387 research outputs found
Feynman integrals for a class of exponentially growing potentials
We construct the Feynman integrands for a class of exponentially growing
time-dependent potentials as white noise functionals. We show that they solve
the Schroedinger equation. The Morse potential is considered as a special case
The truncated discrete moment problem from one to infinite dimensions
The discrete truncated moment problem considers the question whether given a discrete subsets and a sequence of real numbers one can find a measure supported on whose (power) moments are exactly these numbers. The truncated moment is a challenging problem. We derive a minimal set of necessary and sufficient conditions. This simple problem is surprisingly hard and not treatable with known techniques. Applications to the truncated moment problem for point processes, the so-called relizability or representability problem are given. This is a joint work with M. Infusino, J. Lebowitz and E. Speer
Marked Gibbs measures via cluster expansion
We give a sufficiently detailed account on the construction of marked Gibbs
measures in the high temperature and low fugacity regime. This is proved for a
wide class of underlying spaces and potentials such that stability and
integrability conditions are satisfied. That is, for state space we take a
locally compact separable metric space and a separable metric space for
the mark space. This framework allowed us to cover several models of classical
and quantum statistical physics. Furthermore, we also show how to extend the
construction for more general spaces as e.g., separable standard Borel spaces.
The construction of the marked Gibbs measures is based on the method of cluster
expansion.Comment: 51 page
Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems
In this paper we provide a connection between the geometrical properties of a
chaotic dynamical system and the distribution of extreme values. We show that
the extremes of so-called physical observables are distributed according to the
classical generalised Pareto distribution and derive explicit expressions for
the scaling and the shape parameter. In particular, we derive that the shape
parameter does not depend on the chosen observables, but only on the partial
dimensions of the invariant measure on the stable, unstable, and neutral
manifolds. The shape parameter is negative and is close to zero when
high-dimensional systems are considered. This result agrees with what was
derived recently using the generalized extreme value approach. Combining the
results obtained using such physical observables and the properties of the
extremes of distance observables, it is possible to derive estimates of the
partial dimensions of the attractor along the stable and the unstable
directions of the flow. Moreover, by writing the shape parameter in terms of
moments of the extremes of the considered observable and by using linear
response theory, we relate the sensitivity to perturbations of the shape
parameter to the sensitivity of the moments, of the partial dimensions, and of
the Kaplan-Yorke dimension of the attractor. Preliminary numerical
investigations provide encouraging results on the applicability of the theory
presented here. The results presented here do not apply for all combinations of
Axiom A systems and observables, but the breakdown seems to be related to very
special geometrical configurations.Comment: 16 pages, 3 Figure
Translation invariant realizability problem on the d-dimensional lattice: an explicit construction
We consider a particular instance of the truncated realizability problem on the d−dimensional lattice. Namely, given two functions ρ1(i) and ρ2(i,j) non-negative and symmetric on Zd, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any d ≥ 2 when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds
Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables
We consider a smooth one-parameter family
of diffeomorphisms with compact transitive Axiom A attractors , denoting
by the SRB measure of .
Our first result is that for any function in the Sobolev space
, with and , the map is -H\"older continuous for
all . This applies to
(for all ) for and
smooth and the
Heaviside function, if is not a critical value of . Our second result says that
for any such function so that in addition
the intersection of with the support of is
foliated by ``admissible stable leaves'' of , the map
is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.)
Obtaining linear response or fractional response for such observables is motivated
by extreme-value theory
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A novel scaling indicator of early warning signals helps anticipate tropical cyclones
Tipping events in dynamical systems have been studied in many contexts, often modelled by the decay of critical modes, system states which are tending towards bifurcation, characterised by increased return times to stable equilibria. Temporal scaling properties of time series data can be used to detect the presence of a critical mode by estimating the decay rate, and indicators of changes in these properties may therefore be used to provide an early warning signal (EWS) for an impending tipping event. The lag-1 autocorrelation function (ACF(1)) indicator and the detrended fluctuation analysis (DFA) indicator have previously been used in such a way; in this paper we introduce a novel scaling indicator based on the decay rate of the power spectrum (PS). We compare the ACF(1), DFA- and PS-indicators using artificial data; data from a model which includes a bifurcation point; and sea-level pressure data along the paths of 14 tropical cyclones. By using the PS-indicator with such data, we show that the new indicator may be used to provide an EWS in a context where the ACF(1)- and DFA-indicators fail
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