We review here the recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations. The concept is introduced in successive steps through the studies of mapping of such computationally hard problems to the classical spin glass problems. The quantum spin glass problems arise with the introduction of quantum fluctuations, and the annealing behavior of the systems as these fluctuations are reduced slowly to zero. This provides a general framework for realizing analog quantum computation.

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. Analogous to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium probability distribution of money. When the agents do not save, the equilibrium money distribution becomes the usual Gibb's distribution, characteristic of non-interacting agents. However with saving, even for individual self-interest, the dynamics becomes cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as functions of the marginal saving propensity of the agents.

The fiber bundle model describes a collection of elastic fibers under load. The fibers fail sucessively and for each failure, the load distribution among the surviving fibers changes. Even though very simple, this model captures the essentials of failure processes in a large number of materials and settings. We present here a review of the fiber bundle model with different load redistribution mechanisms from the point of view of statistics and statistical physics rather than materials science, with a focus on concepts such as criticality, universality and fluctuations. We discuss the fiber bundle model as a tool for understanding phenomena such as creep, and fatigue, how it is used to describe the behavior of fiber reinforced composites as well as modelling e.g. network failure, traffic jams and earthquake dynamics.

When an interacting many-body system, such as a magnet, is driven in time by an external perturbation, such as a magnetic field, the system cannot respond instantaneously due to relaxational delay. The response of such a system under a time-dependent field leads to many novel physical phenomena with intriguing physics and important technological applications. For oscillating fields, one obtains hysteresis that would not occur under quasistatic conditions in the presence of thermal fluctuations. Under some extreme conditions of the driving field, one can also obtain a non-zero average value of the variable undergoing such "dynamic hysteresis". This non-zero value indicates a breaking of symmetry of the hysteresis loop, around the origin. Such a transition to the "spontaneously broken symmetric phase" occurs dynamically when the driving frequency of the field increases beyond its threshold value which depends on the field amplitude and the temperature. Similar dynamic transitions also occur for pulsed and stochastically varying fields.We present an overview of the ongoing researches in this not-so-old field of dynamic hysteresis and transitions.

We have numerically simulated the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents, distributed widely between the agents (0 ≤ λ < 1). The system remarkably self-organizes to a critical Pareto distribution of money P (m) ∼ m −(ν+1) with ν ≃ 1. We analyse the robustness (universality) of the distribution in the model. We also argue that although the fractional saving ingredient is a bit unnatural one in the context of gas models, our model is the simplest so far, showing selforganized criticality, and combines two century-old distributions: Gibbs (1901) Considerable investigations have already been made to study the nature of income or wealth distributions in various economic communities, in particular, in different countries. For more than a hundred years, it is known that the probability distribution P (m) for income or wealth of the individuals in the market decreases with the wealth m following a power law, known as Pareto law [1]:where the value of the exponent ν is found to lie between 1 and 2 [2,3,4]. It is also known that typically less than 10% of the population in any country possesses about 40% of the wealth and follow the above power law. The rest of the low-income group population, in fact the majority, clearly follows a different law, identified very recently to be the Gibbs distribution [5,6,7]. Studies on real data show that the high income group indeed follow Pareto law, with ν varying from 1.6 for USA [6], to 1.8 − 2.2 in Japan [3]. The value of ν thus seem to vary a little from economy to economy. We have studied here numerically a gas model of a trading market. We have considered the effect of saving propensity of the traders. The saving propensity is assumed to have a randomness. Our observations indicate that Gibbs and Pareto distributions fall in the same category and can appear naturally in the century-old and well-established kinetic theory of gas [8]: Gibbs distribution for no saving and Pareto distribution for agents with quenched random saving propensity. Our model study also indicates the appearance of self-organized criticality [9] in the simplest model so far, namely in the kinetic theory of gas models, when the stability effect of savings [10] is incorporated.We consider an ideal-gas model of a closed economic system where total money M and total number of agents * Electronic address: arnab@cmp.saha.ernet.in N is fixed. No production or migration occurs and the only economic activity is confined to trading. Each agent i, individual or corporate, possess money m i (t) at time t. In any trading, a pair of traders i and j randomly exchange their money [5,7,11], such that their total money is (locally) conserved and none end up with negative money (m i (t) ≥ 0, i.e, debt not allowed):time (t) changes by one unit after each trading. The steady-state (t → ∞) distribution of money is Gibbs one:Hence...