• Kac’s ring: The case of four colours

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    • Keywords


      Stosszahlansatz; $H$-theorem; nonequilibrium statistical mechanics

    • Abstract


      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.

    • Author Affiliations



      1. Department of Physics, University of Mumbai, Mumbai 400 098, India
    • Dates

  • Pramana – Journal of Physics | News

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      Posted on July 25, 2019

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