Structural properties of the Indian Railway network is studied in the light of recent investigations of the scaling properties of different complex networks. Stations are considered as 'nodes' and an arbitrary pair of stations is said to be connected by a 'link' when at least one train stops at both stations. Rigorous analysis of the existing data shows that the Indian Railway network displays small-world properties. We define and estimate several other quantities associated with this network. Given a chance, how would we have possibly organized our train travel? People dislike to change trains to reach their destinations. Therefore an extreme possibility would be to run a single train passing through all stations in the country so that no change of train is needed at all! An obvious disadvantage in this strategy is that the average distance between the stations become very large and so also the time needed for travel. The other limiting situation would be, to run a train between any pair of neighbouring stations and try to travel along the minimal paths. This requires a change of train at every station, which is also clearly not economically viable. Railway networks in no country in the world follow either of the two ways, actually they go mid-way. Like any other transport system the main motivation of railways is to be fast and economic. To achieve it, railways run simultaneously many trains, covering short as well as long routes so that a traveller does not need to change more than only a few trains to reach any arbitrary destination in the country.In this paper we analyse the structure of the Indian Railway network (IRN). This is done in the context of recent investigations of the scaling properties of several complex networks e.g., social, biological, computational networks [1] etc. Identifying the stations as nodes of the network and a train which stops at any two stations as the link between the nodes we measure the average distance between an arbitrary pair of stations and find that it depends only logarithmically on the total number of stations in the country. While from the network point of view this implies the small-world nature of the railway network, in practice a traveller has to change only few trains to reach an arbitrary destination. This implies that over years, the railway network has been evolved with the sole aim in mind to make it fast and economic, eventually its structure has become a small-world network [2].The structure and properties of several social, biological and computational networks like the World-wide web (WWW) [3]
We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect it's arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals τ , and we consider the evolution of the wave function till the time a detection occurs. We study the probabilities of its first detection at some time and conversely the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, consisting of unitary transformations followed by projections, to one described by a nonHermitian Hamiltonian. For some examples, of a particle moving on one and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one-and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and non-trivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.
Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the time of arrival (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider the setup where a detector keeps making instantaneous measurements at regular finite time intervals till it detects the particle at some time t, which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics. PACS numbers: 03.65.-w, 03.65.Ta, 03.65.CaThe problem of defining the time of arrival (TOA) of a particle in quantum mechanics, and determining its probability distribution, has been a difficult and intriguing problem, one that is closely related to the foundations of quantum mechanics. A large body of work has studied this problem using a wide variety of approaches [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. A somewhat older but still relevant review of the various attempts to do so have been described in [15]. Experimentally, time of flight of atoms from source to detector, are routinely measured but these are typically in the semi-classical regime, and making meaning of these measurements in the quantum regime is not straightforward [15].There are several aspects that are involved in discussions of the TOA : (i) First there is the question of the effect of measurements made to detect the particle's arrival. The question of repeated ideal measurements of a quantum system was discussed in the seminal paper of Misra and Sudarshan [16] who studied this question in a general setting and showed the surprising result, the so-called quantum Zeno effect, that the probability of detecting a particle (or decay from the initial state) vanishes in the limit that the time interval between measurements τ → 0 [17,18]. This means that continuous measurements to find the time of arrival leads to the particle being never detected ! The Zeno effect has been experimentally verified [19] though questions of interpretation remain [21]. Hence the question of making measurements at regular finite intervals arises and it becomes necessary to study the effect, that null measurements have, on the time evolution of a quantum system and on the TOA distribution [6]. A related issue is that of defining POVMs corresponding to TOA measurements [14]; (ii) There is then the question of defining a self-adjoint time operator and some progress has been made here [9]. Determining arrival time distributions from these definitions has its own issues [10]; (iii) Finally there is the important que...
We investigate both the local and global persistence behaviour in ANNNI (axial next-nearest neighour Ising) model. We find that when the ratio κ of the second neighbour interaction to the first neighbour interaction is less than 1, P (t), the probability of a spin to remain in its original state upto time t shows a stretched exponential decay. For κ > 1, P (t) has a algebraic decay but the exponent is different from that of the nearest neighbour Ising model. The global persistence behaviour shows similar features. We also conduct some deeper investigations in the dynamics of the ANNNI model and conclude that it has a different dynamical behaviour compared to the nearest neighbour Ising model.
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