Exact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.Comment: 21 pages, 4 figure

Field theories of active particle systems and their entropy production

Active particles that translate chemical energy into self-propulsion can maintain a far-from-equilibrium steady state and perform work. The entropy production measures how far from equilibrium such a particle system operates and serves as a proxy for the work performed. Field theory offers a promising route to calculating entropy production, as it allows for many interacting particles to be considered simultaneously. Approximate field theories obtained by coarse-graining or smoothing that draw on additive noise can capture densities and correlations well, but they generally ignore the microscopic particle nature of the constituents, thereby producing spurious results for the entropy production. As an alternative we demonstrate how to use Doi-Peliti field theories, which capture the microscopic dynamics, including reactions and interactions with external and pair potentials. Such field theories are in principle exact, while offering a systematic approximation scheme, in the form of diagrammatics. We demonstrate how to construct them from a Fokker-Planck equation (FPE) of the single-particle dynamics and show how to calculate entropy production of active matter from first principles. This framework is easily extended to include interaction. We use it to derive exact, compact and efficient general expressions for the entropy production for a vast range of interacting particle systems. These expressions are independent of the underlying field theory and can be interpreted as the spatial average of the local entropy production. They are readily applicable to numerical and experimental data. In general, any pair interaction draws at most on the three point, equal time density and an n-point interaction on the (2n-1)-point density. We illustrate the technique in a number of exact, tractable examples, including some with pair-interaction.Comment: 10 page main text, no figures; 49 pages supplement, two figure

Entropy production of non-reciprocal interactions

Non-reciprocal interactions, in general, break detailed balance. We study an active particle system where activity originates from asymmetric, pairwise interaction forces that result in an injection of energy at the microscopic scale. Using a field theory that captures microscopic dynamics, we calculate correlation functions and the entropy production to characterise the non-equilibrium properties of this many-particle active system in the stationary state. We support our analytical results with numerical simulations.Comment: 15 pages, 5 figure

Run-and-tumble motion in a harmonic potential: field theory and entropy production

Abstract: Run-and-tumble (RnT) motion is an example of active motility where particles move at constant speed and change direction at random times. In this work we study RnT motion with diffusion in a harmonic potential in one dimension via a path integral approach. We derive a Doi-Peliti field theory and use it to calculate the entropy production and other observables in closed form. All our results are exact

Entropy Production in Exactly Solvable Systems.

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms

Time-dependent branching processes: a model of oscillating neuronal avalanches

Funder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266Abstract: Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the co-existence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). Ordinary continuous-time branching processes with constant extinction and branching rates are commonly used as models of neuronal activity, yet they lack any such time-dependence. In the present work, we extend a basic branching process by allowing the extinction rate to oscillate in time as a new model to describe cortical dynamics. By means of a perturbative field theory, we derive relevant observables in closed form. We support our findings by quantitative comparison to numerics and qualitative comparison to available experimental results

Volume explored by a branching random walk on general graphs.

Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding-the branching random walk (BRW)-is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs' dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks

2005年度大学院文学研究科修士論文・文学部卒業論文題目一覧

<p>(a) Comparison of the exact probability of survival, <i>ρ</i>(<i>L</i>), given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161586#pone.0161586.e032" target="_blank">Eq (17)</a>, with the approximations given by the scaling law <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161586#pone.0161586.e038" target="_blank">Eq (22)</a> and by the scaling law with the first correction to scaling, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161586#pone.0161586.e058" target="_blank">Eq (40)</a>, for different <i>m</i> and <i>L</i>. (b) The same taking the <i>y</i>–axis logarithmic. (c) The same data, taking the ratio between the approximation given by the scaling law [], <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161586#pone.0161586.e038" target="_blank">Eq (22)</a>, and the exact value of <i>ρ</i>(<i>L</i>). Larger values of <i>L</i> are included in this case. The program used to draw the figure is provided as <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0161586#pone.0161586.s001" target="_blank">S1 File</a>.</p

Interactions, correlations and collective behaviour in non-equilibrium systems

Non-equilibrium systems, which usually involve a large number of interacting particles, are ubiquitous in nature and society. Interactions, such as volume exclusion or branching, induce correlations in the system and often translate into emergent collec- tive behaviour at macroscopic scales. This is the case, for instance, in active matter, where particles are subject to local non-thermal forces that are transformed into mechanical work. Establishing the relationship between macroscopic patterns and microscopic dynamics analytically is a challenge that has motivated different approaches in the community. In this thesis I have focused on some statistical properties with an emphasis on correlations, both spatial and temporal, of six different non-equilibrium particle systems: the Oslo rice pile model, branching processes and applications, the voter model, and run-and-tumble motion. The first three systems display critical phenomena and the last one is an instance of active motility. The approaches that I have followed are, on the one hand, one-to-one mappings between different, known, stochastic processes and, on the other hand, the Doi-Peliti field theory formalism. The point in common between both approaches is the ability to retain the particle entity, which proves essential when tackling observables that strongly depend on the microscopic dynamics of the system, such as correlation functions. However, the field-theoretic approach is much more systematic, as it becomes apparent in the applicability of this route to different kinds of reaction-diffusion particle systems.Open Acces