Singular Interaction Potentials Research Articles

From many-body quantum dynamics to the Hartree–Fock and Vlasov equations with singular potentials

In a combined mean-field and semiclassical regime, we consider the time evolution of N fermions interacting through singular pair interaction potentials of the form \pm |x-y|^{-a} , which includes the Coulomb and gravitational interactions. We prove that the many-body dynamics of mixed states are well approximated by solutions of the Hartree–Fock and Vlasov equations in terms of Schatten norms. The errors in these approximations are expressed in terms of the expected number of particles, N , and the Planck constant, h . For cases where a\in(0,1/2) , we obtain local-in-time results when N^{-1/2}\ll h \leq N^{-1/3} . Notably, this leads to the derivation of the Vlasov equation with singular potentials. For cases where a\in[1/2,1] , our results hold only within a small time scale or require an N -dependent cut-off. A fundamental ingredient in our analysis is the propagation of regularity for solutions to the Hartree–Fock equation uniformly in the Planck constant, which holds for a\in(0,1] .

Minimizers of 3D anisotropic interaction energies

Abstract We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of the recent work [J. A. Carrillo and R. Shu, Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D, Comm. Pure Appl. Math. (2023), 10.1002/cpa.22162] in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that, for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two-dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one-dimensional vertical collapse is not allowed. Collapse to lower-dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.

Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

AbstractWe study a large family of Riesz‐type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse‐supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse‐supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension d≥3, including the case of a Coulomb singularity in dimension d=3. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant ħ.

Large deviations for configurations generated by Gibbs distributions with energy functionals consisting of singular interaction and weakly confining potentials

We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on $n$-particle configurations, each of which is defined in terms of an inverse temperature $\beta _{n}$ and an energy functional consisting of a (possibly singular) interaction potential and a (possibly weakly) confining potential. Under fairly general assumptions on the potentials, we use a common framework to establish LDPs both with speeds $\beta _{n}/n \rightarrow \infty $, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed $\beta _{n} =n$, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling, simulated annealing and asymptotic convex geometry. Our approach, which uses the weak convergence method developed by Dupuis and Ellis, establishes LDPs with respect to stronger Wasserstein-type topologies. Our results address several interesting examples not covered by previous works, including the case of a weakly confining potential, which allows for rate functions with minimizers that do not have compact support, thus resolving several open questions raised in a work of Chafaï et al.

Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials

First, weak solutions of generalized stochastic Hamiltonian systems (gsHs) are constructed via essential m-dissipativity of their generators on a suitable core. For a scaled gsHs we prove convergence of the corresponding semigroups and tightness of the weak solutions. This yields convergence in law of the scaled gsHs to a distorted Brownian motion. In particular, the results confirm the convergence of the Langevin dynamics in the overdamped regime to the overdamped Langevin equation. The proofs work for a large class of (singular) interaction potentials including, e.g. potentials of Lennard-Jones type.

Particle Dynamics Subject to Impenetrable Boundaries: Existence and Uniqueness of Mild Solutions

We consider the dynamics of interacting particle systems where the particles are confined to a bounded, possibly nonconvex domain $\Omega$. When particles hit the boundary, we consider an instant change in velocity, which turns the system describing the particle dynamics into an ODE with a discontinuous right-hand side. As an alternative to the typical approach of analyzing such a system by using weak solutions to ODEs with multivalued right-hand sides (i.e., applying the theory introduced by Filippov in 1988), we define and construct mild solutions. The merit of mild solutions is that uniqueness of such solutions follows easily from Gronwall's lemma, that certain properties of the solutions are easy to establish, and that they provide a convenient framework for proving many-particle limits. We supplement our theory of mild solutions with an application to gradient flows of interacting particle energies with a singular interaction potential and illustrate its features by means of numerical simulations on various choices for the (nonconvex) domain $\Omega$.

Propagation of Chaos for a Class of First Order Models with Singular Mean Field Interactions

Dynamical systems of $N$ particles in $\Bbb{R}^{D}$ interacting by a singular pair potential of mean field type are considered. The systems are assumed to be of gradient type and the existence of a macroscopic limit in the many particle limit is established for a large class of singular interaction potentials in stochastic as well as deterministic settings. The main assumption on the potentials is an appropriate notion of quasi-convexity. When $D=1$ the convergence result is sharp when applied to strongly singular repulsive interactions and for a general dimension $D$ the result applies to attractive interactions with Lipschitz singular interaction potentials, leading to stochastic particle solutions to the corresponding macroscopic aggregation equations. The proof uses the theory of gradient flows in Wasserstein spaces of Ambrosio, Gigli, and Savaré.

Rate of convergence toward Hartree dynamics with singular interaction potential

We consider a system of N-bosons with a two-body interaction potential V∈L2(R3)+L∞(R3), possibly more singular than the Coulomb interaction. We show that, with H1(R3) initial data, the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order 1/N, for any fixed time. The N-dependence of the bound is optimal.

Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

We consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions. By using a measure transportation technique developed in [3], we show that Wigner measures propagate along the nonlinear Hartree flow. Such property was previously proved only for bounded potentials in our works [5, 6] with a slightly different strategy

AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials. Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝd\{0}.

Continuum models of cohesive stochastic swarms: The effect of motility on aggregation patterns

Mathematical models of swarms of moving agents with non-local interactions have many applications and have been the subject of considerable recent interest. For modest numbers of agents, cellular automata or related algorithms can be used to study such systems, but in the present work, instead of considering discrete agents, we discuss a class of one-dimensional continuum models, in which the agents possess a density ρ(x,t) at location x at time t. The agents are subject to a stochastic motility mechanism and to a global cohesive inter-agent force. The motility mechanisms covered include classical diffusion, nonlinear diffusion (which may be used to model, in a phenomenological way, volume exclusion or other short-range local interactions), and a family of linear redistribution operators related to fractional diffusion equations. A variety of exact analytic results are discussed, including equilibrium solutions and criteria for unimodality of equilibrium distributions, full time-dependent solutions, and transitions between asymptotic collapse and asymptotic escape. We address the behaviour of the system for diffusive motility in the low-diffusivity limit for both smooth and singular interaction potentials and show how this elucidates puzzling behaviour in fully deterministic non-local particle interaction models. We conclude with speculative remarks about extensions and applications of the models.

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space P2(R) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed by Ambrosio et al. (2005) [2]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in P2(R), which was already obtained by Deslippe et al. (2004) [17] and Biler et al. (2010) [6] by different methods. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials.

Schrödinger models for solutions of the Bethe–Salpeter equation in Minkowski space

By application of the ``geometric spectral inversion'' technique, which we have recently generalized to accommodate also singular interaction potentials, we construct from spectral data emerging from the solution of the Minkowski-space formulation of the homogeneous Bethe--Salpeter equation describing bound states of two spinless particles a Schr\"odinger approach to such states in terms of nonrelativistic potential models. This spectrally equivalent modeling of bound states yields their qualitative features (masses, form factors, etc.) without having to deal with the more involved Bethe--Salpeter formalism.

Stability of stationary states of non-local equations with singular interaction potentials

We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin. For locally attractive singular interaction potentials we prove under a linear stability condition local non-linear stability of stationary states consisting of a finite sum of Dirac masses. For singular repulsive interaction potentials we show the stability of stationary states of uniformly bounded solutions under a convexity condition. Finally, we present numerical simulations to illustrate our results.

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.

On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg- picture dynamics of “observables”, thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a “semi-classical” limit.

Notes on possible S-wave resonances in relativistic two-particle systems

Some models of relativistic two-particle systems are examined for possible irregularities in the S-wave continuous spectrum. None are found, though we show that with singular effective interaction potentials it is quite easy to run into artificial resonances originating from inaccurate calculations.

Singular interaction potentials in classical statistical mechanics

Singular interaction potentials in classical statistical mechanics

Some observations on the operator H=−(1/2)d2/dx2 +m2x2/2+g/x2

In the present work we study the differential operator H=−(1/2)d2/dx2 +m2x2/2+g/x2. This operator known as the Hamiltonian of the quantal oscillator has been a matter of study since the beginning of quantum mechanics. Recently, it has become again actual after the paper of Calogero where the correspondent N body problem (developed in many works) is studied. Parisi and the author have used H as Hamiltonian, studying the anomalous dimensions in one-dimensional quantum field theory. Finally, Klauder, using H as a simple degree of freedom example, has studied some qualitative features of quantum theories with singular interaction potentials. In the following work we are going to study H, showing that H is equivalent to ``half an harmonic oscillator'' for the odd and even eigenspaces separately.