253 research outputs found
On stochasticity in nearly-elastic systems
Nearly-elastic model systems with one or two degrees of freedom are
considered: the system is undergoing a small loss of energy in each collision
with the "wall". We show that instabilities in this purely deterministic system
lead to stochasticity of its long-time behavior. Various ways to give a
rigorous meaning to the last statement are considered. All of them, if
applicable, lead to the same stochasticity which is described explicitly. So
that the stochasticity of the long-time behavior is an intrinsic property of
the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic
Convergence of invariant densities in the small-noise limit
This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction
Brownian Simulations and Uni-Directional Flux in Diffusion
Brownian dynamics simulations require the connection of a small discrete
simulation volume to large baths that are maintained at fixed concentrations
and voltages. The continuum baths are connected to the simulation through
interfaces, located in the baths sufficiently far from the channel. Average
boundary concentrations have to be maintained at their values in the baths by
injecting and removing particles at the interfaces. The particles injected into
the simulation volume represent a unidirectional diffusion flux, while the
outgoing particles represent the unidirectional flux in the opposite direction.
The classical diffusion equation defines net diffusion flux, but not
unidirectional fluxes. The stochastic formulation of classical diffusion in
terms of the Wiener process leads to a Wiener path integral, which can split
the net flux into unidirectional fluxes. These unidirectional fluxes are
infinite, though the net flux is finite and agrees with classical theory. We
find that the infinite unidirectional flux is an artifact caused by replacing
the Langevin dynamics with its Smoluchowski approximation, which is classical
diffusion. The Smoluchowski approximation fails on time scales shorter than the
relaxation time of the Langevin equation. We find the unidirectional
flux (source strength) needed to maintain average boundary concentrations in a
manner consistent with the physics of Brownian particles. This unidirectional
flux is proportional to the concentration and inversely proportional to
to leading order. We develop a BD simulation that maintains
fixed average boundary concentrations in a manner consistent with the actual
physics of the interface and without creating spurious boundary layers
A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds
We demonstrate that any self-adjoint coupling in a quantum graph vertex can
be approximated by a family of magnetic Schroedinger operators on a tubular
network built over the graph. If such a manifold has a boundary, Neumann
conditions are imposed at it. The procedure involves a local change of graph
topology in the vicinity of the vertex; the approximation scheme constructed on
the graph is subsequently `lifted' to the manifold. For the corresponding
operator a norm-resolvent convergence is proved, with the natural
identification map, as the tube diameters tend to zero.Comment: 19 pages, one figure; introduction amended and some references added,
to appear in CM
The Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension
We extend the previously developed weak noise scheme, applied to the noisy
Burgers equation in 1D, to the Kardar-Parisi-Zhang equation for a growing
interface in arbitrary dimensions. By means of the Cole-Hopf transformation we
show that the growth morphology can be interpreted in terms of dynamically
evolving textures of localized growth modes with superimposed diffusive modes.
In the Cole-Hopf representation the growth modes are static solutions to the
diffusion equation and the nonlinear Schroedinger equation, subsequently
boosted to finite velocity by a Galilei transformation. We discuss the dynamics
of the pattern formation and, briefly, the superimposed linear modes.
Implementing the stochastic interpretation we discuss kinetic transitions and
in particular the properties in the pair mode or dipole sector. We find the
Hurst exponent H=(3-d)/(4-d) for the random walk of growth modes in the dipole
sector. Finally, applying Derrick's theorem based on constrained minimization
we show that the upper critical dimension is d=4 in the sense that growth modes
cease to exist above this dimension.Comment: 27 pages, 19 eps figs, revte
Systemic Risk and Default Clustering for Large Financial Systems
As it is known in the finance risk and macroeconomics literature,
risk-sharing in large portfolios may increase the probability of creation of
default clusters and of systemic risk. We review recent developments on
mathematical and computational tools for the quantification of such phenomena.
Limiting analysis such as law of large numbers and central limit theorems allow
to approximate the distribution in large systems and study quantities such as
the loss distribution in large portfolios. Large deviations analysis allow us
to study the tail of the loss distribution and to identify pathways to default
clustering. Sensitivity analysis allows to understand the most likely ways in
which different effects, such as contagion and systematic risks, combine to
lead to large default rates. Such results could give useful insights into how
to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P.
Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer
Proceedings in Mathematics and Statistics, Vol. 110 2015
On occurrence of spectral edges for periodic operators inside the Brillouin zone
The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio
Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems
We introduce a new method, allowing to describe slowly time-dependent
Langevin equations through the behaviour of individual paths. This approach
yields considerably more information than the computation of the probability
density. The main idea is to show that for sufficiently small noise intensity
and slow time dependence, the vast majority of paths remain in small space-time
sets, typically in the neighbourhood of potential wells. The size of these sets
often has a power-law dependence on the small parameters, with universal
exponents. The overall probability of exceptional paths is exponentially small,
with an exponent also showing power-law behaviour. The results cover time spans
up to the maximal Kramers time of the system. We apply our method to three
phenomena characteristic for bistable systems: stochastic resonance, dynamical
hysteresis and bifurcation delay, where it yields precise bounds on transition
probabilities, and the distribution of hysteresis areas and first-exit times.
We also discuss the effect of coloured noise.Comment: 37 pages, 11 figure
Fixation, transient landscape and diffusion's dilemma in stochastic evolutionary game dynamics
Agent-based stochastic models for finite populations have recently received
much attention in the game theory of evolutionary dynamics. Both the ultimate
fixation and the pre-fixation transient behavior are important to a full
understanding of the dynamics. In this paper, we study the transient dynamics
of the well-mixed Moran process through constructing a landscape function. It
is shown that the landscape playing a central theoretical "device" that
integrates several lines of inquiries: the stable behavior of the replicator
dynamics, the long-time fixation, and continuous diffusion approximation
associated with asymptotically large population. Several issues relating to the
transient dynamics are discussed: (i) multiple time scales phenomenon
associated with intra- and inter-attractoral dynamics; (ii) discontinuous
transition in stochastically stationary process akin to Maxwell construction in
equilibrium statistical physics; and (iii) the dilemma diffusion approximation
facing as a continuous approximation of the discrete evolutionary dynamics. It
is found that rare events with exponentially small probabilities, corresponding
to the uphill movements and barrier crossing in the landscape with multiple
wells that are made possible by strong nonlinear dynamics, plays an important
role in understanding the origin of the complexity in evolutionary, nonlinear
biological systems.Comment: 34 pages, 4 figure
Elementary derivation of Spitzer's asymptotic law for Brownian windings and some of its physical applications
A simple derivation of Spitzer'z asymptotic law for Brownian windings
[Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations
>.These include the cases of planar Brownian walks interacting with a single
puncture and Brownian walks on a single truncated cone with variable conical
angle interacting with the truncated conical tip.Such situations are typical in
the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They
also have some applications in polymer physic
- …