The Problem of Traffic: A Mathematical Modeling Journey

How can we mathematically model traffic? Specifically we will study the problem of a single lane of cars and the perturbation from equilibrium that occurs when one car brakes, and that braking effect travels down the line of cars, amplifying as it goes along, due to the delayed reaction time of the drivers. The ultimate phenomena we would like to predict is that stop-and-go behaviour where cars don't just travel at a constant speed in rush hour, but alternate between braking and accelerating. However, when building this model our inputs are very microscopic considerations about how an individual car brakes or accelerates based on the following distance and relative velocity of the car ahead of it. This video also aims to be an introduction to broad themes in mathematical modelling of real world problems, where we define a problem, choose the inputs to the system we will consider, make assumptions, build the model, and finally assess the model. Finally, the real piece of mathematics we are going to get out of this are called differential-delay equations, and I'll show you a bit about how to solve such equations at the end. This video is part of the second iteration of the Summer of Math Exposition, hosted by ‪@3blue1brown‬ and ‪@LeiosLabs‬ . There are so many great videos in this in this exposition so definitely check them out by using the hashtag #SoME2. 0:00 The Challenge of Traffic 0:28 #SoME2 0:53 The Modelling Process 1:27 Defining the Problem 2:04 Choosing Which Variables to Consider 4:03 Making Assumptions 5:46 Building the Microscopic Model for Each Car 9:56 Macroscopic Equilibrium 10:34 The Relationship between Density and Velocity 16:23 Maximizing Flux and the Optimal Oensity 20:33 Modelling a Sequence of Cars 24:07 Modelling the First Car 26:05 Full Model: A Differential Delay System 27:06 Assessing the Model Graphically 29:33 Assessing the Model Qualitatively 31:45 Solving Differential Delay Systems Check out my MATH MERCH line in collaboration with Beautiful Equations ► COURSE PLAYLISTS: ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Logic, P...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   ►CALCULUS I:    • Calculus I (Limits, Derivative, Integrals)...   ► CALCULUS II:    • Calculus II (Integration Methods, Series, ...   ►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus (Vect...   ►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line Integra...   ►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)   ►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs   ►GAME THEORY:    • Game Theory   OTHER PLAYLISTS: ► Learning Math Series    • 5 Tips To Make Math Practice Problems Actu...   ►Cool Math Series:    • Cool Math Series   BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based):   / treforbazett   ►Instagram (photography based):   / treforphotographyg