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The nonlinear Schr\"odinger and the Schr\"odinger-Newton equations model many phenomena in various fields. Here, we perform an extensive numerical comparison between splitting methods (often employed to numerically solve these equations) and the integrating factor technique, also called Lawson method. Indeed, the latter is known to perform very well for the nonlinear Schr\"odinger equation, but has not been thoroughly investigated for the Schr\"odinger-Newton equation. Comparisons are made in one and two spatial dimensions, exploring different boundary conditions and parameters values. We show that for the short range potential of the nonlinear Schr\"odinger equation, the integrating factor technique performs better than splitting algorithms, while, for the long range potential of the Schr\"odinger-Newton equation, it depends on the particular system considered.

For a classical system with long-range interactions, a soft mode exists whenever a stationary state spontaneously breaks a continuous symmetry of the Hamiltonian. Besides that, if the corresponding coordinate associated to the symmetry breaking is periodic, the same energy of the different stationary states and finite $N$ thermal fluctuations result in a superdiffusive motion of the center of mass for total zero momentum, that tends to a normal diffusion for very long-times. As examples of this, we provide a two-dimensional self-gravitating system, a free electron laser and the Hamiltonian Mean-Field (HMF) model. For the latter, a detailed theory for the motion of the center of mass is given. We also discuss how the coupling of the soft mode to the mean-field motion of individual particles may lead to strong chaotic behavior for a finite particle number, as illustrated by the HMF model.

We propose different experimental methods to measure the analog of the Debye length in a very large Magneto-Optical Trap, which should characterize the spatial correlations in the atomic cloud. An analytical, numerical and experimental study of the response of the atomic cloud to an external modulation potential suggests that this Debye length, if it exists, is significantly larger than what was expected., Comment: 12 pages, 6 figures. v2 Clarify the paper presentation + correct minor points

We consider the two dimensional motion of a particle into a confining potential, subjected to Brownian forces, associated with two different temperatures on the orthogonal directions. Exact solutions are obtained for an asymmetric harmonic potential in the overdamped and underdamped regimes, whereas perturbative approaches are used for more general potentials. The resulting non equilibrium stationary state is characterized with a nonzero orthoradial mean current, corresponding to a global rotation of the particle around the center. The rotation is due to two symmetry breaking: two different temperatures and a mismatch between the principal axes of the confining asymmetric potential and the temperature axes. We confirm our predictions by performing Brownian dynamics simulation. Finally, we propose to observe this effect on a laser cooled atomic system., Comment: 11 pages, 9 Figures, submitted to PRE

We explore the formation and relaxation of so-called quasi-stationary states (QSS) for particle distributions in three dimensions interacting via an attractive radial pair potential $V(r \rightarrow \infty) \sim 1/r^\gamma$ with $\gamma > 0$, and either a soft-core or hard-core regularization at small $r$. In the first part of the paper we generalize, for any spatial dimension $d \geq 2$, Chandrasekhar's approach for the case of gravity to obtain analytic estimates of the rate of collisional relaxation due to two body collisions. The resultant relaxation rates indicate an essential qualitative difference depending on the integrability of the pair force at large distances: for $\gamma >d-1$ the rate diverges in the large particle number $N$ (mean field) limit, unless a sufficiently large soft core is present; for $\gamma < d-1$, on the other hand, the rate vanishes in the same limit even in the absence of any regularization. In the second part of the paper we compare our analytical predictions with the results of extensive parallel numerical simulations in $d=3$, for a range of different exponents $\gamma$ and soft cores leading to the formation of QSS. We find, just as for the previously well studied case of gravity (which we also revisit), excellent agreement between the parametric dependence of the observed relaxation times and our analytic predictions. Further, as in the case of gravity, we find that the results indicate that, when large impact factors dominate, the appropriate cut-off is the size of the system (rather than, for example, the mean inter-particle distance). Our results provide strong evidence that the existence of QSS is robust only for long-range interactions with a large distance behavior $\gamma < d-1$; for $\gamma \geq d-1$ the existence of such states will be conditioned strongly on the short range properties of the interaction., Comment: 21 pages, 11 figures, submitted to PRE

Systems of particles with long range interactions present two important processes: first, the formation of out-of-equilibrium quasi-stationary states (QSS), and the collisional relaxation towards Maxwell-Boltzmann equilibrium in a much longer timescale. In this paper, we study the collisional relaxation in the Hamiltonian-Mean-Field model (HMF) using the appropriate kinetic equations for a system of $N$ particles at order $1/N$ : the Landau equation when collective effects are neglected and the Lenard-Balescu equation when they are taken into account. We derive explicit expressions for the diffusion coefficients using both equations for any magnetization, and we obtain analytic expressions for highly clustered configurations. An important conclusion is that in this system collective effects are crucial in order to describe the relaxation dynamics. We compare the diffusion calculated with the kinetic equations with simulations set up to simulate the system with or without collective effects, obtaining a very good agreement between theory and simulations., Comment: 15 pages, 10 figures. To be published in Physical Review E

We propose to use a cloud of laser cooled atoms in a quasi two dimensional trap to investigate a non equilibrium collapse phase transition in presence of gravitational-like interaction. Using theoretical arguments and numerical simulations, we show that, like in two dimensional gravity, a transition to a collapsed state occurs below a critical temperature. In addition and as a signature of the non equilibrium nature of the system, persistent particles currents, dramatically increasing close to the phase transition, are observed., Comment: 5 pages, 4 figures

Systems with long-range interactions, such as self-gravitating clusters and magnetically confined plasmas, do not relax to the usual Boltzmann-Gibbs thermodynamic equilibrium, but become trapped in quasi-stationary states (QSS) the life time of which diverges with the number of particles. The QSS are characterized by the lack of ergodicity which can result in a symmetry broken QSS starting from a spherically symmetric particle distribution. We will present a theory which allows us to quantitatively predict the instability threshold for spontaneous symmetry breaking for a class of d-dimensional self-gravitating systems., Comment: 5 pages, 4 figures. Accepted for publication in Physical Review Letters

We study non-linear gravitational clustering from cold gaussian power-law initial conditions in a family of scale-free EdS models, characterized by a free parameter $\kappa$ fixing the ratio between the mass driving the expansion and the mass which clusters. As in the "usual" EdS model, corresponding to $\kappa=1$, self-similarity provides a powerful instrument to delimit the physically relevant clustering resolved by a simulation. Likewise, if stable clustering applies, it implies scale-free non-linear clustering. We derive the corresponding exponent $\gamma_{sc} (n, \kappa)$ of the two point correlation function. We then report the results of extensive N-body simulations, of comparable size to those previously reported in the literature for the case $\kappa=1$, and performed with an appropriate modification of the GADGET2 code. We observe in all cases self-similarity in the two point correlations, down to a lower cut-off which decreases monotonically in time in comoving coordinates. The self-similar part of the non-linear correlation function is fitted well in all cases by a single power-law with an exponent in good agreement with $\gamma_{sc} (n, \kappa)$. Our results thus indicate that stable clustering provides an excellent approximation to the non-linear correlation function over the resolved self-similar scales, at least down to $\gamma_{sc} (n, \kappa) \approx 1$, corresponding to the case $n=-2$ for $\kappa=1$. We conclude, in contrast notably with the results of Smith et al. (2003), that a clear identification of the breakdown of stable clustering in self-similar models - and the possible existence of a "universal" region in which non-linear clustering becomes independent of initial conditions - remains an important open problem, which should be addressed further in significantly larger simulations., Comment: 31 pages, 14 figures, additional results for 256^3 simulations, added references and other small modifications, final version to appear in MNRAS

"Quasi-stationary" states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction V(r \rightarrow \infty) \sim 1/r^a with a > 0, in d>1 dimensions. We generalize analytic calculations known for gravity in d=3 to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for a < d-1, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for a > d-1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential., Comment: 5 pages, 3 figures; final version to appear in Phys. Rev. Lett

We formalize a classification of pair interactions based on the convergence properties of the {\it forces} acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the "usual" thermodynamic limit. For a pair interaction potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it bounded} pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the {\it pair force} is absolutely integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to this case as {\it dynamically short-range}, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the {\it dynamically long-range} case, i.e., a \leq d-1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with a > d-2 (or a < d-2), for which the PDF of the {\it difference in forces} is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system., Comment: 12 pages, 1 figure; significantly shortened and focussed, additional references, version to appear in J. Stat. Phys

The effects of discreteness arising from the use of the N-body method on the accuracy of simulations of cosmological structure formation are not currently well understood. After a discussion of how the relevant discretisation parameters introduced should be extrapolated to recover the Vlasov-Poisson limit, we study numerically, and with analytical methods we have developed recently, the central issue of how finite particle density affects the precision of results. In particular we focus on the power spectrum at wavenumbers around and above the Nyquist wavenumber, in simulations in which the force resolution is taken smaller than the initial interparticle spacing. Using simulations of identical theoretical initial conditions sampled on four different "pre-initial" configurations (three different Bravais lattices, and a glass) we obtain a {\it lower bound} on the real discreteness error. With the guidance of our analytical results, we establish with confidence that the measured dispersion is not contaminated either by finite box size effects or by subtle numerical effects. Our results show notably that, at wavenumbers {\it below} the Nyquist wavenumber, the dispersion increases monotonically in time throughout the simulation, while the same is true above the Nyquist wavenumber once non-linearity sets in. For normalizations typical of cosmological simulations, we find lower bounds on errors at the Nyquist wavenumber of order of a percent, and larger above this scale. The only way this error may be reduced below these levels at these scales, and indeed convergence to the physical limit firmly established, is by extrapolation, at fixed values of the other relevant parameters, to the regime in which the mean comoving interparticle distance becomes less than the force smoothing scale., Comment: 26 pages, 15 figures, minor changes, slightly shortened, version to be published in MNRAS

In two recent articles a detailed study has been presented of the out of equilibrium dynamics of an infinite system of self-gravitating points initially located on a randomly perturbed lattice. In this article we extend the treatment of the early time phase during which strong non-linear correlations first develop, prior to the onset of ``self-similar'' scaling in the two point correlation function. We establish more directly, using appropriate modifications of the numerical integration, that the development of these correlations can be well described by an approximation of the evolution in two phases: a first perturbative phase in which particles' displacements are small compared to the lattice spacing, and a subsequent phase in which particles interact only with their nearest neighbor. For the range of initial amplitudes considered we show that the first phase can be well approximated as a transformation of the perturbed lattice configuration into a Poisson distribution at the relevant scales. This appears to explain the ``universality'' of the spatial dependence of the asymptotic non-linear clustering observed from both shuffled lattice and Poisson initial conditions., Comment: 11 pages, 11 figures, shortened introductory sections and other minor modifications, version to appear in Phys. Rev. E

We apply a recently developed perturbative formalism which describes the evolution under their self-gravity of particles displaced from a perfect lattice to quantify precisely, up to shell crossing, the effects of discreteness in dissipationless cosmological N-body simulations. We give simple expressions, explicitly dependent on the particle density, for the evolution of power in each mode as a function of red-shift. For typical starting red-shifts the effect of finite particle number is to {\it slow down} slightly the growth of power compared to that in the fluid limit (e.g. by about ten percent at half the Nyquist frequency), and to induce also dispersion in the growth as a function of direction at a comparable level. In the limit that the initial red-shift tends to infinity, at fixed particle density, the evolution in fact diverges from that in the fluid limit (described by the Zeldovich approximation). Contrary to widely held belief, this means that a simulation started at a red-shift much higher than the red-shift of shell crossing actually gives a worse, rather than a better, result. We also study how these effects are modified when there is a small-scale regularization of the gravitational force. We show that such a smoothing may reduce the anisotropy of the discreteness effects, but it then {\it increases} their average effect. This behaviour illustrates the fact that the discreteness effects described here are distinct from those usually considered in this context, due to two-body collisions. Indeed the characteristic time for divergence from the collisionless limit is proportional to $N^{2/3}$, rather than $N/ \log N$ in the latter case., Comment: 18 pages, 10 figures, sequel to astro-ph/0410451 (on initial conditions), final version with minor changes, to appear in Phys. Rev. D

The relation between the results of cosmological N-body simulations, and the continuum theoretical models they simulate, is currently not understood in a way which allows a quantification of N dependent effects. In this first of a series of papers on this issue, we consider the quantification of such effects in the initial conditions of such simulations. A general formalism developed in [1] allows us to write down an exact expression for the power spectrum of the point distributions generated by the standard algorithm for generating such initial conditions. Expanded perturbatively in the amplitude of the input (i.e. theoretical, continuum) power spectrum, we obtain at linear order the input power spectrum, plus two terms which arise from discreteness and contribute at large wavenumbers. For cosmological type power spectra, one obtains as expected, the input spectrum for wavenumbers k smaller than that characteristic of the discreteness. The comparison of real space correlation properties is more subtle because the discreteness corrections are not as strongly localised in real space. For cosmological type spectra the theoretical mass variance in spheres and two point correlation function are well approximated {\it above} a finite distance. For typical initial amplitudes this distance is a few times the inter-particle distance, but it diverges as this amplitude (or, equivalently, the initial red-shift of the cosmological simulation) goes to zero, at fixed particle density. We discuss briefly the physical significance of these discreteness terms in the initial conditions, in particular with respect to the definition of the continuum limit of N-body simulations., Comment: Final published version, 22 pages and 16 figures, very significantly revised, one important correction (range of allowed exponents), modified title (now part one of a series)