Articles written in Pramana – Journal of Physics

    • Kac’s ring: The case of four colours


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      We present an instance from nonequilibrium statistical mechanics which combines increase in entropy and finite Poincaré recurrence time. The model we consider is a variation of the well-known Kac’s ring where we consider balls of four colours. As is known, Kac introduced this model where balls arranged between lattice sites, in each time step, move one step clockwise. The colour of the balls change as they cross marked sites. This very simple example rationalize the increase in entropy and recurrence. In our variation, the interesting quantity which counts the difference in the number of balls of different colours is shown to reduce to a set of linear equations if the probability of change of colour is symmetric among a pair of colours. The transfer matrix turns out to be non-Hermitian with real eigenvalues, leading to all colours being equally likely for long times, and a monotonically varying entropy. The new features appearing due to four colours is very instructive.

    • Classical system underlying a diffracting quantum billiard


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      For a point scatterer placed slightly off the centre of a circular enclosure, rays are found which vividly exhibit the effect of diffraction. The Schrödinger equation was mapped in the complex plane by employing a fractional linear transformation which brings the point scatterer to the centre. But the mass of the particle becomes a function of space coordinates, bearing anisotropy. For the transformed problem, the corresponding classical Hamiltonian is written and solved with Snell’s laws on the boundary. The solutions of the Hamilton’s equations thus found constitute, in fact, the ray-manifold underlying the diffraction at the level of the wave description.

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