Abstract
Recently, it has been shown that for out-of-equilibrium systems, there are additional constraints on thermodynamical evolution besides the ordinary second law. These form a new family of second laws of thermodynamics, which are equivalent to the monotonicity of quantum Rényi divergences. In black hole thermodynamics, the usual second law is manifest as the area increase theorem. Hence one may ask if these additional laws imply new restrictions for gravitational dynamics, such as for out-of-equilibrium black holes? Inspired by this question, we study these constraints within the AdS/CFT correspondence. First, we show that the Rényi divergence can be computed via a Euclidean path integral for a certain class of excited CFT states. Applying this construction to the boundary CFT, the Rényi divergence is evaluated as the renormalized action for a particular bulk solution of a minimally coupled gravity-scalar system. Further, within this framework, we show that there exist transitions which are allowed by the traditional second law, but forbidden by the additional thermodynamical constraints. We speculate on the implications of our findings.
Highlights
Hole’s horizon and GN is Newton’s constant [1,2,3,4]
Recently, it has been shown that for out-of-equilibrium systems, there are additional constraints on thermodynamical evolution besides the ordinary second law. These form a new family of second laws of thermodynamics, which are equivalent to the monotonicity of quantum Renyi divergences
Within this framework, we show that there exist transitions which are allowed by the traditional second law, but forbidden by the additional thermodynamical constraints
Summary
The traditional second law has a number of different formulations and interpretations, from Carnot and Clausius, to Boltzmann and Gibbs, and to more modern versions such as the Jarzynski equality [35, 36], and the eigenstate thermalization hypothesis [37, 38]. If we consider the equilibration of a closed system so that the energy is conserved, the decrease in the free energy is equivalent to an increase in the entropy This version of the second law holds for the thermodynamical entropy, and the statistical mechanical entropy SB = log N , where N is the number of microstates, as well as the von Neumann entropy S(ρ) = − tr ρ log ρ (see for example the discussion in [42]), which generalises the statistical mechanical entropy to the case where the system is quantum, and where the probability of being in any microstate is not necessarily equal. These two last possibilities are expressly forbidden by these additional second laws
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