arXiv subjects

Peter Pickl

Publications and source records attributed to Peter Pickl.

At least 37 records · Page 2Linked to original sources

arXiv2018

Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions

We present a microscopic derivation of the defocusing two-dimensional cubic nonlinear Schrödinger 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 $W_β(x)=N^{-1+2 β}W(N^βx)$, for any $β>0$, 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 Schrödinger 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.

math-ph↗

arXiv2018

Mean-field limits of particles in interaction with quantized radiation fields

We report on a simple strategy to treat mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. Extending the method of counting, introduced in [Lett. Math. Phys. 97, 151-164], with ideas inspired by [http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf] and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as a combination of the method of counting and the coherent state approach. It is similar to the coherent state approach but might be slightly better suited to systems in which a fixed number of particles couple to radiation. The strategy is effective and provides explicit error bounds. As an instructional example we derive the Schrödinger-Klein-Gordon system of equations from the Nelson model with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the non-relativistic particles in Sobolev norm. More complicated models like the Pauli-Fierz Hamiltonian can be treated in a similar manner [arXiv:1609.01545].

math-ph↗

arXiv2018

Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian

We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field is rescaled by $1/\sqrt{N}$. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit $N \rightarrow \infty$ the charged particles as well as the photons exhibit condensation, with the time evolution approximately described by the Maxwell-Schrödinger system, which models the coupling of a non-relativistic particle to the classical electromagnetic field. Our result is obtained by an extension of the "method of counting", introduced by Pickl, to condensates of charged particles in interaction with their radiation field.

math-ph↗

arXiv2018

Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations

We 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.

math-ph↗

arXiv2018

Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials

We 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].

math-ph↗

arXiv2017

Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers

Autoinducers 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.

q-bio.PE↗

arXiv2017

Microscopic derivation of the Keller-Segel equation in the sub-critical regime

We derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $χ< 8 π$. The Coulomb interaction force is regularised with a cutoff of size $N^{- α}$, with arbitrary $α\in (0, 1 / 2)$. In particular we obtain a quantitative result for the maximal distance between the real and mean-field $N$-particle trajectories.

math.AP↗

arXiv2016

Free time evolution of a tracer particle coupled to a Fermi gas in the high-density limit

The dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions is studied. We analyze the model for coupling parameter g=1 (i.e., not in the weak coupling regime), and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift which is given at leading order by the spatially homogeneous mean field potential of the gas particles. Here, the mean field approximation turns out to be accurate although the fluctuations of the potential around its mean value can be arbitrarily large. Our result is in contrast to a dense bosonic gas in which the free motion of a tracer particle would be disturbed already on a very short time scale. The proof is based on the use of strong phase cancellations in the deviations of the microscopic dynamics from the mean field time evolution.

math-ph↗

arXiv2016

Bogoliubov corrections and trace norm convergence for the Hartree dynamics

We consider the dynamics of a large number N of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for general initial states, e.g., those that are expected to be ground states of interacting systems.

math-ph↗

arXiv2016

A mean field limit for the Vlasov-Poisson system

We present a probabilistic proof of the mean field limit and propagation of chaos $N$-particle systems in three dimensions with positive (Coulomb) or negative (Newton) $1/r$ potentials scaling like $1/N$ and an $N$-dependent cut-off which scales like $N^{-1/3+ ε}$. In particular, for typical initial data, we show convergence of the empirical distributions to solutions of the Vlasov-Poisson system with either repulsive electrical or attractive gravitational interactions.

math-ph↗

arXiv2016

A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics

We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schroedinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in [Pickl, Lett. Math. Phys., 97(2):151-164, 2011] for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.

math-ph↗

arXiv2015

On Mean Field Limits for Dynamical Systems

We present a purely probabilistic proof of propagation of molecular chaos for $N$-particle systems in dimension $3$ with interaction forces scaling like $1/\vert q\vert^λ$ with $λ<2$ and cut-off at $q = N^{-1/3}$. The proof yields a Gronwall estimate for the maximal distance between exact microscopic and approximate mean-field dynamics. This can be used to show propagation of molecular chaos, i.e. weak convergence of the marginals to the corresponding products of solutions of the respective mean-field equation without cut-off in a quantitative way. Our results thus lead to a derivation of the Vlasov equation from the microscopic $N$-particle dynamics with force term arbitrarily close to the physically relevant Coulomb- and gravitational forces.

math-ph↗

arXiv2015

Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction

We study the time evolution of a system of $N$ spinless fermions in $\mathbb{R}^3$ which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schr{ö}dinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.

math-ph↗

arXiv2010

A simple derivation of mean field limits for quantum systems

We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean field description of a many particle quantum system directly into a mathematical algorithm. It is effective because the strategy yields with lesser effort better results than previously achieved. As an instructional example we treat a simple model for the time dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean field scalings leading e.g. to the Gross-Pitaevskii equation can also be treated.

math-ph↗

arXiv2010

Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields

Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.

math-ph↗

arXiv2009

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

We consider the time evolution of a system of $N$ identical bosons whose interaction potential is rescaled by $N^{-1}$. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit $N \to \infty$ the quantum $N$-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum $N$-body dynamics to the Hartree dynamics.

math-ph↗

arXiv2009

Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction

Using a new method it is possible to derive mean field equations from the microscopic $N$ body Schrödinger evolution of interacting particles without using BBGKY hierarchies. In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii equation which is usually derived assuming positivity of the interaction. The new method for dealing with mean field limits presented in [6] allows us to relax this condition. The price we have to pay for this relaxation is however that we have to restrict the scaling behavior to $β<1/3$ and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function $μ^{Ψ_0}$ to a pure state $|ϕ_0><ϕ_0|$.

math-ph↗

arXiv2008

On the Time Dependent Gross Pitaevskii- and Hartree Equation

We are interested in solutions $Ψ_t$ of the Schrödinger 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 $\mathcal{N}^{ϕ_0}$ be the particle number operator with respect to some $ϕ_0\in L^2(\mathbb{R}^3\to\mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}< Ψ_0,\mathcal{N}^{ϕ_0}Ψ_0>$ converges to one as $N\to\infty$. We shall show that we can find a $ϕ_t\in L^2(\mathbb{R}^3\to\mathbb{C})$ such that $\lim_{N\to\infty}N^{-1}< Ψ_t,\mathcal{N}^{ϕ_t}Ψ_t>=1$ and that $ϕ_t$ 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 $ϕ_t$ 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 $Ψ_t$.

math-ph↗