A rigourous demonstration of the validity of Boltzmann's scenario for the spatial homogenization of a freely expanding gas and the equilibration of the Kac ring

Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number $N$ of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, apparently forever. We provide here rigourous new results that mathematically prove the basic features of Boltzmann's scenario for two classical models: a simple boundary-free model for the spatial homogenization of a non-interacting gas of point particles, and the well-known Kac ring model. Our results, based on concentration inequalities that go back to Hoeffding, and which focus on the typical behavior of individual macroscopic systems, improve upon previous results by providing estimates, exponential in $N$, of probabilities and time scales involved

Random Walk Access Times on Partially-Disordered Complex Networks: an Effective Medium Theory

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are different for steps across lattice bonds from the rates $f$ across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially-disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.Comment: 12 pages, 8 figures; added new results and discussion; added appendix on numerical procedures. To appear in PR

Dynamical mechanisms leading to equilibration in two-component gases

Demonstrating how microscopic dynamics cause large systems to approach thermal equilibrium remains an elusive, longstanding, and actively-pursued goal of statistical mechanics. We identify here a dynamical mechanism for thermalization in a general class of two-component dynamical Lorentz gases, and prove that each component, even when maintained in a non-equilibrium state itself, can drive the other to a thermal state with a well-defined effective temperature.Comment: 5 pages, 5 figure

Traversal Times for Random Walks on Small-World Networks

We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate F that is different from the rate f for steps across small-world connections. When f >> F, the mean time to traverse the network exhibits a transition associated with percolation of the random graph (i.e., small-world) part of the network, and a collapse of the data onto a universal curve. This transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps. We develop a simple self-consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition, as is characteristic of most effective medium theories.Comment: 9 pages, 5 figure

Rigorous results on approach to thermal equilibrium, entanglement, and nonclassicality of an optical quantum field mode scattering from the elements of a non-equilibrium quantum reservoir

Rigorous derivations of the approach of individual elements of large isolated systems to a state of thermal equilibrium, starting from arbitrary initial states, are exceedingly rare. This is particularly true for quantum mechanical systems. We demonstrate here how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum system, one that can be viewed as a natural quantum analog of several previously studied classical models. In particular, we consider an optical mode passing through a reservoir composed of a large number of sequentially-encountered modes of the same frequency, each of which it interacts with through a beam splitter. We then analyze the dependence of the asymptotic state of this mode on the assumed stationary common initial state $\sigma$ of the reservoir modes and on the transmittance $\tau=\cos\lambda$ of the beam splitters. These results allow us to establish that at small $\lambda$ such a mode will, starting from an arbitrary initial system state $\rho$, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized. We show in addition that, when the initial states are pure, the asymptotic state of the optical mode is maximally entangled with the reservoir and exhibits less nonclassicality than the state of the reservoir modes

Adiabatic-Nonadiabatic Transition in the Diffusive Hamiltonian Dynamics of a Classical Holstein Polaron

We study the Hamiltonian dynamics of a free particle injected onto a chain containing a periodic array of harmonic oscillators in thermal equilibrium. The particle interacts locally with each oscillator, with an interaction that is linear in the oscillator coordinate and independent of the particle's position when it is within a finite interaction range. At long times the particle exhibits diffusive motion, with an ensemble averaged mean-squared displacement that is linear in time. The diffusion constant at high temperatures follows a power law D ~ T^{5/2} for all parameter values studied. At low temperatures particle motion changes to a hopping process in which the particle is bound for considerable periods of time to a single oscillator before it is able to escape and explore the rest of the chain. A different power law, D ~ T^{3/4}, emerges in this limit. A thermal distribution of particles exhibits thermally activated diffusion at low temperatures as a result of classically self-trapped polaronic states.Comment: 15 pages, 4 figures Submitted to Physical Review

Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks

We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds ($F$) and across network shortcuts ($f$). For fast shortcuts ($f/F\gg 1$) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.Comment: 8 pages, 6 figures; expanded discussions, added figures and references. To appear in J Stat Phy

Chaotic Dynamics of a Free Particle Interacting Linearly with a Harmonic Oscillator

We study the closed Hamiltonian dynamics of a free particle moving on a ring, over one section of which it interacts linearly with a single harmonic oscillator. On the basis of numerical and analytical evidence, we conjecture that at small positive energies the phase space of our model is completely chaotic except for a single region of complete integrability with a smooth sharp boundary showing no KAM-type structures of any kind. This results in the cleanest mixed phase space structure possible, in which motions in the integrable region and in the chaotic region are clearly separated and independent of one another. For certain system parameters, this mixed phase space structure can be tuned to make either of the two components disappear, leaving a completely integrable or completely chaotic phase space. For other values of the system parameters, additional structures appear, such as KAM-like elliptic islands, and one parameter families of parabolic periodic orbits embedded in the chaotic sea. The latter are analogous to bouncing ball orbits seen in the stadium billiard. The analytical part of our study proceeds from a geometric description of the dynamics, and shows it to be equivalent to a linked twist map on the union of two intersecting disks.Comment: 17 pages, 11 figures Typos corrected to display section label

An open-source solution for advanced imaging flow cytometry data analysis using machine learning

Imaging flow cytometry (IFC) enables the high throughput collection of morphological and spatial information from hundreds of thousands of single cells. This high content, information rich image data can in theory resolve important biological differences among complex, often heterogeneous biological samples. However, data analysis is often performed in a highly manual and subjective manner using very limited image analysis techniques in combination with conventional flow cytometry gating strategies. This approach is not scalable to the hundreds of available image-based features per cell and thus makes use of only a fraction of the spatial and morphometric information. As a result, the quality, reproducibility and rigour of results are limited by the skill, experience and ingenuity of the data analyst. Here, we describe a pipeline using open-source software that leverages the rich information in digital imagery using machine learning algorithms. Compensated and corrected raw image files (.rif) data files from an imaging flow cytometer (the proprietary .cif file format) are imported into the open-source software CellProfiler, where an image processing pipeline identifies cells and subcellular compartments allowing hundreds of morphological features to be measured. This high-dimensional data can then be analysed using cutting-edge machine learning and clustering approaches using “user-friendly” platforms such as CellProfiler Analyst. Researchers can train an automated cell classifier to recognize different cell types, cell cycle phases, drug treatment/control conditions, etc., using supervised machine learning. This workflow should enable the scientific community to leverage the full analytical power of IFC-derived data set. It will help to reveal otherwise unappreciated populations of cells based on features that may be hidden to the human eye that include subtle measured differences in label free detection channels such as bright-field and dark-field imagery

Enhancing the relevance of Shared Socioeconomic Pathways for climate change impacts, adaptation and vulnerability research

This paper discusses the role and relevance of the shared socioeconomic pathways (SSPs) and the new scenarios that combine SSPs with representative concentration pathways (RCPs) for climate change impacts, adaptation, and vulnerability (IAV) research. It first provides an overview of uses of social–environmental scenarios in IAV studies and identifies the main shortcomings of earlier such scenarios. Second, the paper elaborates on two aspects of the SSPs and new scenarios that would improve their usefulness for IAV studies compared to earlier scenario sets: (i) enhancing their applicability while retaining coherence across spatial scales, and (ii) adding indicators of importance for projecting vulnerability. The paper therefore presents an agenda for future research, recommending that SSPs incorporate not only the standard variables of population and gross domestic product, but also indicators such as income distribution, spatial population, human health and governance