Finite Number Of Particles Research Articles

Elastic fields and effective moduli of particulate nanocomposites with the Gurtin–Murdoch model of interfaces

A complete solution has been obtained for periodic particulate nanocomposite with the unit cell containing a finite number of spherical particles with the Gurtin–Murdoch interfaces. For this purpose, the multipole expansion approach by Kushch et al. [Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L., 2011. Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59, 1702–1716] has been further developed and implemented in an efficient numerical algorithm. The method provides accurate evaluation of local fields and effective stiffness tensor with the interaction effects fully taken into account. The displacement vector within the matrix domain is found as a superposition of the vector periodic solutions of Lamé equation. By using local expansion of the total displacement and stress fields in terms of vector spherical harmonics associated with each particle, the interface conditions are fulfilled precisely. Analytical averaging of the local strain and stress fields in matrix domain yields an exact, closed form formula (in terms of expansion coefficients) for the effective elastic stiffness tensor of nanocomposite. Numerical results demonstrate that elastic stiffness and, especially, brittle strength of nanoheterogeneous materials can be substantially improved by an appropriate surface modification.

Branching diffusion in inhomogeneous media

We investigate the long-time evolution of branching diffusion processes (starting with a finite number of particles) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical regime, we describe the asymptotics of the number of particles in a given domain. In the sub-critical and critical regimes, we show that the limiting number of particles is finite and describe its distribution.

The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

In this paper we find explicit formulas for: (1) Green's function for a system of one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.

Complex Brownian motion representation of the Dyson model

Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of $h$-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.

A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin isℋ1=L2(M),whereMis ad-dimensional unit torusM=ℝd/ℤdwith a flat metric. The phase space ofkspins isℋk=L2sym(Mk), the subspace ofL2(Mk)formed by functions symmetric under the permutations of the arguments. The Fock spaceH=⊕k=0,1,…ℋkyields the phase space of a system of a varying (but finite) number of particles. We associate a spaceH≃H(i)with each vertexi∈Γof a graph(Γ,ℰ)satisfying a special bidimensionality property. (Physically, vertexirepresents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i)-Δ/2, the minus a half of the Laplace operator onM, responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentialsU(1)(x),x∈M, describing a field generated by a heavy atom, (b) two-body potentialsU(2)(x,y),x,y∈M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentialsV(x,y),x,y∈M, scaled along the graph distanced(i,j)between verticesi,j∈Γ, which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie groupGacts onM, represented by a Euclidean space or torus of dimensiond'≤d, preserving the metric and the volume inM. Furthermore, we suppose that the potentialsU(1),U(2), andVareG-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian isG-invariant, provided that the thermodynamic variables (the fugacityzand the inverse temperatureβ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.

On a Vlasov–Euler system for 2D sprays with gyroscopic effects

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect in two space dimensions. The system couples a Vlasov-type equation and an Euler- type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin-type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from the paper On the motion of a small body immersed in a two-dimensional incompressible perfect fluid, accepted for publication in Bulletin de la SMF where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.

Quantum information approach to the description of quantum phase transitions

The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.

Particle addition and subtraction channels and the behavior of composite particles

Composite particles made of elementary fermions can exhibit a wide range of behavior ranging from fermionic to bosonic depending on the quantum state of the fermions and the experimental situation considered. This behavior is captured by the fundamental operations of single-particle addition and subtraction and two-particle interference. We analyze the quantum channels that implement the physical operations of addition and subtraction of indistinguishable particles. In particular, we construct optimal Kraus operators to implement these probabilistic operations for systems of a finite number of particles. We then use these to measure the quality of bosonic and fermionic behavior in terms of single-particle addition and subtraction and two-particle interference. For the specific case of composite particles made of two distinguishable fermions, we find a transition from fermionic to bosonic behavior as a function of the entanglement between the two constituents. We also apply these considerations to composite particles of two distinguishable bosons and identify the relation between entanglement and bosonic behavior for these systems.

Thermodynamics of an Ideal Bose Gas with a Finite Number of Particles Confined in a Three-Dimensional Quartic Trap

Within an exact canonical-ensemble treatment, we investigate the thermodynamics for a finite number of ideal bosons confined in a three-dimensional quartic trap. We calculate several physical quantities including the specific heat C N , chemical potential μ, condensate fraction 〈n 0〉/N, root-mean-square fluctuations δn 0 of the condensate population, and transition temperature T c . We discuss the particle-number dependence of T c through proposing three T c definitions, which are compared with ones derived in the grand canonical ensemble.

Product vacua with boundary states

We introduce a family of quantum spin chains with nearest-neighbor interactions that can serve to clarify and refine the classification of gapped quantum phases of such systems. The gapped ground states of these models can be described as a product vacuum with a finite number of particles bound to the edges. The numbers of particles, ${n}_{L}$ and ${n}_{R}$, that can bind to the left and right edges of the finite chains serve as indices of the particular phase a model belongs to. All these ground states, which we call product vacua with boundary states (PVBS), can be described as matrix product states (MPS). We present a curve of gapped Hamiltonians connecting the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to its representative PVBS model, which has indices ${n}_{L}={n}_{R}=1$. We also present examples with ${n}_{L}={n}_{R}=J$, for any integer $J\ensuremath{\ge}1$, that are related to a recently introduced class of $\text{SO}(2J+1)$-invariant quantum spin chains.

Thermal phase transitions for Dicke-type models in the ultrastrong-coupling limit

We consider the Dicke model in the ultrastrong-coupling limit to investigate thermal phase transitions and their precursors at finite particle numbers N for bosonic and fermionic systems. We derive partition functions with degeneracy factors that account for the number of configurations and derive explicit expressions for the Landau free energy. This allows us to discuss the difference between the original Dicke (fermionic) and the bosonic case. We find a crossover between these two cases that shows up, for example, in the specific heat.

On some properties of Markov chain Monte Carlo simulation methods based on the particle filter

Andrieu et al. (2010) prove that Markov chain Monte Carlo samplers still converge to the correct posterior distribution of the model parameters when the likelihood estimated by the particle filter (with a finite number of particles) is used instead of the likelihood. A critical issue for performance is the choice of the number of particles. We add the following contributions. First, we provide analytically derived, practical guidelines on the optimal number of particles to use. Second, we show that a fully adapted auxiliary particle filter is unbiased and can drastically decrease computing time compared to a standard particle filter. Third, we introduce a new estimator of the likelihood based on the output of the auxiliary particle filter and use the framework of Del Moral (2004) to provide a direct proof of the unbiasedness of the estimator. Fourth, we show that the results in the article apply more generally to Markov chain Monte Carlo sampling schemes with the likelihood estimated in an unbiased manner.

Reciprocal Time Relation of Noncolliding Brownian Motion with Drift

We consider an $N$-particle system of noncolliding Brownian motion starting from $x_1 \leq x_2 \leq ... \leq x_N$ with drift coefficients $\nu_j, 1 \leq j \leq N$ satisfying $\nu_1 \leq \nu_2 \leq ... \leq \nu_N$. When all of the initial points are degenerated to be zero, $x_j=0, 1 \leq j \leq N$, the equivalence is proved between a dilatation with factor $1/t$ of this drifted process and the noncolliding Brownian motion starting from $\nu_1 \leq \nu_2 \leq ... \leq \nu_N$ without drift observed at reciprocal time $1/t$, for arbitrary $t > 0$. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.

Single-file diffusion of particles in a box: Transient behaviors

We consider a finite number of particles with soft-core interactions, subjected to thermal fluctuations and confined in a box with excluded mutual passage. Using numerical simulations, we focus on the influence of the longitudinal confinement on the transient behavior of the longitudinal mean squared displacement. We exhibit several power laws for its time evolution according to the confinement range and to the rank of the particle in the file. We model the fluctuations of the particles as those of a chain of springs and point masses in a thermal bath. Our main conclusion is that actual system dynamics can be described in terms of the normal oscillation modes of this chain. Moreover, we obtain complete expressions for the physical observables, in excellent agreement with our simulations. The correct power laws for the time dependency of the mean squared displacement in the various regimes are recovered, and analytical expressions of the prefactors according to the relevant parameters are given.

Almost complete coherent state subsystems and partial reconstruction of wavefunctions in the Fock–Bargmann phase-number representation

We provide (partial) reconstruction formulas and discrete Fourier transforms for wavefunctions in standard Fock–Bargmann (holomorphic) phase-number representation from a finite number N of phase samples {θk = 2πk/N}N − 1k = 0 for a given mean number p of particles. The resulting coherent state subsystem is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, is ‘almost complete’ (a pseudo-frame) and partial reconstruction formulas are provided along with a study of the accuracy of the approximation, which tends to be exact when p < N and/or N → ∞.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

A heuristic approach to BEC self-trapping in double wells beyond the mean field

We present a technically simple treatment of self-trapping of Bose–Einstein condensates in double well traps based on intuitive semiclassical approximations. Our analysis finally leads to a convenient closed-form approximation for the time-averaged population imbalance valid in both the mean-field case and in the case of finite particle numbers for short times.

Stochastic self-assembly of incommensurate clusters

Nucleation and molecular aggregation are important processes in numerous physical and biological systems. In many applications, these processes often take place in confined spaces, involving a finite number of particles. Analogous to treatments of stochastic chemical reactions, we examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation. We enumerate the highest probability steady states, and derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Upon comparison with results obtained from the associated mass-action Becker-Döring equations, we find striking differences between the two corresponding equilibrium mean cluster concentrations. These differences depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass "incommensurability" arises, a single remainder particle can "emulsify" the system by significantly broadening the equilibrium mean cluster size distribution. This discreteness-induced broadening effect is periodic in the total mass of the system but arises even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite. Ironically, classic mass-action equations are fairly accurate in the coarsening regime, before equilibrium is reached, despite the presence of large stochastic fluctuations found via kinetic Monte-Carlo simulations. Our findings define a new scaling regime in which results from classic mass-action theories are qualitatively inaccurate, even in the limit of large total system size.

Entanglement entropy of quantum wire junctions

We consider a fermion gas on a star graph modeling a quantum wire junction and derive the entanglement entropy of one edge with respect to the rest of the junction. The gas is free in the bulk of the graph, the interaction being localized in its vertex and described by a non-trivial scattering matrix. We discuss all point-like interactions, which lead to a unitary time evolution of the system. We show that for a finite number of particles N, the Rényi entanglement entropies of one edge grow as ln N with a calculable prefactor, which depends not only on the central charge, but also on the total transmission probability from the considered edge to the rest of the graph. This result is extended to the case with a harmonic potential in the bulk.

Hamiltonian Traffic Dynamics in Microfluidic-Loop Networks

Recent microfluidic experiments revealed that large particles advected in a fluidic loop display long-range hydrodynamic interactions. However, the consequences of such couplings on the traffic dynamics in more complex networks remain poorly understood. In this Letter, we focus on the transport of a finite number of particles in one-dimensional loop networks. By combining numerical, theoretical, and experimental efforts, we evidence that this collective process offers a unique example of Hamiltonian dynamics for hydrodynamically interacting particles. In addition, we show that the asymptotic trajectories are necessarily reciprocal despite the microscopic traffic rules explicitly break the time-reversal symmetry. We exploit these two remarkable properties to account for the salient features of the effective three-particle interaction induced by the exploration of fluidic loops.

Critical temperature of a Rashba spin-orbit-coupled Bose gas in a harmonic trap

We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived based on a semiclassical approximation to the single-particle-energy spectrum and density of states and are compared with exact results obtained by explicitly summing discrete energy levels for a small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to finite spin-orbit coupling. For a large coupling strength and a finite number of particles $N$, the critical temperature scales as ${N}^{2/5}$ and ${N}^{2/3}$ in three and two dimensions, respectively, contrasted to the predictions of ${N}^{1/3}$ and ${N}^{1/2}$ in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.