Related papers: Shadowing-based reliability decay in softened n-bo…
A {\it shadow} is an exact solution to a chaotic system of equations that remains close to a numerically computed solution for a long time, ending in a {\it glitch}. We study the distribution of shadow durations at low dimension and how…
We present a new time-stepping criterion for N-body simulations that is based on the true dynamical time of a particle. This allows us to follow the orbits of particles correctly in all environments since it has better adaptivity than…
Using the Ricci and scalar curvatures of the configuration manifold of gravitational N-body systems, we study the exponential instability in their trajectories. It is found that the exponentiation time-scale for isotropic Plummer spheres…
We revisit the r\^{o}le of discreteness and chaos in the dynamics of self-gravitating systems by means of $N$-body simulations with active and frozen potentials, starting from spherically symmetric stationary states and considering the…
This paper proves that shadowing solutions can be almost surely nonphysical. This finding invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior. This theoretical…
Much of standard galaxy dynamics rests on the implicit assumption that the corresponding N-body problem is (near) integrable. This notion although leading to great simplification is by no means a fact. It is therefore important to develop…
Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative…
Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the…
The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full $N$-body problem and that of trajectories moving in the gravitational field of non-interacting background…
A criterion for the choice of optimal softening length $\epsilon$ and time-step $dt$ for $N$-body simulations of a collisionless stellar system is analyzed. Plummer and Hernquist spheres are used as models to follow how changes in various…
Molecular dynamics refers to the computer simulation of a material at the atomic level. An open problem in numerical analysis is to explain the apparent reliability of molecular dynamics simulations. The difficulty is that individual…
This work discusses the main analogies and differences between the deterministic approach underlying most cosmological N-body simulations and the probabilistic interpretation of the problem that is often considered in mathematics and…
Numerical solutions to Newton's equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and…
Simulations of purely self-gravitating N-body systems are often used in astrophysics and cosmology to study the collisionless limit of such systems. Their results for macroscopic quantities should then converge well for sufficiently large…
We consider the motion of a particle, taking into account its interaction with environmental degrees of freedom. The dephasing time is determined by the nature of the environment, and depends on the particle velocity. Our interest is in the…
Consider a system of N identical hard spherical particles moving in a d-dimensional box and undergoing elastic, possibly multi-particle, collisions. We develop a new algorithm that recovers the pre-collision state from the post-collision…
The Schroedinger equation for N particles interacting through effective pair potentials is derived from the massless Nelson model with ultraviolet cutoffs. We consider a scaling limit where the particles are slow and heavy, but, in contrast…
Dynamical simulations are a fundamental tool for studying the secular evolution of disc galaxies. Even at their maximum resolution, they still follow a limited number of particles and typically resolve scales of the order of a few tens of…
We investigate the old problem of the fast relaxation of collisionless $N$-body systems which are collapsing or perturbed, emphasizing the importance of (non-collisional) discreteness effects. We integrate orbit ensembles in fixed external…
A novel approach to account for hard-body interactions in (overdamped) Brownian dynamics simulations is proposed for systems with non-vanishing force fields. The scheme exploits the analytically known transition probability for a Brownian…