scholarly journals Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 645
Author(s):  
Paolo De Gregorio ◽  
Lamberto Rondoni
Keyword(s):  
Three Dimensional ◽  
Intermediate Stage ◽  
Optimal Number ◽  
Three Dimensions ◽  
Mathematical Work ◽  
Limiting Factor ◽  
One Dimensional ◽  
Occupation Numbers ◽  
Bose Einstein ◽  
Number Of Particles

From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of ( 2 / 3 ) -rds in one dimension, and ( 3 / 5 ) -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.

scholarly journals Topological correlators and surface defects from equivariant cohomology

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Rodolfo Panerai ◽  
Antonio Pittelli ◽  
Konstantina Polydorou
Keyword(s):  
Quantum Mechanics ◽  
Equivariant Cohomology ◽  
Surface Defects ◽  
Star Product ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
The Novel ◽  
One Dimensional ◽  
Dual Quantum

Abstract We find a one-dimensional protected subsector of $$ \mathcal{N} $$ N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $$ \mathcal{N} $$ N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

On the Recovery of 3D Spatial Statistics of Particles from 1D Measurements: Implications for Airborne Instruments

2014 ◽  
Vol 31 (10) ◽  
pp. 2078-2087 ◽  
Author(s):  
Michael L. Larsen ◽  
Clarissa A. Briner ◽  
Philip Boehner
Keyword(s):  
Point Processes ◽  
Pair Correlation ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
Cloud Droplets ◽  
Sampling Variability ◽  
One Dimensional ◽  
Natural Sampling ◽  
The One

Abstract The spatial positions of individual aerosol particles, cloud droplets, or raindrops can be modeled as a point processes in three dimensions. Characterization of three-dimensional point processes often involves the calculation or estimation of the radial distribution function (RDF) and/or the pair-correlation function (PCF) for the system. Sampling these three-dimensional systems is often impractical, however, and, consequently, these three-dimensional systems are directly measured by probing the system along a one-dimensional transect through the volume (e.g., an aircraft-mounted cloud probe measuring a thin horizontal “skewer” through a cloud). The measured RDF and PCF of these one-dimensional transects are related to (but not, in general, equal to) the RDF/PCF of the intrinsic three-dimensional systems from which the sample was taken. Previous work examined the formal mathematical relationship between the statistics of the intrinsic three-dimensional system and the one-dimensional transect; this study extends the previous work within the context of realistic sampling variability. Natural sampling variability is found to constrain substantially the usefulness of applying previous theoretical relationships. Implications for future sampling strategies are discussed.

scholarly journals IDEAL-MODIFIED BOSONIC GAS TRAPPED IN AN ARBITRARY THREE-DIMENSIONAL POWER-LAW POTENTIAL

2012 ◽  
Vol 27 (31) ◽  
pp. 1250181 ◽  
Author(s):  
E. CASTELLANOS ◽  
C. LÄMMERZAHL
Keyword(s):  
Dispersion Relation ◽  
Power Law ◽  
Three Dimensional ◽  
Particle Dispersion ◽  
Einstein Condensate ◽  
Gravity Models ◽  
Condensation Temperature ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Number Of Particles

We analyze the effects caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose–Einstein condensate trapped in a generic three-dimensional power-law potential. We prove that the shift in the condensation temperature, caused by a deformed dispersion relation, described as a non-trivial function of the number of particles and the shape associated to the corresponding trap, could provide bounds for the parameters associated to such deformation. In addition, we calculate the fluctuations in the number of particles as a criterium of thermodynamic stability for these systems. We show that the apparent instability caused by the anomalous fluctuations in the thermodynamic limit can be suppressed considering the lowest energy associated to the system in question.

Estimating Three-Dimensional Space from Multiple Two-Dimensional Views

1993 ◽  
Vol 2 (1) ◽  
pp. 44-53 ◽  
Author(s):  
Kristinn R. Thorisson
Keyword(s):  
Visual Search ◽  
Eye Movement ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Three Dimensions ◽  
Limiting Factor ◽  
Single Subject ◽  
Search Performance ◽  
Two Dimensional

The most common visual feedback technique in teleoperation is in the form of monoscopic video displays. As robotic autonomy increases and the human operator takes on the role of a supervisor, three-dimensional information is effectively presented by multiple, televised, two-dimensional (2-D) projections showing the same scene from different angles. To analyze how people go about using such segmented information for estimations about three-dimensional (3-D) space, 18 subjects were asked to determine the position of a stationary pointer in space; eye movements and reaction times (RTs) were recorded during a period when either two or three 2-D views were presented simultaneously, each showing the same scene from a different angle. The results revealed that subjects estimated 3-D space by using a simple algorithm of feature search. Eye movement analysis supported the conclusion that people can efficiently use multiple 2-D projections to make estimations about 3-D space without reconstructing the scene mentally in three dimensions. The major limiting factor on RT in such situations is the subjects' visual search performance, giving in this experiment a mean of 2270 msec (SD = 468; N = 18). This conclusion was supported by predictions of the Model Human Processor (Card, Moran, & Newell, 1983), which predicted a mean RT of 1820 msec given the general eye movement patterns observed. Single-subject analysis of the experimental data suggested further that in some cases people may base their judgments on a more elaborate 3-D mental model reconstructed from the available 2-D views. In such situations, RTs and visual search patterns closely resemble those found in the mental rotation paradigm (Just & Carpenter, 1976), giving RTs in the range of 5-10 sec.

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

2013 ◽  
Vol 14 (1) ◽  
pp. 219-241 ◽  
Author(s):  
Linghua Kong ◽  
Jialin Hong ◽  
Jingjing Zhang
Keyword(s):  
Energy Conservation ◽  
Computational Cost ◽  
Three Dimensional ◽  
Schrodinger Equations ◽  
Pitaevskii Equation ◽  
Schrödinger Equations ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Bose Einstein ◽  
Energy Conservation Law

AbstractThe local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

Diffraction of Electrons by Two and Three Mutually Orthogonal Standing Light Waves

1975 ◽  
Vol 53 (2) ◽  
pp. 157-164 ◽  
Author(s):  
F. Ehlotzky
Keyword(s):  
Electron Scattering ◽  
Light Wave ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Standing Light Wave ◽  
Light Waves ◽  
The One ◽  
Rectangular Lattices

The one-dimensional problem of electron scattering by a standing light wave, known as the Kapitza–Dirac effect, is shown to be easily extendable to two and three dimensions, thus showing all characteristics of diffraction of electrons by simple two- and three-dimensional rectangular lattices.

scholarly journals Stability analysis of three-dimensional breather solitons in a Bose–Einstein condensate

2005 ◽  
Vol 461 (2063) ◽  
pp. 3561-3574 ◽  
Author(s):  
M Matuszewski ◽  
E Infeld ◽  
G Rowlands ◽  
M Trippenbach
Keyword(s):  
Three Dimensional ◽  
Einstein Condensate ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Stability Properties ◽  
Bose Einstein Condensate ◽  
The Stability ◽  
Bose Einstein ◽  
Dimensional Soliton

We investigated the stability properties of breather soliton trains in a three-dimensional Bose–Einstein condensate (BEC) with Feshbach-resonance management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is confined only by a one-dimensional optical lattice and we consider strong, moderate and weak confinement. By strong confinement we mean a situation in which a quasi two-dimensional soliton is created. Moderate confinement admits a fully three-dimensional soliton. Weak confinement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in effective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully confirmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully three-dimensional analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We find that in the three-dimensional case the stability region splits into two parts. However, as we tighten the confinement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for the quasi two-dimensional case. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust three-dimensional solitons in experiments in a BEC in a one-dimensional lattice.

A Unified Correction Method for the Acoustic Refraction (UCMAR) Caused by a Three Dimensional Shear Layer

2019 ◽  
Vol 105 (5) ◽  
pp. 732-742
Author(s):  
Lican Wang ◽  
Rongqian Chen ◽  
Yancheng You ◽  
Wenjun Wu ◽  
Ruofan Qiu
Keyword(s):  
Wind Tunnel ◽  
Temperature Gradient ◽  
Shear Layer ◽  
Ray Tracing ◽  
Three Dimensional ◽  
Correction Method ◽  
Three Dimensions ◽  
Acceptable Solution ◽  
One Dimensional ◽  
Planar Shear

The acoustic refraction induced by the shear layer in an open-jet wind tunnel causes a source shift when estimating the source location with beamforming. Traditional correction methods of the shear layer refraction are achieved through a computational eff ort or limited using one-dimensional or planar shear layer. In this paper, the unified correction method for acoustic refraction (UCMAR) is suitable for the three dimensions that covers several traditional forms. Meanwhile, the UCMAR can consider more general configurations, such as the temperature gradient on both sides of the shear layer and the off -axis source in a circular wind tunnel. These configurations are validated through a ray tracing technique and a benchmark example. In addition, the principle of time reverse is integrated with UCMAR. This results in a reverse UCMAR, which can quickly attain an acceptable solution.

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