Finite Number Of Particles Research Articles

Chaos and relaxation to equilibrium in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in nonequilibrium stationary states. For a finite number of particles a nonequilibrium state has a finite lifetime, so that eventually a system will relax to thermodynamic equilibrium. The time that a system remains trapped in a quasistationary state (QSS) scales with the number of particles as N(δ), with δ>0, and diverges in the thermodynamic limit. In this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a QSS. We discover that chaos, measured by the Lyapunov exponents, favors faster relaxation to equilibrium. Surprisingly, weak chaos favors faster relaxation than strong chaos.

Secular diffusion in discrete self-gravitating tepid discs

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is described using the inhomogeneous Balescu-Lenard equation. Assuming that only tightly wound transient spirals are present in the disc, a WKB approximation provides a simple and tractable quadrature for the corresponding drift and diffusion coefficients. It provides insight into the physical processes at work during the secular diffusion of a self-gravitating discrete disc and makes quantitative predictions on the initial variations of the distribution function in action space. When applied to the secular evolution of an isolated stationary self-gravitating Mestel disc, this formalism predicts initially the importance of the corotation resonance in the inner regions of the disc leading to a regime involving radial migration and heating. It predicts in particular the formation of a "ridge like" feature in action space, in agreement with simulations, but over-estimates the timescale involved in its appearance. Swing amplification is likely to resolve this discrepancy. In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limit may also describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic center.

Comment on “Bose-Einstein condensation with a finite number of particles in a power-law trap”

In Jaouadi et al. [Phys. Rev. A 83, 023616 (2011)] the authors derive an analytical finite-size expansion for the Bose-Einstein condensation critical temperature of an ideal Bose gas in a generic power-law trap. In the case of a harmonic trap, this expansion adds higher-order terms to the well-known first-order correction. We point out a delicate point in connection to these results, showing that the claims of Jaouadi et al. should be treated with caution. In particular, for a harmonic trap, the given expansion yields results that, depending on what is considered to be the critical temperature of the finite system, do not generally improve on the established first-order correction. For some nonharmonic traps, the results differ at first order from other results in the literature.

Reply to “Comment on ‘Bose-Einstein condensation with a finite number of particles in a power-law trap’ ”

In this reply we show that the criticisms raised by J. Noronha are based on a misapplication of the model we have proposed in [A. Jaouadi, M. Telmini, E. Charron, Phys. Rev. A 83, 023616 (2011)]. Here we explicitly discuss the range of validity of the approximations underlying our analytical model. We also show that the discrepancies pointed out for very small atom numbers and for very anisotropic traps are not surprising since these conditions exceed the range of validity of the model.

Quantum-mechanical engines working with an ideal gas with a finite number of particles confined in a power-law trap

Based on quantum thermodynamic processes, we make a quantum-mechanical (QM) extension of the typical heat engine cycles, such as the Carnot, Brayton, Otto, Diesel cycles, etc., with no introduction of the concept of temperature. When these QM engine cycles are implemented by an ideal gas confined in an arbitrary power-law trap, a relation between the quantum adiabatic exponent and trap exponent is found. The differences and similarities between the efficiency of a given QM engine cycle and its classical counterpart are revealed and discussed.

Collective modes of a harmonically trapped one-dimensional Bose gas: The effects of finite particle number and nonzero temperature

Following the idea of the density functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap, by using a local chemical potential - calculated with the Lieb-Liniger exact solution - as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit, to the mean-field Thomas-Fermi regime, and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective mode frequencies, we make comparisons of our results with the recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.

Scaling limit for Brownian motions with one-sided collisions

We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the $\mathrm{Airy}_{1}$ process.

Phase diagrams of systems of two and three levels in the presence of a radiation field

We study the structure of the phase diagram for systems consisting of two- and three-level particles dipolarly interacting with a one-mode electromagnetic field, inside a cavity, paying particular attention to the case of a finite number of particles, and showing that the divergences that appear in other treatments are a consequence of the mathematical approximations employed and can be avoided by studying the system in an exact manner quantum-mechanically or via a catastrophe formalism with variational trial states that satisfy the symmetries of the appropriate Hamiltonians. These variational states give an excellent approximation not only to the exact quantum phase space, but also to the energy spectrum and the expectation values of the atomic and field operators. Furthermore, they allow for analytic expressions in many of the cases studied. We find the loci of the transitions in phase space from one phase to the other, and the order of the quantum phase transitions are determined explicitly for each of the configurations, with and without detuning. We also derive the critical exponents for the various systems, and the phase structure at the triple point present in the Ξ-configuration of three-level systems is studied.

Exact solution of the van der Waals model in the critical region

The celebrated van der Waals model describes simple fluids in the thermodynamic limit and predicts the existence of a critical point associated to the gas–liquid phase transition. However the behavior of critical isotherms according to the equation of state, where a gas–liquid phase transition occurs, significantly departs from experimental observations. The correct critical isotherms are heuristically re-established via the Maxwell equal areas rule. A long standing open problem in mean field theory is concerned with the analytic description of van der Waals isotherms for a finite size system that is consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired by the theory of nonlinear conservation laws, we propose a novel mean field approach, based on statistical mechanics, that allows to calculate the van der Waals partition function for a system of large but finite number of particles N. Our partition function naturally extends to the whole space of thermodynamic variables, reproduces, in the thermodynamic limit N→∞, the classical results outside the critical region and automatically encodes Maxwell’s prescription. We show that isothermal curves evolve in the space of thermodynamic variables like nonlinear breaking waves and the criticality is explained as the mechanism of formation of a classical hydrodynamic shock.

Separation versus diffusion in a two species system

We consider a finite number of particles that move in $\mathbb{Z}$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$-particle becomes a $b$-particle at constant rate, while the leftmost $b$-particle becomes $a$-particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a nonlinear system of two PDE’s with free boundaries.

Relations Among Two Methods for Computing the Partition Function of the Two-Dimensional One-Component Plasma

The two-dimensional one-component plasma ---2dOCP--- is a system composed by $n$ mobile particles with charge $q$ over a neutralizing background in a two-dimensional surface. The Boltzmann factor of this system, at temperature $T$, takes the form of a Vandermonde determinant to the power $\Gamma = q^2/(2\pi\varepsilon k_BT)$, where $\Gamma$ is the coupling constant of this Coulomb system. The partition function of the model has been computed exactly for the even values of the coupling constant $\Gamma$, and a finite number of particles $n$, by two means: 1) by recognizing that the Boltzmann factor is the square of a Jack polynomial and expanding it in an appropriate monomial base, and 2) by mapping the system onto a 1-dimensional chain of interacting fermions. In this work the connection among the two methods is derived, and some properties of the expansion coefficients for the power of the Vandermonde determinant are explored.

Thomson rings in a disk.

We discuss the basic principles of self-organization of a finite number of charged particles interacting via the 1/r Coulomb potential in disk geometry. The analysis is based on the cyclic symmetry and periodicity of the Coulomb interaction between particles located on several rings. As a result, a system of equations is derived, which allows us readily to determine with high accuracy the equilibrium configurations of a few hundred charged particles. For n≳200, we predict the formation of a hexagonal core and valence circular rings for the centered configurations.

Nano-beam X-ray microscopy of dried colloidal films.

We report on a nano-beam small angle X-ray scattering study on densely-packed, dried binary films made out of spherical silica particles with radii of 11.2 and 19.3 nm. For these three-dimensional thin films prepared by drop casting, only a finite number of colloidal particles contributes to the scattering signal due to the small beam size of 400 × 400 nm(2). By scanning the samples, the structure and composition of the silica particle films are determined spatially resolved revealing spatial heterogeneities in the films. Three different types of domains were identified: regions containing mainly large particles, regions containing mainly small particles, and regions where both particle species are mixed. Using the new angular X-ray cross-correlations analysis (XCCA) approach, spatial maps of the local type and degree of orientational order within the silica particle films are obtained. Whereas the mixed regions have dominant two-fold order, weaker four-fold and marginal six-fold order, regions made out of large particles are characterized by an overall reduced orientational order. Regions of small particles are highly ordered showing actually crystalline order. Distinct differences in the local particle order are observed by analyzing sections through the intensity and XCCA maps. The different degree of order can be understood by the different particle size polydispersities. Moreover, we show that preferential orientations of the particle domains can be studied by cross-correlation analysis yielding information on particle film formation. We find patches of preferential order with an average size of 8-10 μm. Thus, by this combined X-ray cross-correlation microscopy (XCCM) approach the structure and orientational order of films made out of nanometer sized colloids can be determined. This method will allow to reveal the local structure and order of self-assembled structures with different degree of order in general.

Evolution of bubble size distribution in particle stabilised bubble dispersions: Competition between particle adsorption and dissolution kinetics

It is shown that in systems containing bubbles stabilised by nanoparticles, the time scales for the dissolution of small microbubbles can be comparable with those involving the transport and adsorption of the stabilising nanoparticles onto the surface of the bubbles. We have studied the evolution of model bubble size distribution functions in the light of this effect and also the competition between different sized bubbles for the finite number of available particles. It is found that for dispersions moderately rich in nanoparticles, the width of the final distribution function can become broader than the initial one, whereas for cases deficient in particles the reverse is observed. For each given bubble size, there exists a particle to bubble concentration ratio above which the final size of a bubble of this radius is no longer affected by the presence of other bubbles. In a system deficient in particles, this can still hold true for bubbles in the lower end of the size distribution range, but not the ones at the upper end. By considering simple cases consisting of just two bubbles sizes, we show that the degree of shrinkage of the bigger bubbles is significantly increased in the presence of a small amount of gas in the form of smaller bubbles. In contrast, the final bubble size of the smaller bubbles is found to be largely insensitive to the amount of gas included within larger bubbles. The implications of these results for the final fraction of retained gas, in these types of particle stabilised bubble systems, are also discussed.

Renormalized entropy of entanglement in relativistic field theory

Entanglement is defined between subsystems of a quantum system, and at fixed time two regions of space can be viewed as two subsystems of a relativistic quantum field. The entropy of entanglement between such subsystems is ill-defined unless an ultraviolet cutoff is introduced, but it still diverges in the continuum limit. This behaviour is generic for arbitrary finite-energy states, hence a conceptual tension with the finite entanglement entropy typical of nonrelativistic quantum systems. We introduce a novel approach to explain the transition from infinite to finite entanglement, based on coarse graining the spatial resolution of the detectors measuring the field state. We show that states with a finite number of particles become localized, allowing an identification between a region of space and the nonrelativistic degrees of freedom of the particles therein contained, and that the renormalized entropy of finite-energy states reduces to the entanglement entropy of nonrelativistic quantum mechanics.

Simulations of plasma obeying Coulomb's law and the formation of suprathermal ion tails in the solar wind

AbstractWe present a model for the velocity distribution function (VDF) of the suprathermal ion tail in the solar wind and its commonly observed spectrum, a power law in velocity of index −5. The mechanism for our model is acceleration from extremely small scale electric fields caused by the finite number of particles in the solar wind (or any plasma). The simulations produce a VDF with a broken power law at short times and a type of κ distribution VDF at later times. The simulation uses very few assumptions as it is based solely on Coulomb's law, and the results are robust to a range of parameters that, although outside the range of the solar wind at 1 AU, are close to those in the lower solar atmosphere. Furthermore, we provide predictions based on the following mechanism: the probability distribution function for the electric field (the electric field distribution function or EDF) dictates the final state of the plasma, which means that the VDF at any time is a convolution of the initial VDF and the EDF. We have also shown that by varying the power law index of the electrostatic force law, the relationship between the two is such that the EDF power law index is exactly 5 for Coulomb's law, harder for shorter‐range forces, and softer for longer‐range forces, converging to a power law of index 3 in the former limit and to a Maxwellian in the latter.

Kinetics of aggregation with a finite number of particles and application to viral capsid assembly.

The formation of a viral capsid can be seen as a time-dependent process generated by the arrival of aggregates of various sizes at a nucleation site. Based on a model where aggregates arrive at a Poissonian rate to form a capsid particle, we develop kinetics assembly equations, that account for a finite size cluster and thus for rejection of too large aggregate. The model is derived under the assumption that the aggregate distribution has reached an exponential steady state. To account for the stochastic nature of the aggregates arrival, we also derive a stochastic equation to compute the mean time for a cluster to be formed. We find that this time has a minimum for a unique aggregate distribution. We obtain asymptotic expression for this time that we compare with numerically simulations. Finally, we find the mean size of the largest aggregate forming a viral capsid.

On the objective origin of the phase transitions and metastability in many-particle systems

The modern equilibrium statistical mechanics, intended to link microscopic dynamics to thermodynamic laws for macroscopic quantities, is faced with significant difficulties, as applied to the description of the macroscopic properties of real condensed media within wide thermodynamic ranges, including the vicinities of the phase transition points. A particular problem is the absence of metastable states in the Gibbs statistical mechanics of the systems composed of finite number of particles. Nevertheless, accordance between equilibrium statistical mechanics and thermodynamics of condensed media is achievable if we take account of the mutual correlation (the feedback) between the microscopic properties of molecules and the macrostate of the corresponding medium. This can be done via usage of the "enhanced" Hamilton operator of the considered many-particle system, which contains some temperature-dependent term(s), and the following introduction of the generalized equilibrium distribution over microstates. For illustration of the reasonableness of the proposed approach and of its practical availability in the applications to computing the macroscopic characteristics of condensed media, a cell model of melting/crystallization and metastable supercooled liquid for a water-like medium is presented.

Particle propagator of the spin Calogero–Sutherland model

Explicit-exact expressions for the particle propagator of the spin 1/2 Calogero–Sutherland model are derived for the system of a finite number of particles and for that in the thermodynamic limit. Derivation of the expression in the thermodynamic limit is also presented in detail. Combining this result with the hole propagator obtained in earlier studies, we calculate the spectral function of the single particle Green's function in the full range of the energy and momentum space. The resultant spectral function exhibits power-law singularity characteristic to correlated particle systems in one dimension.

Statistical distribution of bonding distances in a unidimensional solid

We study a Fermi–Pasta–Ulam-like chain with Lennard-Jones potentials to model a unidimensional solid in contact with heat baths at a given temperature. We formulate an explicit analytical expression for the probability density of bonding distances between neighboring particles, which depends on temperature similarly to the distribution of velocities. For a finite number of particles, its validity is verified with high accuracy through molecular dynamics simulations. We also provide a theoretical framework which is consistent with the numerical findings. We give an analytic expression of the mean bond distance and elastic constant in the case of the square-well and harmonic interparticle potentials: we outline the role played by the hard-core repulsion. We also calculate the same quantities in the case of series expansions of Lennard-Jones potential truncated at different, even series power.