In this work, we study the relativistic oscillators in a noncommutative space and in a magnetic field. It is shown that the effect of the magnetic field may compete with that of the noncommutative space and that is able to vanish the effect of the noncommutative space.

We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose-Einstein condensed state with the Hawking temperature Tc = TH if the particle number of the system be equal to the number of quantum bits of space-time N ≃ A/lp 2 . Entropy of the gas is proportional to the area of the horizon (A) by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics.

It is known that the emission rate of entropy from a Schwarzschild black hole is exactly the same as that of a one dimensional quantum channel [1]. We calculate the dimension of entropy emission from a D dimensional pure Lovelock black holes. Our results indicate that the dimension of transmission for odd D dimensional space-times is equal to D and for even D dimensional spacetimes, the dimension of quantum channel becomes 1 + ǫ(Λ), where Λ is cosmological constant. It is interesting that cosmological constant may put some constraint on dimension of quantum channel in even dimensional space-times. The effect of Generalized Uncertainty Principle (GUP) on the dimension of transmission of entropy for a Schwarzschild black hole is also investigated.

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Recently, Kawai, Parrondo, and Van den Broeck have related dissipation to time-reversal asymmetry. We generalized the result by considering a protocol where the physical system is driven away from an initial thermal equilibrium state with temperature β 0 to a final thermal equilibrium state at a different temperature. We illustrate the result using a model with an exact solution, i.e., a particle in a moving one-dimensional harmonic well.