# Browsing by Author "Pickl, Peter"

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Item Open Access Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentialsJeblick, Maximilian; Pickl, PeterWe present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].Item Open Access Derivation of the Time Dependent Gross-Pitaevskii Equation in Two DimensionsJeblick, Maximilian; Leopold, Nikolai; Pickl, PeterWe present a microscopic derivation of the defocusing two-dimensional cubic nonlinear Schr\"odinger equation as a mean field equation starting from an interacting $N$-particle system of Bosons. We consider the interaction potential to be given either by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$, for any $\beta>0$, or to be given by $V_N(x)=e^{2N} V(e^N x)$, for some spherical symmetric, positive and compactly supported $W,V \in L^\infty(\mathbb{R}^2,\mathbb{R})$. In both cases we prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schr\"odinger equation in trace norm. For the latter potential $V_N$ we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.Item Open Access Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers.(eLife, 2017-07-25) Bauer, Matthias; Knebel, Johannes; Lechner, Matthias; Pickl, Peter; Frey, ErwinAutoinducers are small signaling molecules that mediate intercellular communication in microbial populations and trigger coordinated gene expression via 'quorum sensing'. Elucidating the mechanisms that control autoinducer production is, thus, pertinent to understanding collective microbial behavior, such as virulence and bioluminescence. Recent experiments have shown a heterogeneous promoter activity of autoinducer synthase genes, suggesting that some of the isogenic cells in a population might produce autoinducers, whereas others might not. However, the mechanism underlying this phenotypic heterogeneity in quorum-sensing microbial populations has remained elusive. In our theoretical model, cells synthesize and secrete autoinducers into the environment, up-regulate their production in this self-shaped environment, and non-producers replicate faster than producers. We show that the coupling between ecological and population dynamics through quorum sensing can induce phenotypic heterogeneity in microbial populations, suggesting an alternative mechanism to stochastic gene expression in bistable gene regulatory circuits.Item Open Access Existence of Spontaneous Pair CreationPickl, PeterA prove of the existence of spontaneous pair creation as an external field problem in second quantized Dirac theory. PHD ThesisItem Open Access Generalized Eigenfunctions for critical potentials with small perturbationsPickl, PeterWe estimate the behavior of the generalized eigenfunctions of critical Dirac operators (which are Dirac operators with eigenfunctions and/or resonances for $E=m$) plus small perturbations in the potential. The results also apply for other differential operators (for example Schr\"odinger operators).Item Open Access Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populationsFrey, Erwin; Knebel, Johannes; Pickl, PeterWe prove a mean-field equation for the dynamics of quorum-sensing microbial populations. In the stochastic many-particle process, individuals of a population produce public good molecules to different degrees. Individual production is metabolically costly such that non-producers replicate faster than producers. In addition, individuals sense the average production level in the well-mixed population and adjust their production in response ("quorum sensing"). Here we prove that the temporal evolution of such quorum-sensing populations converges to a macroscopic mean-field equation for increasing population sizes. To prove convergence, we introduce an auxiliary stochastic mean-field process that mimics the dynamics of the mean-field equation and that samples independently the individual's production degrees between consecutive update steps. This way, the law of large numbers is separated from the propagation of errors due to correlations. Our developed method of an auxiliary stochastic mean-field process may help to prove mean-field equations for other stochastic many-particle processes.Item Open Access Microscopic derivation of the Keller-Segel equation in the sub-critical regimeGarcía, Ana Cañizares; Pickl, PeterWe derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff of size $N^{- \alpha}$, with arbitrary $\alpha \in (0, 1 / 2)$. In particular we obtain a quantitative result for the maximal distance between the real and mean-field $N$-particle trajectories.Item Open Access On the Time Dependent Gross Pitaevskii- and Hartree EquationPickl, PeterWe are interested in solutions $\Psi_t$ of the Schr\"odinger equation of $N$ interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with $N$ in such a way, that the interaction energy per particle stays more or less constant. Let $\mathcal{N}^{\phi_0}$ be the particle number operator with respect to some $\phi_0\in L^2(\mathbb{R}^3\to\mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}< \Psi_0,\mathcal{N}^{\phi_0}\Psi_0>$ converges to one as $N\to\infty$. We shall show that we can find a $\phi_t\in L^2(\mathbb{R}^3\to\mathbb{C})$ such that $\lim_{N\to\infty}N^{-1}< \Psi_t,\mathcal{N}^{\phi_t}\Psi_t>=1$ and that $\phi_t$ is -- dependent of the scaling of the range of the interaction -- solution of the Gross-Pitaevskii or Hartree equation. We shall also show that under additional decay conditions of $\phi_t$ the limit can be taken uniform in $t<\infty$ and that convergence of the relative particle number implies convergence of the $k$-particle density matrices of $\Psi_t$.