship).
Something that bothers me about a lot of these papers though is they often perform simulations with periodic boundary conditions. I get why they do that, but it seems to me that that will cause spatial correlations to show up that wouldn't happen in real life. Would the rate of lane formation be different in a system with different boundary conditions?
Another limitation mentioned in the paper that makes this model difficult to apply to crowds of pedestrians is that they assume the interaction between 2 pedestrians depends only on their relative displacement. However, it's known that they key parameter in pedestrian interactions is actually the Time To Collision--i.e., pedestrian collision avoidance is fundamentally anticipatory [1]. Presumably this would complicate the model too much though (now you have to take into account displacement and relative velocity).
>As someone who works on pedestrian simulation software
I'm very curious what software that is.
>Would the rate of lane formation be different in a system with different boundary conditions?
Wouldn't it have to? Given that formation is the (density * speed * wiggleness) I assume boundary variability will have a logistic relationship with density right?
> Given that formation is the (density * speed * wiggleness) I assume boundary variability will have a logistic relationship with density right?
I'm not sure what you mean by this--boundary variability meaning the variance of velocity? I'm also not 100% sure what I mean about my own concern :D--really I'm just confused. I would feel like I understood the significance of the simulations more if, instead of having the same agents pop up at the bottom as just exited the top (i.e. periodic boundaries), agents simply left the simulation entirely at the top and new ones entered at the bottom (with correspondingly new velocities). Left-right boundaries are trickier I guess, not sure what would be reasonable there. Maybe just no boundaries.
There are good reasons why periodic boundaries are used (ensures density is well-defined and consistent, makes analytical calculations easier, etc.). It just strikes me as a obviously non-physical model, and one that could make emergent structures easier to form.
>We introduce a theoretical approach that uses temporal coarse-graining akin to Einstein’s kinetic theory of Brownian motion
Since when did Brownian motion smoke and air particles have eyes and need to stand on two legs?
I sometimes think scientists need to get out more, maybe attend Glastonbury, where you are walking from the Pyramid Stage to the Other Stage, after its rained heavily and Eavis hasnt given the order to put any or enough straw down, so everyone is concentrating on staying upright by looking down at the mud swamp that will consume their next step and wellington boot whilst also briefly but hesitantly looking at the person or persons immediately in front of them in order to facilitate their escape.
Sure arena's are less likely to be water logged mud swamps, but outdoor puddles introduce their own obstacles as do cambers of the surface. Get past ticket control, enter big atriums and newbies will pause in awe of the size whilst also getting their bearings, regulars will know where to go. I see this behaviour in every major train station or airport, especially as the signs are generally up high, diverting their gaze from taking the next step, thus slowing down the flow of people.
I dont see any of these factors being accounted for in this paper.
And its for the above mentioned reasons, namely not accounting for the proper metrics that count, that some self driving vehicle companies are not going to succeed any time soon at level 5, because their algorithm weighting is all wrong.
> We introduce a theoretical approach that uses temporal coarse-graining akin to Einstein’s kinetic theory of Brownian motion (34). Our averaging scheme is valid in the case of nonjamming mixtures of hard particles, where the dynamics is dominated by pairwise interactions, which is a good approximation for typical pedestrian flows as well as dilute colloids. We recover and unify in a systematic manner the fundamental insights of Helbing and Vicsek (22) as well as Vissers et al. (11) and Klymko et al. (23) by showing that undulation-induced drift and diffusion can both contribute to lane nucleation. We also demonstrate that diffusive processes suppress the formation of very narrow lanes, thereby providing a dynamical selection mechanism that favors the nucleation of lanes of a particular width. We provide explicit formulas for the propensity of a given system to nucleate lanes, and we present a simple approximate rule that lanes emerge at a rate proportional to the product of agent speed, density, and an effective parameter related to the average magnitude of lateral displacement in agent-agent collisions.
So, there are 2 main mechanisms proposed in the literature on lane formation: "drift" and "diffusion". Drift arises from the tendency of people to have a preferred direction to turn to avoid a collision (i.e. right- or left-bias). When facing an opposing flow, repeated conflicts will cause you to slowly drift sideways in your preferred direction. Diffusion is the random Brownian Motion-like jostling that arises from conflicts with opposing pedestrians, which has no directional bias.
In both cases, you're getting jostled around more when there's an opposing flow in front of you than when you're in a lane of people going in the same direction, so there will be a tendency for lanes to form spontaneously ("nucleate"), even from a completely homogeneous initial configuration.
The point of the paper is that they derive, with minimal assumptions, a quantitative relationship that incorporates both mechanisms, and makes specific non-obvious predictions about the shapes and widths of lanes. It also predicts that the rate of lane formation has a simple form (density * speed * average jostle displacement from conflicts with other pedestrians). They do some real-world experiments to validate some of their predictions.
As someone who works on pedestrian simulation software, I don't see how to use this to write the simulation code, but it could be an interesting way to validate the software (i.e. verify that lane formation obeys the predicted relationship).
Something that bothers me about a lot of these papers though is they often perform simulations with periodic boundary conditions. I get why they do that, but it seems to me that that will cause spatial correlations to show up that wouldn't happen in real life. Would the rate of lane formation be different in a system with different boundary conditions?
Another limitation mentioned in the paper that makes this model difficult to apply to crowds of pedestrians is that they assume the interaction between 2 pedestrians depends only on their relative displacement. However, it's known that they key parameter in pedestrian interactions is actually the Time To Collision--i.e., pedestrian collision avoidance is fundamentally anticipatory [1]. Presumably this would complicate the model too much though (now you have to take into account displacement and relative velocity).
[1] http://motion.cs.umn.edu/PowerLaw/