scholarly journals GROSS–PITAEVSKII MODEL FOR NONZERO TEMPERATURE BOSE–EINSTEIN CONDENSATES

2006 ◽  
Vol 16 (09) ◽  
pp. 2713-2719 ◽  
Author(s):  
KESTUTIS STALIUNAS
Keyword(s):  
Power Spectra ◽  
Pitaevskii Equation ◽  
Nonzero Temperature ◽  
Ginzburg Landau ◽  
Long Wavelength ◽  
Occupation Numbers ◽  
Ginzburg Landau Equations ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical

Momentum distributions and temporal power spectra of nonzero temperature Bose–Einstein condensates are calculated using a Gross–Pitaevskii model. The distributions are obtained for micro-canonical ensembles (conservative Gross–Pitaevskii equation) and for grand-canonical ensembles (Gross–Pitaevskii equation with fluctuations and dissipation terms). Use is made of equivalence between statistics of the solutions of conservative Gross–Pitaevskii and dissipative complex Ginzburg–Landau equations. In all cases the occupation numbers of modes follow a 〈Nk〉 ∝ k-2 dependence, which corresponds in the long wavelength limit (k → 0) to Bose–Einstein distributions. The temporal power spectra are of 1/fα form, where: α = 2 - D/2 with D the dimension of space.

PROPAGATION OF SOLITARY WAVES ON LOSSY NONLINEAR TRANSMISSION LINES

2009 ◽  
Vol 23 (01) ◽  
pp. 1-18 ◽  
Author(s):  
E. KENGNE ◽  
R. VAILLANCOURT
Keyword(s):  
Solitary Waves ◽  
Transmission Lines ◽  
Small Amplitude ◽  
Reductive Perturbation Method ◽  
Ginzburg Landau ◽  
Nonlinear Transmission Lines ◽  
Nonlinear Transmission ◽  
Long Wavelength ◽  
Reductive Perturbation ◽  
Ginzburg Landau Equations

We present a lossy nonlinear transmission RLC line and show how the coupled Ginzburg–Landau equations can be derived in the small amplitude and long wavelength limit using a standard reductive perturbation method and complex expansion. Soliton-like solution of the simplified equation was searched and the instability of a class of phase-winding solutions was explored.

The perfect Bose-Einstein gas in the theory of the quantum-mechanical grand canonical ensembles

1952 ◽  
Vol 212 (1111) ◽  
pp. 552-558 ◽  
Keyword(s):  
Equation Of State ◽  
Energy Levels ◽  
Einstein Condensation ◽  
Quantum Mechanical ◽  
Condensation Phenomenon ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical ◽  
Different Parts ◽  
Number Of Particles

The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particles in the system, JV, goes to infinity. It is also shown that for finite N at the critical volume ∂ n p/∂v n is of the order N1/3 (n-2) in accordance with a result obtained by Wergeland & Hove-Storhoug.

scholarly journals PATTERN FORMING DYNAMICAL INSTABILITIES OF BOSE–EINSTEIN CONDENSATES

2004 ◽  
Vol 18 (05n06) ◽  
pp. 173-202 ◽  
Author(s):  
P. G. KEVREKIDIS ◽  
D. J. FRANTZESKAKIS
Keyword(s):  
Modulational Instability ◽  
Mean Field ◽  
Three Dimensional ◽  
Two Dimensions ◽  
One Dimension ◽  
Pitaevskii Equation ◽  
Long Wavelength ◽  
Pattern Forming ◽  
Bose Einstein ◽  
Dynamical Instabilities

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.

scholarly journals Fluctuations of observables for free fermions in a harmonic trap at finite temperature

2018 ◽  
Vol 4 (3) ◽  
Author(s):  
Aurélien Grabsch ◽  
Satya Majumdar ◽  
Grégory Schehr ◽  
Christophe Texier
Keyword(s):  
Numerical Simulations ◽  
Finite Temperature ◽  
General Relation ◽  
Harmonic Trap ◽  
Free Fermions ◽  
Linear Statistics ◽  
Occupation Numbers ◽  
Positive Axis ◽  
Canonical Ensembles ◽  
Grand Canonical

We study a system of 1D non-interacting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of certain observables of the form of sums of a function of the fermions’ positions, \mathcal{L}=\sum_n h(x_n)ℒ=∑nh(xn). Such observables are also called linear statistics of the positions. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions \mathcal{N}_+𝒩+ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations. We discuss the universality of the results with respect to the nature of the confinement.

scholarly journals Exponential Attractor for Coupled Ginzburg-Landau Equations Describing Bose-Einstein Condensates and Nonlinear Optical Waveguides and Cavities

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Gui Mu ◽  
Jun Liu
Keyword(s):  
Optical Waveguides ◽  
Nonlinear Optical ◽  
Lipschitz Continuity ◽  
Initial Boundary ◽  
Exponential Attractor ◽  
Ginzburg Landau ◽  
Exponential Attractors ◽  
Squeezing Property ◽  
Ginzburg Landau Equations ◽  
Bose Einstein

The existence of the exponential attractors for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities with periodic initial boundary is obtained by showing Lipschitz continuity and the squeezing property.

Statistical mechanics

2017 ◽  
Author(s):  
Michael P. Allen ◽  
Dominic J. Tildesley
Keyword(s):  
Statistical Mechanics ◽  
Potential Model ◽  
Quantum Corrections ◽  
Inhomogeneous Systems ◽  
Hard Constraints ◽  
Complex Liquids ◽  
Fluid Membranes ◽  
Dynamical Properties ◽  
Canonical Ensembles ◽  
Grand Canonical

This chapter contains the essential statistical mechanics required to understand the inner workings of, and interpretation of results from, computer simulations. The microcanonical, canonical, isothermal–isobaric, semigrand and grand canonical ensembles are defined. Thermodynamic, structural, and dynamical properties of simple and complex liquids are related to appropriate functions of molecular positions and velocities. A number of important thermodynamic properties are defined in terms of fluctuations in these ensembles. The effect of the inclusion of hard constraints in the underlying potential model on the calculated properties is considered, and the addition of long-range and quantum corrections to classical simulations is presented. The extension of statistical mechanics to describe inhomogeneous systems such as the planar gas–liquid interface, fluid membranes, and liquid crystals, and its application in the simulation of these systems, are discussed.

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