We present an analytical and numerical study of the velocity distribution function of self gravitating collisionless particles, which include dark matter and star clusters. We show that the velocity distribution derived through the Eddington's formula is identical to the analytical one derived directly from the generalized entropy of non-extensive statistical mechanics. This implies that self gravitating collisionless structures are to be described by non-extensive thermo-statistics. We identify a connection between the density slope of dark matter structures, γ, from ρ ∼ r −γ , and the entropic index, q, from the generalized entropy, S q . Our numerical result confirms the analytical findings of earlier studies and clarifies which is the correct connection between the density slope and the entropic index. We use this result to conclude that from a fundamental statistical mechanics point of view the central density slope of self gravitating collisionless dark matter structures is not constrained, and even cored dark matter structures are allowed with γ = 0. We find that the outer density slope is bounded by γ = 10/3.
This paper generalizes the recently proposed approaches for calculating the derivative couplings between adiabatic states in density-functional theory (DFT) based on a Slater transition-state density to transitions such as singlet-singlet excitations, where a single-determinant ansatz is insufficient. The proposed approach is based on restricted open-shell Frank et al. [J. Chem. Phys. 108, 4060 (1998)] theory used to describe a spin-adapted Slater transition state. To treat the dependence of electron-electron interactions on the nuclear positions, variational linear-response density-functional perturbation theory is generalized to reference states with an orbital-dependent Kohn-Sham Hamiltonian and nontrivial occupation patterns. The methods proposed in this paper are not limited to the calculation of derivative coupling vectors, but can also be used for the calculation of other transition matrix elements. Moreover, they can be used to calculate the linear response of open-shell systems to arbitrary external perturbations in DFT.
In the present paper we consider the motion of a very heavy tracer particle in a medium of a very dense, non-interacting Bose gas. We prove that, in a certain mean-field limit, the tracer particle will be decelerated and come to rest somewhere in the medium. Friction is caused by emission of Cerenkov radiation of gapless modes into the gas. Mathematically, a system of semilinear integro-differential equations, introduced in [FSSG10], describing a tracer particle in a dispersive medium is investigated, and decay properties of the solution are proven. This work is an extension of [FGS10]; it is an extension because no weak coupling limit for the interaction between tracer particle and medium is assumed. The technical methods used are dispersive estimates and a contraction principle. * danegli@itp.phys.ethz.ch † zhougang@itp.phys.ethz.ch 1 2 −δ dr .(2.1)By numerical simulation we find that there exists a constant δ * ≃ 0.66 such that Ω(δ * ) = 1.Moreover, for any constant δ < δ * , we have Ω(δ) < 1.For the system of equations (1.4) we prove the following main theorem,
Abstract. We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.
In memory of V.S. Buslaev, a scientist and a friend Will appear in St.Petersburg Math Journal (issue dedicated to V.S. Buslaev)" AbstractWe investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic aggregation of the organisms. Our results are confirmed by numerical simulations and the formula we derive coincides with the formula of Herrero and Velázquez for specially constructed solutions. IntroductionIn this paper we analyze the aggregation dynamics in the (reduced) Keller-Segel model of chemotaxis. Chemotaxis is the directed movement of organisms in response to the concentration gradient of an external chemical signal and is common in biology. The chemical signals can come from external sources or they can be secreted by the organisms themselves.Chemotaxis is believed to underly many social activities of micro-organisms, e.g. social motility, fruiting body development, quorum sensing and biofilm formation. A classical example is the dynamics and the aggregation of Escherichia coli colonies under starvation conditions [17]. Another example is the Dictyostelium amoeba , where single cell bacterivores, when challenged by adverse conditions, form multicellular structures of ∼ 10 5 cells [15,23]. Also, endothelial cells of humans react to vascular endothelial growth factor to form blood vessels through aggregation [22].Consider organisms moving and interacting in a domain Ω ⊆ R d , d = 1, 2 or 3. Assuming that the organism population is large and the individuals are small relative to the domain Ω, Keller and Segel derived a system of reaction-diffusion equations governing the organism density ρ : Ω × R + → R + and chemical concentration c : Ω × R + → R + . The equations are of the form ∂ t ρ = D ρ ∆ρ − ∇ · f (ρ)∇c ∂ t c = D c ∆c + αρ − βc.(1)
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