Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality.

We define the Q factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability p is increased, the Q factor for the system size L grows systematically to its maximum value Q_{max}(L) at a specific value p_{max}(L) and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of p_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices.

Finding critical points and correlation length exponents using finite size scaling of Gini index.

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.

Inequality of avalanche sizes in models of fracture.

Prediction of an imminent catastrophic event in a driven disordered system is of paramount importance-from the laboratory scale controlled fracture experiment to the largest scale of mechanical failure, i.e., earthquakes.

Kinetic Models of Wealth Distribution with Extreme Inequality: Numerical Study of Their Stability against Random Exchanges.

In view of some recent reports on global wealth inequality, where a small number (often a handful) of people own more wealth than 50% of the world's population, we explored if kinetic exchange models of markets could ever capture features where a significant fraction of wealth can concentrate in the hands of a few as the market size approaches infinity. One existing example of such a kinetic exchange model is the Chakraborti or Yard-Sale model; in the absence of tax redistribution, etc., all wealth ultimately condenses into the hands of a single individual (for any value of ), and the market dynamics stop.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini () index and the Kolkata () index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as , indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'.

Quantum annealing and computation: challenges and perspectives.

In the introductory article of this theme issue, we provide an overview of quantum annealing and computation with a very brief summary of the individual contributions to this issue made by experts as well as a few young researchers. We hope the readers will get the touch of the excitement as well as the perspectives in this unusually active field and important developments there. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Quantum annealing: an overview.

In this review, after providing the basic physical concept behind quantum annealing (or adiabatic quantum computation), we present an overview of some recent theoretical as well as experimental developments pointing to the issues which are still debated. With a brief discussion on the fundamental ideas of continuous and discontinuous quantum phase transitions, we discuss the Kibble-Zurek scaling of defect generation following a ramping of a quantum many body system across a quantum critical point. In the process, we discuss associated models, both pure and disordered, and shed light on implementations and some recent applications of the quantum annealing protocols.

Stochastic Learning in Kolkata Paise Restaurant Problem: Classical and Quantum Strategies.

We review the results for stochastic learning strategies, both classical (one-shot and iterative) and quantum (one-shot only), for optimizing the available many-choice resources among a large number of competing agents, developed over the last decade in the context of the Kolkata Paise Restaurant (KPR) Problem. Apart from few rigorous and approximate analytical results, both for classical and quantum strategies, most of the interesting results on the phase transition behavior (obtained so far for the classical model) uses classical Monte Carlo simulations. All these including the applications to computer science [job or resource allotments in Internet-of-Things (IoT)], transport engineering (online vehicle hire problems), operation research (optimizing efforts for delegated search problem, efficient solution of Traveling Salesman problem) will be discussed.

Kinetic exchange income distribution models with saving propensities: inequality indices and self-organized poverty level.

We report the numerical results for the steady-state income or wealth distribution [Formula: see text] and the resulting inequality measures (Gini [Formula: see text] and Kolkata [Formula: see text] indices) in the kinetic exchange models of market dynamics. We study the variations of [Formula: see text] and of the indices [Formula: see text] and [Formula: see text] with the saving propensity [Formula: see text] of the agents, with two different kinds of trade (kinetic exchange) dynamics. In the first case, the exchange occurs between randomly chosen pairs of agents and in the next, one of the agents in the chosen pair is the poorest of all and the other agent is randomly picked up from the rest of the population (where, in the steady state, a self-organized poverty level or SOPL appears).

Social inequality analysis of fiber bundle model statistics and prediction of materials failure.

Inequalities are abundant in a society with a number of agents competing for a limited amount of resources. Statistics on such social inequalities are usually represented by the Lorenz function L(p), where p fraction of the population possesses L(p) fraction of the total wealth, when the population is arranged in ascending order of their wealth. Similarly, in scientometrics, such inequalities can be represented by a plot of the citation count versus the respective number of papers by a scientist, again arranged in ascending order of their citation counts.

Development of Econophysics: A Biased Account and Perspective from Kolkata.

We present here a somewhat personalized account of the emergence of econophysics as an attractive research topic in physical, as well as social, sciences. After a rather detailed storytelling about our endeavors from Kolkata, we give a brief description of the main research achievements in a simple and non-technical language. We also briefly present, in technical language, a piece of our recent research result.

Phase transition in the Kolkata Paise Restaurant problem.

A novel phase transition behavior is observed in the Kolkata Paise Restaurant problem where a large number (N) of agents or customers collectively (and iteratively) learn to choose among the N restaurants where she would expect to be alone that evening and would get the only dish available there (or may get randomly picked up if more than one agent arrive there that evening). The players are expected to evolve their strategy such that the publicly available information about past crowds in different restaurants can be utilized and each of them is able to make the best minority choice. For equally ranked restaurants, we follow two crowd-avoiding strategies: strategy I, where each of the n(t) number of agents arriving at the ith restaurant on the tth evening goes back to the same restaurant the next evening with probability [n(t)], and strategy II, with probability p, when n(t)>1.

Flory-like statistics of fracture in the fiber bundle model as obtained via Kolmogorov dispersion for turbulence: A conjecture.

It has long been conjectured that (rapid) fracture propagation dynamics in materials and turbulent motion of fluids are two manifestations of the same physical process. The universality class of turbulence (Kolmogorov dispersion, in particular) is conjectured to be identifiable with the Flory statistics for linear polymers (self-avoiding walks on lattices). These help us to relate fracture statistics to those of linear polymers (Flory statistics).

Statistical physics of fracture and earthquakes.

Manifestations of emergent properties in stressed disordered materials are often the result of an interplay between strong perturbations in the stress field around defects. The collective response of a long-ranged correlated multi-component system is an ideal playing field for statistical physics. Hence, many aspects of such collective responses in widely spread length and energy scales can be addressed by the tools of statistical physics.

Hydrodynamic descriptions for surface roughness in fracture front propagation.

Fracture is ubiquitous in a crystalline material. Inspired by the observed phenomenological similarities between the spatial profile of a fractured surface and velocities in hydrodynamic turbulence, we set up a hydrodynamic description for the dynamics of fracture surface propagation mode I or opening fracture front. We consider several related continuum hydrodynamic models and use them to extract the similarities between the profile of a fractured surface and velocities in hydrodynamic turbulence.

Possible ergodic-nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model and quantum annealing.

We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamiltonian at finite temperatures and that at zero temperature obtained using the exact diagonalization method. Our numerical results indicate the existence of a low- but finite-temperature quantum-fluctuation-dominated ergodic region along with the classical fluctuation-dominated high-temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases.

Universality of Citation Distributions for Academic Institutions and Journals.

Citations measure the importance of a publication, and may serve as a proxy for its popularity and quality of its contents. Here we study the distributions of citations to publications from individual academic institutions for a single year. The average number of citations have large variations between different institutions across the world, but the probability distributions of citations for individual institutions can be rescaled to a common form by scaling the citations by the average number of citations for that institution.

Classical-to-quantum crossover in the critical behavior of the transverse-field Sherrington-Kirkpatrick spin glass model.

We study the critical behavior of the Sherrington-Kirkpatrick model in transverse field (at finite temperature) using Monte Carlo simulation and exact diagonalization (at zero temperature). We determine the phase diagram of the model by estimating the Binder cumulant. We also determine the correlation length exponent from the collapse of the scaled data.

Zipf's law in city size from a resource utilization model.

We study a resource utilization scenario characterized by intrinsic fitness. To describe the growth and organization of different cities, we consider a model for resource utilization where many restaurants compete, as in a game, to attract customers using an iterative learning process. Results for the case of restaurants with uniform fitness are reported.

Self-organized dynamics in local load-sharing fiber bundle models.

We study the dynamics of a local load-sharing fiber bundle model in two dimensions under an external load (which increases with time at a fixed slow rate) applied at a single point. Due to the local load-sharing nature, the redistributed load remains localized along the boundary of the broken patch. The system then goes to a self-organized state with a stationary average value of load per fiber along the (increasing) boundary of the broken patch (damaged region) and a scale-free distribution of avalanche sizes and other related quantities are observed.

Noise-induced rupture process: phase boundary and scaling of waiting time distribution.

A bundle of fibers has been considered here as a model for composite materials, where breaking of the fibers occur due to a combined influence of applied load (stress) and external noise. Through numerical simulation and a mean-field calculation we show that there exists a robust phase boundary between continuous (no waiting time) and intermittent fracturing regimes. In the intermittent regime, throughout the entire rupture process avalanches of different sizes are produced and there is a waiting time between two consecutive avalanches.

Continuous transition of social efficiencies in the stochastic-strategy minority game.

We show that in a variant of the minority game problem, the agents can reach a state of maximum social efficiency, where the fluctuation between the two choices is minimum, by following a simple stochastic strategy. By imagining a social scenario where the agents can only guess about the number of excess people in the majority, we show that as long as the guessed value is sufficiently close to the reality, the system can reach a state of full efficiency or minimum fluctuation. A continuous transition to less efficient condition is observed when the guessed value becomes worse.

Phase transitions in crowd dynamics of resource allocation.

We define and study a class of resource allocation processes where gN agents, by repeatedly visiting N resources, try to converge to an optimal configuration where each resource is occupied by at most one agent. The process exhibits a phase transition, as the density g of agents grows, from an absorbing to an active phase. In the latter, even if the number of resources is in principle enough for all agents (g<1), the system never settles to a frozen configuration.

Threshold-induced phase transition in kinetic exchange models.

We study an ideal-gas-like model where the particles exchange energy stochastically, through energy-conserving scattering processes, which take place if and only if at least one of the two particles has energy below a certain energy threshold (interactions are initiated by such low-energy particles). This model has an intriguing phase transition in the sense that there is a critical value of the energy threshold below which the number of particles in the steady state goes to zero, and above which the average number of particles in the steady state is nonzero. This phase transition is associated with standard features like "critical slowing down" and nontrivial values of some critical exponents characterizing the variation of thermodynamic quantities near the threshold energy.

Opinion formation in kinetic exchange models: spontaneous symmetry-breaking transition.

We propose a minimal multiagent model for the collective dynamics of opinion formation in the society by modifying kinetic exchange dynamics studied in the context of income, money, or wealth distributions in a society. This model has an intriguing spontaneous symmetry-breaking transition to polarized opinion state starting from nonpolarized opinion state. In order to analyze the model, we introduce an iterative map version of the model, which has very similar statistical characteristics.