# Browsing by Author "Pickl, P"

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Item Open Access A Mean Field Limit for the Vlasov–Poisson System(Archive for Rational Mechanics and Analysis, 2017-09) Lazarovici, D; Pickl, P© 2017, Springer-Verlag Berlin Heidelberg. 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.Item Open Access A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics(Mathematical Physics, Analysis and Geometry, 2016-03) Petrat, S; Pickl, P© 2016, The Author(s). We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger 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.Item Open Access A Simple Derivation of Mean Field Limits for Quantum Systems(Letters in Mathematical Physics, 2011-08) Pickl, PWe 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, with less effort, the strategy yields 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 (Pickl in Derivation of the time dependent Gross Pitaevskii equation with external fields, preprint; Pickl in Derivation of the time dependent Gross Pitaevskii equation without positivity condition on the interaction, preprint). © 2011 Springer.Item Open Access Bogoliubov corrections and trace norm convergence for the Hartree dynamicsMitrouskas, D; Petrat, S; Pickl, PWe 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.Item Open Access Derivation of the Bogoliubov Time Evolution for Gases with Finite Speed of SoundPetrat, S; Pickl, P; Soffer, AThe derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both the ground state properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically allow a convergence result for the reduced density matrix only, one gets norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In the present paper we consider an interacting Bose gas in the ground state with slight perturbations. We consider the case where the volume of the gas - in units of the support of the excitation - and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (non-zero) speed of sound in the gas. We show that the difference between the N-body description and the Bogoliubov description is small in $L^2$ as the density of the gas tends to infinity. In this situation the ratio of the occupation number of the ground-state and the excitation forming the fluctuations will influence the leading order of the dynamics of the system. In this sense we show the validity of the Bogoliubov time evolution in a situation where the temperature has an effect on the dynamics of the system.Item Open Access Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz HamiltonianLeopold, N; Pickl, PWe 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-Schr\"odinger 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.Item Open Access Derivation of the Time Dependent Gross-Pitaevskii Equation Without Positivity Condition on the Interaction(Journal of Statistical Physics, 2010-07) Pickl, PUsing a new method (Pickl in A simple derivation of mean field limits for quantum systems, 2010) 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 (Erdös et al. in Commun. Pure Appl. Math. 59(12):1659-1741, 2006; Invent. Math. 167:515-614, 2007). The new method for dealing with mean field limits presented in Pickl (2010) 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 of the interaction and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function μ Ψ0 to a pure state {pipe}φ 0 〉〈φ 0 {pipe}. © 2010 Springer Science+Business Media, LLC.Item Open Access Derivation of the time dependent Gross–Pitaevskii equation with external fields(Reviews in Mathematical Physics, 2015-02) Pickl, P© 2015 World Scientific Publishing Company. Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective. Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross-Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction. Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross-Pitaevskii scaling.Item Open Access Derivation of the Time Dependent Two Dimensional Focusing NLS EquationJeblick, M; Pickl, PIn this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schr\"odinger equation starting from an interacting $N$-particle system of Bosons. The interaction potential we consider is given by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$ for some bounded and compactly supported $W$. We assume the $N$-particle Hamiltonian fulfills stability of second kind. The class of initial wave functions is chosen such that the variance in energy is small. We then 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 either Sobolev trace norm, if the external potential is in some $L^p$ space, $p \in ]2, \infty]$, or in trace norm, for more general external potentials.Item Open Access Dynamics of sound waves in an interacting Bose gas(Advances in Mathematics, 2016-04) Deckert, DA; Fröhlich, J; Pickl, P; Pizzo, A© 2016 Elsevier Inc. We consider a non-relativistic quantum gas of N bosonic atoms confined to a box of volume Λ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, ρ=NΛ, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Λ and small ratio Λρ. The initial state of the gas is assumed to be close to a product state of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of Λ. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio Λρ. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.Item Open Access Effective dynamics of a tracer particle in a dense homogeneous quantum gasJeblick, M; Mitrouskas, D; Petrat, S; Pickl, PWe investigate the mean field regime of the dynamics of a tracer particle in a homogenous quantum gas. For a bosonic gas, we show that this regime is constrained by the well known requirement of an appropriate mean field scaling of the interaction. For fermions, however, we find an important qualitative difference. Not only are fermions much more homogeneously distributed than bosons but also deviations from the mean are due only to fast degrees of freedom in the gas. This observation leads to an explanation of why a tracer particle behaves freely in the dense homogeneous fermion gas despite of a non-scaled interaction, i.e., despite of non-vanishing statistical fluctuations. Finally, we indicate how the gained insight can be rigorously justified.Item Open Access Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas(Communications in Mathematical Physics, 2014-06) Deckert, DA; Fröhlich, J; Pickl, P; Pizzo, AWe study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density → ∞). Our estimates allow us to interchange the thermodynamic with the mean-field limit. © 2014 Springer-Verlag Berlin Heidelberg.Item Open Access Free Time Evolution of a Tracer Particle Coupled to a Fermi Gas in the High-Density Limit(Communications in Mathematical Physics, 2017-11) Jeblick, M; Mitrouskas, D; Petrat, S; Pickl, P© 2017, Springer-Verlag GmbH Germany. 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.Item Open Access Kinetic energy estimates for the accuracy of the time-dependent Hartree–Fock approximation with Coulomb interaction(Journal de Mathématiques Pures et Appliquées, 2016-01) Bach, V; Breteaux, S; Petrat, S; Pickl, P; Tzaneteas, T© 2015 The Authors. We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger'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.Item Open Access On Mean Field Limits for Dynamical Systems(Journal of Statistical Physics, 2016-07) Boers, N; Pickl, P© 2015, Springer Science+Business Media New York. We present a purely probabilistic proof of propagation of molecular chaos for N-particle systems in dimension 3 with interaction forces scaling like 1 / | q| 3 λ - 1 with λ smaller but close to one 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 weak convergence of the one-particle marginals to 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.Item Open Access On the mean-field limit for the Vlasov-Poisson-Fokker-Planck systemHuang, H; Liu, JG; Pickl, PWe devise and study a random particle blob method for approximating the Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show that maximal distance between the exact microscopic and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size $N^{-\delta}$ ($\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability $1-N^{-\alpha}$ for any $\alpha>0$, which improves the cut-off in [10]. Our result thus leads to a derivation of VPFP equations from the microscopic $N$-particle system. In particular we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.Item Open Access Spontaneous Pair Creation RevisitedPickl, P; Duerr, DRecently the so called Spontaneous Pair Creation of electron positron pairs in a strong external field has been rigorously established. We give here the heuristic core of the proof, since the results differ from those given in earlier works.