Mean-Field PDE Model

Available identifiers

MaRDI QIDQ6775428

for large population sizes particle-based dynamics may be approximated by a mean-field PDE model

Description

These equations describe the spatiotemporal evolution of the concentration of particles, depending on position and time, and approximate the empirical measure of the particle-based dynamics in the case of large N.

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List of mathematical expressions with quantities

Convolution Between Interaction Force And Density Formulation	$(F^{\prime }*c(\cdot ,t))(x):=\underset{\mathbb{𝕋}}{\int }{F}^{\prime }(x-y)c(y,t)dy$
$(F^{\prime }*c(\cdot ,t))$ symbol represents Convolution Between Interaction Force And Density
$F^{\prime }$ symbol represents Interaction Force
$c$ symbol represents Concentration Of Particles
$t$ symbol represents Time
Evolution Of The Concentration Of Particles PDE	${\partial }_{t}c(x,t)={\partial }_x(c(x,t)(F^{\prime }*c(\cdot ,t))(x))+\frac{{\sigma }^2}{2}{\partial }_{xx}c(x,t)$
$(F^{\prime }*c(\cdot ,t))$ symbol represents Convolution Between Interaction Force And Density
$F^{\prime }$ symbol represents Interaction Force
$\sigma$ symbol represents Diffusion Coefficient
$c$ symbol represents Concentration Of Particles
$t$ symbol represents Time