Finite Number Of Particles Research Articles

Parallel Simulation of Cellular Flow in Microvessels Using a Particle Method

We developed a numerical method for large-scale simulations of cellular flow in microvessels. We employed a particle method, where all blood components were modeled using a finite number of particles. Red blood cell deformation was modeled by a spring network of membrane particles. A domain decomposition method was used for parallel implementation on distributed memory systems. In a strong scaling test up to 64 CPU cores, we obtained a linear speedup with the number of CPU cores, and demonstrated that our model can simulate O(10 3 ) red blood cells in vessels a few tens of micrometers in diameter. For quantitative validation, we analyzed the Fahraeus effect and the formation of a cell-depleted peripheral layer. Simulations were performed for tube hematocrit ranging from 20 to 45%, and microvessel diameters from 9 to 50 μm. Our numerical results were in good agreement with previous experimental results both for the discharge hematocrit and cell-depleted peripheral layer thickness.

Stability of Discrete Breathers in Diatomic Fermi-Pasta-Ulam Type Lattices

We consider the discrete breathers in one-dimensional diatomic Fermi-Pasta-Ulam type lattices. A discrete breather in the limit of zero mass ratio, i.e., the anti-continuous limit, consists of a finite number of in-phase or anti-phase excited light particles, separated by particles at rest. Existence of the discrete breathers is proved for small mass ratio by continuation from the anti-continuous limit. We prove that the discrete breather is linearly stable if it is continued from a solution consisting of alternating anti-phase excited particles, otherwise it is linearly unstable, near the anti-continuous limit.

Coagulation–fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus

We develop a coagulation–fragmentation model to study a system composed of a small number of stochastic objects moving in a confined domain, that can aggregate upon binding to form local clusters of arbitrary sizes. A cluster can also dissociate into two subclusters with a uniform probability. To study the statistics of clusters, we combine a Markov chain analysis with a partition number approach. Interestingly, we obtain explicit formulas for the size and the number of clusters in terms of hypergeometric functions. Finally, we apply our analysis to study the statistical physics of telomeres (ends of chromosomes) clustering in the yeast nucleus and show that the diffusion–coagulation–fragmentation process can predict the organization of telomeres.

THE EFFECTS OF A FINITE NUMBER OF PARTICLES ON TWO TRAPPED QUANTUM GASES

The effects of a finite number of particles on the thermodynamic properties of ideal Bose and Fermi gases trapped in any-dimensional harmonic potential are investigated. The orders of relative corrections to the thermodynamic quantities due to the finite number of particles are estimated in different situations. The results obtained for the two trapped quantum gases are compared, and consequently, it is shown that the finite-particle-number effects for the condensed Bose gas (a Bose gas with Bose–Einstein Condensation (BEC) occurring in the system) are much more significant than those for the Fermi gas and normal Bose gas (a Bose gas without BEC).

Bose–Einstein condensation of a [formula omitted]-deformed boson system in a harmonic potential trap

Within the framework of the theory of q-deformed bosons, we investigate Bose–Einstein condensation of a q-deformed boson system, which has finite particle number and is trapped in a harmonic potential trap, by using the approach of modified density of states. For such a trapped q-deformed boson system, we obtain the expressions of the effective critical temperature Tc, the ground state population N0/N and the specific heat C. Further numerical calculation shows that Bose–Einstein condensation of such a system has some particular and interesting features which are different from those of a trapped standard boson system. We consider that this work may provide much insight into the theory of q-deformed bosons, and may also be useful for the further study on Bose–Einstein condensation of a concrete trapped q-deformed boson system.

Isobars of an ideal Bose gas within the grand canonical ensemble

We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble, for a large yet finite number of particles, N. After solving the equation of the spinodal curve, we derive precise formulae for the supercooling and the superheating temperatures which reveal an N^{-1/3} or N^{-1/4} power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if N is greater than or equal to 14393. In particular, for the Avogadro's number of particles, the volume expands discretely about 10^5 times. Our results quantitatively agree with a previous study on the canonical ensemble within 0.1% error.

How close to two dimensions does a Lennard-Jones system need to be to produce a hexatic phase?

We report on a computer simulation study of a Lennard-Jones liquid confined in a narrow slit pore with tunable attractive walls. In order to investigate how freezing in this system occurs, we perform an analysis using different order parameters. Although some of the parameters indicate that the system goes through a hexatic phase, other parameters do not. This shows that to be certain whether a system of a finite particle number has a hexatic phase, one needs to study not only a large system, but also several order parameters to check all necessary properties. We find that the Binder cumulant is the most reliable one to prove the existence of a hexatic phase. We observe an intermediate hexatic phase only in a monolayer of particles confined such that the fluctuations in the positions perpendicular to the walls are less than 0.15 particle diameters, i.e., if the system is practically perfectly 2D.

On quantum statistics for ensembles with a finite number of particles

The well-known Bose–Einstein and Fermi–Dirac quantum distributions can be considered as stationary solutions of kinetic equations for the mean occupation numbers in an ideal gas of an arbitrary finite number of identical particles.

Phase transitions for an ideal Bose condensate trapped in a quartic potential

Based on the classification scheme of phase transitions, we study the phase transitions for an ideal Bose gas with a finite number of particles confined in a three-dimensional quartic trap. We show that the phase transition of an ideal Bose gas in the three-dimensional quartic trap is of third order for finite particle numbers, quite different from the fact that the phase transition is of first order in the thermodynamic limit. We discuss the effects of finite particle numbers on the nature of the phase transitions, and determine the dependence of transition temperature on particle number.

Soliton solutions of a Calogero model in a harmonic potential

A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of the ‘soliton’ solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit, the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite-dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.

Condensate fluctuations of interacting Bose gases within a microcanonical ensemble

Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.

Numerical implementation of mixing and molecular transport in LES/PDF studies of turbulent reacting flows

Probability Density Function (PDF) methods in combination with Large Eddy Simulations (LES) are a powerful tool for studying turbulent reacting flow problems and we are interested in the implementation of mixing and molecular transport in LES/PDF methods. The numerical methodology used for solution procedure is the hybrid particle/mesh method and a fractional step scheme is used to solve for transport, reaction and mixing sequentially. Mixing is modeled using the Interaction by Exchange with the Mean (IEM) model and the effects of molecular transport are incorporated as a mean drift term in the mixing step. This methodology avoids spurious production of scalar variance and also allows direct incorporation of differential diffusion effects. In this study, various numerical implementations of mixing and molecular transport are presented and evaluated, using the Method of Manufactured Solutions (MMS), for (1) accuracy, (2) detailed conservation, (3) realizability, and (4) stability. Moreover, the methodology is shown to be successful in capturing the effects of differential diffusion accurately with the additional property of ensuring realizability of species mass fractions. Finally and most importantly, we describe a new variance reduction technique by way of an implicit smoothing methodology. This smoothing scheme is shown to satisfy conservation, boundedness and regularity criteria. Moreover, for an appropriate choice of the smoothing length scale, significant improvements in accuracy can be achieved for an incremental increase in computational cost. Also, it is shown that with smoothing on a length scale greater than the grid size, the bias and statistical errors due to there being a finite number of particles in the Lagrangian Monte Carlo simulations scale as N tot - 1 and N tot - 1 / 2 respectively, where N tot is the total number of particles in the computational domain, whereas without smoothing these errors scale as N pc - 1 and N pc - 1 / 2 , where N pc is the much smaller number of particles in a computational cell.

Application of green functions to finite Fermi systems with fixed numbers of particles and strong pairing

The single-particle Green function formalism (Gor’kov’s method) is applied to calculate the energies of finite superconducting Fermi systems with a finite number of particles. It is shown that strong pairing leads to equations similar to those from the Bardeen-Cooper-Schrieffer theory.

Bose-Einstein condensation with a finite number of particles in a power-law trap

Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number $N$ of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature $T_c$ in terms of a power series in $x_0=\epsilon_0/k_BT_c$, where $\epsilon_0$ denotes the zero-point energy of the trapping potential. This expression, which applies in cartesian, cylindrical and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of $x_0$ up to the second order. We show that, for a harmonic trap, the well known first order shift of the critical temperature $\Delta T_c/T_c \propto N^{-1/3}$ is inaccurate when $N \leqslant 10^{5}$, the next order (proportional to $N^{-1/2}$) being significant. We also show that finite size effects on the condensation temperature cancel out in a cubic trapping potential, e.g. $V(\mathbi{r}) \propto r^3$. Finally, we show that in a generic power-law potential of higher order, e.g. $V(\mathbi{r}) \propto r^\alpha$ with $\alpha > 3$, the shift of the critical temperature becomes positive. This effect provides a large increase of $T_c$ for relatively small atom numbers. For instance, an increase of about +40% is expected with $10^4$ atoms in a $V(\mathbi{r}) \propto r^{12}$ trapping potential.

Condensation in the Inclusion Process and Related Models

We study condensation in several particle systems related to the inclusion process. For an asymmetric one-dimensional version with closed boundary conditions and drift to the right, we show that all but a finite number of particles condense on the right-most site. This is extended to a general result for independent random variables with different tails, where condensation occurs for the index (site) with the heaviest tail, generalizing also previous results for zero-range processes. For inclusion processes with homogeneous stationary measures we establish condensation in the limit of vanishing diffusion strength in the dynamics, and give several details about how the limit is approached for finite and infinite systems. Finally, we consider a continuous model dual to the inclusion process, the so-called Brownian energy process, and prove similar condensation results.

A characteristic length scale in confined elastic buckling of a force chain

We examine the confined elastic buckling of a single N-particle force chain. For a given set of material parameters, a characteristic load-carrying capacity is attained for a clearly defined range of force chain lengths N ≥ L*. The rotation and translation of particles along periodically located segments of the critical buckling mode, each of length L*, are on average consistent with those for particles inside the shear band. Preliminary results from postbuckling analysis show that the critical buckling mode is unstable and that buckling culminates in a localized response over one of these segments where observed shear band kinematics prevail. Thus, the localized buckling response is characterized by the rotation of a finite number of particles with a clearly defined length L*. For a wide range of material properties, L* is around eight particle diameters—the observed shear band thickness for many granular materials, most notably sand.

Counter-ions at charged walls: Two-dimensional systems

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double layer), or between two such lines (interacting double layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant [Formula: see text] is small, is briefly recapitulated (the coupling constant is defined as [Formula: see text] [Formula: see text] [Formula: see text] e (2) , where [Formula: see text] is the inverse temperature, and e the counter-ion charge). The opposite limit ( [Formula: see text] [Formula: see text] ∞ is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. These two limiting results are compared at intermediary values of the coupling constant [Formula: see text] = 2[Formula: see text] ([Formula: see text] = 1, 2, 3) , to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at [Formula: see text] = 2 and up to relatively large finite N at [Formula: see text] = 4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at [Formula: see text] = 6 . The like-charge attraction regime, possible for large [Formula: see text] but precluded at mean-field level, survives for [Formula: see text] = 4 and 6, but disappears at [Formula: see text] = 2 .

По поводу квантовых статистик для ансамблей с конечным числом частиц

По поводу квантовых статистик для ансамблей с конечным числом частиц

Existence and stability of discrete breathers in diatomic Fermi–Pasta–Ulam type lattices

We consider discrete breathers in one-dimensional diatomic Fermi–Pasta–Ulam type lattices. A discrete breather in the limit of zero mass ratio, i.e. the anti-continuous limit, consists of a finite number of in-phase or anti-phase excited light particles, separated by particles at rest. The existence of discrete breathers is proved for small mass ratio by continuation from the anti-continuous limit. We prove that the discrete breathers are all unstable near the anti-continuous limit, except for those continued from solutions consisting of alternating anti-phase excited particles.

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈 n 0〉/ N and specific heat C N around the transition temperature T c ( i. e., t = T/ T c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ≃ 0.068.