Stochastic Kinematic flows arising from interacting particle systems

Jeremy Worsfold

Supervised by Tim Rogers and Paul Milewski

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

Things we want to Model

Cars

Traffic models

Fish

Swarming Models

Macroscopic

Mesoscopic

Microscopic

Mesoscopic

  • Easy to simulate ✔
  • Ideal for reaction diffusion ✔
  • Fundamentally change the system ✖
  • Can then be hard to relate back to micro/macro models ✖

Can we avoid discretising space and still understand how systems behave with low particle numbers?

How we go about modelling

Cars/Fish/Whatever are $N$ particles with positions $X_n\in[0,2\pi)$ for $n=1,\dots,N$

$$ \mathrm{d}X_n = \frac{1}{N}\sum_{m=1}^N f(X_m-X_n) \mathrm{d}t + \frac{\sqrt{2D}}{N}\sum_{m=1}^N g(X_m-X_n)\mathrm{d}W_n $$

From small scale to large scale

$$ \mathrm{d}X_n = \frac{1}{N}\sum_{m=1}^N f(X_m-X_n) \mathrm{d}t + \frac{\sqrt{2D}}{N}\sum_{m=1}^N g(X_m-X_n)\mathrm{d}W_n $$

Introduce a concept of density of particles: $$ \rho(x,t) = \frac{1}{N}\sum_{n=1}^N \delta(x-X_n(t)) $$

From small scale to large scale

$$ \rho_t + \underbrace{\left(\rho(f\ast\rho)\right)_x}_{\text{velocity}} - \underbrace{D\left(\rho\left(g\ast\rho\right)^2\right)_{xx}}_{\text{diffusion}} + \underbrace{\sqrt{\frac{2D}{N}}\left((g\ast\rho)\rho^{\frac{1}{2}}\eta\right)_x}_{\text{noise}} = 0 $$

with $$ (f\ast\rho)(x,t) = \int_0^{2\pi} f(x-y)\rho(y,t) \mathrm{d}y, \quad \langle \eta(x,t)\eta(y,s) \rangle = \delta(x-y)\delta(t-s) $$

If $g(x)=1$, this reduces to a known case (Dean 1996) $$ \rho_t + \left(\rho(f\ast\rho)\right)_x - D\rho_{xx} + \sqrt{\frac{2D}{N}}(\rho^{\frac{1}{2}}\eta)_x = 0 $$

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

Whitham Model of Traffic

$$ \rho_{t}+ \left(\rho v(\rho)\right)_x - D\rho_{xx}=0 $$ with $v(\rho)=v_0\left(1-\frac{\rho}{\rho_{max}}\right)$

Without diffusion term we get shock fronts.

Whitham Model of Traffic

Our particles describe the flow of cars along a single lane road

$$ \rho_{t}+ \left(\rho v(\rho)\right)_x - D\rho_{xx}=0 $$ with $v(\rho)=v_0\left(1-\frac{\rho}{\rho_{max}}\right)$

$$ \rho_t + \left(\rho(f\ast\rho)\right)_x - D\left(\rho\left(g\ast\rho\right)^2\right)_{xx} = 0 $$

$$ \begin{aligned} f(x) & = v_0 - \frac{v_0}{\rho_{max}}\delta(x) \\ g(x) & = 1 \end{aligned} $$

Swarming Model (Milewski 2008)

Our particles describe the swarming of some organism which only looks ahead of itself

$$ \rho_{t}+\left(\rho(K\ast\rho)\right)_x - D(\rho^3)_{xx}=0 $$

$$ K(x) = \begin{cases} e^{-x}, x\geq 0\\ 0, x \le 0 \end{cases} $$

$$ \rho_t + \left(\rho(f\ast\rho)\right)_x - D\left(\rho\left(g\ast\rho\right)^2\right)_{xx} = 0 $$

$$ \begin{aligned} \Rightarrow f(x) & = K(x) \\ g(x) & = \delta(x) \end{aligned} $$

Swarming Model (Milewski 2008)

PDE

Our Model

Density dependent diffusion


If we have $g(x)=\delta(x)$ then we have $$ \rho_t = D\left(\rho^3\right)_{xx}. $$ The solution to this is a semi-ellipse which grows horizontally over time. $$ \rho(x,t) = \frac{2}{\pi r(t)}\sqrt{1-\left(\frac{x}{r(t)}\right)^2} $$ with $r(t) = 2\left(3Dt/\pi^2\right)^{1/4}$.

Diffusion Term

$$ \rho_t = D\left(\rho\left(g\ast\rho\right)^2\right)_{xx} \quad , \quad \mathrm{d}X_n = \frac{1}{N}\sum_{m=1}^N g(X_m-X_n) \mathrm{d}W_n $$
$g(x)=1$ $$ \rho_t = D\rho_{xx} $$

Fickian

$g(x)=\delta(x)$ $$ \rho_t = D\left(\rho^3\right)_{xx} $$

Porous Medium

$g(x)=1-\alpha\cos(x)$ $$ \begin{aligned} \rho_t & = D\left(\rho - 2\alpha\rho(\cos * \rho) \right.\\ & \left.\quad\quad +\alpha^2\rho(\cos * \rho)^2\right)_{xx} \end{aligned} $$

Strange...

Cosine Noise Coupling

$g(x)=1-\alpha\cos(x)$ $$ \begin{aligned} \rho_t & = D\left(\rho - 2\alpha\rho(\cos * \rho) \right.\\ & \left.\quad\quad +\alpha^2\rho(\cos * \rho)^2\right)_{xx} \end{aligned} $$

Two possible stable steady states: $$ \rho(x,t) = \begin{cases} \frac{1}{2\pi}, & 0 < \alpha < 1 \\ \frac{1}{4\pi}\left(\delta(x-\Delta) + \delta(x+\Delta)\right), & \alpha>1 \end{cases} $$ with $\cos^2(\Delta) = 1/\alpha$.

$\boldsymbol{\alpha}\boldsymbol{=}\mathbf{2.5}$

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

Quantifying the Fluctuations

The Whitham model has a stable, homogeneous state, $\rho(x,t)=\rho_0$. $$ \rho_{t}+ \left(\rho v(\rho)\right)_x - D\rho_{xx} + \sqrt{\frac{2D}{N}}\left(\rho^{\frac{1}{2}}\eta\right)_x = 0 $$

We can characterise the frequency, spacing and strength of these fluctuations.

$$ \rho(x,t) = \frac{1}{2\pi} + \frac{1}{\sqrt{N}}\xi(x,t) $$

Look at the the fourier series in space and FT in time of the density

$$ \varrho_k(\omega) = \frac{1}{2\pi}\delta_{k,0}\delta(\omega) + \frac{1}{\sqrt{N}}\hat{\xi}_k(\omega) $$

Quantifying the Fluctuations

$$ \varrho_k(\omega) = \frac{1}{2\pi}\delta_{k,0}\delta(\omega) + \frac{1}{\sqrt{N}}\hat{\xi}_k(\omega) $$

$$ \langle |\hat{\xi}_k(\omega)|^2 \rangle =\frac{D \tilde{g}_{0}^{2} k^{2}}{2 \pi^{2}\left|\left(-J_k^*-i\omega\right)\right|^{2}} $$ $$ J_{k}=ik\left(\tilde{f}_{0}+\tilde{f}_{-k}\right)+k^{2} D \tilde{g}_{0}\left(\tilde{g}_{0}+2 g_{-k}\right) $$

Ellipse Fluctuations


$$ \rho_{t} - D(\rho^3)_{xx} + \sqrt{\frac{2D}{N}}\left(\rho^{\frac{3}{2}}\eta\right)_x = 0 $$ We can also analyse this in a similar way $$ \rho(x,t) = \rho^*(x,t) + \frac{1}{\sqrt{N}}\xi(x,t) $$

Ellipse Fluctuations


$$ \xi_t = \frac{1}{4t}\left[(r^2(t)-x^2)\xi_{xx} - 4x\xi_x - 2\xi\right] + \left((\rho^*)^{3/2}\eta\right)_x $$ Cumulative moments: $$ M_n(t) := \int_{-\pi}^{\pi} x^n \rho(x,t) \mathrm{d}x $$

Fluctuations in the cumulative moments: $$ Z_n(t) := \sqrt{N}\left(M_n(t) - M_n^*(t)\right) $$


Fluctuations in the moments

$$ Z_n(t) := \sqrt{N}\left(M_n(t) - M_n^*(t)\right) $$

$$ Z_n'(t) = \frac{n(n-1)}{4t}\left[r^2 Z_{n-2} - Z_n\right] + \int_0^{2\pi} \zeta_n \mathrm{d}x $$

$$ \begin{aligned} \langle Z_1(t)\rangle & = 0 , \quad \langle Z_1^2(t)\rangle = \frac{\sqrt{3Dt}}{\pi} \\ \langle Z_2(t)\rangle & = 0 , \quad \langle Z_2^2(t)\rangle = \frac{t}{4\pi^2} \end{aligned} $$

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

  • Modelling Paradigm
  • Macroscopic Examples
  • Finite-sized Effects
  • Direction of Future Work

Recap

  • Created a model for 1D interacting particles
  • Showed how it behaves in the infinite particle limit
  • Related it to examples of kinematic flows
  • This could give insight into the underlying mechanics of the interactions
  • Showed how we can characterise the stochastic effects

Incorporating Velocity

If we want to explore more complex and interesting examples, we will have to look at particles with stochastic velocities. This might look like $$ \mathrm{d}X_n = U_n \mathrm{d}t $$ $$ \mathrm{d}U_n = \frac{1}{N}\sum_{m=1}^N f(X_m-X_n) \mathrm{d}t + \frac{1}{N}\sum_{m=1}^N g(X_m-X_n)\mathrm{d}W_n $$

Thank you for listening!

Does anyone have any questions?

Please keep them to yourselves