We demonstrate how Sakharov’s idea of induced gravity allows one to explain the statistical-mechanical origin of the entropy of a black hole. According to this idea, gravity becomes dynamical as the result of quantum effects in the system of heavy constituents of the underlying theory. The black hole entropy is related to the properties of the vacuum in the induced gravity in the presence of th...

The Asymptotic Equipartition Property (see, e.g., [3]) implies that, if we choose the characters of a string s of length n independently and according to the same probability distribution P over the alphabet then, for large values of n, the 0th-order empirical entropy H0(s) of s (see, e.g., [4]) will almost certainly be close to the entropy H(P ) of P . Batu, Dasgupta, Kumar and Rubinfeld [1] s...

Theorem 1 (Jewett-Krieger). Let (X,A , μ, S) be an ergodic dynamical system. Then there is a compact space Y and a homeomorphism T on Y , which is uniquely ergodic and minimal, such that if ν denotes the unique T -invariant probability measure on Y and B the Borel σalgebra of Y , the ergodic dynamical system (Y,B, ν, T ) is equivalent to (X,A , μ, S). In addition, (Y,B, ν, T ) can be constructe...

We study mixing of isothermal fluids by controlling the global hydrodynamic entropy . In particular, based on the statistical coupling between the evolution of and the global viscous dissipation <ϵ> , we analyze stirring protocols such that ∼t(α)⇔<ϵ>∼t(α-1), with 0<α≤1. For a model array of vortices [Fukuta and Murakami, Phys. Rev. E 57, 449 (1998)], we show that: (i) feedback contro...

In this paper, the two parameter ADK entropy, as a generalized of Re'nyi entropy, is considered and some properties of it, are investigated. We will see that the ADK entropy for continuous random variables is invariant under a location and is not invariant under a scale transformation of the random variable. Furthermore, the joint ADK entropy, conditional ADK entropy, and chain rule of this ent...

In the Letter [1] (also [2]) there is a claim that the generalised second law of thermodynamics (entropy increase) for black holes provides some limits on the rate of variation of the fundamental constants of nature (electric charge e, speed of light c, etc.). We have come to a different conclusion. The results in [1, 2] are based on assumption that mass of a black hole does not change without ...

Extremal black holes in M-theory compactification on M × S are microscopically represented by fivebranes wrapping P × S, where M is a Calabi-Yau threefold and P is a four-cycle in M . Additional spacetime charges arise from momentum around the S and expectation values for the self-dual three-form field strength in the fivebrane. The microscopic entropy of the fivebrane as a function of all the ...

We demonstrate how Sakharov’s idea of induced gravity allows one to explain the statistical-mechanical origin of the entropy of a black hole. According to this idea, gravity becomes dynamical as the result of quantum effects in the system of heavy constituents of the underlying theory. The black hole entropy is related to the properties of the vacuum in the induced gravity in the presence of th...

We prove that for the relative entropy of faithful normal states φ and ω on the von Neumann algebra M the formula S(φ,ω) = sup{ω(Λ)-logφ(/): h = h*eM] holds. In general von Neumann algebras the relative entropy was defined and investigated by Araki [1, 3]. After Lieb had proved the joint convexity of the relative entropy in the type / case [10] several proofs appeared in the literature and they...

To any periodic and full C∗–dynamical system (A, α,R) a certain invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius theorem asserts the existence of KMS states at inverse temperatures the logarithm of the inner and outer spectral radii of s. Such KMS st...