549 research outputs found
Application of thermodynamics to driven systems
Application of thermodynamics to driven systems is discussed. As particular
examples, simple traffic flow models are considered. On a microscopic level,
traffic flow is described by Bando's optimal velocity model in terms of
accelerating and decelerating forces. It allows to introduce kinetic,
potential, as well as total energy, which is the internal energy of the car
system in view of thermodynamics. The latter is not conserved, although it has
certain value in any of two possible stationary states corresponding either to
fixed point or to limit cycle in the space of headways and velocities. On a
mesoscopic level of description, the size n of car cluster is considered as a
stochastic variable in master equation. Here n=0 corresponds to the fixed-point
solution of the microscopic model, whereas the limit cycle is represented by
coexistence of a car cluster with n>0 and free flow phase. The detailed balance
holds in a stationary state just like in equilibrium liquid-gas system. It
allows to define free energy of the car system and chemical potentials of the
coexisting phases, as well as a relaxation to a local or global free energy
minimum. In this sense the behaviour of traffic flow can be described by
equilibrium thermodynamics. We find, however, that the chemical potential of
the cluster phase of traffic flow depends on an outer parameter - the density
of cars in the free-flow phase. It allows to distinguish between the traffic
flow as a driven system and purely equilibrium systems.Comment: 9 pages, 6 figures. Eur. Phys. J. B (2007) to be publishe
How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?
An analogy of the Fokker-Planck equation (FPE) with the Schr\"odinger
equation allows us to use quantum mechanics technique to find the analytical
solution of the FPE in a number of cases. However, previous studies have been
limited to the Schr\"odinger potential with a discrete eigenvalue spectrum.
Here, we will show how this approach can be also applied to a mixed eigenvalue
spectrum with bounded and free states. We solve the FPE with boundaries located
at x=\pm L/2 and take the limit L\rightarrow\infty, considering the examples
with constant Schr\"{o}dinger potential and with P\"{o}schl-Teller potential.
An oversimplified approach was proposed earlier by M.T. Araujo and E. Drigo
Filho. A detailed investigation of the two examples shows that the correct
solution, obtained in this paper, is consistent with the expected Fokker-Planck
dynamics.Comment: 13 pages, 5 figure
Space shuttle: Basic supersonic force data for a Grumman delta wing orbiter configuration ROS-NB1
Supersonic force data for scale model of space shuttle delta wing orbite
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 escape from the cluster whose rate depends on the cluster size directly.
The latter is justified using the available experimental data for the
correlation properties of the synchronized mode. We write the appropriate
master equation converted then into the Fokker-Plank equation for the cluster
distribution function and analyze the formation of the critical car cluster due
to the climb over a certain potential barrier. The further cluster growth
irreversibly gives rise to the jam formation. Numerical estimates of the
obtained characteristics and the experimental data of the traffic breakdown are
compared. In particular, we draw a conclusion that the characteristic intrinsic
time scale of the breakdown phenomenon should be about one minute and explain
the case why the traffic volume interval inside which traffic breakdown is
observed is sufficiently wide.Comment: RevTeX 4, 14 pages, 10 figure
Zero range model of traffic flow
A multi--cluster model of traffic flow is studied, in which the motion of
cars is described by a stochastic master equation. Assuming that the escape
rate from a cluster depends only on the cluster size, the dynamics of the model
is directly mapped to the mathematically well-studied zero-range process.
Knowledge of the asymptotic behaviour of the transition rates for large
clusters allows us to apply an established criterion for phase separation in
one-dimensional driven systems. The distribution over cluster sizes in our
zero-range model is given by a one--step master equation in one dimension. It
provides an approximate mean--field dynamics, which, however, leads to the
exact stationary state. Based on this equation, we have calculated the critical
density at which phase separation takes place. We have shown that within a
certain range of densities above the critical value a metastable homogeneous
state exists before coarsening sets in. Within this approach we have estimated
the critical cluster size and the mean nucleation time for a condensate in a
large system. The metastablity in the zero-range process is reflected in a
metastable branch of the fundamental flux--density diagram of traffic flow. Our
work thus provides a possible analytical description of traffic jam formation
as well as important insight into condensation in the zero-range process.Comment: 10 pages, 13 figures, small changes are made according to finally
accepted version for publication in Phys. Rev.
Equilibrium distributions in thermodynamical traffic gas
We derive the exact formula for thermal-equilibrium spacing distribution of
one-dimensional particle gas with repulsive potential V(r)=r^(-a) (a>0)
depending on the distance r between the neighboring particles. The calculated
distribution (for a=1) is successfully compared with the highway-traffic
clearance distributions, which provides a detailed view of changes in
microscopical structure of traffic sample depending on traffic density. In
addition to that, the observed correspondence is a strong support of studies
applying the equilibrium statistical physics to traffic modelling.Comment: 5 pages, 6 figures, changed content, added reference
Long-lived states of oscillator chain with dynamical traps
A simple model of oscillator chain with dynamical traps and additive white
noise is considered. Its dynamics was studied numerically. As demonstrated,
when the trap effect is pronounced nonequilibrium phase transitions of a new
type arise. Locally they manifest themselves via distortion of the particle
arrangement symmetry. Depending on the system parameters the particle
arrangement is characterized by the corresponding distributions taking either a
bimodal form, or twoscale one, or unimodal onescale form which, however,
deviates substantially from the Gaussian distribution. The individual particle
velocities exhibit also a number of anomalies, in particular, their
distribution can be extremely wide or take a quasi-cusp form. A large number of
different cooperative structures and superstructures made of these formations
are found in the visualized time patterns. Their evolution is, in some sense,
independent of the individual particle dynamics, enabling us to regard them as
dynamical phases.Comment: 8 pages, 5 figurs, TeX style of European Physical Journa
Spring-block model for a single-lane highway traffic
A simple one-dimensional spring-block chain with asymmetric interactions is
considered to model an idealized single-lane highway traffic. The main elements
of the system are blocks (modeling cars), springs with unidirectional
interactions (modeling distance keeping interactions between neighbors), static
and kinetic friction (modeling inertia of drivers and cars) and spatiotemporal
disorder in the values of these friction forces (modeling differences in the
driving attitudes). The traveling chain of cars correspond to the dragged
spring-block system. Our statistical analysis for the spring-block chain
predicts a non-trivial and rich complex behavior. As a function of the disorder
level in the system a dynamic phase-transition is observed. For low disorder
levels uncorrelated slidings of blocks are revealed while for high disorder
levels correlated avalanches dominates.Comment: 6 pages, 7 figure
Relationship between a Non-Markovian Process and Fokker-Planck Equation
We demonstrate the equivalence of a non-Markovian evolution equation with a linear memory-coupling and a Fokker-Planck equation (FPE). In case the feedback term offers a direct and permanent coupling of the current probability density to an initial distribution, the corresponding FPE offers a non-trivial drift term depending itself on the diffusion parameter. As the consequence the deterministic part of the underlying Langevin equation is likewise determined by the noise strength of the stochastic part. This memory induced stochastic behavior is discussed for different, but representative initial distributions. The analytical calculations are supported by numerical results. © 2006 Elsevier B.V. All rights reserved.The authors (S.T. and K.Z.) acknowledge support by the DFG (SFB 418) as well as by DAAD (S. Tatur)
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