Table of contents

We have analytically explored both the zero temperature and the finite temperature scaling theory for the collapse of an attractively interacting 3D harmonically trapped Bose gas in a synthetic magnetic field. We have considered short ranged (contact) attractive inter-particle interactions and Hartree–Fock approximation for the same. We have separately studied the collapse of both the condensate and the thermal cloud below and above the condensation point, respectively. We have obtained an anisotropy, artificial magnetic field, and temperature dependent critical number of particles for the collapse of the condensate. We have found a dramatic change in the critical exponent (from α = 1 to 0) of the specific heat ($C_v\propto|T-T_\mathrm{c}|^{\alpha}$) when the thermal cloud is about to collapse with the critical number of particles ($N = N_\mathrm{c}$) just below and above the condensation point. All the results obtained by us below and around the condensation point are experimentally testable within the present-day experimental set-up for the ultracold systems in the magneto-optical traps.

Even in a simple stochastic process, the study of the full distribution of time-integrated observables can be a difficult task. This is the case of a much-studied process such as the Ornstein–Uhlenbeck process where, recently, anomalous dynamical scaling of large deviations of time-integrated functionals has been highlighted. Using the mapping of a continuous stochastic process to a continuous time random walk via the 'excursions technique', we introduce a comprehensive formalism that enables the calculation of the complete distribution of the time-integrated observable $A = \int_0^T v^n(t) \mathrm dt$, where n is a positive integer and v(t) is the random velocity of a particle following Ornstein–Uhlenbeck dynamics. We reveal an interesting connection between the anomalous rate function associated with the observable A and the statistics of the area under the first-passage functional during an excursion. The rate function of the latter, analyzed here for the first time, exhibits anomalous scaling behavior and a critical point in its dynamics, both of which are explored in detail. The case of the anomalous scaling of large deviations, originally associated with the presence of an instantonic solution in the weak noise regime of a path integral approach, is here produced by a so-called 'big jump effect', in which the contribution to rare events is dominated by the largest excursion. Our approach, which is quite general for continuous stochastic processes, allows us to associate a physical meaning with the anomalous scaling of large deviations through the big jump principle.

In physics, with the advent of topological materials, and in chemistry, within the scope of chemical graph theory, topological invariants and/or indices have been considered to successfully characterize innumerous systems. In particular, strong links have been identified (both numerically and analytically) between properties of the Ising model on a lattice L and features of the so-called spanning trees (STs) of L. Nontheless, studies exploring this connection tend to address only a handful of cases given the demands of the necessary calculations. But examining only a few instances prevents one from looking for general trends across numerous L's, which could eventually reveal universal traits. In this contribution, we present the most comprehensive investigation to date, analyzing the Ising-ST relation for all the L's belonging to the families $\mathfrak{F}_K$ of $1 \unicode{x2A7D} K \unicode{x2A7D} 6$–uniform periodic tiling of the plane, in a total of 1248 lattices. With this goal, we develop optimized protocols (taking advantage of a recently proposed $\mathbb{Z}^4$ representation for $\mathfrak{F}_K$) to compute for each L its ST constant λ and the Kac–Ward matrix. The determinant of the latter yields the Ising model free energy and consequently the critical temperature $T_\mathrm{c}$. Then, considering the relatively large sample generated, we use machine learning techniques, which disclose a general correlation between the Ising critical temperature and the ST constant, described by a simple quadratic polynomial function P. As a benchmark, we test P for some arbitrary lattices (outside $\mathfrak{F}_K$), finding rather satisfactory fittings. These results point to a useful classification scheme for Ising $T_\mathrm{c}$ in 2D, demonstrating that λ can be a relevant topological concept to investigate lattice models. Finally, as a positive 'side-effect' of our computations, for these $\mathfrak{F}_K$'s we confirm (and even improve) a conjectured inequality associating λ and the effective coordinator number κ of a lattice.

The Nishimori point of the random bond Ising model is a prototype of renormalization group fixed points with strong disorder. We show that the exact correlation length and crossover critical exponents at this point can be identified in two and three spatial dimensions starting from properties of the Nishimori line. These are the first exact exponents for frustrated random magnets, a circumstance to be also contrasted with the fact that the exact exponents of the Ising model without disorder are not known in three dimensions. Our considerations extend to higher dimensions and models other than Ising.

We examine the behavior of a single impurity particle embedded within a totally asymmetric simple exclusion process (TASEP). By analyzing the impurity's dynamics, characterized by two arbitrary hopping parameters α and β, we investigate both its macroscopic impact on the system and its individual trajectory, providing new insights into the interaction between the impurity and the TASEP environment. We classify the induced hydrodynamic limit shapes based on the initial densities to the left and right of the impurity, along with the values of the parameters α, β. We develop a new method that enables the analysis of the impurity's behavior within an arbitrary density field, thereby generalizing the traditional coupling technique used for second-class particles. With this tool, we extend to the impurity case under certain parameter conditions, Ferrari and Kipnis's results on the distribution of the asymptotic speed of a second-class particle within a rarefaction fan.

In recent years, simulations of pedestrians using multi-agent reinforcement learning (MARL) have been studied. This study considers roads in a grid-world environment and implements pedestrians as MARL agents using an echo-state network and the least squares policy iteration method. In this environment, the ability of these agents to learn to move forward by avoiding other agents is investigated. Specifically, we consider two types of tasks: the choice between a narrow direct route and a broad detour and the bidirectional pedestrian flow in a corridor. The simulation results indicate that the learning is successful when the density of agents is not that high.

In this paper, we present a detailed statistical analysis related to the characterization of spatial and temporal fluctuations present in the rainfall patterns of the North-East region ($26.05^{\circ}\,\mathrm{N}$–26.95$^{\circ}\,\mathrm{N}$, $88.05^{\circ}\,\mathrm{E}$–94.95$^{\circ}\,\mathrm{E}$) of India using half hourly rainfall data over the last two decades from 2001–2020. We analyze the nature of the rain distribution by computing the mean, second moment of fluctuation, skewness and kurtosis of the temporal rainfall data that indicate the presence of heavy tails in the right skewed distribution, a typical feature of the presence of rare events. We find that the temporal distribution of rainfall data follows a multiplicative Log-Normal probability distribution. Further, we compute the spatial and temporal correlation of rainfall in this region, indicating that rainfall events are correlated in the spatial direction within about 70 km. The power spectral density of the temporal rainfall data shows power law behavior with frequency with an exponent ${\sim}-1.5$ close to the Kolmogorov exponent (${\sim}-1.67$) exhibited by the turbulent passive scalar driven by the mean flow. Our wavelet analysis reveals evidence of multiple frequencies in rainfall patterns which can be attributed to different short and long range factors responsible for rainfall. We have also used the Hilbert Huang transformation to identify the frequencies corresponding to the fluctuating and quasi-periodic parts of the rainfall time series. Finally, we report the multifractal nature of rainfall patterns using multifractal detrended fluctuation analysis and using the width of multifractal spectrum we have identified the low and high fluctuation dominated rainfall region.