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The commands are as follows: % % 1) latex aipsamp % 2) bibtex aipsamp % 3) latex aipsamp % 4) latex aipsamp % % Use this file as a source of example code for your aip document. % Use the file aiptemplate.tex as a template for your document. \documentclass[% aip, % jmp, % bmf, % sd, % rsi, amsmath,amssymb, %preprint,% reprint,% %author-year,% %author-numerical,% % Conference Proceedings ]{revtex4-1} \usepackage{graphicx}% Include figure files \usepackage{dcolumn}% Align table columns on decimal point \usepackage{bm}% bold math %\usepackage[mathlines]{lineno}% Enable numbering of text and display math %\linenumbers\relax % Commence numbering lines \usepackage{xcolor} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{mathptmx} \usepackage{etoolbox} \usepackage{breqn} \usepackage{hyperref} %% Apr 2021: AIP requests that the corresponding %% email to be moved after the affiliations \makeatletter \def\@email#1#2{% \endgroup \patchcmd{\titleblock@produce} {\frontmatter@RRAPformat} {\frontmatter@RRAPformat{\produce@RRAP{*#1\href{mailto:#2}{#2}}}\frontmatter@RRAPformat} {}{} }% \makeatother \begin{document} \preprint{AIP/123-QED} \title{Polarised crowd in motion: Insights into statistical and dynamical behavior} \author{A. Pratikshya Jena} \email{pratikshyajena.rs.phy20@iitbhu.ac.in} \altaffiliation[Also at ]{Indian Institute of Technology} %Lines break automatically or can be forced with \\ \author{B. Shradha Mishra}% \email{smishra.phy@iitbhu.ac.in} \affiliation{ Department of Physics$, IIT (BHU), Varanasi, U.P. India - 221005$%\\This line break forced with \textbackslash\textbackslash }% \date{\today}% It is always \today, today, % but any date may be explicitly specified \begin{abstract} %The collection of active self-propelling agents has a tendency to show interesting statistical and dynamical properties. Such %characteristics are more significant in the context of the human crowd. Considering this, in the present study, we develop a %minimal computational model to mimic the crowd in a marathon race. We aim to examine the influence of frontliners %on crowd dynamics by comparing the simulated races with and without their presence. Our study's primary outcome revealed that the participants' local velocity and density exhibit a wave pattern similar to what is observed in %actual races. The traveling wave in the crowd consistently propagates with a constant speed, irrespective of the system %size under consideration. The participants' dynamic in the longitudinal direction mainly contributes to the velocity %fluctuation, and the fluctuation in the transverse direction is suppressed. In the absence of frontliners, the fluctuations %in density and velocity weaken without significantly influencing the crowd's other statistical and dynamic characteristics. Through this work, we aim %to enhance our understanding of crowd motion, which can inform the development of effective crowd management %strategies and contribute to successfully controlling such events. {The collection of active agents often exhibits intriguing statistical and dynamical properties, particularly when considering human crowds. In this study, we have developed a computational model to simulate the recent experiment on real marathon races by Bain {\em et al.,} [Science {\bf 363}, 46–49 (2019)]. Our primary goal is to investigate the impact of race staff on crowd dynamics. By comparing simulated races with and without the presence of race staff, our study reveals that the local velocity and density of participants display a wave pattern akin to real races for both the cases. The observed traveling wave in the crowd consistently propagates at a constant speed, regardless of the system size under consideration. The participants' dynamics in the longitudinal direction primarily contribute to velocity fluctuations, while fluctuations in the transverse direction are suppressed. In the absence of race staff, density and velocity fluctuations weaken without significantly affecting other statistical and dynamic characteristics of the crowd. Through this research, we aim to deepen our understanding of crowd motion, providing insights that can inform the development of effective crowd management strategies and contribute to the successful control of such events. } \end{abstract} \maketitle \section{\label{sec:level1} INTRODUCTION\protect\\ \lowercase{} } %Active matter \cite{ramaswamy2010mechanics, marchetti2013hydrodynamics,toner2005hydrodynamics,bechinger2016active,vicsek1995novel,li2023nonequilibrium,al2023chiral,PhysRevResearch.5.043246} is an interesting class of nonequilibirum systems consist of interacting agents that consume energy individually %and generate motion and mechanical stress collectively. %Such systems span a broad %range of length scales, starting from microscopic scales of few micrometers sized, viz. molecular motors, bacterial colonies, cells, cytoskeleton %filaments, etc. to the macroscopic scales of few meters and kilometers, %such as bird flock, human crowd, and animal herd \cite{cichos2020machine, needleman2017active,foster2017connecting, fodor2018statistical, balasubramaniam2022active, feder2007statistical, bottinelli2017jammed}. Collective behaviour of agents in these systems show interesting dynamical and statistical properties. Most of the recent studies have focused on the systems on small scales. But there are very few studies on the collection of active agents of macroscopic scales \cite{hueschen2023wildebeest,cavagna2010scale,viscido2004individual,parrish1999complexity,buhl2006disorder}, especially the human crowd in motion. \\ Comprehending the behavior or dynamics of the human crowd is valuable for optimizing both our everyday professional operations and our individual well-being. This understanding can significantly enhance managerial effectiveness as well. %Nowadays %city-scale public events are gaining significance in the global competition for socio-economic growth \cite{ritchie2009resident,liu2007effects,popescu2012role}. Events like sports competitions, exhibitions, and national celebrations exemplify the growing demands for appropriate technological solutions and support to effectively monitor and manage large crowds \cite{gong2020crowd, sharma2018review, wijermans2016landscape, duran2021crowd}. %In the present day, crowd management has become increasingly challenging. To prevent stampedes \cite{helbing2002simulation, helbing2007dynamics} and ensure safe and enjoyable events, crowd monitoring has become essential. %The rapid urbanization further emphasizes the need for proper support in planning public infrastructure for high-density crowds. %The effectiveness of crowd management is heavily influenced by pedestrian behavior within the crowd. By gaining insights how individuals behave within a crowd, appropriate strategies can be implemented to ensure safety and optimize the crowd flow. %Efficient crowd management demands the collaboration of diverse disciplines and practices, encompassing sociology, theoretical physics, applied mathematics, computer sciences, and more \cite{sime1995crowd,zeitz2009crowd,borch2013crowd,zhang2018physics,khozium2012proposed,bellomo2022towards}. \\ Numerous studies are available on crowd management that include real-world events involving camera tracking and data collection, as well as computer-simulated scenarios to investigate crowd behavior \cite{summers1982middle,costill1971determinants,james2015survival,daamen2003experimental,kretz2006experimental,hoogendoorn2003extracting,isobe2003many,helbing2003lattice,helbing1995social,yu2005centrifugal,burstedde2001simulation,barbosa2018human,helbing2015saving,karamouzas2014universal,fujita2019traffic,helbing2007crowd,duan2020crowd,garcimartin2015flow,johansson2008crowd,yi2020simulation,helbing2002simulation,liu2019simulation,wang2013simulation}. \\ % These dual approaches empower researchers to gain valuable insights, leading to effective crowd management strategies and safety measures. %A major portion of crowd management studies centers on crowd disasters, stampedes, large gatherings at national and spiritual events, as well as city-marathon races \cite{summers1982middle,costill1971determinants,james2015survival}. %Recent studies emphasize on experimental investigations of pedestrian flow \cite{daamen2003experimental,kretz2006experimental,hoogendoorn2003extracting}. In addition to the experimental studies, numerous computational and statistical models \cite{isobe2003many,helbing2003lattice,helbing1995social,yu2005centrifugal,burstedde2001simulation,barbosa2018human,helbing2015saving,karamouzas2014universal,fujita2019traffic,helbing2007crowd} have been developed to explore and analyze pedestrian flow patterns. {\bf include the citations from next part here}\\ %To investigate crowd disasters and stampede situations, a significant number of experimental studies \cite{duan2020crowd,garcimartin2015flow,johansson2008crowd} are conducted, relying on empirical methods and computer simulations \cite{yi2020simulation,helbing2002simulation,liu2019simulation,wang2013simulation}. \\ Another prominent event is the marathon race, which demands special attention when addressing crowd control measures. %Currently, marathon is a global cultural and sports event, representing personal accomplishment. It provides opportunity for community unit and also enhances local tourism by showcasing culture and landmarks. Indeed, there are numerous studies focused on city marathons \cite{lin2018empirical,hallmann2010event,billat2009detection,oficial2021effect,sabhapandit2008crowding,bain2019dynamic,rodriguez2014convection}, which aim to capture and analyze crowd dynamics during the races where these studies utilize various methods, including video analysis, data tracking, and participant surveys to understand how the crowd behave and interact during marathon events. The computer-simulated numerical models \cite{xu2014crowd,kwong2019modelling,zach2017motivation,strnad1985physics} for marathon attempt to replicate the real events. These models use mathematical equations and algorithms to represent the interactions and dynamics of the crowd during the marathon.\\ In a recent study\cite{bain2019dynamic}, the authors have utilized tens of thousand road-race participants in four starting corals: Chicago 2016, Chicago 2017, Paris 2017, Atlanta 2017 \cite{bain2019dynamic} to explain the flowing behaviour of polarized crowds by examining it's response to boundary motion. The outcomes of this experimental observations as well the continuum description \cite{bain2019dynamic}, elucidates the crowd dynamics specifically concerning velocity and density waves and show that the stimulations from side boundaries are inefficient and that optimal information transfer is obtained from guiding the crowd from its forefront. %influenced by the presence of race staff (frontliners) at the boundaries. The longitudinal velocity wave propagates upstream at a constant speed which is a characteristics of information transfer in the polarized crowd in all races. However, the orientational fluctuations are suppressed locally in transverse direction. \\ %Most of the previous computational models are based on the social interaction among the individuals of the crowd \cite{helbing1995social,chen2021modeling,moussaid2010walking,askarizad2020influence}. %The previous computational models of human crowd are based on the special types of social interaction among the individuals of the crowd \cite{helbing1995social,chen2021modeling,moussaid2010walking,askarizad2020influence}. Through this work, our objective is to develop a microscopic agent based model that can be applicable to polarized crowds. Here, we consider a collection of participants of the crowd on a two-dimensional track. The position and velocity of the individual participant are updated using agent based equations of motion to capture the essential dynamics of runners during a marathon event. %It's outcome are certainly comparable with the real-world data obtained from video tracking, enhancing its practical applicability. By providing this simplified model we aim to enhance our understanding of crowd behavior and dynamics, making it easier to understand to researchers and practitioners alike. \\ Motivated by the real-world situations where the crowd can be guided by some forefront (race staff) \cite{bain2019dynamic} or completely independent crowd free to move on a desired track, we provide the model with and without race staff at the forefront. %frontliner in our model and study the system with and without frontliners. Where the frontliners are defined as the race staff which always lead the crowd. % The race staffs play a vital role in organizing and managing the events to conduct them successfully and more professionally. In large races, the race staff form boundary to monitor the crowd motion and spectator areas, ensuring safety and the prevention of interference among the participants \cite{patel2022chemical}. The races without race staff may lack proper organization and management but are still enjoyable with a higher degree of freedom \cite{helbing2007crowd,johansson2008crowd}. \\ % Our model incorporates the presence or absence of frontliners in the simulated races as a key parameter. % Using this model, we examined various observables , including velocity, density profiles, speed of wave, velocity and density distribution of the crowd at a specific time, and velocity as well as density auto-correlation functions. % Most of the these observable can be measured by collecting the data on real crowds. The analysis of these data provide essential information about the effects of presence and absence of frontliners on these different aspects of crowd behavior. \\ The principal results of this paper are the propagation of hybrid coupled velocity-density wave throughout the system opposite to the crowd motion, as observed in recent experimental work \cite{bain2019dynamic}. %However, for without frontliners %it appears that the spreading of the profile is more prominent but the basic structure remains the same. The appearance of such coupled velocity-density wave is not limited to the participants who are macroscopic in size, but also present on the mesoscopic scale; like the collective motion of bacteria. In the work of \cite{park2003influence}, it is found that the environmental topology of complex structures is used by Escherichia coli to create traveling waves of high cell density, a prelude to quorum sensing. Additionally, the speed distribution depicts the propagating wave has a constant speed irrespective of the number of participants in the races considered. Based on the observation from the distribution for longitudinal and transverse velocity, it is found that the velocity shows the large fluctuations in longitudinal direction, whereas the fluctuations are suppressed in transverse direction. %The distribution of density illustrates the initial dispersion of the individuals in different direction gradually diminishes and the spreading become more uniform and stabilizes into a consistent pattern as individuals find their preferred positions and this is consistent for both the system with and without frontliners. The density and velocity propagate periodically through the system. %but the density wave travels faster than that of velocity. %{\bf we will include some points, how it can be utilised to spread some information in polarised crowd, rest of the introduction is okay !} \section{MODEL AND NUMERICAL METHODOLOGY} We model a collection of participants on a two-dimensional passage. We use the agent based dynamical equations of motion to describe the motion of runners. It takes into account the position ${\bf{r}}_{i}=(x_{i},y_{i})$ and velocity ${\bf{V}}_{i}=(V_{xi}, V_{yi})$ of the ${\it i}^{th}$ participant. We choose the $x-$ and $y-$ directions as direction parallel {$\parallel$} (longitudinal) and {$\perp$} (transverse) to the direction of moving crowd respectively as shown in the schematic of the model Fig. \ref{fig:modelfig} . The velocity of the each participant is updated by \begin{equation} \frac{ d {\bf{V}}_{i}}{d t} ={\bf{h}}+{\bf{F}}_{i} + {\bar{\eta}_{i}}({\bf r}, t) \label{eqn.1} \end{equation} where left hand side is simply the inertial term, the mass of the participant is taken as unity (because the Gravity is unimportant for the motion in a plane). The three terms on the right hand side are different forces: (i) biased drive to polarise the crowd along the track of the path $+x $ direction with ${\bf h} = (h_0, 0)$, (ii) The short ranged soft-repulsive interaction among the participant denoted by ${\bf{F}}_{i}= \sum_{j=1}^{N} {\bf{f}}_{ij}$, where ${\bf{f}}_{ij}=(r_{ij}-2 r_0)\hat{r}_{ij}$ if $r_{ij} \le 2r_0$ otherwise it is zero. Such force accounts for the mutual exclusion among the participants of size $r_0$, here $r_{ij}=|{\bf{r}}_{j}-{\bf{r}}_{i}|$ , $\hat{r}_{ij}= \frac{{\bf{r}}_{j}-{\bf{r}}_{i}} {r_{ij}}$ and (iii) small random uncorrelated thermal noise ${\bar{\eta}}$ having strengths $(\Delta_{\eta_x}, \Delta_{\eta_y})$ taking care of perturbation arises due to any kind of random fluctuations present in the system. \\ Further particles move in direction of their velocity obeying the following rule; \begin{equation} \frac{d {x_i}}{d t}={v_{0}}{V}_{xi}+{{\zeta}}_{xi}(t) \label{eqn.2} \end{equation} \begin{equation} \frac{d y_i}{d t}=V_{yi}+{{\zeta}}_{yi}(t) \label{eqn.3} \end{equation} In the $y-$direction participants simply follow the velocity in $y-$direction, whereas in the $x-$direction the step size of the participants $v_0$ is modified using a Gaussian distribution $P(v_0)=\frac{1}{\sigma\sqrt{2\pi}}\exp(\frac{v_0-\mu}{2\sigma})^2$ with mean $\mu = 0.6$ and variance $\sigma = 0.03$, where at every time and for different participants it is obtained independently. Motivated with the recent experiments on the dense bacterial solution \cite{meacock2021bacteria, lisicki2019swimming}, we introduced the distribution of speed $P(v_0)$ of the participants in the longitudinal direction. As reported in \cite{lisicki2019swimming}, the range of microswimmers maintained a common distribution in their speed, moreover in \cite{meacock2021bacteria}, it was found that the collective migration increases in bacterial solution by moving slowly. Thus, introducing a distribution in speed, where we have finite number of participants moving slowly may help the crowd propagation. Also introducing a distribution of speed in longitudinal direction avoids overcrowding. In the transverse direction, since the mean of the velocity is always zero, no such modification is required. The similar mechanism can also be obtained by introducing a distribution for $h_0$. In both directions, velocity is further modified with an additional random uncorrelated noise ${\bar{\zeta}}$ of strengths ($\Delta_{\zeta_x}, \Delta_{\zeta_x})$ to account for any kind of random fluctuations present in the system.\\ In the experiment \cite{bain2019dynamic} the participants in the race are guided towards the starting line by chains of staff members or {\em frontliners}. To incorporate the presence of such race staff we introduce race staff in our model, which act like moving forefront for the crowd. To introduce the fronliners, we marked $50$ participants present at the forefront of the crowd as frontliners when race starts, shown inside the rectangular box in Fig. \ref{fig:modelfig}(a). The velocity and position update of the participants and the frontliners are as given in Eq. \ref{eqn.1}, \ref{eqn.2} and \ref{eqn.3}. The frontliners have no constraints in their dynamic in the longitudinal direction, whereas the participants experience the frontliners as moving wall at their front and cannot cross it. We also studied the race without frontliners, by treating all the participants equivalent and they do not feel any moving forefront as shown in Fig. \ref{fig:modelfig}(b). The two systems in the presence and absence of frontliners are named as system {\em with} and {\em without} frontliners (WF and WOF respectively).\\ %They can significantly influence the behavior and performance of the participants behind them. By simulating races with and without frontliners, we can evaluate the impact of their presence on crowd dynamics, including aspects such as speed distribution, density and velocity wave propagation, and the overall flow of the crowd. \\ We choose effective size of the participants $r_0 = 0.8 $ as the intrinsic length scale in the system and the ratio of $\frac{r_0}{\mu} = {0.8}/{0.6} = 4/3 = \tau$ is the intrinsic time scale in the system. This is a typical time on an average a participant takes to cross its own size. Further we re-scaled all the lengths and times in the system in terms of $r_0$ and $\tau$. We consider races with varying numbers of participants ($N$) within the crowd. Specifically, we choose values of $N$ in the range $500-4000$ (typical number of participants in a standard race). These different numbers allow us to examine the impact of crowd size on the observed dynamics. \\ For all $N$ values, the participants are initially placed in narrow width $W = 62 r_0 $ in the transverse direction with reflecting boundaries. Initial number density $\rho = N/(W \times L_0)$ of the participants is $1.0$, where $L_0$ is the initial spread of the participants in the longitudinal direction and it is varied from $12.5 r_0$ to $125 r_0$ for different size of the races. Later, the participants are allowed to move according to the update equations given in Eq. \ref{eqn.1}, \ref{eqn.2} and \ref{eqn.3}. We have performed the simulation by taking the small time step $\Delta t=\frac{2}{3}\times 10^{-3} \tau$. One simulation step is counted when all the participants as well as the frontliners are updated once. The simulation is performed for total time $T= 500 \tau$. Motion in longitudinal direction is in open space, and there is no direct attraction among the participants, hence spread of the crowd in the longitudinal direction $L(t)$ increases with time. The spread is faster for the system WOF in comparison to system WF. Therefore, our most of the measurements are performed for time upto $t= 300 \tau $, such that density is not very low and interaction among the participants is relevant. In the experiment \cite{bain2019dynamic} as well the crowd motion is observed for $100$ to $500$ seconds such that a finite mean density is maintained in the system. \\ We choose the strengths of the four random fluctuations $\Delta_{\eta_x} = \Delta_{\eta_y} = \Delta_{\zeta_x} = \Delta_{\zeta_y} = 10^{-4}$ small. The strength of the small polarized field $h_0$ is fixed to $0.1$. Hence, participants experience a constant flow in the direction of ${\bf h}$. Averaging over $50$ independent realisations are performed for good statistics. \\ % The velocity update of the participants as well as the frontliners are the same whereas the participants position are updated in such a manner that they cannot cross the frontliners in longitudinal direction. Whereas the frontliners have no contariend in their dynamic in longitudinal direction. \begin{figure*} \centering \includegraphics[scale=0.7]{modelm.png} \caption{ The schematic of our model for the races (a) with frontliners and (b) without frontliners . The images are generated from the simulation. The circles depict the participants in the races in both the scenarios and color of the circle represent the local $x-$ component of velocity $V_x $ of the participants. {\bf{h}} is the external drive to polarize the crowd. In Figure (a), the squares inside the rectangle represent the frontliners. $W$ is the width of the track in the transverse direction.} \label{fig:modelfig} \end{figure*} \label{MODEL AND NUMERICAL METHODOLOGY} \section{Results} We first analyse the impact of frontliners on the dynamics of participants in the races. Movies \href{https://drive.google.com/file/d/15j-177RRyJTkbk4p0_bYLBEgjvM948ke/view?usp=sharing}{SM1} and \href{https://drive.google.com/file/d/1iuzD5HgZajoBYtlJ4kmLfbKSeRANHpuP/view?usp=sharing}{SM2}capture the animation of the system WF and WOF for a particular crowd size by setting N = 1000. respectively. The corresponding figures at time $t \simeq 100\tau$ are shown in Fig. \ref{fig:modelfig} (a-b). The circles represent the participants and the colors of the circles represent the magnitude of $x-$ component of velocity $V_x (t)$ of the participants. %The $x-$ coordinate of the participants is shifted by the mean location of the frontliners. We can very clearly observe that starting from the initial homogeneous and random velocity of particles, density and velocity patterns are formed. The detail results of the two systems with and without frontliners will be discussed later. \begin{figure} \centering \includegraphics[scale=0.45]{mod.png} \caption{The picture depicts the track is divided into $n$ rectangular cells. Here, the x-axis represents $x_{0}-x$, where x and $x_{0}$ are the co-ordinates of participants and the mean $x$-coordinates of the frontliners respectively. The zoomed picture presents a single cell inside which the coarse-grained density $\rho(x, t)$ and velocity {\bf $v_x$}$(x, t)$ are calculated. The color of the circles have the same meaning as in Fig. \ref{fig:modelfig}. } \label{fig:vrho} \end{figure} \begin{figure*} \centering \includegraphics[scale=0.38]{rhovx.png} \caption{The plot (a-c) and (g-i) showcase the coarse-grained density $\rho(x, t)$ at three times $t= (30, 75, 150)\tau$ in the presence and absence of frontliners respectively. The plot (d-f) and (j-l) represent coarse-grained velocity {\bf $v_{x}$}(x, t) for the same. The vertical downward arrow represents the direction of crowd propagation.} \label{fig:rhovx} \end{figure*} \begin{figure*} \centering \includegraphics[scale=0.40]{wf.eps} \includegraphics[scale=0.40]{wof.eps} \caption{(A) The plots (a-d) illustrate the density propagation at different times $ t =(15, 30, 45, 60)\tau$, while the plots (e-h) depict the corresponding propagation of velocity at those specific moments, for the systems with frontliners (WF). (B) The plots (a-d) illustrate the density propagation, while the plots (e-h) depict the propagation of velocity for the systems without frontliners (WOF) at the exact instances for the with frontliners (WF) scenario. The blue shaded region in all plot shows the region of high density band, with location of maximum density is represented by vertical dashed line in Fig. A (a-d) and B. (a-d). The vertical dashed line in Fig. A (e-h) shows the location of secondary maxima of velocity $v_x(x, t)$ for system WF. In Fig. B (e-h), the vertical dashed line depict the location where velocity shows abrupt change for the system WOF. The arrow represents the direction of crowd propagation in both the cases.} \label{fig:1dwave} \end{figure*} \begin{figure*} \centering \includegraphics[scale=0.45]{rvtn.eps} \caption{The plots (a-b) showcase the local density $(\rho)$ vs. local velocity $(v_{x})$ at time $t=(15, 30, 45, 60)\tau $ for the race having 1000 participants in the presence and absence of frontliners respectively.} \label{fig:rvt} \end{figure*} \begin{figure} \centering \includegraphics[scale=0.22]{wh.eps} \caption{(color online) The plots (a-b) depicts the speed distribution of the participants at time $t=225\tau$ for four different system sizes $N=1000, 1500, 2500, 3000$. The inset of plot (a) shows that the speed distribution of participants with frontliners fits well with log-normal distribution.} \label{fig:speed} \end{figure} \begin{figure} \centering \includegraphics[scale=0.28]{pxy.png} \caption{(color online) (a-b) illustrates distribution of longitudinal and transverse components of crowd's velocity $P(V_{i,j})$ at a particular instant ($t\simeq 225\tau$) for different $N= 1500$ and $3000$ respectively in the presence of frontliners. (c-d) shows replicated plots in the absence of frontliners. } \label{fig:pxpy} \end{figure} \begin{figure*} \centering \includegraphics[scale=0.38]{density_hist.png} \caption{ In the plots (a-c), the histograms display the density at three different times $t = (15, 45, 60)\tau$ for a race with $N = 3000$ individuals in presence of the frontliners. Here the $x-$axis represents the local $\rho$ and the $y-$axis represents the distribution of density $P(\rho)$. The red dotted lines represent the density of the frontliners. The plots (d-f) showcase the same in the absence of frontliners.} \label{fig:denhist} \end{figure*} \begin{figure*} \centering \includegraphics[scale=0.4]{den2.png} \caption{The figures (a-c), (d-f) depict histograms of density at a specific time ($t=60 \tau$), showing the distribution of individuals within different system sizes ($N=1000$, $3000$ and $4000$) in the presence and absence of frontliners respectively. The red dotted lines represent the density of the frontliners.} \label{fig:denN} \end{figure*} % \begin{figure} % \centering % \includegraphics[scale=0.27]{vt.eps} % \caption{Caption} % \label{fig:enter-label} % \end{figure} % \begin{figure} % \centering % \includegraphics[scale=0.6]{cv1.eps} % \caption{Color plot (online) (a-b) depicts the averaged velocity auto-correlation $(C_{v}(t))$ vs. time(t) and averaged density auto-correlation $(C_{\rho}(t))$ vs. time(t) of the participants in the races having participants $N=1000,2500,3000,4000$. } % \label{fig:acf} % \end{figure} \begin{figure} \centering \includegraphics[scale=0.2]{vracf.eps} \caption{The plots (a-b) depict the velocity auto-correlation $(C_{P_x}(t))$ vs. time ($t$) and density auto-correlation $(C_{\rho}(t))$ vs. time ($t$) in the presence and absence of frontliners respectively.} \label{fig:cvr} \end{figure} % \begin{figure*} % \centering % \includegraphics[scale=0.45]{rvt2.png} % \caption{The plots (a-b),(c-d) showcase the one-dimensional coarse-grained density $\rho(x)$ and velocity $p_x(x)$ at three different times $t$ = $(5, 10, ..)\times 6.25\tau$ for system size $N=1000$ in presence and absence of frontliners respectively.} % \label{fig:rvt2} % \end{figure*} %{\bf write few sentences to motivate the main observable you have and physical significance of those observable.} Based on the experiments on the real races \cite{lin2018empirical,pycke2022marathon,bain2019dynamic,kwong2019modelling}, various observable such as velocity and density waves, speed and velocity distribution of the participants, distribution of density in the system as well as density and velocity auto-correlation can be examined to explore the dynamics and characteristics of the moving crowd. These observable are valuable tools for characterizing the dynamic behavior of participants during the races.\\ To get the results from our model, we assume the crowd as a continuum and investigate the dynamics of the crowd by measuring the local coarse-grained density $\rho(x, t)$ and velocity {\bf $v_{x}$}(x, t). The movement of individual participants during the races generates pressure, leading to emergence of hybrid wave of velocity-density which propagates through the system. We further examine the participants speed ($u = \sqrt{(V_x^2+V_y^2)}$) distribution $P(u)$ and velocity distribution $P(V_{i, j})$ (where (i, j) represent x and y components of velocity) to understand the characteristics of the travelling wave. To interpret the spreading and squeezing of crowd during the races, we also analyze the density distribution $P(\rho)$ of the crowd in races having different numbers of participants and at different times. This can be beneficial to understand the crowd distribution to manage and control the stampede like situation during the marathon events. We additionally calculate the velocity and density auto-correlation functions $C_{v_{x}}(t)$ and $C_{\rho}(t)$ respectively to get the information about how the velocity and density are correlated with respect to time. The decay of the velocity auto-correlation provides information about characteristic time-scale associated with the relaxation of the system. Below the discussion of each observable are provided one by one from the result of our model.\\ %\subsubsection{Velocity and density profile} {\em Velocity and density profile}:- To employ a continuum approach to analyze the large-scale motion within the crowd, we plot the local coarse-grained density, denoted as $\rho(x, t)$, and the velocity, represented as $v_x(x, t)$, at different spatial positions $x$ and time $t$. To define the coarse-grained local density and velocity, we divide the whole track on which the race goes on into $n = 500$ rectangular cells of size $\Delta x = 10 r_0$ in $x-$direction and breadth equal to the full length of the track in $y-$direction. The $x=0$ line is always fixed at the mean of the $x-$coordinates of the frontliners $(x_0)$. \ Later, the distances in the longitudinal direction for both the models system WF and WOF are measured with respect to mean $x-$coordinates of the frontliners and front of the wave respectively. Then we calculate the density in each cell $\rho(x, t)$ by counting the number of participants in each cell and dividing by the area of the cell. To determine the coarse-grained velocity $v_x(x, t)$, we add the longitudinal velocities of the participants in the cell. Fig. \ref{fig:vrho} shows the schematic of the above procedure explained to determine the coarse grain density $\rho(x, t)$ and velocity $v_x(x, t)$. \\ Fig. \ref{fig:rhovx} provides a visual representation of crowd flow by showing the propagation of coarse-grained density $\rho(x, t)$ and velocity $v_x(x, t)$ at three times. The direction of flow of crowd is downward and vertical spread of the track is in the horizontal direction. %The Fig. \ref{fig:rhovx} (a-f), shows the plots for the case, races for the scenario involving frontliners at three different times t=$(0, 50, 100)\times 0.625\tau$ respectively. And (g-l) depicts the plots for the case without frontliners. In Fig. \ref{fig:rhovx} (a-c) for the local density $\rho(x, t)$ and (d-f) for the local velocity $v_x(x, t)$ at three different times $t \simeq (30, 75, 150)\tau$ respectively for the system with frontliners. Similarly Fig. \ref{fig:rhovx} (g-i) for the local density $\rho(x, t)$ and (j-l) for the local velocity $v_x(x, t)$ at three different times $t\simeq (30, 75, 150)\tau$ respectively for the system without frontliners. Starting from the early time with a constant density in a small region, the density and velocity profile undergoes a propagation inferring a hybrid coupled wave of density and velocity transmitting within the system. It is clearly noticeable that with time an initial homogeneous density of participants splits into density wave, which moves in the direction opposite to the direction of crowd. \\ In Fig. \ref{fig:1dwave}(A-B) we plot the one-dimensional coarse-grained density $\rho(x,t)$ and velocity $v_x(x,t)$ for four different times $t\simeq (15, 30, 45, 60)\tau$ for the system WF and WOF respectively. Distinctly, the density pattern shows a peak and the peak propagates in the direction opposite to the direction of crowd motion (as shown by arrow in the figure). With the vertical dashed line, we mark the position of the peak of density and similarly for the velocity (excluding the large spike near the frontliners) for the system WF. For the system WOF, velocity does not show a peak, but a vertical dashed line is drawn at the position where velocity shows an abrupt decay to zero. We find that both the density and the velocity waves travel in the same direction with some phase difference [also see the corresponding animations for the particles and coarse-grained density and velocity in \href{https://drive.google.com/file/d/15j-177RRyJTkbk4p0_bYLBEgjvM948ke/view?usp=sharing}{SM1} and \href{https://drive.google.com/file/d/1iuzD5HgZajoBYtlJ4kmLfbKSeRANHpuP/view?usp=sharing}{SM2} for the system WF and WOF respectively]. \\ To further find the relation between density and velocity, in Fig. \ref{fig:rvt}(a-b) we showcase the local density $\rho(x)$ vs. local velocity $v_x(x)$ at three different time $t\simeq (15,75,180)\tau$ for the race having $1000$ participants in the presence and absence of frontliners respectively. The relation between density and velocity field is highly non-monotonic in nature. For small velocity: first density increases with $v_x(x)$ indicating the tail of the crowd which is away from the forefront. This part of the crowd is also shown by the region left to the blue shaded region in the one-dimensional plot of density and shown in Fig. \ref{fig:1dwave}(A-B). On further increasing velocity, relatively large number of participants move with a moderate velocity, where the plot shows a peak, that mimics the density band in the system. The same is shown by small shaded region with blue in Fig. \ref{fig:1dwave}(A-B). On further increasing velocity, density again suppresses, but the plot shows the abrupt change for system with frontliners \ref{fig:rvt}(a), whereas the change is gradual for system WOF. For the system WF, there is a thin layer of participants move with high velocity are responsible for such behaviour. The density-velocity plot is narrow for the system WF, whereas it is wider for the system WOF, which can also be seen by the one-dimensional plot of density in Fig. \ref{fig:1dwave}(A-B), which is narrow for system WF and wider for the system WOF. As we increase time, the nature of density vs. velocity plot remains the same, but slowly crowd disperse and both density and velocity shifts towards the smaller values. This non-monotonic nature is intrinsic to the collective behaviour of participants moving in different races as found in previous experiments \cite{lin2018empirical}. The presence of band of high density is responsible for such non-monotonic behaviour.\\ Till now, we focused on the dynamic nature of the crowd, density and velocity pattern in the system. Now, we focus our attention on the characteristics of the moving participants. % the distribution shows that relatively large number of participants are moving at a moderate velocity,which then sharply for high velocities. That suggests the presence of frontliners, creating a depletion layer and the participants slow down their velocities as they experiences some barrier. In the case of WOF Fig. \ref{fig:rvt}(b), the distribution is wider and more symmetric than the previous one. This implies that comparatively large number of participants are moving with moderate velocity and the decrease in participants for higher velocities is smoother compared to the WF case indicating still there are finite number of participants moving at high velocity as they do not encounter any barrier as in the case of WF.\\ % For regions with lower velocity encourages the increase in local density, but after some moderate velocity density starts to decrease. Hence the local density $\rho(p_x)$ has a non-monotonic dependence on local velocity as shown in Fig. \ref{fig:rvt}(a). % % % %Likewise, for the system without frontliners (WOF) too. % Based on the observation for WOF it appears that the pattern of the local density and velocity follows the same trend as for the system with frontliners, only the spread of density and velocity is more and intensity of the wave weakens. The relation between local density $\rho(p_x)$ on velocity is non-monotonic in nature as shown in Fig. \ref{fig:rvt}(b). The key difference in the dependence of local density on local velocity for the two cases is that for the system WOF it follow a nearly Gaussian distribution with range of distribution wider than that for the system with frontliners. Whereas, for the system WF, the $\rho(p_x)$ deviates from the Gaussian with smaller range of $p_x$. The Gaussian dependence of the local density $\rho$ of the crowd on the local velocity is previously found in the study of \cite{lin2018empirical}, where the authors has investigated empirically the velocity distributions of finishers in New York City marathon, American Chicago marathon, Berlin marathon and London marathon without any race staffs. \\ {\em Speed distribution}:- The Fig. \ref{fig:speed} illustrates the distribution of speeds $u$ of participants for both the cases, the presence and the absence of frontliners. The distribution of local speed $P(u)$ is obtained by calculating distribution of speeds of all the participants. Fig. \ref{fig:speed}(a) depicts the speed distribution of participants in the presence of frontliners at a specific time $t\simeq 225\tau$. The figure showcases the impact of different system sizes $(N=1000, 1500, 2500, 3000)$ represented by distinct colors. In the inset of Fig. \ref{fig:speed}(a), we show the lines are the fit to the log-normal function $P(u) =\frac{1}{ua\sqrt{2\pi}}exp(-(log(u)-b)^2/2a^2)$, (where $a$ = 0.4 , $b$ = 0.65 for $N = 1500$). The mean of the speed distribution is non-zero for all $N$'s. The crowd motion is happening through the system with almost same mean speed ranges from (0.5-0.8) for different system sizes. We further calculated the speed distributions for other times $t\simeq 100\tau$ and $t\simeq 150\tau$ and found that the mean of the speed distribution remains invariant with time too. Hence, initially moving crowd relaxes quickly and then it propagates with a constant speed. Further, since the mean speed does not depend on the size of the crowd, hence the traveling wave in the crowd is non-dispersive in nature, unlike the normal waves in any media \cite{griffiths2021introduction} .\\ %{\bf we need to write the consequences of it in the discussion section}. %propagating crowd moves almost with the constant speed.\\ %Interestingly, the $P(u)$ remains unchanged irrespective of the system sizes under consideration.\\ In Fig. \ref{fig:speed}(b), the distribution $P(u)$ is shown for the scenario without frontliners. The distribution is wider in comparison to the system WF, with no tail at larger speed. Once again as found for the system WF, the mean speed for all sizes of the race lies in the small range (0.7-0.9). This suggests that the presence or absence of frontliners does not have a significant impact on the mean of the travelling speed of the wave indicating system's robustness in response to external influencers/perturbations. The two distributions of the system WF and WOF have one key difference that the $P(u)$ for the system WF is $\log-$normal and have a long tail, whereas no such tail is observed for system WOF. For the system with frontliners few participants are moving with relatively higher speed. Participants adjacent to the frontliners are mainly contributing for such high speeds [see \href{https://drive.google.com/file/d/15j-177RRyJTkbk4p0_bYLBEgjvM948ke/view?usp=sharing}{SM1} for a visual demonstration of the occurrence]. %\subsubsection{Velocity distribution} {\em Velocity distribution}:- Further, the fluctuations in the velocity of participants in longitudinal and transverse directions are measured by calculating the distribution of the two components of velocities of participants $V_x$ and $V_y$ for three different sizes of the crowd. Fig. \ref{fig:pxpy} (a-b) depicts the longitudinal and transverse component of crowd's velocity distribution for a specific time $t=225\tau$ for the system with frontliners and for the three different system sizes $(N= 1500, 3000)$. Similarly, (c-d) represents the identical plots without fronliners in the same sequence. From Fig. \ref{fig:pxpy} , it can be observed that the transverse component has zero mean, whereas the longitudinal component exhibits a peak at some non-zero value. These findings suggest that there is no such significant movement in the transverse direction and the propagation of velocity wave primarily occurs in the longitudinal direction. The distribution is narrow for the transverse direction, whereas it is broader for the longitudinal direction suggesting that the moving crowd has larger fluctuations in longitudinal direction in comparison to transverse direction, which is completely opposite to what has been observed for the polar flocks \cite{toner1998flocks,toner1995long,kaiser2017flocking,bastien2020model,schaerf2017effects}. The large fluctuations in longitudinal direction is also observed in real marathon races \cite{bain2019dynamic}, whereas we expect the suppression of velocity fluctuations in the longitudinal direction, due to the presence of a constant driving force ${\bf h}$. The confinement present in the transverse direction might be responsible for the suppressed fluctuation. The result of increase in the size of the flock increases the role of confinement and lead to more narrower distribution of velocity in the transverse direction as can be clearly seen in Figs. \ref{fig:pxpy}(a-d). \\ The two distributions $P(V_x)$ and $P(V_y)$ have some overlap for the system with frontliners, whereas there is no overlap for system without frontliners. This is due to the presence of strong density band, which moves backward and slow down the motion of the participants near to it for the system WF. This can be also seen from Fig. \ref{fig:1dwave}A (a-d): density shows a long-tail in the direction opposite to the direction of crowd propagation, and participants moving there have smaller speed as shown in Fig. \ref{fig:1dwave}A(e-h). But, for the system WOF, both density and velocity drop abruptly. Since, the intensity of the density band is dilute for system WOF, hence we do not see small velocity of participants in this case and no overlap. The distribution $P(V_x)$ also shows a long tail, that is mainly due to the presence of a moving forefront in the system with frontliners and it results in bigger range of $V_x$ in the system. Restricted motion of the participants due to the presence of frontliners in the direction of moving crowd leads to relatively less broadness for the distribution in longitudinal direction in comparison to the participants without frontliners. That further lead to an overlap in the two distributions for the system with frontliners. \\ % In fig \ref{fig:pxpy}(a-c), the common value observed in the longitudinal and transverse components suggests that there is some correlation between both components indicating their interdependence while in fig\ref{fig:pxpy}(d-f) no such mutual dependency is observed from the data. {\bf what is the reason that velocity distributions in two directions have some overlap for system with FL and no overlap for system without FL} %\subsubsection{Density distribution} {\em Density distribution}:- To further quantify the presence of band of high density and its stability over different realisations, we plot the distribution of local density $P(\rho)$ in the system. %Till now we have focused on the characteristics of the crowd based on the velocity wave, but density of the moving crowd also have interesting properties as it was clear from the snapshots shown in Fig. \ref{fig:rhovx} and animations in SM1 and SM2. Hence, we now focus on the characteristics of the crowd based on the distribution of local density $P(\rho)$. Fig. \ref{fig:denhist} (a-c) and (d-f), illustrates the density histogram of the system at three different times $t\simeq (15, 45, 60)\tau$, comparing the two cases, the presence and absence of frontliners. As shown in Fig. \ref{fig:denhist}(a), at the early time the initial distribution is wide with a large range of density with a bimodal character, represents the presence of a very narrow band with density almost close to the density of frontliners (as shown by the vertical line in each panel). The distribution contracts with time in Fig. \ref{fig:denhist}(b-c) and retains its bimodal nature as well as the difference from the density close to frontliners density increases. Fig. \ref{fig:denhist}(d-f) depicts the same for time steps $t\simeq (15, 45, 60)\tau$ for the system without frontliners. For this case, we see more clear bimodal distribution of $P(\rho)$ for all times. Which is due to the relatively wider band in the system WOF in comparison to the WF. For both the cases, the distribution shrinks to the lower density with respect to time, due to the spread of the crowd in the longitudinal direction. \\ Further, we also calculated the $P(\rho)$ at fixed time and for different system size $N=1000, 1500, 2500$ for both the cases system WF and WOF as shown in Figs. \ref{fig:denN}(a-c) and (d-f) respectively. For the system WF, the $P(\rho)$ shows the bimodal structure for all system sizes. Also for the system WOF, the nature of the distribution remains the same for all system size, it shows two small peaks at low and high density with a dip for intermediate density. The bimodal nature becomes more prominent for system WOF, due to wider band in comparison to narrow band for the system WF. This implies the characteristics of the density remains invariant with respect to the system size. The one small bar at high density for the system WF shows the density of participants close to the frontliners. The frontliners' density is marked with vertical dashed line in Fig. \ref{fig:denN}(a-c). The shift in the mean density for both the cases is due to the different relaxation time for different $N$.\\ %which are depicted by the finite densities at which probability $P(\rho)$ is nearly zero {\bf check}. %and stabilizes into a consistent pattern as individuals find their preferred positions. % \subsubsection{Non-dispersive wave} %\subsubsection{Auto-correlation functions} {\em Speed of density and velocity waves}:- To further quantify the characteristics of the traveling density and velocity wave, we calculate the auto-correlation functions of the fluctuations of density and velocity defined as; $C_{v_x}(t)=\langle {\delta v_x}(t) { \delta v_x}(t+\delta t) \rangle$ and $C_{\rho}(t)=\langle \delta\rho(t) \delta\rho(t+\delta t) \rangle$ respectively. The fluctuations in velocity and density are defined as: $\delta v_x(t) = (v_x(t) - \bar{v})$ and $\delta \rho(t) = (\rho(t) - \rho_0)$, where $\bar{v}$ and $\rho_0$ is mean value of $v_x(t)$ and $\rho(t)$ over the time. The $C_{v_x}(t)$ and $C_{\rho}(t)$ are averaged over 50 realizations and four different sizes of the race and many reference times $t_0$. Fig. \ref{fig:cvr}(a-b) show the plots of the auto-correlation functions of the density $C_\rho(t)$ and the velocity $C_{v_x}(t)$ for the system WF and WOF respectively. Both the $C_\rho(t)$ and the $C_{v_x}(t)$ show the early time exponential decay with time. Both the curves almost overlap with each other. Hence, the velocity and density wave travels almost with the same speed. \label{Results} \section{Discussion} We provide an agent based model to replicate the crowd in a marathon race and examined the properties of the moving crowd. Through this study, we try to provide a supporting computational model to replicate the results obtained in recent experiment on real marathon races \cite{bain2019dynamic}. The races considered in \cite{bain2019dynamic}, are guided by the race staff at the front of the race. Races without this oversight may be less organized but offer a more liberating experiences. Hence, we introduce a model that incorporates both scenarios to accurately simulate the actual marathon experience. \\ Most of the outcomes of our study, qualitatively fit with the observables which are reported in \cite{bain2019dynamic} for different races by using the empirical data obtained by video-tracking. Additionally our study introduce a model for race without any race staff and give the detailed comparison for the two cases. The main result of our paper is the propagation of hybrid coupled velocity-density wave in the direction opposite to direction of crowd motion as found in real races \cite{bain2019dynamic}. The speed distribution depicts the propagating wave has a constant speed irrespective of the number of participants in the races considered. That makes the characteristic of the travelling wave in polarised crowd very different from the usual wave moving through a medium, which is dispersive in nature. % \bf trava the small note on the non-dispersive nature of the travelling wave of crowd and its comparison with the other real world wave.} The distribution of longitudinal and transverse velocity shows that the fluctuation of velocity primarily occurs in the longitudinal direction and velocity fluctuations in transverse direction are highly suppressed, unlike the velocity fluctuation in polar flock \cite{toner1998flocks,toner1995long,kaiser2017flocking,bastien2020model,schaerf2017effects}. We observed a non-monotonic relation between the local density and velocity in the system. This plot shows a largest density with a most-probable speed, which reflects the presence of density band in both the cases with and without frontliners. We also found that the key characteristics of the moving crowd are similar for both types of races. Only the contrast of density is weak for the system without fronliners due to the completely free motion in the direction of crowd propagation. The results reported in \cite{bain2019dynamic} mainly focused on the characteristics of velocity of participants. In our present study we also provide a detailed correspondence of density and velocity fields. Any information about the density of the participants and its relation with their velocity can be useful for the control and predict the behaviour of the system. \\ The appearance of the density wave in the system, is applicable for wide range of active systems starting from microscopic scales like collection of bacteria, molecular motors, etc. to the macroscopic scales like animal herd. The key result of our study: the presence of density wave is already reported in a system on a much smaller scale like collection of bacteria sliding on patterned substrate \cite{park2003influence}. Our findings lead to future direction of research, focusing on examining the impacts of various types of boundaries positioned perpendicular to the passage. Another prospective pathway of future research involves the introduction of some external perturbations to investigate the response i.e. stampede like situation \cite{de2019human,barbosa2018human,kudrolli2008swarming} . \label{Discussion} % \section{Supplementary Material} % SM1 and SM2 movies capture the animations of the systems WF (With frontiliners) and WOF (Without Frontliners) respectively for a particular crowd size by setting N = 1000. % \label{Supplementary Material} \begin{acknowledgments} P.J. and S. M. thank Prof. Jacques Prost for useful discussions. P.J. gratefully acknowledge the DST INSPIRE fellowship for funding this project. The support and the resources provided by PARAM Shivay Facility under the National Supercomputing Mission, Government of India at the Indian Institute of Technology, Varanasi are gratefully acknowledged by all authors. S.M. thanks DST-SERB India, ECR/2017/000659, CRG/2021/006945 and MTR/2021/000438 for financial support. P.J. and S.M. also thank the Centre for Computing and Information Services at IIT (BHU), Varanasi. \end{acknowledgments} \section*{Data availability} The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. \label{Data availability} \nocite{*} \bibliography{citation}% Produces the bibliography via BibTeX. \begin{thebibliography}{84}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} }% \providecommand \@ifnum [1]{% \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi }% \providecommand \@ifx [1]{% \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi }% \providecommand \natexlab [1]{#1}% \providecommand \enquote [1]{``#1''}% \providecommand \bibnamefont [1]{#1}% \providecommand \bibfnamefont [1]{#1}% \providecommand \citenamefont [1]{#1}% \providecommand \href@noop [0]{\@secondoftwo}% \providecommand \href [0]{\begingroup \@sanitize@url \@href}% \providecommand 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\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont {Hoskin}}, \bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont {Jones}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Laing}}, \emph {et~al.},\ }\bibfield {title} {\enquote {\bibinfo {title} {Survival with newly diagnosed metastatic prostate cancer in the “docetaxel era”: data from 917 patients in the control arm of the stampede trial (mrc pr08, cruk/06/019)},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {European urology}\ }\textbf {\bibinfo {volume} {67}},\ \bibinfo {pages} {1028--1038} (\bibinfo {year} {2015})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Daamen}\ and\ \citenamefont {Hoogendoorn}(2003)}]{daamen2003experimental}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Daamen}}\ and\ \bibinfo {author} {\bibfnamefont {S.~P.}\ \bibnamefont {Hoogendoorn}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Experimental research of pedestrian walking behavior},}\ 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{\bibfnamefont {W.}~\bibnamefont {Yu}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Chen}}, \bibinfo {author} {\bibfnamefont {L.-Y.}\ \bibnamefont {Dong}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Dai}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Centrifugal force model for pedestrian dynamics},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical Review E}\ }\textbf {\bibinfo {volume} {72}},\ \bibinfo {pages} {026112} (\bibinfo {year} {2005})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Burstedde}\ \emph {et~al.}(2001)\citenamefont {Burstedde}, \citenamefont {Klauck}, \citenamefont {Schadschneider},\ and\ \citenamefont {Zittartz}}]{burstedde2001simulation}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Burstedde}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Klauck}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Schadschneider}}, \ and\ \bibinfo {author} {\bibfnamefont 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{Q.}~\bibnamefont {Chen}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Simulation of pedestrian evacuation in stampedes based on a cellular automaton model},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Simulation Modelling Practice and Theory}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo {pages} {102147} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Liu}, \citenamefont {Liu},\ and\ \citenamefont {Wei}(2019)}]{liu2019simulation}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Liu}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Liu}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Wei}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Simulation of crowd evacuation behaviour in outdoor public places: A model based on shanghai stampede},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {International Journal of Simulation Modelling}\ }\textbf {\bibinfo {volume} {18}},\ 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{2021})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Sabhapandit}, \citenamefont {Majumdar},\ and\ \citenamefont {Redner}(2008)}]{sabhapandit2008crowding}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Sabhapandit}}, \bibinfo {author} {\bibfnamefont {S.~N.}\ \bibnamefont {Majumdar}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Redner}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Crowding at the front of marathon packs},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Journal of Statistical Mechanics: Theory and Experiment}\ }\textbf {\bibinfo {volume} {2008}},\ \bibinfo {pages} {L03001} (\bibinfo {year} {2008})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bain}\ and\ \citenamefont {Bartolo}(2019)}]{bain2019dynamic}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Bain}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Bartolo}},\ }\bibfield {title} {\enquote 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{Volpe}}]{bechinger2016active}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Bechinger}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Di~Leonardo}}, \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {L{\"o}wen}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Reichhardt}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Volpe}}, \ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Volpe}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Active particles in complex and crowded environments},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Reviews of Modern Physics}\ }\textbf {\bibinfo {volume} {88}},\ \bibinfo {pages} {045006} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Vicsek}\ \emph {et~al.}(1995)\citenamefont {Vicsek}, \citenamefont {Czir{\'o}k}, \citenamefont {Ben-Jacob}, \citenamefont {Cohen},\ and\ \citenamefont {Shochet}}]{vicsek1995novel}% \BibitemOpen \bibfield {author} 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{\bibfnamefont {B.}~\bibnamefont {Mehlig}}, \ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Volpe}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning for active matter},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature Machine Intelligence}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {94--103} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Needleman}\ and\ \citenamefont {Dogic}(2017)}]{needleman2017active}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Needleman}}\ and\ \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Dogic}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Active matter at the interface between materials science and cell biology},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature reviews materials}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {1--14} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Foster}\ \emph {et~al.}(2017)\citenamefont {Foster}, \citenamefont {Yan}, \citenamefont {F{\"u}rthauer}, \citenamefont {Shelley},\ and\ \citenamefont {Needleman}}]{foster2017connecting}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont {Foster}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yan}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {F{\"u}rthauer}}, \bibinfo {author} {\bibfnamefont {M.~J.}\ \bibnamefont {Shelley}}, \ and\ \bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont {Needleman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Connecting macroscopic dynamics with microscopic properties in active microtubule network contraction},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {New Journal of Physics}\ }\textbf {\bibinfo {volume} {19}},\ \bibinfo {pages} {125011} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Fodor}\ and\ \citenamefont {Marchetti}(2018)}]{fodor2018statistical}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {{\'E}.}~\bibnamefont {Fodor}}\ and\ \bibinfo {author} {\bibfnamefont {M.~C.}\ \bibnamefont {Marchetti}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The statistical physics of active matter: From self-catalytic colloids to living cells},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physica A: Statistical Mechanics and its Applications}\ }\textbf {\bibinfo {volume} {504}},\ \bibinfo {pages} {106--120} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Balasubramaniam}, \citenamefont {M{\`e}ge},\ and\ \citenamefont {Ladoux}(2022)}]{balasubramaniam2022active}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Balasubramaniam}}, \bibinfo {author} {\bibfnamefont {R.-M.}\ \bibnamefont {M{\`e}ge}}, \ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Ladoux}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Active nematics across scales from 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{author} {\bibfnamefont {J.}~\bibnamefont {Silverberg}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Jammed humans in high-density crowd disasters},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {APS March Meeting Abstracts}}},\ Vol.\ \bibinfo {volume} {2017}\ (\bibinfo {year} {2017})\ pp.\ \bibinfo {pages} {H14--004}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Viscido}, \citenamefont {Parrish},\ and\ \citenamefont {Gr{\"u}nbaum}(2004)}]{viscido2004individual}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~V.}\ \bibnamefont {Viscido}}, \bibinfo {author} {\bibfnamefont {J.~K.}\ \bibnamefont {Parrish}}, \ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Gr{\"u}nbaum}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Individual behavior and emergent properties of fish schools: a comparison of observation and theory},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Marine Ecology Progress Series}\ }\textbf {\bibinfo {volume} {273}},\ 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{\bibfnamefont {N.}~\bibnamefont {Perozo}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Garnier}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Helbing}}, \ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Theraulaz}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The walking behaviour of pedestrian social groups and its impact on crowd dynamics},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {PloS one}\ }\textbf {\bibinfo {volume} {5}},\ \bibinfo {pages} {e10047} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Askarizad}\ and\ \citenamefont {Safari}(2020)}]{askarizad2020influence}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Askarizad}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Safari}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The influence of social interactions on the behavioral patterns of the people in urban spaces (case study: The pedestrian zone of 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