Dispersion Law for a One-Dimensional Weakly Interacting Bose Gas with Zero Boundary Conditions

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Linget al. 2016Physica D327, 13–29 (doi:10.1016/j.physd.2016.03.012); Fenget al. 2016Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).

By taking into account two types of boundary conditions (BC), free boundary condition (FBC) and zero boundary condition (ZBC), an improved transverse Ising model with the consideration of surface transition layers (STL) is used to describe the polarization properties of ferroelectric thin films in the framework of the mean field approximation. Functions representing the intra-layer and inter-layer couplings are introduced to characterize the BC and STL, which reflect the structure variation of film surface. Comparing the results obtained by employing FBC and ZBC, some effects of BC are derived in the model, which demonstrated that the BC play important roles on the properties of ferroelectric thin films. It is shown that the effect of ZBC can extend to deeper film than that of FBC. When different BC are adopted, competition between the fields (depolarization field and transverse field) and BC induces some different and interesting phenomena. Some theoretical results in this paper are found to be in reasonable accordance with experimental data and have some application importance.

We determine the ground-state energy and the effective dispersion law for a one-dimensional system of point bosons under zero boundary conditions. The ground-state energy is close to the value for a periodic system. However, the dispersion law is essentially different from that for a periodic system, if the coupling is weak (weak interaction or high concentration) or intermediate. We propose also a new method for construction of the thermodynamics for a gas of point bosons. It turns out that the difference in the dispersion laws of systems with periodic and zero boundary conditions does not lead to a difference in the thermodynamic quantities. In addition, under zero boundary conditions, the microscopic sound velocity does not coincide with the macroscopic one. This means that either the method of determination of k in the dispersion law E(k) is unsuitable or the low-energy excitations are not phonons.

The momentum-shell renormalization group (RG) is used to study the condensation of interacting Bose gases without and with disorder. First of all, for the homogeneous disorder-free Bose gas the interaction-induced shifts in the critical temperature and chemical potential are determined up to second order in the scattering length. The approach does not make use of dimensional reduction and is thus independent of previous derivations. Secondly, the RG is used together with the replica method to study the interacting Bose gas with delta-correlated disorder. The flow equations are derived and found to reduce, in the high-temperature limit, to the RG equations of the classical Landau-Ginzburg model with random-exchange defects. The random fixed point is used to calculate the condensation temperature under the combined influence of particle interactions and disorder.

An interacting Bose gas atT=0, with a pair Hamiltonian where there exists a generalized condensation in state is investigated up to an upper limitk0 in momentum space. Corresponding to the idea that an ideal Bose gas can be represented by a coherent state, each operator is replaced by a coherent field operator. It is then found that for a particular choice of the potential, the transformations and procedures of Bogoliubov type lead to a stable energy spectrum with a gap at finite volume, but to a photon spectrum at infinite volume. Though a major portion of the interaction in momentum space is negative, a minimum point in the energy-density diagram is indicated.

These Ecole polytechnique federale de Lausanne EPFL, n° 248 (1976) Reference doi:10.5075/epfl-thesis-248Print copy in library catalog Record created on 2005-03-16, modified on 2016-08-08

We present the description of thermodynamic properties of a Bose gas by considering the hierarchy of equations satisfied by imaginary-time Green functions. In the case of an ideal gas the appearance of the off-diagonal long-range order (presence of condensate) is derived in a particularly simple way. The simplicity of the analysis persists at the level of the mean-field theory. Using Kac's scaling of the repulsive binary potential we then determine asymptotic corrections to the mean-field theory for thermodynamic states far from the region of the mean-field condensation. Our predictions are in accordance with rigorous results obtained previously, whereas the Hartree-Fock approximation is shown to neglect a whole class of equally important contributions.

Fluctuations in Ideal and Interacting Bose–Einstein Condensates: From the Laser Phase Transition Analogy to Squeezed States and Bogoliubov Quasiparticles

A nonlinear elliptic-hyperbolic system of partial differential equations which is a simplified form of the equations of viscous, compressible, barotropic flow at steady-state is studied. A boundary value problem for the system on a strip $D = (0,\alpha ) \times ( - \infty ,\infty )$ is considered. Zero boundary conditions for the velocities u and v on the sides of the strip D are imposed, and for pressure $p(0,y) = p_0 (y)$ is imposed, where $p_0 (y)$ has a jump at $y = 0$. Jump conditions for the system show that u and v are continuous. However, their derivatives and pressure have a curve of discontinuity. With sufficient small width of the strip D the Schauder fixed-point theorem gives a solution with a curve of discontinuity. The results suggest that there are two-dimensional, viscous, barotropic, steady-state compressible flows with discontinuity.

I demonstrate that $^4$He adsorbates in carbon nanotube materials can be treated as one-dimensional interacting gas of spinless bosons for temperatures below 8 K and for coverages such that all the adsorbates are in the groove positions of the carbon nanotube bundles. The effects of adsorbate-adsorbate interactions are studied within the scheme of virial expansion approach. The theoretical predictions for the specific heat of the interacting adsorbed gas are given.

We study the condensation of a homogeneous interacting Bose gas in the presence of a superimposed weak periodic potential. In contrast to, e.g., a harmonic trap, the periodic potential does not reduce the significance of long-wavelength fluctuations and thus influences the phase transition in a more profound way. We present general thermodynamic descriptions of the system above condensation in terms of mean-field as well as renormalization-group theory. Our approaches are based on describing the interplay between the effects of the interactions and the potential by means of a renormalization of the potential amplitude. These theories are then used to study the critical chemical potential and particle density and their dependence on the properties of the applied potential. A main result of our investigations concerns the characteristic nonperturbative momentum scale kc of the critical interacting Bose gas. It is shown that the existence of this momentum scale becomes macroscopically manifest in the behavior of the critical thermodynamic properties.

We theoretically investigate the equation of state and Tan's contact of a non-degenerate three dimensional Bose gas near a broad Feshbach resonance, within the framework of large-$N$ expansion. Our results agree with the path-integral Monte Carlo simulations in the weak-coupling limit and recover the second-order virial expansion predictions at strong interactions and high temperatures. At resonance, we find that the chemical potential and energy are significantly enhanced by the strong repulsion, while the entropy does not change significantly. With increasing temperature, the two-body contact initially increases and then decreases like $T^{-1}$ at large temperature, and therefore exhibits a peak structure at about $4T_{c0}$, where $T_{c0}$ is the Bose-Einstein condensation temperature of an ideal, non-interacting Bose gas. These results may be experimentally examined with a non-degenerate unitary Bose gas, where the three-body recombination rate is substantially reduced. In particular, the non-monotonic temperature dependence of the two-body contact could be inferred from the momentum distribution measurement.

Light-tissue interactions are commonly studied for near infra-red (NIR) spectroscopy for imaging and diagnosis. While imaging is bound to the surface due to the high tissue scattering, many diagnosis methods, such as the photoplethysmograph (PPG) and pulse oximeter, focus on sensing instead of imaging. There are two approaches for the investigation of light-tissue interactions: one numerical and the other analytical. The most common numerical method is the Monte Carlo (MC) simulation, which is a statistical study of photon migration from the optical properties of the different tissue regions. However, the yield of such statistical approaches is limited. Analytical methods are commonly based on the diffusion theory, yet they are inaccurate near the light source. There are several solution methods, where extrapolated boundary conditions lead to a more accurate solution. Previously, we proposed measuring the full scattering profile (FSP), which is the angular intensity distribution, of cylindrical tissues. MC simulations revealed that the FSP has a fixed intensity point, named the iso-pathlength (IPL) point, which does not depend on the reduced scattering coefficient. The location of this point depends solely on the geometry. In this study, we derived the FSP from the steady state diffusion theory based on the extrapolated zeroboundary condition. A comparison of different reduced scattering coefficients revealed the IPL point, where the intensity remains constant. Furthermore, a MC simulation was performed under the same geometry and optical properties. We show that the position of the IPL according to the diffusion theory is in agreement with the MC simulation.

The specific heat of an ideal semi-infinite ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) Bose gas is calculated numerically using a number of boundary conditions. The results provide a suitable comparison with the analytical calculations of Fisher and Barber and of Pathria. The agreement is very good except for the Neumann boundary condition where the numerical result shows deviation from that of Pathria. However, the earlier assumption of the existence of a "correlation length" appears to have little physical basis. For the hard-wall case, the anomaly in the specific heat is probably due to constraint on particle density. The distinctive effect of such a boundary condition (${\ensuremath{\Psi}}_{s}=0$) is explored more thoroughly in maximal assemblies ($L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$) where its effect on specific heat shows up prominently.

We consider the path-integral representation of the ideal Bose gas under various boundary conditions. We show that Bose--Einstein condensation occurs at the famous critical density threshold, by proving that its $1$-particle-reduced density matrix exhibits off-diagonal long-range order above that threshold, but not below. Our proofs are based on the well-known Feynman--Kac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting Poisson--Dirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (a part from a standard eigenvalue expansion), using elementary tools like Markov's inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets and asymptotics for random walks with Pareto-distributed steps.