We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle number and suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.
The Landau-Pekar equations describe the dynamics of a strongly coupled polaron. Here we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau-Pekar equations and their derivation from the Froehlich model obtained in [8, 7] to larger times.
We prove an adiabatic theorem for the Landau-Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant $α$. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau-Pekar equations until times short compared to $α^2$.
We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced densi…
We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar [10].
We discuss a one-dimensional version of the Landau-Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear adiabatic theorem. Dispersive estimates for solutions of the Schrödinger equation with time-dependent potential are a key technical ingredient in our proof.
A polaron is an electron interacting with a polar crystal, which is able to form a bound state by using the distortions of the crystal induced by its own density of charge. In this paper we derive Pekar's famous continuous model for polarons (in which the crystal is replaced by a simple effective Coulomb self-attraction) by studying the macroscopic limit of the reduced Hartree-Fock theory of the crystal. The macroscopic density of the polaron converges to that of Pekar's nonlinear model, with a possibly anisotropic dielectric matrix. The polaron also exhibits fast microscopic oscillations whic…
Fröhlich's polaron Hamiltonian describes an electron coupled to the quantized phonon field of an ionic crystal. We show that in the strong coupling limit the dynamics of the polaron is approximated by an effective non-linear partial differential equation due to Landau and Pekar, in which the phonon field is treated as a classical field.
We consider a system of N particles interacting via a short-range smooth potential, in a intermediate regime between the weak-coupling and the low-density. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path.Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limi…
In a recent paper we give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Here we present our results in the simplified case of a one-dimensional system of particles interacting via a delta-potential.
We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly non-local interaction, we derive diffusive corrections …
We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimatin…
We study the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. Assuming that the particles are initially in a Bose-Einstein condensate and that the excitations of the phonon field are initially in a coherent state we provide an effective dynamics which approximates the time evolved many-body state in norm, provided that the number of particles is large. The approximation is given by a product state which evolves according to the Landau-Pekar equations and which is corrected by a Bogoliubov dynamics. In addition…
We consider the Aharonov-Bohm magnetic potential and study the transition from normal to superconducting solutions within the Ginzburg-Landau model of superconductivity. We obtain oscillations consistent with the Little-Parks effect. We study the same problem but for a regularization of the Aharonov-Bohm potential, which leads to an interesting Aharonov-Bohm like magnetic field, and we prove that the transition between superconducting and normal solutions is not monotone too. Our results show a mechanism to derive the Aharonov-Bohm magnetic potential starting from a step magnetic field thereby…
We examine a family of microscopic models of plasmas, with a parameter $α$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $α$, for the whole range (0, 1/2], by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusi…
We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.
We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.
In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics and can be reformulated as a non-Markovian hyperbolic equation, which converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov's derivation of the kinetic equation by means of a multi…
This paper consists of three parts. In part I, we microscopically derive Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then couples $n$ order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure $d$-wave order parameters and (b) mixed $(s+d)$-…
We present a new derivation of the Onsager-Joyce-Montgomery (OJM) equilibrium statistical theory for point vortices on the plane, using the Bogoliubov-Feynman inequality for the free energy, Gibbs entropy function and Landau's approximation. This formulation links the heuristic OJM theory to the modern variational mean field theories. Landau's approximation is the physical counterpart of a large deviation result, which states that the maximum entropy state does not only have maximal probability measure but overwhelmingly large measure relative to other macrostates.
We study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian r…
The derivation of macroscopic irreversible dynamics of nonequilibrium systems from microscopic equations was recently revisited from the point of view of infinitely extended quantum systems. Here we have briefly reviewed the $C^*$ algebra and its application to equilibrium systems as well as introduced some recent results on NESS. In addition, we have demonstrated the derivation of Landauer formula rigorously for quadratic systems but using a more physical presentation.
We consider the dynamics of nonlinear Schrödinger equations with strong constant magnetic fields. In an asymptotic scaling limit the system exhibits a purely magnetic confinement, based on the spectral properties of the Landau Hamiltonian. Using an averaging technique we derive an associated effective description via an averaged model of nonlinear Schrödinger type. In a special case this also yields a derivation of the LLL equation.
Some relevant transport properties of solids do not depend only on the spectrum of the electronic Hamiltonian, but on finer properties preserved only by unitary equivalence, the most striking example being the conductance. When interested in such properties, and aiming to a simpler model, it is mandatory to check that the simpler effective Hamiltonian is approximately unitarily equivalent to the original one, in the appropriate asymptotic regime. In this paper, we consider the Hamiltonian of an electron in a 2-dimensional periodic potential (e.g. generated by the ionic cores of a crystalline s…
Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations [Schmuck et al., Proc. R. Soc. A 468:3705-3724, 2012.], we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large Péclet number. As for classical convection-diffusion problems, we obtain systematically diffusion-disp…
A rigorous derivation of macroscopic spin-wave equations is demonstrated. We introduce a macroscopic mean-field limit and derive the so-called Landau-Lifshitz equations for spin waves. We first discuss the ferromagnetic Heisenberg model at T=0 and finally extend our analysis to general spin hamiltonians for the same class of ferromagnetic ground states.
