Collective behavior of a large number of self-propelled particles (SPP) can result in spontaneously developping ordered motion governed by changes in some control parameter(s). Many groups of living beings (from cells and bacteria to fish, birds, mammals and even humans) exhibit this specific kind of motion under particular conditions. From the point of view of statistical physics, the occurrence of collective motion is a non-equilibrium phase transition which has attracted much attention of the community in the last decades (for more details see the reviews Castellano et al. (2009); Vicsek and Zafeiris (2012); Ginelli (2016)).
The first step in the understanding of this complex behavior has been to propound simple but non-trivial models, such as the Vicsek Model (VM) Vicsek et al. (1995) in the middle of the nineties. In the VM, point particles with constant speed interact with each-other only by trying to align their direction of motion with their nearest-neighbors, with an uncertainty of this process represented by an external noise . This simple rule (explained in detail in section II.1) is enough to reproduce flocking behavior at low values of , as commonly observed in nature.
However, the complexity of collective motion often requires to go beyond a point particle model by taking into account short- and long- range particle-particle interactions and interactions with the environment. In particular, it is possible to model a crowd as a system of particles by considering person-person and person-wall interactions. One of the first models of this type proposed to describe pedestrian motion was the so-called Social Force Model (SFM) Helbing and Molnár (1995). Unlike the simplistic Vicsek Model, the SFM introduces the idea of social interactions by modeling the individual reaction to the effect of environment (either other pedestrians or borders), and a preferential direction of motion. This model will be explained in detail in Sec. II.2.
The main idea of our work is to analyze by using statistical mechanical tools the evacuation of people along a hallway in a non-panic regime, such as occurs in schools, hospitals or airports. To model this situation, it is important to take into account a series of considerations such as: the excluded-volume and mass of human-particles; the interpersonal interactions that make them want to keep their own space; the intent to remain away from the walls; the existence of a desired direction of motion; but also the idea of being influenced by the motion of nearest-neighbors. Keeping this in mind, we propose in the present work a new approach by introducing a model a combination of the standard VM and SFM, which takes into account the particle-interactions as in the SFM and additionally a Vicsek-like alignment modulated by a noise . We call this the SFM+VM model, and it will be described in detail in Sec. II.3.
In order to identify the influence of the different parts of the model, we have compared the VM, the SFM+VM, and the SFM in the stationary configuration of particles moving through a corridor-like system in a normal evacuation situation (slow speed regime). We have studied all models under the same external conditions (number of particles , system-size , particle speed , external noise , etc.). In particular, we have analyzed the VM under different boundary conditions, with the purpose of introducing the effect of walls into the VM, and the idea of a desired direction of motion, such as it is defined in the SFM, in order to flesh out the comparison between models, and we have studied the effect of these variations on the order-disorder phase transition. Finally, it is worth mentioning that even though the SFM has been widely studied (see for example Helbing and Johansson (2011); Chen et al. (2018) and references therein), the statistical physics issues related to phase transitions of social models have been less extensively studied (see e.g. Cambui et al. (2017); Baglietto and Parisi (2011)).
ii.1 Vicsek Model (VM)
The Vicsek Model Vicsek et al. (1995) describes the dynamics of SPP characterized at time by their position and velocity (), and in its simplest version all particles are considered to have the same speed (). At each time step, particle assumes the average direction of motion of its neighbors (within an interaction circle of radius ) distorted by the existence of an external noise of amplitude (). The simple update rules in the 2D case are given by
where is the average of the direction of motion of all the nearest-neighbors of the th particle (whose distance ), and
is a scalar noise uniformly distributed in the range. The update rules given by Eqs. 1 and 3 are known in the literature as angular noise Vicsek et al. (1995) and forward update Chaté et al. (2008), respectively. By choosing as the time unit and as the length unit, the only control parameters of the model are the noise amplitude , the speed and the density of particles , where is the volume of the 2D system.
These simple rules are enough to reproduce a fundamental aspect of collective behavior: the existence of a phase transition between a disordered state and an ordered phase (where the direction of motion is the same for all particles) as the noise intensity decreases. The average velocity of the system, defined as
is the appropriate order parameter to describe this phase transition Vicsek et al. (1995); Vicsek and Zafeiris (2012). In the disordered phase while for the ordered phase. A critical line separates the disordered from the ordered states. In order to ensure the independence from initial conditions, has to be averaged over time after it reaches a stationary regime. This value is called . The order of this phase transition is still a matter of controversy Chaté et al. (2008); Baglietto and Albano (2009). Until now the consensus seems to indicate that depending on how the noise is applied angular noise or vectorial noise the transition can be continuous or first-order like, respectively (for more details see Vicsek and Zafeiris (2012)).
ii.2 The Social Force Model
Contemporary to the VM, the Social Force Model (SFM) was proposed by Helbing and Molnar Helbing (1991); Helbing and Molnár (1995) to describe the behavior of pedestrians. Since then, the SFM has been widely studied and applied for different situations (see for example Helbing et al. (2000, 2001); Helbing and Johansson (2011)).
The SFM determines the direction of motion for each particle by taking into account three interactions: the "Desire Force", the "Social Force", and the "Granular Force", which are defined as
The "Desire Force" () represents the desire of SPP to march in a given direction; if we are modeling the evacuation of a crowd, the target of the desire will be the exit. This force involves the idea of a desired speed of motion , and it is given by
where is the particle mass, is the current velocity,
the unit vector pointing to the target direction andis the relaxation time of the particle velocity towards . In the present work, the desired speed has been taken as equal to the Vicsek-particle speed .
The "Social Force" () takes into account the fact that people like to move without bodily contact with other individuals. The "private space" wish is represented as a long range repulsive force based on the distance between the center of mass of the individual and its neighbor . The complete expression is the following:
where the constants and define the strength and range of the social force, is the normalized vector pointing from pedestrian to , and is the pedestrian diameter when one considers identical particle sizes.
The "Granular Force" () is considered when the pedestrians are in contact each other. It is a repulsive force inspired by granular interactions, includes compression and friction terms, and is expressed as
where is zero when and otherwise. The first term represents a compressive force, its strength given by the constant , which acts in the direction. The second term in Eq. 7 related to friction acts in the tangential direction (orthogonal to ), and it depends on the difference multiplied by the constant .
The interaction pedestrianwall is defined analogously by means of social () and granular forces (). If denotes the distance between the -pedestrian and the wall, and is the wall normal pointing to the particle, the ”Social force” is defined as
Similarly, denominating as the direction tangential (orthogonal to ), the "Granular force" is expressed as
For the force constants in the interactions between the particles and the walls we choose the same values as for the interparticle forces.
By considering all the forces described above, the equation of motion for pedestrian of mass is given by
ii.3 The combined model (SFM+VM)
Realistic evacuations in a non-panic situation have different behavior depending on geometry and average speed. However, and even in the case of similar boundary conditions, it is expected that one observes differences between evacuations in a school, hospital or airport. An important factor here is the existence of intrinsic fluctuations in the moving-interacting particles, such as children, passengers or patients. We propose to introduce these fluctuations including in the standard SFM the external noise parameter as in the Vicsek model.
The central idea is to take into account not only the social interactions described above but the influence of nearest-neighbors in the Vicsek-style, and to include noise as an external parameter, which modulates this interaction. We will refer to this new model as SFM+VM. In this way, in the SFM+VM the velocity of particle is given by
Here is the velocity of particle given by Eq.(2).
Iii Simulation Details
As was previously mentioned, we are going to compare all models under equal external conditions. An important point here is to match all relevant physical units in the models. Since and are common variables, we can extend their role as time and length units from the VM to all models studied. In order to match the simulations to real systems, we have assumed that one length-unit () is equivalent to one meter (in SI), and one second corresponds to one time step . After this assumption, it is possible to define the mass and force variables in Eq.(10) as kilograms and Newtons, respectively, as the SFM requires.
Monte Carlo simulations were performed in a system of self-propelled particles moving in corridor of size , with and . Periodic boundary conditions were applied in the horizontal direction at , so that circulation of particles occurred in a loop. A size of was then chosen to make sure that the fastest particles do not meet the stragglers.
To correlate pedestrians with particles, in the case of SFM+VM, we have considered particles with (in units of ) and 80 kg of mass. The characteristic parameters of SFM interactions (Eqs. 5–7) have been taken from previous works Helbing and Molnár (1995); Frank and Dorso (2011), specifically = 2000 N, = 0.08 m, kg s and kg/(m s). These values correspond to a typical crowd.
In every case studied, several tests have been performed to assure the reliability of the data, that are not shown here for the sake of space. We made sure that starting from different initial conditions of particle distribution in position and direction of motion, reached the same value in the stationary regime (). Also, the run-to-run fluctuations in the profile were not drastic. To determine the number of reasonable runs for the simulations, a first study was made observing how the average value of and its uncertainty varied according to how many runs were taken in the average. As a conclusion, we considered runs for each set of simulation parameters.
As was mentioned, the VM is a simple model that does not take into account short-range interactions such as excluded-volume or friction between particles and with walls. Moreover, taking into account that the aim of this work is to analyze the collective motion of individuals moving through a corridor, we introduce a series of variations on the VM. On the one hand, we relaxed the standard periodic boundary conditions (PBC) to a bouncing-back condition (BbC) in the y-direction. In this way, we simulate impenetrable walls at and , where particles rebound without losing energy.
On the other hand, we have introduced a desired-direction (DD) of motion, in such away that the direction of motion at time given by Eq.(1) is modified by the addition of a desired-angle as
with in our case, indicating that particles prefer to move to the end of the corridor. This variation of the VM allows to introduce the existence of a preferred direction of motion, in the same sense as it is introduced in the SFM.
Finally, particles were inserted randomly in the first 0.5 of the corridor with the intention to move towards the end of the corridor. The speed was considered in the range (m/s), consistent with normal evacuations in schools, hospitals, cinemas, etc.
For this reason, we have explored the range of (m) in order to represent the typical widths of corridors.
Iv Results and discussion
A full set of simulations was performed by taking into account the considerations mentioned above. In Figure 1 we show several snapshots of stationary configurations in the ordered phase for all models with the same external parameters ( and ) and different boundary conditions.
As can be seen, in the VM (Figure 1 (a–c)), stationary-ordered-states correspond to the existence of bands perpendicular to the direction of motion. This fact has been widely studied and reported in the literature (see Vicsek and Zafeiris (2012); Ginelli (2016), and reference therein). However, it is worth to mention that different boundary conditions seem to affect the local order within the band of particles (e.g., PBC in Figure 1(a) in comparison with BbBC in Figure 1(b)). Following with the qualitative analysis of the VM cases, the most ordered configuration corresponds to the VM+BbBC+DD, as it is expected. Here, the incorporation of a preferential direction of motion can be interpreted as an external field applied in the system promoting order. On the other hand, in both SFM+VM at low noise (Figure 1(d)) and SFM (Figure 1(e)), particles are ordered in a horizontal cluster configuration trying to keep a distance between each other and with the walls.
The similarity in behavior observed in snapshot configurations between low noise SFM+VM (Figure 1(d)) and SFM (Figure 1(e)) is broken when the noise increases. This is explicitly observed in Figure 2, where, as it is expected, disorder increases with noise .
To appreciate in detail the influence of Vicsek interactions on the spatial configuration in the SFM, we have studied the evolution of particle clusters for several noise strengths. For this purpose, we define the density profile in the x-direction as
and consequently can be interpreted as a histogram, with bins of width . In our case, we have fixed so that we have the chance to find many particles in the bin. The size of the bin is such that it is not too small to allow the cluster to be described as continuous (no empty bins in the middle) but also not so large that it cannot be appreciated how the size changes over time.
In this way, the width of the density profile at time () can be defined as the distance between the maximum and the minimum values of for which , properly normalized by considering the PBC applied in the -direction. With this idea, is directly associated with the extension of the cluster of particles as a function of time.
The obtained results show that at low noise the stationary cluster keeps its form and is constant in time. On the other hand, when the noise increases particles spread and therefore the cluster width grows with time. This effect is most relevant for . The dynamic dependence of (Figure 3 (b)) suggest a power-law behavior of the form
where has a strong dependence on noise. In fact, a least-squares fit of the data gives for and the SFM, for , and for . The behavior observed for is compatible with a diffusive-like spread of the particle front with the typical Einstein exponent . For higher noise it seems that the competition between random movement (given by the noise) and social-force interactions gives as a result a sub-diffusive behavior with . The behavior observed for is reminiscent of the freezing by heating effect observed in the SFM in the panic-regime Helbing et al. (2000, 2002). In this case, the existence of increasing fluctuations in the system, or nervousness of pedestrians, produces a blocking effect and even when they are in a disordered state particles can not move in the desired direction of motion.
Let us now turn to the question of an underlying non-equilibrium phase transition in the different models as a function of the noise strength by evaluating the stationary state
and its variance.
The dependence of both quantities ( and ) as a function of for fixed speed () and lattice size () can be observed in Figure 4. As a first comment, it should be noticed that the application of BbBC seems to move the VM transition (maximum in Fig. 4(b)) to higher values of in comparison with PBC. This is in agreement with the snapshots of Figure 1 (a-b); at a given noise () the PBC case is more disordered than the BbBC one. Even when in both cases the stripe geometry confines particle movement, the existence of impenetrable walls in the BbBC increases significantly the confinement effects, and therefore it is expected that VM-alignment should be more relevant than for PBC.
The application of a desired direction of motion (DD) in the VM has substantially different consequences. Unlike in the standard VM (with both PBC and BbBC), VM+DD prevents the existence of a disordered phase as it is expected. This appears reflected in the fact that , and is maximal when .
In the case of the SFM+VM the presence of an external noise clearly modifies the behavior of motion in comparison with SFM in a non-panic regime in a corridor (when trivially one expects ). Because in the SFM+VM repulsive interactions between particles make the formation of a condensed cluster more difficult, it is observed that for every . The existence of a maximum in (Fig. 4(b)) suggests to analyze the dependence of this behavior on external noise upon a variation of speed and system size . To check the reliability of the SFM+VM outcomes, we have performed the same analysis in VM cases, where the maximum is related to the existence of a phase transition.
Our results are shown in Fig. 5. As can be seen, in the VM (Fig. 5(a-b)) the peak of the susceptibility defined as becomes narrower and higher as both and system size are increased. This behavior is not observed in the case of VM+DD (Fig. 5(c)), as expected. In the case of the SFM+VM the –rounded– peak of as a function of increases in height as and the system size grow, however, it sharpens only very slowly. This is indicative of an underlying phase transition but to clearly establish its existence it would be necessary to study much larger system sizes at fixed density than is possible within the scope of the present work. However, it is noteworthy that although the effect of external noise in the model is less important than in the VM, it seems enough to break the symmetry imposed by SFM interactions.
Similar behavior is observed for the dependence of on speed (see Fig. 6). The maximum diminishes as the speed increases, and its position seems to move to for larger . As can also be seen in this figure, for a given speed the maximum values of are higher in the case of the VM+BbBC than the SFM+VM.
Finally, substantial information can be gleaned from level plots of for several values and noise values in the range (Fig. 7(a-c)). Because in these plots we are keeping and fixed, the vertical axis is an indirect representation of the density . From these plots it can be appreciated that although the previous analysis was presented for a given value of and , similar behavior is observed for and (Fig. 7). In this way, our conclusions can be extended to a wide range of densities and speeds within the non-panic regime.
V Summary and Conclusions
We have studied the role of interactions in the behavior of pedestrians moving in a corridor-like system. For this purpose, we have introduced a new model that we have called SFM+VM, as a combination of both the well-known Vicsek and Social Force models. To check its performance, we have started our analysis with the Vicsek model in a confined geometry with different boundary conditions applied. In particular, we have studied the effects of bouncing-back boundary conditions that reproduce the effects of walls, and the existence of a desired direction of motion -the end of the corridor. We have compared these results to those obtained with the SFM, and the new SFM+VM.
In the first place, particle configurations in the ordered-steady-state are qualitatively different between the models analyzed. While in the VM, particles move in a more or less compact band perpendicular to the direction of motion, in the SFM and the SFM+VM particles exhibit some horizontal stripes parallel to the direction of motion. This effect is a consequence of the repulsive interactions between particles and with the walls present in those two models (Figures 1 and 2). In order to compare both SFM and SFM+VM, we have analyzed the density-profiles in the direction of motion () and determined its width as a function of time. Our results indicate that the width in the SFM+VM case has a power-law behavior with a dynamical exponent , which depends on the external noise . In particular, we have determined that at and at (Figure 3). We have associated this change from an expected diffusive-like to a sub-diffusive behavior to the competition between VM-like interactions and social interactions. At high noise, fluctuations have a freezing by heating effect, that has been reported only in the panic-regime before Helbing et al. (2000, 2002). In our case, this effect appears in the system even at low-speed values, as a consequence of the introduction of the external noise.
In the second place, we have analyzed the order parameter , defined as the average velocity of the system, and its variance as function of external noise for the diverse models described above (Figures 45). In the VM, the application of different boundary conditions (periodic and bouncing-back) moves the critical value of the order-disorder phase transition to higher values of . In contrast, the existence of a desired direction of motion in the VM promotes the order even at high values of external noise, annihilating the phase transition as expected. In the SFM+VM, the existence of an external noise that modulates the Vicsek-like interactions brakes the SFM symmetry (Figure 6). As a consequence, both the order parameter and its variance are sensitive to this effect. Finally, we consider it important to remark that these outcomes can be observed in the whole range of densities and speeds studied, which encompass reasonable values of evacuations in a non-panic regime (Figure 7).
Based on these results, we can conclude that the SFM+VM is a successful model to describe the pedestrian motion along a corridor in a non-panic regime. This new model allows us to elucidate the role of the competition between social and alignment interactions, characteristics of the SFM and the VM, respectively. In the SFM+VM, alignment interactions are tuned by the external noise same as for the VM and this allows us to address questions on the existence of a non-equilibrium order-disorder transition controlled by this parameter. Our results are qualitatively compatible with the existence of such a transition, however, the approach to the thermodynamic limit seems to be very slow putting a final quantitative conclusion beyond the scope of this work.
Acknowledgements.This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de La Plata (Argentina), and Universidad Nacional de Quilmes (Argentina). Simulations were done on the cluster of Unidad de Cálculo, IFLYSIB. We also thank Martin-Luther-University Halle-Wittenberg and Alexander von Humboldt Foundation for financial support.
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