On the Energy-Momentum Spectrum of a Homogeneous Fermi Gas

We consider translation-invariant quantum systems in thermodynamic limit. We argue that their energy-momentum spectra should have shapes consistent with effective models involving quasiparticles. Our main example is second quantized homogeneous interacting Fermi gas in a large cubic box with periodic boundary conditions, at zero temperature. We expect that its energy-momentum spectrum has a positive energy gap and a positive critical velocity.     

Smolukhowski problem for degenerate Bose gases

We construct a kinetic equation simulating the behavior of degenerate quantum Bose gases with the collision rate proportional to the molecule velocity. We obtain an analytic solution of the half-space boundary-value Smolukhowski problem of the temperature jump at the interface between the degenerate Bose gas and the condensed phase. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 498–511, June, 2008.     

Quantum Phase Transition in an Interacting Fermionic Chain

We rigorously analyze the quantum phase transition between a metallic and an insulating phase in (non-solvable) interacting spin chains or one-dimensional fermionic systems. In particular, we prove the persistence of Luttinger liquid behavior in the presence of an interaction even arbitrarily close to the critical point, where the Fermi velocity vanishes and the two Fermi points coalesce. The analysis is based on two different multiscale analysis; the analysis of the first regime provides gain factors which compensate the small divisors due to the vanishing Fermi velocity.     

Large data solutions to the viscous quantum hydrodynamic model with barrier potential

We discuss analytically the stationary viscous quantum hydrodynamic model including a barrier potential, which is a nonlinear system of partial differential equations of mixed order in the sense of Douglis–Nirenberg. Combining a reformulation by means of an adjusted Fermi level, a variational functional, and a fixed point problem, we prove the existence of a weak solution. There are no assumptions on the size of the given data or their variation. We also provide various estimates of the solution that are independent of the quantum parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Gas Near a Wall: Shortened Mean Free Path,Reduced Viscosity,and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall and compute the Navier–Stokes solution of the velocity of a shear flow parallel to the wall. Under the simplifying assumption of constant temperature of the gas, the velocity profile becomes an explicit nonlinear function of the distance from the wall and exhibits a Knudsen boundary layer near the wall. To verify the validity of the obtained formula, we perform the Direct Simulation Monte Carlo computations for the shear flow of argon and nitrogen at normal density and temperature. We find excellent agreement between our velocity approximation and the computed DSMC velocity profiles both within the Knudsen boundary layer and away from it.     

Anomalous Behavior in an Effective Model of Graphene with Coulomb Interactions

We analyze by exact Renormalization Group (RG) methods the infrared properties of an effective model of graphene, in which two-dimensional (2D) massless Dirac fermions propagating with a velocity smaller than the speed of light interact with a 3D quantum electromagnetic field. The fermionic correlation functions are written as series in the running coupling constants, with finite coefficients that admit explicit bounds at all orders. The implementation of Ward Identities in the RG scheme implies that the effective charges tend to a line of fixed points. At small momenta, the quasi-particle weight tends to zero and the effective Fermi velocity tends to a finite value. These limits are approached with a power law behavior characterized by non-universal critical exponents.     

Level shift operators for open quantum systems

Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.     

Linear response theory for magnetic Schrödinger operators in disordered media

We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the well-known Kubo-Str?eda formula for the quantum Hall conductivity at zero temperature.     

Boundary Problems for a Quantum Fermi Gas

We derive a relaxation kinetic equation describing the behavior of Fermi gases. We consider the Kramers problem for the isothermal slipping in a half-space and obtain an analytic solution of the problem and the explicit form of the distribution function for particles moving toward the wall. We analyze the dependence of the equation itself and the slipping velocity on the parameter, the ratio between the chemical potential and the product of the Boltzmann constant and the absolute temperature.     

Fermion martingales

Summary We show that strictly quasi-free Fermion martingales may be expressed as a sum of quantum stochastic integrals with respect to the Fermi creation and annihilation processes and a multiple of the identity.     

The Green–Kubo Formula for Locally Interacting Fermionic Open Systems

We consider a model describing finitely many free Fermi gas reservoirs coupled by local interactions and prove the Green–Kubo formulas and the Onsager reciprocity relations for heat and charge fluxes generated by temperature and chemical potential differentials. Submitted: September 2, 2006. Accepted: November 21, 2006.     

Quantum statistical systems in <Emphasis Type="Italic">D</Emphasis>-dimensional space using a fractional derivative

We investigate the thermodynamic properties of some quantum statistical systems with a fractional Hamiltonian in D-dimensional space. We calculate the partition function of the system of N fractional quantum oscillators and the thermodynamic quantities associated with it. We consider the thermal and critical properties of both Bose and Fermi gases in the context of the fractional energy and described by a fractional derivative.     

Quasi-Relativism,the Narrow-Gap Property,and Forced Electron Dynamics in Solids

Narrow-gap semiconductors, used in quantum network engineering, are characterized by small effective electron masses on the Fermi level and hence by high electron mobility in the lattice. We construct an explicitly solvable model that clarifies one possible mechanism for small effective masses to appear. Another mathematical model constructed here describes a possible mechanism for using a traveling wave to control an alternating quantum current in a one-dimensional lattice.     

Thomas-Fermi近似问题求解

该文将Ｔｈｏｍａｓ Ｆｅｒｍｉ近似问题分解为一个带奇点的常微分方程边值问题和一个最优化问题，讨论了解的存在唯一性和解的性质，给出了Ｔｈｏｍａｓ Ｆｅｒｍｉ近似问题求解的具体步骤．     

Nonisothermic flow of gas mixture in a channel at intermediate Knudsen numbers

Solution of the problem of gas mixture flow in a plane channel at intermediate Knudsen numbers is considered on the basis of the 20-moment approximation as a function of distribution. The applied method consists of averaging moment equations valid throughout the flow region (including the Knudsen layers) with the determination of boundary values of macroscopic parameters on the wall using the approximate Loyalka method /1,2/. Expressions are obtained for a binary mixture for the mean molar velocity averaged over the channel cross section, difference of component velocities, and the relative heat flux in the presence of longitudinal gradients of partial pressures, and for the temperature gradients. Respective kinetic coefficients of the Onsager matrix are calculated. Dependence of these coefficients on the Knudsen number, and the properties of molecule scatter on the channel wall are analyzed in detail in the case of one-component gas and of a binary mixture with small relative difference of mass and diameters of molecule scatter.     

On the Semiclassical Asymptotics of the Current and Magnetic Moment of a Non-Interacting Electron Gas at Zero Temperature in a Strong Constant Magnetic Field

We calculate the asymptotic form of the quantum current and magnetic moment of a non-interacting electron gas at zero temperature. The calculation uses coherent states and a novel commutator identity for the current operator.     

On the Properties of a Quasihydrodynamic System of Equations for a Homogeneous Gas Mixture with a Common Regularizing Velocity

Differential Equations - We study a quasihydrodynamic system of equations for a homogeneous (with common velocity and temperature) multicomponent gas mixture in the absence of chemical reactions...     

Global regularity of generalized magnetic Benard problem

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermal transpiration for the linearized Boltzmann equation

The phenomena of thermal transpiration due to the boundary temperature gradient is studied on the level of the linearized Boltzmann equation for the hard‐sphere model. We construct such a flow for a highly rarefied gas between two plates and also in a circular pipe. It is shown that the flow velocity parallel to the plates is proportional to the boundary temperature gradient. For a highly rarefied gas, that is, for a sufficiently large Knudsen number κ, the flow velocity between two plates is of the order of log κ, and the flow velocity in a pipe is of finite order. Our analysis is based on certain pointwise estimates of the solutions of the linearized Boltzmann equation. © 2006 Wiley Periodicals, Inc.     

Long-time behavior of solutions for full compressible quantum model in R3

In this paper, long-time behavior of solutions for full compressible quantum flows in three-dimensional whole space is studied. We establish the optimal time decay rates for higher-order spatial derivatives of density, velocity and temperature, which will improve the work of Pu and Guo (2016).