Peter Pickl
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View article: On the mean-field limit for the Vlasov-Poisson system in two dimensions
On the mean-field limit for the Vlasov-Poisson system in two dimensions Open
We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in two dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^2}$ and $N$-dependent cut-off at $|q|>N…
View article: Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems
Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems Open
The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the time-d…
View article: On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations
On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations Open
The derivation of effective descriptions for interacting many-body systems is an important branch of applied mathematics. We prove a propagation of chaos result for a system of $N$ particles subject to Newtonian time evolution with or with…
View article: Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas
Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas Open
We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum $$k_{\text {F}}$$ , we prove that the effective dynami…
View article: Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas
Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas Open
We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum $k_\text{F}$, we prove that the effective dynamics is gene…
View article: Exponential decay of the number of excitations in the weakly interacting Bose gas
Exponential decay of the number of excitations in the weakly interacting Bose gas Open
We consider N trapped bosons in the mean-field limit with coupling constant λN = 1/(N − 1). The ground state of such systems exhibits Bose–Einstein condensation. We prove that the probability of finding ℓ particles outside the condensate w…
View article: Exponential decay of the number of excitations in the weakly interacting Bose gas
Exponential decay of the number of excitations in the weakly interacting Bose gas Open
We consider $N$ trapped bosons in the mean-field limit with coupling constant $λ_N=1 / (N-1)$. The ground state of such systems exhibits Bose--Einstein condensation. We prove that the probability of finding $\ell$ particles outside the con…
View article: Microscopic derivation of Vlasov equation with compactly supported pair potentials
Microscopic derivation of Vlasov equation with compactly supported pair potentials Open
We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with compactly supported pair potentials of the form $N^{3β-1} ϕ(N^βx)$ for $β\in\left[0,\frac{1}{7}\right)$ and $…
View article: Beyond Bogoliubov dynamics
Beyond Bogoliubov dynamics Open
We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are …
View article: Effective pair interaction between impurity particles induced by a dense Fermi gas
Effective pair interaction between impurity particles induced by a dense Fermi gas Open
We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a $d$-dimensional box. The impurities interact with the fermions via a two-body potential $λv(x)$ where $λ$ is a coupling constant and $v(x)$ f…
View article: Combined mean field limit and non-relativistic limit of Vlasov–Maxwell particle system to Vlasov–Poisson system
Combined mean field limit and non-relativistic limit of Vlasov–Maxwell particle system to Vlasov–Poisson system Open
In this paper, we consider the mean field limit and non-relativistic limit of the relativistic Vlasov–Maxwell particle system to the Vlasov–Poisson equation. With the relativistic Vlasov–Maxwell particle system being a starting point, we c…
View article: Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons
Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons Open
In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N $$d$$ -dimensional bosons for large N . Th…
View article: Derivation of the Maxwell--Schrödinger Equations from the Pauli--Fierz Hamiltonian
Derivation of the Maxwell--Schrödinger Equations from the Pauli--Fierz Hamiltonian Open
Presented at the QMath13 Conference: Mathematical Results in Quantum Theory, October 8-11, 2016 at the Clough Undergraduate Learning Commons, Georgia Tech.
View article: Bogoliubov corrections and trace norm convergence for the Hartree dynamics
Bogoliubov corrections and trace norm convergence for the Hartree dynamics Open
We consider the dynamics of a large number [Formula: see text] 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…
View article: A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication
A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication Open
We present a probabilistic approach for derivation of the kinetic Cucker-Smale (C-S) equation from the particle C-S model with singular communication. For the system we are considering, it is impossible to validate effective description fo…
View article: On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system
On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system Open
We rigorously justify the mean-field limit of a $N$-particle system subject to the Brownian motion and interacting through a Newtonian potential in $\mathbb{R}^3$. Our result leads to a derivation of the Vlasov-Poisson-Fokkker-Planck (VPFP…
View article: Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations
Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations Open
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 meta…
View article: Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials
Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials Open
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 t…
View article: Derivation of the Bogoliubov Time Evolution for a Large Volume\n Mean-field Limit
Derivation of the Bogoliubov Time Evolution for a Large Volume\n Mean-field Limit Open
The derivation of mean-field limits for quantum systems at zero temperature\nhas attracted many researchers in the last decades. Recent developments are the\nconsideration of pair correlations in the effective description, which lead to\na…
View article: Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit
Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit Open
The 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 mu…
View article: Derivation of the bogoliubov time evolution for gases with finite speed of sound
Derivation of the bogoliubov time evolution for gases with finite speed of sound Open
The 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 mu…
View article: Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers
Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers Open
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, t…
View article: Author response: Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers
Author response: Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers Open
Article Figures and data Abstract eLife digest Introduction Set-up of the quorum-sensing model Results of numerical simulations Results of mathematical analysis Discussion Materials and methods Appendix 1 Appendix 2 Appendix 3 References D…
View article: Microscopic derivation of the Keller-Segel equation in the sub-critical\n regime
Microscopic derivation of the Keller-Segel equation in the sub-critical\n regime Open
We derive the two-dimensional Keller-Segel equation from a stochastic system\nof $N$ interacting particles in the case of sub-critical chemosensitivity $\\chi\n< 8 \\pi$. The Coulomb interaction force is regularised with a cutoff of size\n…