Noise-Induced Phase Transitions

Comment on “How skew distributions emerge in evolving systems” by Choi M. Y. et al.

Power-law distributions and other skew distributions, observed in various models and real systems, are considered. As an example, critical exponents determined from highly accurate experimental data very close to the λ-transition point in liquid helium are discussed in some detail. A model, describing evolving systems with increasing number of elements, is considered to study the distribution over element sizes. Stationary power-law distributions are found. Certain non-stationary skew distributions are obtained and analyzed, based on exact solutions. Validerad; 2010; 20100908 (weber)

Scaling behavior of an airplane-boarding model

An airplane-boarding model, introduced earlier by Frette and Hemmer [Phys. Rev. E 85, 011130 (2012)], is studied with the aim of determining precisely its asymptotic power-law scaling behavior for a large number of passengers $N$. Based on Monte Carlo simulation data for very large system sizes up to $N={2}^{16}=65\phantom{\rule{0.16em}{0ex}}536$, we have analyzed numerically the scaling behavior of the mean boarding time $\ensuremath{\langle}{t}_{b}\ensuremath{\rangle}$ and other related quantities. In analogy with critical phenomena, we have used appropriate scaling Ans\"atze, which include the leading term as some power of $N$ (e.g., $\ensuremath{\propto}$${N}^{\ensuremath{\alpha}}$ for …

The Master Equation

The Langevin Equation

Lorentzian-geometry-based analysis of airplane boarding policies highlights "slow passengers first" as better.

We study airplane boarding in the limit of large number of passengers using geometric optics in a Lorentzian metric. The airplane boarding problem is naturally embedded in a 1+1 dimensional space-time with a flat Lorentzian metric. The duration of the boarding process can be calculated based on a representation of the one-dimensional queue of passengers attempting to reach their seats, into a two-dimensional space-time diagram. The ability of a passenger to delay other passengers depends on their queue positions and row designations. This is equivalent to the causal relationship between two events in space-time, whereas two passengers are time-like separated if one is blocking the other, an…

Nucleation in Supersaturated Vapors

Metastability of Traffic Flow in Zero-Range Model

The development of traffic jams in vehicular flow is an everyday example of the occurence of phase separation in low-dimensional driven systems, a topic which has attracted much recent interest [1–4]. In [5] the existence of phase separation is related to the size-dependence of domain currents and a quantitative criterion is obtained by considering the zero-range process (ZRP) as a generic model for domain dynamics. We use zero-range picture to study the phase separation in traffic flow in the spirit of the probabilistic (master equation) description of transportation [6]. Significantly, we find [7] that prior to condensation studied in previous works [8, 9] the system can exist in a homoge…

Probabilistic description of traffic flow

Abstract A stochastic description of traffic flow, called probabilistic traffic flow theory, is developed. The general master equation is applied to relatively simple models to describe the formation and dissolution of traffic congestions. Our approach is mainly based on spatially homogeneous systems like periodically closed circular rings without on- and off-ramps. We consider a stochastic one-step process of growth or shrinkage of a car cluster (jam). As generalization we discuss the coexistence of several car clusters of different sizes. The basic problem is to find a physically motivated ansatz for the transition rates of the attachment and detachment of individual cars to a car cluster…

One-Dimensional Diffusion

Probabilistic description of traffic breakdowns

We analyze the characteristic features of traffic breakdown. To describe this phenomenon we apply to the probabilistic model regarding the jam emergence as the formation of a large car cluster on highway. In these terms the breakdown occurs through the formation of a certain critical nucleus in the metastable vehicle flow, which enables us to confine ourselves to one cluster model. We assume that, first, the growth of the car cluster is governed by attachment of cars to the cluster whose rate is mainly determined by the mean headway distance between the car in the vehicle flow and, may be, also by the headway distance in the cluster. Second, the cluster dissolution is determined by the car …

The Ornstein-Uhlenbeck Process

From Random Walker to Vehicular Traffic: Motion on a Circle

Driving of cars on a highway is a complex process which can be described by deterministic and stochastic forces. It leads to equations of motion with asymmetric interaction and dissipation as well as to new energy flow law already presented at previous TRAFFIC AND GRANULAR FLOW meetings. Here we consider a model, where motion of an asymmetric random walker on a ring with periodic boundary conditions takes place. It is related to driven systems with active particles, energy input and depot. This simple model can be further developed towards more complicated ones, describing vehicular or pedestrian traffic. Three particular cases are considered, starting with discrete coordinate and time, the…

Models for highway traffic and their connections to thermodynamics

Models for highway traffic are studied by numerical simulations. Of special interest is the spontaneous formation of traffic jams. In a thermodynamic system the traffic jam would correspond to the dense phase (liquid) and the free flowing traffic would correspond to the gas phase. Both phases depending on the density of cars can be present at the same time. A model for a single lane circular road has been studied. The model is called the optimal velocity model (OVM) and was developed by Bando, Sugiyama, et al. We propose here a reformulation of the OVM into a description in terms of potential energy functions forming a kind of Hamiltonian for the system. This will however not be a globally …

Longitudinal and Transverse Correlation Functions in the  4 Model below and near the Critical Point

We have extended our method of grouping Feynman diagrams (GFD theory) to study the transverse and longitudinal correlation functions G⊥(k) and G‖(k) in φ model below the critical point (T < Tc) in the presence of an infinitesimal external field. Our method allows a qualitative analysis without cutting the perturbation series. The long-wave limit k → 0 has been studied at T < Tc, showing that G⊥(k) a k−λ⊥ and G‖(k) b k−λ‖ with exponents d/2 < λ⊥ < 2 and λ‖ = 2λ⊥−d are the physical solutions of our equations at the spatial dimensionality 2 < d < 4, which coincides with the asymptotic solution at T → Tc as well as with a nonperturbative renormalization group (RG) analysis provided in our paper…

Scaling Regimes and the Singularity of Specific Heat in the 3D Ising Model

AbstractThe singularity of specific heat CV of the three-dimensional Ising model is studied based on Monte Carlo data for lattice sizes L≤1536. Fits of two data sets, one corresponding to certain value of the Binder cumulant and the other — to the maximum of CV, provide consistent values of C0 in the ansatz CV(L)=C0+ALα/ν at large L, if α/ν=0.196(6). However, a direct estimation from our data suggests that α/ν, most probably, has a smaller value (e.g., α/ν= 0.113(30)). Thus, the conventional power-law scaling ansatz can be questioned because of this inconsistency. We have found that the data are well described by certain logarithmic ansatz.

Correlation Functions and Finite–Size Effects in Granular Media

A model is considered, where the local order parameter is an n–component vector. This model allows us to calculate correlation functions, describing the correlations between local order parameter at different spatial coordinates. The longitudinal and transverse Fourier–transformed two–point correlation functions \(G_{\parallel }(\mathbf{k})\) and \(G_{\perp }(\mathbf{k})\) in presence of an external field h are considered in some detail. In the thermodynamic limit, these correlation functions exhibit the so-called Goldstone mode singularities below certain critical temperature at an infinitesimal external field \(h = +0\). The actual model can be applied to granular media, in which case it …

Dynamics and Thermodynamics of Traffic Flow

Application of thermodynamics to traffic flow is discussed. On a microscopic level, traffic flow is described by Bando’s optimal velocity model in terms of accelerating and decelerating forces. It allows us to introduce kinetic, potential, as well as a total energy, which is the internal energy of the car system in view of thermodynamics. The total energy is however not conserved, although it has a certain value in any of the two possible stationary states corresponding either to a fixed point or to a limit cycle solution in the space of headways and velocities.

Parallelization of the Wolff single-cluster algorithm.

A parallel [open multiprocessing (OpenMP)] implementation of the Wolff single-cluster algorithm has been developed and tested for the three-dimensional (3D) Ising model. The developed procedure is generalizable to other lattice spin models and its effectiveness depends on the specific application at hand. The applicability of the developed methodology is discussed in the context of the applications, where a sophisticated shuffling scheme is used to generate pseudorandom numbers of high quality, and an iterative method is applied to find the critical temperature of the 3D Ising model with a great accuracy. For the lattice with linear size L=1024, we have reached the speedup about 1.79 times …

Vehicular Motion and Traffic Breakdown: Evaluation of Energy Balance

Microscopic traffic models based on follow–the–leader behaviour are strongly asymmetrically interacting many–particle systems. The well–known Bando’s optimal velocity model includes the fact that (firstly) the driver is always looking forward interacting with the lead vehicle and (secondly) the car travels on the road always with friction. Due to these realistic assumptions the moving car needs petrol for the engine to compensate dissipation by rolling friction. We investigate the flux of mechanical energy to evaluate the energy balance out of the given nonlinear dynamical system of vehicular particles. In order to understand the traffic breakdown as transition from free flow to congested t…

Air Traffic, Boarding and Scaling Exponents

The air traffic is a very important part of the global transportation network. In distinction from vehicular traffic, the boarding of an airplane is a significant part of the whole transportation process. Here we study an airplane boarding model, introduced in 2012 by Frette and Hemmer, with the aim to determine precisely the asymptotic power–law scaling behavior of the mean boarding time 〈t b 〉 and other related quantities for large number of passengers N. Our analysis is based on an exact enumeration for small system sizes N ≤ 14 and Monte Carlo simulation data for very large system sizes up to \(N = 2^{16} = 65,536\). It shows that the asymptotic power–law scaling 〈t b 〉 ∝ N α holds with…

Applications to Traffic Breakdown on Highways

During the last years researches into properties of vehicle ensembles on highways form a new branch of physics, called physics of traffic flow. On macroscopic scales the vehicle ensembles exhibit a wide class of phenomena like phase separation and phase transformations widely met in physical systems. Due to the steadily increasing traffic volume in cities and on highways, the mathematical modelling of these phenomena has attracted a great interest. Particularly, the topic of car—following has become of increased importance in traffic engineering and safety research.

Monte Carlo estimation of transverse and longitudinal correlation functions in the model

Abstract Monte Carlo simulations of the three-dimensional O ( 4 ) model in the ordered phase are performed to study the Goldstone mode effects. Our data show a distinct scaling region, where the Fourier-transformed transverse correlation function behaves as ∝ k − λ ⊥ with λ ⊥ 2 ( λ ≃ 1.95 ), in disagreement with the standard theoretical prediction λ ⊥ = 2 .

Many-Particle Systems

Bounded Drift-Diffusion Motion

The Fokker-Planck Equation