TY - JOUR
AB - We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations.
AU - Leopold, Nikolai K
AU - Petrat, Sören P
ID - 6788
IS - 10
JF - Annales Henri Poincare
SN - 1424-0637
TI - Mean-field dynamics for the Nelson model with fermions
VL - 20
ER -
TY - JOUR
AB - We discuss thermodynamic properties of harmonically trapped
imperfect quantum gases. The spatial inhomogeneity of these systems imposes
a redefinition of the mean-field interparticle potential energy as compared
to the homogeneous case. In our approach, it takes the form a
2N2 ωd, where
N is the number of particles, ω—the harmonic trap frequency, d—system’s
dimensionality, and a is a parameter characterizing the interparticle interaction.
We provide arguments that this model corresponds to the limiting case of
a long-ranged interparticle potential of vanishingly small amplitude. This
conclusion is drawn from a computation similar to the well-known Kac scaling
procedure, which is presented here in a form adapted to the case of an isotropic
harmonic trap. We show that within the model, the imperfect gas of trapped
repulsive bosons undergoes the Bose–Einstein condensation provided d > 1.
The main result of our analysis is that in d = 1 the gas of attractive imperfect
fermions with a = −aF < 0 is thermodynamically equivalent to the gas of
repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill
the relation aB + aF = . This result supplements similar recent conclusion
about thermodynamic equivalence of two-dimensional (2D) uniform imperfect
repulsive Bose and attractive Fermi gases.
AU - Mysliwy, Krzysztof
AU - Napiórkowski, Marek
ID - 6840
IS - 6
JF - Journal of Statistical Mechanics: Theory and Experiment
TI - Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps
VL - 2019
ER -
TY - JOUR
AB - We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.
AU - Boccato, Chiara
AU - Brennecke, Christian
AU - Cenatiempo, Serena
AU - Schlein, Benjamin
ID - 6906
JF - Communications in Mathematical Physics
SN - 0010-3616
TI - Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime
VL - 376
ER -
TY - JOUR
AB - We modify the "floating crystal" trial state for the classical homogeneous electron gas (also known as jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground-state energy of jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in density functional theory. Finally, in the third system each electron interacts with a periodic image of itself; that is, periodic boundary conditions are imposed on the interaction potential.
AU - Lewin, Mathieu
AU - Lieb, Elliott H.
AU - Seiringer, Robert
ID - 7015
IS - 3
JF - Physical Review B
SN - 2469-9950
TI - Floating Wigner crystal with no boundary charge fluctuations
VL - 100
ER -
TY - JOUR
AB - We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density 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 VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
AU - Jeblick, Maximilian
AU - Leopold, Nikolai K
AU - Pickl, Peter
ID - 7100
IS - 1
JF - Communications in Mathematical Physics
SN - 0010-3616
TI - Derivation of the time dependent Gross–Pitaevskii equation in two dimensions
VL - 372
ER -