18,096 research outputs found
About Dynamical Systems Appearing in the Microscopic Traffic Modeling
Motivated by microscopic traffic modeling, we analyze dynamical systems which
have a piecewise linear concave dynamics not necessarily monotonic. We
introduce a deterministic Petri net extension where edges may have negative
weights. The dynamics of these Petri nets are well-defined and may be described
by a generalized matrix with a submatrix in the standard algebra with possibly
negative entries, and another submatrix in the minplus algebra. When the
dynamics is additively homogeneous, a generalized additive eigenvalue may be
introduced, and the ergodic theory may be used to define a growth rate under
additional technical assumptions. In the traffic example of two roads with one
junction, we compute explicitly the eigenvalue and we show, by numerical
simulations, that these two quantities (the additive eigenvalue and the growth
rate) are not equal, but are close to each other. With this result, we are able
to extend the well-studied notion of fundamental traffic diagram (the average
flow as a function of the car density on a road) to the case of two roads with
one junction and give a very simple analytic approximation of this diagram
where four phases appear with clear traffic interpretations. Simulations show
that the fundamental diagram shape obtained is also valid for systems with many
junctions. To simulate these systems, we have to compute their dynamics, which
are not quite simple. For building them in a modular way, we introduce
generalized parallel, series and feedback compositions of piecewise linear
concave dynamics.Comment: PDF 38 page
Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz
We present a general method for constructing integrable stochastic processes,
with two-step discrete time Floquet dynamics, from the transfer matrix
formalism. The models can be interpreted as a discrete time parallel update.
The method can be applied for both periodic and open boundary conditions. We
also show how the stationary distribution can be built as a matrix product
state. As an illustration we construct a parallel discrete time dynamics
associated with the R-matrix of the SSEP and of the ASEP, and provide the
associated stationary distributions in a matrix product form. We use this
general framework to introduce new integrable generalized exclusion processes,
where a fixed number of particles is allowed on each lattice site in opposition
to the (single particle) exclusion process models. They are constructed using
the fusion procedure of R-matrices (and K-matrices for open boundary
conditions) for the SSEP and ASEP. We develop a new method, that we named
"fused" matrix ansatz, to build explicitly the stationary distribution in a
matrix product form. We use this algebraic structure to compute physical
observables such as the correlation functions and the mean particle current.Comment: 33 pages, to appear in Nuclear Physics
Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
The foundations of the chaotic scattering theory for transport and
reaction-rate coefficients for classical many-body systems are considered here
in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is
employed to obtain an expression for the escape-rate for a phase space
trajectory to leave a finite open region of phase space for the first time.
This expression relates the escape rate to the difference between the sum of
the positive Lyapunov exponents and the K-S entropy for the fractal set of
trajectories which are trapped forever in the open region. This result is well
known for systems of a few degrees of freedom and is here extended to systems
of many degrees of freedom. The formalism is applied to smooth hyperbolic
systems, to cellular-automata lattice gases, and to hard sphere sytems. In the
latter case, the goemetric constructions of Sinai {\it et al} for billiard
systems are used to describe the relevant chaotic scattering phenomena. Some
applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file.
Figures are available on request from [email protected]
The Berger-Wang formula for the Markovian joint spectral radius
The Berger-Wang formula establishes equality between the joint and
generalized spectral radii of a set of matrices. For matrix products whose
multipliers are applied not arbitrarily but in accordance with some Markovian
law, there are also known analogs of the joint and generalized spectral radii.
However, the known proofs of the Berger-Wang formula hardly can be directly
applied in the case of Markovian products of matrices since they essentially
rely on the arbitrariness of appearance of different matrices in the related
matrix products. Nevertheless, as has been shown by X. Dai the Berger-Wang
formula is valid for the case of Markovian analogs of the joint and the
generalized spectral radii too, although the proof in this case heavily
exploits the more involved techniques of multiplicative ergodic theory. In the
paper we propose a matrix theory construction allowing to deduce the Markovian
analog of the Berger-Wang formula from the classical Berger-Wang formula.Comment: 13 pages, 29 bibliography references; minor corrections; accepted for
publication in Linear Algebra and its Application
Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues
We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality due to Muckenhoupt
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
New algorithms for computing of asymptotic expansions for stationary
distributions of nonlinearly perturbed semi-Markov processes are presented. The
algorithms are based on special techniques of sequential phase space reduction,
which can be applied to processes with asymptotically coupled and uncoupled
finite phase spaces.Comment: 83 page
Adaptive Resolution Simulation in Equilibrium and Beyond
In this paper, we investigate the equilibrium statistical properties of both
the force and potential interpolations of adaptive resolution simulation
(AdResS) under the theoretical framework of grand-canonical like AdResS
(GC-AdResS). The thermodynamic relations between the higher and lower
resolutions are derived by considering the absence of fundamental conservation
laws in mechanics for both branches of AdResS. In order to investigate the
applicability of AdResS method in studying the properties beyond the
equilibrium, we demonstrate the accuracy of AdResS in computing the dynamical
properties in two numerical examples: The velocity auto-correlation of pure
water and the conformational relaxation of alanine dipeptide dissolved in
water. Theoretical and technical open questions of the AdResS method are
discussed in the end of the paper
Erratum: Signal propagation in proteins and relation to equilibrium fluctuations (PLoS Computational Biology (2007) 3, 9, (e172) DOI: 10.1371/journal.pcbi.0030172))
Elastic network (EN) models have been widely used in recent years for describing protein dynamics, based on the premise that the motions naturally accessible to native structures are relevant to biological function. We posit that equilibrium motions also determine communication mechanisms inherent to the network architecture. To this end, we explore the stochastics of a discrete-time, discrete-state Markov process of information transfer across the network of residues. We measure the communication abilities of residue pairs in terms of hit and commute times, i.e., the number of steps it takes on an average to send and receive signals. Functionally active residues are found to possess enhanced communication propensities, evidenced by their short hit times. Furthermore, secondary structural elements emerge as efficient mediators of communication. The present findings provide us with insights on the topological basis of communication in proteins and design principles for efficient signal transduction. While hit/commute times are information-theoretic concepts, a central contribution of this work is to rigorously show that they have physical origins directly relevant to the equilibrium fluctuations of residues predicted by EN models
Developments in the theory of randomized shortest paths with a comparison of graph node distances
There have lately been several suggestions for parametrized distances on a
graph that generalize the shortest path distance and the commute time or
resistance distance. The need for developing such distances has risen from the
observation that the above-mentioned common distances in many situations fail
to take into account the global structure of the graph. In this article, we
develop the theory of one family of graph node distances, known as the
randomized shortest path dissimilarity, which has its foundation in statistical
physics. We show that the randomized shortest path dissimilarity can be easily
computed in closed form for all pairs of nodes of a graph. Moreover, we come up
with a new definition of a distance measure that we call the free energy
distance. The free energy distance can be seen as an upgrade of the randomized
shortest path dissimilarity as it defines a metric, in addition to which it
satisfies the graph-geodetic property. The derivation and computation of the
free energy distance are also straightforward. We then make a comparison
between a set of generalized distances that interpolate between the shortest
path distance and the commute time, or resistance distance. This comparison
focuses on the applicability of the distances in graph node clustering and
classification. The comparison, in general, shows that the parametrized
distances perform well in the tasks. In particular, we see that the results
obtained with the free energy distance are among the best in all the
experiments.Comment: 30 pages, 4 figures, 3 table
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