Finite-size effects in a D-dimensional ideal Fermi gas

By using the Euler-MacLaurin formula, this paper studies the thermodynamic properties of an ideal Fermi gas confined in a D-dimensional rectangular container. The general expressions of the thermodynamic quantities with the finite-size corrections are given explicitly and the effects of the size and shape of the container on the properties of the system are discussed. It is shown that the corrections of the thermodynamic quantities due to the finite-size effects are significant to be considered for the case of strong degeneracy but negligible for the case of weak degeneracy or non-degeneracy. It is important to find that some familiar conclusions under the thermodynamic limit are no longer valid for the finite-size systems and there are some novel characteristics resulting from the finite-size effects, such as the nonextensivity of the system, the anisotropy of the pressure, and so on.

Finite-size analysis of continuous-variable quantum key distribution with entanglement in the middle

Continuous-variable quantum key distribution(CVQKD) protocols with entanglement in the middle(EM) enable long maximal transmission distances for quantum communications. For the security analysis of the protocols, it is usually assumed that Eve performs collective Gaussian attacks and there is a lack of finite-size analysis of the protocols. However,in this paper we consider the finite-size regime of the EM-based CVQKD protocols by exposing the protocol to collective attacks and coherent attacks. We differentiate between the collective attacks and the coherent attacks while comparing asymptotic key rate and the key rate in the finite-size scenarios. Moreover, both symmetric and asymmetric configurations are collated in a contrastive analysis. As expected, the derived results in the finite-size scenarios are less useful than those acquired in the asymptotic regime. Nevertheless, we find that CVQKD with entanglement in the middle is capable of providing fully secure secret keys taking the finite-size effects into account with transmission distances of more than 30 km.

Assessment of force models on finite-sized particles at finite Reynolds numbers

Finite-sized inertial spherical particles are fully-resolved with the immersed boundary projection method(IBPM)in the turbulent open-channel flow by direct numerical simulation(DNS).The accuracy of the particle surface force models is investigated in comparison with the total force obtained via the fully-resolved method.The results show that the steady-state resistance only performs well in the streamwise direction,while the fluid acceleration force,the added-mass force,and the shear-induced Saffman lift can effectively compensate for the large-amplitude and high-frequency characteristics of the particle surface forces,especially for the wall-normal and spanwise components.The modified steady-state resistance with the correction effects of the acceleration and the fluid shear can better represent the overall forces imposed on the particles,and it is a preferable choice of the surface force model in the Lagrangian point-particle method.

Transition and self-sustained turbulence in dilute suspensions of finite-size particles

We study the transition to turbulence of channel flow of finite-size particle suspensions at low volume fraction, i.e., φ ≈0.001. The critical Reynolds number above which turbulence is sustained reduces to Re ≈ 1675, in the presence of few particles, independently of the initial condition, a value lower than that of the corresponding single-phase flow, i.e., Re ≈1775. In the dilute suspension, the initial arrangement of the particles is important to trigger the transition at a fixed Reynolds number and particle volume fraction. As in single phase flows, streamwise elongated disturbances are initially induced in the flow. If particles can induce oblique disturbances with high enough energy within a certain time, the streaks breakdown, flow experiences the transition to turbulence and the particle trajectories become chaotic, Otherwise, the streaks decay in time and the particles immigrate towards the channel core in a laminar flow.

Effects of finite-size neutrally buoyant particles on the turbulent channel flow at a Reynolds number of 395

A direct-forcing fictitious domain(DFFD) method is used to perform fully resolved numerical simulations of turbulent channel flows laden with large neutrally buoyant particles. The effects of the particles on the turbulence(including the mean velocity,the root mean square(RMS) of the velocity fluctuation, the probability density function(PDF) of the velocity, and the vortex structures) at a friction Reynolds number of 395 are investigated. The results show that the drag-reduction effect caused by finite-size spherical particles at low particle volumes is negligibly small. The particle effects on the RMS velocities at Re_τ = 395 are significantly smaller than those at Re_τ = 180, despite qualitatively the same effects, i.e., the presence of particles decreases the maximum streamwise RMS velocity near the wall via weakening the large-scale streamwise vortices,and increases the transverse and spanwise RMS velocities in the vicinity of the wall by inducing smaller-scale vortices. The effects of the particles on the PDFs of the fluid fluctuating velocities normalized with the RMS velocities are small, regardless of the particle size, the particle volume fraction, and the Reynolds number.

Finite-size analysis of eight-state continuous-variable quantum key distribution with the linear optics cloning machine

We propose a method to improve the secret key rate of an eight-state continuous-variable quantum key distribution（CVQKD） by using a linear optics cloning machine（LOCM）. In the proposed scheme, an LOCM is exploited to compensate for the imperfections of Bob＇s apparatus, so that the generated secret key rate of the eight-state protocol could be well enhanced. We investigate the security of our proposed protocol in a finite-size scenario so as to further approach the practical value of a secret key rate. Numeric simulation shows that the LOCM with reasonable tuning gain λ and transmittance τcan effectively improve the secret key rate of eight-state CVQKD in both an asymptotic limit and a finite-size regime.Furthermore, we obtain the tightest bound of the secure distance by taking the finite-size effect into account, which is more practical than that obtained in the asymptotic limit.

Influences of finite-size effects on the self-organized critical-ity of forest-fire model

The influences of finite-size effects on the self-organized criticality (SOC) of the traditional forest-fire model are investigated by means of a new method. The forest size is originally set to a value much greater than the correla-tion length of the forest. Finite-size effects are then studied by equally dividing the forest into more and more separate subsystems on condition that the forest size, igniting prob-ability and planting probability are invariant. A new phe-nomenon, i.e. the finite-size effects with one-side frequency peak, is observed. The boundary between two neighboring subsystems can be regarded as a firebreak. The concept of ’separation ability’ is introduced to represent the probability for the firebreak to block off the fire successfully. Restrain-ing effects of separation ability on finite-size effects are ana-lyzed. Finite-size effects and separation ability, as well as their relations are found to have practical importance to the actual forest-fire protection.

Modeling of finite-size droplets and particles in multiphase flows

The conventional point-particle approach for treating the dispersed phase in a continuous flowfield is extended by taking into account the effect of finite particle size, using a Gaussian interpolation from Lagrangian points to the Eulerian field.The inter-phase exchange terms in the conservation equations are distributed over the volume encompassing the particle size, as opposed to the Dirac delta function generally used in the point-particle approach.The proposed approach is benchmarked against three different flow configurations in a numerical framework based on large eddy simulation（LES） turbulence closure.First, the flow over a circular cylinder is simulated for a Reynolds number of 3900 at 1 atm pressure.Results show good agreement with experimental data for the mean streamwise velocity and the vortex shedding frequency in the wake region.The calculated flowfield exhibits correct physics, which the conventional point-particle approach fails to capture.The second case deals with diesel jet injection in quiescent environment over a pressure range of 1.1–5.0 MPa.The calculated jet penetration depth closely matches measurements.It decreases with increasing chamber pressure, due to enhanced drag force in a denser fluid environment.Finally, water and acetone jet injection normal to air crossflow is studied at1 atm.The calculated jet penetration and Sauter mean diameter of liquid droplets compare very well with measurements.

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|^(-(d-2+η))and a finite-size scaling function of the variables r/L and tL^(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.

Size effect on light propagation modulation near band edges in one-dimensional periodic structures

Periodic photonic structures can provide rich modulation in propagation of light due to well-defined band structures.Especially near band edges,light localization and the effect of near-zero refractive index have attracted wide attention.However,the practically fabricated structures can only have finite size,i.e.,limited numbers of periods,leading to changes of the light propagation modulation compared with infinite structures.Here,we study the size effect on light localization and near-zero refractive-index propagation near band edges in one-dimensional periodic structures.Near edges of the band gap,as the structure's size shrinks,the broadening of the band gap and the weakening of the light localization are discovered.When the size is small,an added layer on the surface will perform large modulation in the group velocity.Near the degenerate point with Dirac-like dispersion,the zero-refractive-index effects like the zero-phase difference and near-unity transmittance retain as the size changes,while absolute group velocity fluctuates when the size shrinks.

Improving source-in-the-middle continuous-variable quantum key distribution using a heralded hybrid linear amplifier

A hybrid linear amplifier is inserted at the output of the source-in-the-middle distribution protocol to overcome the shortcomings of the transmission distance.The modified protocol aims to maintain a high key rate for long-distance transmission under high noise.It has the potential to significantly broaden the application range of the continuous variable quantum key distribution protocol.The effects of amplifier parameters and noise on the modified protocol are analyzed in detail with regard to applying it to a practical system.To make the simulation more realistic,the effect of finite size on the new protocol is taken into account.It will serve as a guideline for the future use of hybrid linear amplifiers.Different parameters can be adjusted to achieve the best performance for key rates of different quantum channels.

Finite size specimens with cracks of icosahedral Al Pd Mn quasicrystals

Icosahedral quasicrystals are the most important and thermodynamically stable in all about 200 kinds of quasicrystals currently observed. Beyond the scope of classical elasticity, apart from a phonon displacement field, there is a phason displacement field in the elasticity of the quasicrystal, which induces an important effect on the mechanical properties of the material and makes an analytical solution difficult to obtain. In this paper, a finite element algorithm for the static elasticity of icosahedral quasicrystals is developed by transforming the elastic boundary value problem of the icosahedral quasicrystals into an equivalent variational problem. Analytical and numerical solutions for an icosahedral A1-Pd-Mn quasicrystal cuboid subjected to a uniaxial tension with different phonon-phason coupling parameters are given to verify the validity of the numerical approach. A comparison between the analytical and numerical solutions of the specimen demonstrates the accuracy and efficiency of the present algorithm. Finally, in order to reveal the fracture behavior of the icosahedral A1-Pd-Mn quasicrystal, a cracked specimen with a finite size of matter is investigated, both with and without phonon-phason coupling. Meanwhile, the geometry factors are calculated, including the stress intensity factor and the crack opening displacement for the finite-size specimen. Computational results reveal the importance of pbonon-phason coupling effect on the icosahedral A1-Pd-Mn quasicrystal. Furthermore, the finite element procedure can be used to solve more complicated boundary value problems.

Continuum percolation of porous media via random packing of overlapping cube-like particles

The pore configuration in porous medium is assumed to be the randomly distributed cube-like particles which can overlap each other in the periodic cubic domain, and the impact of particle characteristics on the percolation property of these cube-like particle packing systems is analyzed.Firstly, by combining the percolation models and finite-size scaling analysis, three numerical parameters(i.e., percolation transition width △L, local percolation threshold ψ_c(L), and correlation length exponent v) for the cube-like particle systems with shape parameter s in[1.0, +∞] are derived successively. Then, based on the relation between the percolation thresholdψ_c in infinite space and the local percolation threshold ψ_c(L), the corresponding ψ_c with s in[1.0, +∞] are further determined. It is shown from the study that the characteristics of cube-like particles have significant influence on the global percolation threshold ψ_c of the particle packing systems. As the parameter s increases from 1.0 to +∞, the percolation threshold ψ_c will go down persistently. When the surface of cube-like particles is cubical and spherical, respectively, the minimum and maximum thresholds ψ_c,min and ψ_c,max are obtained.

Quantum Monte Carlo study on the phase transition for a generalized two-dimensional staggered dimerized Heisenberg model

In order to gain a deeper understanding of the quantum criticality in the explicitly staggered dimerized Heisenberg models, we study a generalized staggered dimer model named the J0 J1 J2 model, which corresponds to the staggered j-j′ model on a square lattice and a honeycomb lattice when J1/J0 equals 1 and O, respectively. Using the quantum Monte Carlo method, we investigate all the quantum critical points of these models with J1/J0 changing from 0 to 1 as a function of coupling ratio a = J2/J0. We extract all the critical values of the coupling ratio ac for these models, and we also obtain the critical exponents v,β/ν, and η using different finite-size scaling ansatz,. All these exponents are not consistent with the three-dimensional Heisenberg universality class, indicating some unconventional quantum ciriteial points in these models.

Three-party reference frame independent quantum key distribution protocol

We present a three-party reference frame independent quantum key distribution protocol which can be implemented without any alignment of reference frames between the sender and the receiver.The protocol exploits entangled states to establish a secret key among three communicating parties.We derive the asymptotic key rate for the proposed protocol against collective attacks and perform a finite-size key security analysis against general attacks in the presence of statistical fluctuations.We investigate the impact of reference frame misalignment on the stability of our protocol,and we obtain a transmission distance of 180 km,200 km,and 230 km for rotation of reference framesβ=π/6,β=π/8 andβ=0,respectively.Remarkably,our results demonstrate that our proposed protocol is not heavily affected by an increase in misalignment of reference frames as the achievable transmission distances are still comparable to the case where there is no misalignment in reference frames(whenβ=0).We also simulate the performance of our protocol for a fixed number of signals.Our results demonstrate that the protocol can achieve an effective key generation rate over a transmission distance of about 120 km with realistic 107 finite data signals and approximately achieve 195 km with 109 signals.Moreover,our proposed protocol is robust against noise in the quantum channel and achieves a threshold error rate of 22.7%.

Thermodynamic properties of a finite Bose gas in a harmonic trap

We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.

Numerical study of anomalous dynamic scaling behaviour of (1+1)-dimensional Das Sarma-Tamborenea model

In order to discuss the finite-size effect and the anomalous dynamic scaling behaviour of Das Sarma-Tamborenea growth model, the （1＋1）-dimensional Das Sarma-Tamborenea model is simulated on a large length scale by using the kinetic Monte-Carlo method. In the simulation, noise reduction technique is used in order to eliminate the crossover effect. Our results show that due to the existence of the finite-size effect, the effective global roughness exponent of the （1＋1）-dimensional Das Sarma-Tamborenea model systematically decreases with system size L increasing when L 〉 256. This finding proves the conjecture by Aarao Reis[Aarao Reis F D A 2004 Phys. Rev. E 70 031607]. In addition, our simulation results also show that the Das Sarma-Tamborenea model in 1＋1 dimensions indeed exhibits intrinsic anomalous scaling behaviour.

Non-geometrical effects on Gaussian beams transmitting through a thin dielectric slab

It is shown that a Gaussian light beam transmitting through a planar thin dielectric slab in the air undergoes four different effects, i.e. lateral Goos-Hanchen-like （GHL） displacement, angular deflection, width modification and longitudinal focal shift as compared with the results predicted by geometrical optics. According to the Taylor expansion of the exponent of transmission coefficient when expressed as an exponential form, the lateral GHL displacement and the angular deflection are the first-order effects and can be negative or positive. The width modification and the longitudinal focal shift are the second-order effects and can also be positive or negative. Owing to the waist-width dependent term, the non-geometrical effects of transmitted beam are not identical with the non-specular effects of reflected beam. The conditions for the validity of those effects are suggested and numerical simulations are also given.

Optical responses of metallic plasmonic arrays under the localized excitation

The metallic plasmonic array that can support both propagating surface plasmon polaritons(PSPPs)and localized surface plasmon resonance(LSPR)possesses rich optical properties and remarkable optical performance,making it a powerful platform for applications in photonics,chemistry,and materials.For practical applications,the excitation spot is usually smaller than the area of metal arrays.It is thus imperative to address“how many array units are enough?”towards a rational design of plasmonic nanostructures.Herein,we employed focused ion beam(FIB)to precisely fabricate a series of plasmonic array structures with increased unit number.By utilizing photoluminescence(PL)and surface-enhanced Raman spectroscopy(SERS),we found that the array units outside the excitation spot still have a significant impact on the optical response within the spot.Combined with the numerical simulation,we found that the boundary of the finite array leads to the loss of PSPP outside the excitation point,which subsequently affects the coupling of PSPP and LSPR in the excitation spot,leading to variations in PL and SERS intensity.Based on the findings,we further tuned the LSPR mode of the metal arrays by electrodeposition to obtain strong near-field enhancement without any influence on the PSPP mode.This work advances the understanding of near-field and far-field optical behavior in finite-size array structures and provides guidance for designing highly-efficient photonic devices.

Experimental quantum secret sharing based on phase encoding of coherent states

Quantum secret sharing(QSS)is one of the basic communication primitives in future quantum networks which addresses part of the basic cryptographic tasks of multiparty communication and computation.Nevertheless,it is a challenge to provide a practical QSS protocol with security against general attacks.A QSS protocol that balances security and practicality is still lacking.Here,we propose a QSS protocol with simple phase encoding of coherent states among three parties.Removing the requirement of impractical entangled resources and the need for phase randomization,our protocol can be implemented with accessible technology.We provide the finite-key analysis against coherent attacks and implement a proof-of-principle experiment to demonstrate our scheme’s feasibility.Our scheme achieves a key rate of 85.3 bps under a 35 d B channel loss.Combined with security against general attacks and accessible technology,our protocol is a promising candidate for practical multiparty quantum communication networks.