Symbolic Dynamics for the Anisotropic N-Centre Problem at Negative Energies

The planar N-centre problem describes the motion of a particle moving in the plane under the action of the force fields of N fixed attractive centres: $$\begin{aligned} \ddot{x}(t)=\sum _{j=1}^N\nabla V_j(x(t)-c_j). \end{aligned}$$ x ¨ ( t ) = ∑ j = 1 N ∇ V j ( x ( t ) - c j ) . In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials $$V_j$$ V j are positive, anisotropic and homogeneous of degree $$-\alpha _j$$ - α j : $$\begin{aligned} V_j(x)=|x|^{-\alpha _j}V_j\left( \frac{x}{|x|}\right) . \end{aligned}$$ V j ( x ) = | x | - α j V j x | x | . The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis’ functional. Compared with the classical N-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

An energy and momentum conserving collisional bracket for the guiding-centre Vlasov–Maxwell–Landau model

This paper proposes a metric bracket for representing Coulomb collisions in the so-called guiding-centre Vlasov–Maxwell–Landau model. The bracket is manufactured to preserve the same energy and momentum functionals as does the Vlasov–Maxwell part and to simultaneously satisfy a revised version of the H-theorem, where the equilibrium distributions with respect to collisional dynamics are identified as Maxwellians. This is achieved by exploiting the special projective nature of the Landau collision operator and the simple form of the system's momentum functional. A discussion regarding a possible extension of the results to electromagnetic drift-kinetic and gyrokinetic systems is included. We anticipate that energy conservation and entropy dissipation can always be manufactured whereas guaranteeing momentum conservation is a delicate matter yet to be resolved.

Roaming pathways and survival probability in real-time collisional dynamics of cold and controlled bialkali molecules

Perfectly controlled molecules are at the forefront of the quest to explore chemical reactivity at ultra low temperatures. Here, we investigate for the first time the formation of the long-lived intermediates in the time-dependent scattering of cold bialkali $$^{23}\hbox {Na}^{87}$$ 23 Na 87 Rb molecules with and without the presence of infrared trapping light. During the nearly 50 nanoseconds mean collision time of the intermediate complex, we observe unconventional roaming when for a few tens of picoseconds either NaRb or $$\hbox {Na}_2$$ Na 2 and $$\hbox {Rb}_2$$ Rb 2 molecules with large relative separation are formed before returning to the four-atom complex. We also determine the likelihood of molecular loss when the trapping laser is present during the collision. We find that at a wavelength of 1064 nm the $$\hbox {Na}_2\hbox {Rb}_2$$ Na 2 Rb 2 complex is quickly destroyed and thus that the $$^{23}\hbox {Na}^{87}$$ 23 Na 87 Rb molecules are rapidly lost.

Small-N collisional dynamics – V. From N ≲ 10 to N ≳ 103

ABSTRACT Direct collisions between finite-sized particles occur commonly in many areas of astrophysics. Such collisions are typically mediated by chaotic, bound gravitational interactions involving small numbers of particles. An important application is stellar collisions, which occur commonly in dense star clusters, and their relevance for the formation of various types of stellar exotica. In this paper, we return to our study of the collision rates and probabilities during small-number chaotic gravitational interactions ($N\, \lesssim$ 10), moving beyond the small-number particle limit and into the realm of larger particle numbers ($N\, \gtrsim$ 103) to test the extent of validity of our analytic model as a function of the particle properties and the number of interacting particles. This is done using direct N-body simulations of stellar collisions in dense star clusters, by varying the relative numbers of particles with different particle masses and radii. We compute the predicted rate of collisions using the mean free path approximation, adopting the point-particle limit and using the sticky-star approximation as our collision criterion. We evaluate its efficacy in the regime where gravitational focusing is important by comparing the theoretical rates to numerical simulations. Using the tools developed in previous papers in this series, in particular Collision Rate Diagrams, we illustrate that our predicted and simulated rates are in excellent agreement, typically consistent with each other to within 1 standard deviation.

Spontaneous Negative Entropy Increments in Granular Flows

The entropy inequality, commonly taken as an axiom of continuum mechanics, is found to be spontaneously violated in macroscopic granular media undergoing collisional dynamics. The result falls within the fluctuation theorem of nonequilibrium thermodynamics, which is known to replace the Second Law for finite systems. This phenomenon amounts to the system stochastically displaying negative increments of entropy. The focus is on granular media in Couette flows, consisting of monosized circular disks (with 10 to 104 disks of diameters 0.01 m to 1 m) with frictional-Hookean contacts simulated by molecular dynamics accounting for micropolar effects. Overall, it is determined that the probability of negative entropy increments diminishes with the Eulerian velocity gradient increasing, while it tends to increase in a sigmoidal fashion with the Young modulus of disks increasing. This behavior is examined for a very wide range of known materials: from the softest polymers to the stiffest (i.e., carbyne). The disks’ Poisson ratio is found to have a weak effect on the probability of occurrence of negative entropy increments.

Boltzmann–Grad asymptotic behavior of collisional dynamics

This paper discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.