64,757 research outputs found
Mean-Field Limits Beyond Ordinary Differential Equations
16th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2016, Bertinoro, Italy, June 20-24, 2016, Advanced LecturesInternational audienceWe study the limiting behaviour of stochastic models of populations of interacting agents, as the number of agents goes to infinity. Classical mean-field results have established that this limiting behaviour is described by an ordinary differential equation (ODE) under two conditions: (1) that the dynamics is smooth; and (2) that the population is composed of a finite number of homogeneous sub-populations, each containing a large number of agents. This paper reviews recent work showing what happens if these conditions do not hold. In these cases, it is still possible to exhibit a limiting regime at the price of replacing the ODE by a more complex dynamical system. In the case of non-smooth or uncertain dynamics, the limiting regime is given by a differential inclusion. In the case of multiple population scales, the ODE is replaced by a stochastic hybrid automaton
Self-Consistent Projection Operator Theory in Nonlinear Quantum Optical Systems: A case study on Degenerate Optical Parametric Oscillators
Nonlinear quantum optical systems are of paramount relevance for modern
quantum technologies, as well as for the study of dissipative phase
transitions. Their nonlinear nature makes their theoretical study very
challenging and hence they have always served as great motivation to develop
new techniques for the analysis of open quantum systems. In this article we
apply the recently developed self-consistent projection operator theory to the
degenerate optical parametric oscillator to exemplify its general applicability
to quantum optical systems. We show that this theory provides an efficient
method to calculate the full quantum state of each mode with high degree of
accuracy, even at the critical point. It is equally successful in describing
both the stationary limit and the dynamics, including regions of the parameter
space where the numerical integration of the full problem is significantly less
efficient. We further develop a Gaussian approach consistent with our theory,
which yields sensibly better results than the previous Gaussian methods
developed for this system, most notably standard linearization techniques.Comment: Comments are welcom
The nature of spacetime singularities
Present knowledge about the nature of spacetime singularities in the context
of classical general relativity is surveyed. The status of the BKL picture of
cosmological singularities and its relevance to the cosmic censorship
hypothesis are discussed. It is shown how insights on cosmic censorship also
arise in connection with the idea of weak null singularities inside black
holes. Other topics covered include matter singularities and critical collapse.
Remarks are made on possible future directions in research on spacetime
singularities.Comment: Submitted to 100 Years of Relativity - Space-Time Structure: Einstein
and Beyond, A. Ashtekar (ed.
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