Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems

We study, using both theory and molecular dynamics simulations, the relaxation dynamics of a microcanonical two dimensional self-gravitating system. After a sufficiently large time, a gravitational cluster of N particles relaxes to the Maxwell-Boltzmann distribution. The time to reach the thermodynamic equilibrium, however, scales with the number of particles. In the thermodynamic limit, $N\to\infty$ at fixed total mass, equilibrium state is never reached and the system becomes trapped in a non-ergodic stationary state. An analytical theory is presented which allows us to quantitatively described this final stationary state, without any adjustable parameters

Linear theory and violent relaxation in long-range systems: a test case

In this article, several aspects of the dynamics of a toy model for longrange Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian Mean Field (HMF). For special cases, exact finite-N linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, $N \rightarrow \infty$, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. All the numerical simulations show a very good agreement with the different theoretical predictions. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports it in the vicinity of the equilibrium state within some linear e-folding times

Energy ejection in the collapse of a cold spherical self-gravitating cloud

When an open system of classical point particles interacting by Newtonian gravity collapses and relaxes violently, an arbitrary amount of energy may in principle be carried away by particles which escape to infinity. We investigate here, using numerical simulations, how this released energy and other related quantities (notably the binding energy and size of the virialized structure) depends on the initial conditions, for the one parameter family of starting configurations given by randomly distributing N cold particles in a spherical volume. Previous studies have established that the minimal size reached by the system scales approximately as N^{-1/3}, a behaviour which follows trivially when the growth of perturbations (which regularize the singularity of the cold collapse in the infinite N limit) are assumed to be unaffected by the boundaries. Our study shows that the energy ejected grows approximately in proportion to N^{1/3}, while the fraction of the initial mass ejected grows only very slowly with N, approximately logarithmically, in the range of N simulated. We examine in detail the mechanism of this mass and energy ejection, showing explicitly that it arises from the interplay of the growth of perturbations with the finite size of the system. A net lag of particles compared to their uniform spherical collapse trajectories develops first at the boundaries and then propagates into the volume during the collapse. Particles in the outer shells are then ejected as they scatter through the time dependent potential of an already re-expanding central core. Using modified initial configurations we explore the importance of fluctuations at different scales, and discreteness (i.e. non-Vlasov) effects in the dynamics.Comment: 20 pages, 27 figures; revised version with small changes and corrections, to appear in MNRA

Outdoor learning spaces: the case of forest school

© 2017 The Author. Area published by John Wiley & Sons Ltd on behalf of Royal Geographical Society (with the Institute of British Geographers). This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.This paper contributes to the growing body of research concerning use of outdoor spaces by educators, and the increased use of informal and outdoor learning spaces when teaching primary school children. The research takes the example of forest school, a form of regular and repeated outdoor learning increasingly common in primary schools. This research focuses on how the learning space at forest school shapes the experience of children and forest school leaders as they engage in learning outside the classroom. The learning space is considered as a physical space, and also in a more metaphorical way as a space where different behaviours are permitted, and a space set apart from the national curriculum. Through semi-structured interviews with members of the community of practice of forest school leaders, the paper seeks to determine the significance of being outdoors on the forest school experience. How does this learning space differ from the classroom environment? What aspects of the forest school learning space support pupils’ experiences? How does the outdoor learning space affect teaching, and the dynamics of learning while at forest school? The research shows that the outdoor space provides new opportunities for children and teachers to interact and learn, and revealed how forest school leaders and children co-create a learning environment in which the boundaries between classroom and outdoor learning, teacher and pupil, are renegotiated to stimulate teaching and learning. Forest school practitioners see forest school as a separate learning space that is removed from the physical constraints of the classroom and pedagogical constraints of the national curriculum to provide a more flexible and responsive learning environment.Peer reviewe

Relaxation to thermal equilibrium in the self-gravitating sheet model

We revisit the issue of relaxation to thermal equilibrium in the so-called "sheet model", i.e., particles in one dimension interacting by attractive forces independent of their separation. We show that this relaxation may be very clearly detected and characterized by following the evolution of order parameters defined by appropriately normalized moments of the phase space distribution which probe its entanglement in space and velocity coordinates. For a class of quasi-stationary states which result from the violent relaxation of rectangular waterbag initial conditions, characterized by their virial ratio R_0, we show that relaxation occurs on a time scale which (i) scales approximately linearly in the particle number N, and (ii) shows also a strong dependence on R_0, with quasi-stationary states from colder initial conditions relaxing much more rapidly. The temporal evolution of the order parameter may be well described by a stretched exponential function. We study finally the correlation of the relaxation times with the amplitude of fluctuations in the relaxing quasi-stationary states, as well as the relation between temporal and ensemble averages.Comment: 37 pages, 24 figures; some additional discussion of previous literature and other minor modifications, final published versio

Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential

We consider a system of particles interacting via a screened Newtonian potential and study phase transitions between homogeneous and inhomogeneous states in the microcanonical and canonical ensembles. Like for other systems with long-range interactions, we obtain a great diversity of microcanonical and canonical phase transitions depending on the dimension of space and on the importance of the screening length. We also consider a system of particles in Newtonian interaction in the presence of a ``neutralizing background''. By a proper interpretation of the parameters, our study describes (i) self-gravitating systems in a cosmological setting, and (ii) chemotaxis of bacterial populations in the original Keller-Segel model

Environmental DNA for the enumeration and management of Pacific salmon

Pacific salmon are a keystone resource in Alaska, generating annual revenues of well over ~US$500 million/yr. Due to their anadromous life history, adult spawners distribute amongst thousands of streams, posing a huge management challenge. Currently, spawners are enumerated at just a few streams because of reliance on human counters and, rarely, sonar. The ability to detect organisms by shed tissue (environmental DNA, eDNA) promises a more efficient counting method. However, although eDNA correlates generally with local fish abundances, we do not know if eDNA can accurately enumerate salmon. Here we show that daily, and near‐daily, flow‐corrected eDNA rate closely tracks daily numbers of returning sockeye and coho spawners and outmigrating sockeye smolts. eDNA thus promises accurate and efficient enumeration, but to deliver the most robust numbers will need higher‐resolution stream‐flow data, at‐least‐daily sampling, and a focus on species with simple life histories, since shedding rate varies amongst jacks, juveniles, and adults

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. The equilibrium states correspond to polytropic distributions. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n and determine their stability by using turning points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.

Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions

We address the thermodynamics (equilibrium density profiles, phase diagram, instability analysis...) and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, we derive the analytic form of the density scaling profile which decays as f(x)=x^{-\alpha}, with alpha=2. In the microcanonical ensemble, we show that f decays as f(x)=x^{-\alpha_{max}}, where \alpha_{max} is a non-trivial exponent. We derive exact expansions for alpha_{max} and f in the limit of large D. Finally, we solve the problem in D=2, which displays rather rich and peculiar features

Spatial regularity of InAs-GaAs quantum dots: quantifying the dependence of lateral ordering on growth rate.

The lateral ordering of arrays of self-assembled InAs-GaAs quantum dots (QDs) has been quantified as a function of growth rate, using the Hopkins-Skellam index (HSI). Coherent QD arrays have a spatial distribution which is neither random nor ordered, but intermediate. The lateral ordering improves as the growth rate is increased and can be explained by more spatially regular nucleation as the QD density increases. By contrast, large and irregular 3D islands are distributed randomly on the surface. This is consistent with a random selection of the mature QDs relaxing by dislocation nucleation at a later stage in the growth, independently of each QD's surroundings. In addition we explore the statistical variability of the HSI as a function of the number N of spatial points analysed, and we recommend N > 10(3) to reliably distinguish random from ordered arrays