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Peter Pickl

Publications and source records attributed to Peter Pickl.

At least 19 recordsLinked to original sources
arXiv2026

Effective Dynamics for the Bose Polaron in the Large-Volume Mean-Field Limit

We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density $ρ$ and large volume $Λ$ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint $Λ^3 \ll ρ$, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.

math-ph↗
arXiv2025

On the mean-field limit for the Vlasov-Poisson system in two dimensions

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 $|q|>N^{-2}$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.

math.AP↗
arXiv2025

Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems

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-dependent Hartree equations as the large-$N$ limit of the microscopic Schrödinger dynamics of $N$ fermions confined to a volume of order one and interacting via strong pair potentials. A central step in our analysis is the implementation of time-dependent gauge transformations, which eliminate the dominant contribution from the interaction potential in both the Schrödinger and Hartree evolutions.

math-ph↗
arXiv2025

On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations

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 without additional white noise influencing the velocities of the particles. We assume that the particles interact according to a regularized Coulomb-interaction with a regularization parameter that vanishes in the $N\to\infty$ limit. The respective effective description is the so called Vlasov-Poisson-Fokker-Planck (VPFP), respectively the Vlasov-Poisson (VP) equation in the case of no or sub-dominant white noise. To obtain our result we combine the relative entropy method from \cite{jabinWang2016} with the control on the difference between the trajectories of the true and the effective description provided in \cite{HLP20} for the VPFP case respectively in \cite{LP} for the VP case. This allows us to prove strong convergence of the marginals, i.e. convergence in $L^1$.

math-ph↗
arXiv2025

On the mean-field limit for the Vlasov-Poisson system

We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+σ}$ where $σ>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.

math-ph↗
arXiv2025

Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas

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 generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Moreover, we prove that the effective dynamics can be approximated by an explicit coupled coherent state. Our method is applicable to a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order $k_\text{F}^{-1}$.

math-ph↗
arXiv2023

Microscopic derivation of Vlasov equation with compactly supported pair potentials

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 $ϕ\in L^{\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Dirac-Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Dirac-Benney equation from the microscopic $N$-particle dynamics with a strong short range force.

math-ph↗
arXiv2022

Beyond Bogoliubov Dynamics

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 given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions.

math-ph↗
arXiv2022

Symplectic Reduction of Classical Mechanics on Shape Space

One of the foremost goals of research in physics is to find the most basic and universal theories that describe our universe. Many theories assume the presence of an absolute space and time in which the physical objects are located and physical processes take place. However, it is more fundamental to understand time as relative to the motion of another object, e.g. the number of swings of a pendulum, and the position of an object primarily as relative to other objects. The goals of this paper is to explain, how using the principle of relationalism (to be introduced below), classical mechanics can be formulated on a most elementary space, which is freed from absolute entities: shape space. On shape space only the relative orientation and length of subsystems are taken into account. In order to find out how the shape of a classical system evolves in time, the method of "symplectic reduction of Hamiltonian systems" is extended to include scale transformations, and in this way the reduction of a classical system with respect to the full similarity group is achieved. A necessary requirement for the validity of the principle of relationalism is that changing the length scale of a system, all parameters of the theory that depend on the length, get changed accordingly. In particular, the principle of relationalism requires a proper transformation of the coupling constants of the interaction potentials in Classical Physics. This leads consequently to a transformation in Planck's measuring units, which enables us to derive a metric on shape space in a unique way. Later in this paper, we will explain the derivation of the reduced Hamiltonian and symplectic form on shape space.

physics.hist-ph↗
arXiv2021

Dynamics of a tracer particle interacting with excitations of a Bose-Einstein condensate

We consider the quantum dynamics of a large number $N$ of interacting bosons coupled a tracer particle, i.e. a particle of another kind, on a torus. We assume that in the initial state the bosons essentially form a homogeneous Bose-Einstein condensate, with some excitations. With an appropriate mean-field scaling of the interactions, we prove that the effective dynamics for $N\to \infty$ is generated by the Bogoliubov-Fröhlich Hamiltonian, which couples the tracer particle linearly to the excitation field.

math-ph↗
arXiv2021

Effective pair interaction between impurity particles induced by a dense Fermi gas

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 $v(x)$ for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum $k_F$ we consider the regime of high density of the fermions, that is, $k_F$ large compared to one. For coupling constants that scale like $λ^2 \sim k_F^{(2-d)}$ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.

math-ph↗
arXiv2020

On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system

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) equation from the microscopic $N$-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and trajectories following the the mean-field is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size $N^{-δ}$ ($\frac{1}{3}\leqδ<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability $1-N^{-α}$ for any $α>0$. Moreover, 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.

math.AP↗
arXiv2019

Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit

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 much more precise description of both spectral properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically lead to convergence for the reduced density matrix only, one obtains norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In this article we consider an interacting Bose gas in the case where both the volume 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. In our first main result we show that the difference between the N-body and the Bogoliubov description is small in $L^2$ as the density of the gas tends to infinity and the volume does not grow too fast. This describes the dynamics of delocalized excitations of the order of the volume. In our second main result we consider an interacting Bose gas near the ground state with a macroscopic localized excitation of order of the density. We prove that the microscopic dynamics of the excitation coming from the N-body Schrödinger equation converges to an effective dynamics which is free evolution with the Bogoliubov dispersion relation. The main technical novelty are estimates for all moments of the number of particles outside the condensate for large volume, and in particular control of the tails of their distribution.

math-ph↗
arXiv2019

Higher order corrections to the mean-field description of the dynamics of interacting bosons

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 $N$. The bosons initially form a Bose-Einstein condensate and interact with each other via a pair potential of the form $(N-1)^{-1}N^{dβ}v(N^β\cdot)$ for $β\in[0,\frac{1}{4d})$. We derive a sequence of $N$-body functions which approximate the true many-body dynamics in $L^2(\mathbb{R}^{d N})$-norm to arbitrary precision in powers of $N^{-1}$. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.

math-ph↗
arXiv2019

Combined Mean Field Limit and Non-relativistic Limit of Vlasov-Maxwell Particle System to Vlasov-Poisson System

In this paper we consider the mean field limit and non-relativistic limit of relativistic Vlasov-Maxwell particle system to Vlasov-Poisson equation. With the relativistic Vlasov-Maxwell particle system being a starting point, we carry out the estimates (with respect to $N$ and $c$) between the characteristic equation of both Vlasov-Maxwell particle model and Vlasov-Poisson equation, where the probabilistic method is exploited. In the last step, we take both large $N$ limit and non-relativistic limit (meaning $c$ tending to infinity) to close the argument.

math-ph↗
arXiv2018

A mean field limit for the Hamiltonian Vlasov system

The derivation of effective equations for interacting many body systems has seen a lot of progress in the recent years. While dealing with classical systems, singular potentials are quite challenging, comparably strong results are known to hold for quantum systems. In this paper, we wish to show how techniques developed for the derivation of effective descriptions of quantum systems can be used for classical ones. While our future goal is to use these ideas to treat singularities in the interaction, the focus here is to present how quantum mechanical techniques can be used for a classical system and we restrict ourselves to regular two-body interaction potentials. In particular we compute a mean field limit for the Hamilton Vlasov system in the sense of Fröhlich, Knowles, Schwarz, and, more recently, Neiss, that arises from classical dynamics. The structure reveals strong analogy to the bosonic quantum mechanical ensemble of the many-particle Schrödinger equation and the Hartree equation as its mean field limit.

math-ph↗
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↗