Efficiency of one-dimensional active transport conditioned on motility

By conditioning a stochastic process on the value of an observable, one obtains a new stochastic process with different properties. We apply this idea in the context of active matter, and condition interacting self-propelled particles on their individual motility. Using the effective process formalism from dynamical large deviations theory, we derive the interactions that actuate the imposed mobility against jamming interactions in two toy models---the totally asymmetric exclusion process and run-and-tumble particles, \emil{in the case of two or three particles}. We provide a framework which takes into account the energy-consumption required for self-propulsion, and address the question of how energy-efficient the emergent interactions are. Upon conditioning, run-and-tumble particles develop an alignment interaction and achieve a higher gain in efficiency than TASEP particles. A point of diminishing returns in efficiency is reached beyond a certain level of conditioning. With recourse to a general formula for the change in energy efficiency upon conditioning, we conclude that the most significant gains occur when there are large fluctuations in mobility to exploit. From a detailed comparison of the emergent effective interaction in a two- versus a three-body scenario, we discover evidence of a screening effect which suggests that conditioning can produce topological rather than metric interactions.Comment: 11 pages, 9 figure

Exact solutions to the long-time statistics of nonequilibrium processes

Nonequilibrium statistical mechanics deals with noisy systems whose dynamics breaks time-reversal symmetry. Stochastic Markov processes form a mathematical framework for a unified theory of nonequilibrium phenomena, particularly at the microscale where fluctuations play a dominant role. The development of both theory and useful applications has been aided by minimal, exactly solvable models, where the logical connection between model feature and behaviour can be ascertained. In this thesis I consider a number of biologically inspired models in relation to two aspects of the long-time statistics of nonequilibrium processes: the attainment of a nonequilibrium steady state, and steady-state fluctuations using dynamical large deviation theory. For the first topic, I consider two models of particles moving stochastically on a ring under no-crossing interactions. The first is of two lattice run-and-tumble particles, each of which moves in a persistent direction interspersed by `tumble' reorientation events. I extend the previously known steady-state solution to a solution for all time, in the sense of obtaining a diagonalization of the Markov generator of the process. The spectrum exhibits eigenvalue crossings at exceptional points of the tumbling rate, which leads to a singular dependence of the relaxation time to the steady state on this parameter. In the second model of heterogeneous single-file diffusion, I solve for the steady state of N driven particles with individual diffusion properties. This reveals an inter-particle ratchet effect by which the particle current is disproportionately affected by slow-diffusing, rather than slow- or fast-driven, particles. The model generalizes to higher dimension without compromising the solution structure, if a key property of quasi-one-dimensionality is maintained. In both models, the relation of key model features to generalized notions of reversibility forms an overarching theme. For the second topic, I first consider a random walker on a linear lattice and the effect that adding internal states to the walker has on the emergence of singular behaviour in the fluctuations of either the velocity observable or the time spent at a given site. In particular, I use generalizations of the run-and-tumble particle and probe the trajectories associated with the different fluctuations regimes that this model can exhibit. I show that internal states can either have a drastic influence on the likelihood of a large deviation, or none at all. I then extend the dynamical large deviation formalism for diffusions to the case of reflective boundaries and current-like observables. In particular, this allows the large deviations of the particle current in the heterogeneous single-file diffusion to be obtained analytically. These are found to coincide with those of a single diffusive particle with certain effective parameters, in interesting contrast to comparable studies on the lattice

Inter-particle ratchet effect determines global current of heterogeneous particles diffusing in confinement

In a model of $N$ volume-excluding spheres in a $d$-dimensional tube, we consider how differences between particles in their drift velocities, diffusivities, and sizes influence the steady state distribution and axial particle current. We show that the model is exactly solvable when the geometrical constraints prevent any particle from overtaking every other -- a notion we term quasi-one-dimensionality. Then, due to a ratchet effect, the current is biased towards the velocities of the least diffusive particles. We consider special cases of this model in one dimension, and derive the exact joint gap distribution for driven tracers in a passive bath. We describe the relationship between phase space structure and irreversible drift that makes the quasi-one-dimensional supposition key to the model's solvability.Comment: 26 pages, 7 figure

Exact spectral solution of two interacting run-and-tumble particles on a ring lattice

Exact solutions of interacting random walk models, such as 1D lattice gases, offer precise insight into the origin of nonequilibrium phenomena. Here, we study a model of run-and-tumble particles on a ring lattice interacting via hardcore exclusion. We present the exact solution for one and two particles using a generating function technique. For two particles, the eigenvectors and eigenvalues are explicitly expressed using two parameters reminiscent of Bethe roots, whose numerical values are determined by polynomial equations which we derive. The spectrum depends in a complicated way on the ratio of direction reversal rate to lattice jump rate, $\omega$. For both one and two particles, the spectrum consists of separate real bands for large $\omega$, which mix and become complex-valued for small $\omega$. At exceptional values of $\omega$, two or more eigenvalues coalesce such that the Markov matrix is non-diagonalizable. A consequence of this intricate parameter dependence is the appearance of dynamical transitions: non-analytic minima in the longest relaxation times as functions of $\omega$ (for a given lattice size). Exceptional points are theoretically and experimentally relevant in, e.g., open quantum systems and multichannel scattering. We propose that the phenomenon should be a ubiquitous feature of classical nonequilibrium models as well, and of relevance to physical observables in this context.Comment: 29 pages, 7 figures, revised submission to J. Stat. Mec

A comparison of dynamical fluctuations of biased diffusion and run-and-tumble dynamics in one dimension

We compare the fluctuations in the velocity and in the fraction of time spent at a given position for minimal models of a passive and an active particle: an asymmetric random walker and a run-and-tumble particle in continuous time and on a 1D lattice. We compute rate functions and effective dynamics conditioned on large deviations for these observables. While generally different, for a unique and non-trivial choice of rates (up to a rescaling of time) the velocity rate functions for the two models become identical, whereas the effective processes generating the fluctuations remain distinct. This equivalence coincides with a remarkable parity of the spectra of the processes' generators. For the occupation-time problem, we show that both the passive and active particles undergo a prototypical dynamical phase transition when the average velocity is non-vanishing in the long-time limit.Comment: 27 pages, 10 figure

Game as a Means of Developing Children's Cognitive Interests at Preschool

Maģistra darba tēma: “Rotaļa kā bērnu izziņas interešu attīstības līdzeklis pirmsskolas pedagoģiskajā procesā”. Darba apjoms 102 lpp., 149 bibliogrāfiskie nosaukumi, 10 tabulas, 6 zīmējumi, 15 diagrammas, 12 pielikumi. Pētījuma mērķis: pētīt bērnu izziņas interešu attīstību sekmējošus līdzekļus. Maģistra darbs sastāv no2 daļām: 1.Darba teorētiskajā daļā aplūkotas pētījumos sastopamās pamatnostādnes par izziņas intereses, kā vispārīgā „intereses” fenomena sastāvdaļu, tās vēsturiskie, filozofiskie, psiholoģiskie un pedagoģiskie aspekti. Analizēti pētnieku J. F. Herbarta, V. Džeimsa, H. Spensera, S. Anaņina darbi, kā arī mūsdienu autori, kuri pētījuši motivāciju, - S. Rubinšteins, J. Iļjins, L. Božoviča, G. Ščukina, K. Bardins u.c. Aplūkotas arī J. Komenska, F. Frēbela, M. Montesori, K. Dēķēna, J. Studenta un daudzu citu autoru pedagoģiskās idejas par bērnu intereses veicināšanu un rotaļas nozīmi šai procesā. Apskatītas arī pedagoģiskā procesa metodes, kas veicina piecgadīgu un sešgadīgu bērnu izziņas interešu attīstību. 2. Darba eksperimentālā daļa veltīta bērnu izziņas interešu attīstīšanai. Izmantojot izstrādātos metodiskos materiālus, tika pārbaudīta hipotēze: pirmsskolas vecuma bērnu izziņas intereses attīstās efektīvāk, ja pedagoģiskajā procesā izmantojot rotaļu. Pētījuma gaitā hipotēze apstiprinājās: vecākā pirmsskolas vecuma bērni izrāda zinātkāri, izziņas attieksmi pret īstenību, ja bērnu iepazīstina ar dažādām dzīves jomām (priekšmetisko pasauli, dabas pasauli, savstarpējo attiecību pasauli, sava „Es” izzināšanu), iesaistot rotaļā.MA paper on a theme “Games as the Means of Children’s Cognitive Interest Development at Kindergarten” includes 102 pages of text, 149 bibliography titles, 10 tables, 6 drawings, 15 charts and 12 appendixes. Research objective was to explore the means encouraging the development of children’s cognitive interests. The paper consists of 2 parts. 1The theoretical part analyses the basic studies of the cognitive interest within the general phenomenon of interest. The historical, the philosophical, the psychological and the educational aspects are touched upon. This part of the paper is based both on the historical educational works by J.F.Herbart, W.James, H.Spenser and S.Ananyn, and on modern authors studying the children’s motivation – S.Rubinstein, Y.Ilyin, L.Bozhovich, G.Tschukin, K.Bardin and others. The educational ideas by J.Komensky, F.FrĻbel, M.Montessori, K.Dēķens, J.Students and many other’s concerning the development of children’s cognitive interest and the role of the games are analysed. Methods and educational approaches encouraging the cognitive interest of five- and six-year children are presented. 2The experimental part examines the development of children’s cognitive interest. A designed resource material was used to verify the hypothesis: including games into the educational process advances the development of preschoolers’ cognitive interest. The hypothesis was approved: older preschoolers manifest more inquisitiveness and react cognitively to the reality when introduced to different fields of life (things, nature, relationships, self-actualization) by game activities

Plasma 25-Hydroxyvitamin D Levels and Fracture Risk in a Community-Based Cohort of Elderly Men in Sweden

Context: Blood levels of 25-hydroxyvitamin D [25(OH)D] is the generally accepted indicator of vitamin D status, but no universal reference level has been reached. Objective: The objective of the study was to determine the threshold at which low plasma 25(OH)D levels are associated with fractures in elderly men and clarify the importance of low levels on total fracture burden. Design and Participants: In the Uppsala Longitudinal Study of Adult Men, a population-based cohort (mean age, 71 yr, n = 1194), we examined the relationship between 25(OH)D and risk for fracture. Plasma 25(OH)D levels were measured with high-pressure liquid chromatography-mass spectrometry. Setting: The study was conducted in the municipality of Uppsala in Sweden, a country with a high fracture incidence. Main Outcome Measure: Time to fracture was measured. Results: During follow-up (median 11 yr), 309 of the participants (26%) sustained a fracture. 25(OH)D levels below 40 nmol/liter, which corresponded to the fifth percentile of 25(OH)D, were associated with a modestly increased risk for fracture, multivariable-adjusted hazard ratio 1.65 (95% confidence interval 1.09-2.49). No risk difference was detected above this level. Approximately 3% of the fractures were attributable to low 25(OH)D levels in this population. Conclusions: Vitamin D insufficiency is not a major cause of fractures in community-dwelling elderly men in Sweden. Despite the fact that cutaneous synthesis of previtamin D during the winter season is undetectable at this northern latitude of 60 degrees , only one in 20 had 25(OH)D levels below 40 nmol/liter, the threshold at which the risk for fracture started to increase. Genetic adaptations to limited UV light may be an explanation for our findings