Entropy Bounds  

In General > s.a. boundaries in field theory; holography.
* Idea: An upper limit for the amount of entropy a system can have; Depending on the context or motivation, it can be given in terms of the system's energy (Bekenstein) or size, for example surface area; From considerations based on a fundamental length, it is normally assumed that quantum gravity implies an upper limit on the entropy for a bounded region.
* Remark: One possible objection to the existence of such a bound, the "species problem", is that as the number of fields considered in a theory grows, so does the entropy of a region, apparently unboundedly; Addessed by remarking that the number of fields actually present is (probably) finite, and that if one changes the number of fields in a theory one can also change the values of the constants in the bound.

Bekenstein Bound
* Idea: For a system with energy E and typical size L, the entropy is bounded by S 2 EL/c.
@ General references: Bekenstein PRD(94)gq/93; Marolf & Roiban JHEP(04)ht [open issue]; Gibbons et al PRD(06)ht [in AdS spacetime]; Casini CQG(08)-a0804 [and relative entropy]; Pesci a0903 [and strength of gravity]; Schmitt a0901.
@ Violations: Marolf & Sorkin PRD(04)ht/03 [hyperentropic objects, Hawking radiation]; Bekenstein PRD(04)ht [re hyperentropic objects].

Holographic / Covariant Bound > s.a. Relaxation [universal bound on relaxation time]; twistors.
* Holographic bound: If A is the area of a circumscribing surface, the entropy of a matter system

Smatter A / 4Gc3 ;

This formula breaks down in large curvature/gravity situations.
* Covariant bound: (Bousso) Formulated in terms of the fields intercepted by the ingoing light sheet from a surface, under some assumptions and up to where the surface has caustics, if any, as

Slight sheet A / 4Gc3 .

@ General references: Bousso JHEP(99)ht, comment Lowe JHEP(99); Bekenstein PLB(00)ht, ht/00-MG9 [second law]; Flanagan et al PRD(00)ht/99; Das et al PRD(01)ht/00 [isolated horizons]; Low CQG(02)gq/01; Casini CQG(03)gq/02 [geometric, and spacetime cutoff]; Bousso RMP(02)ht; Mayo CQG(02); Yurtsever PRL(03)gq [from local quantum field theory], comment Aste ht/06; Husain PRD(04)gq/03 [Einstein-scalar examples]; Ling & Zhang gq/06 [high-order corrections]; Pesci CQG(08)-a0803 [statistical-mechanical meaning]; Ashtekar & Wilson-Ewing PRD(08)-a0805 [and loop quantum cosmology].
@ Sufficient conditions: Bousso et al PRD(03)ht; Gao & Lemos PRD(05)gq.
@ Applications: Danielsson JCAP(03) [and inflation]; Gao & Lemos JHEP(04)ht [and collapse]; Hogan ap/07 [uncertainty principle and possible measurement]; Gersl a0804 [ideal gas of massive particles]; He & Zhang a0805-GRF [on the dynamical horizon].

Cardy-Verlinde Formula > s.a. gravitational thermodynamics.
* Idea: A holography-inspired bound on field entropy in cosmology, in terms of the Casimir energy, which in some limit can be expressed in terms of the Hubble constant.
@ General references: Verlinde ht/00; Cai PRD(01) [Anti-de Sitter black holes]; Youm PLB(02); Nojiri & Odintsov PLB(02) [in Yang-Mills theory]; L:in & Cai PLB(06) [re AdS black holes].
@ Corrections: Nojiri et al MPLA(01); Momen & Sarkar PLB(02) [super-Yang-Mills]; Setare PLB(03) [topological Reissner-Nordström-dS], PRD(04)ht [from generalized uncertainty principle]; Setare IJMPA(06)gq, IJTP(07) [non-commutativity], IJMPA(08)-a0807 [Kerr black hole].

References > s.a. laws of black-hole thermodynamics.
@ General references: Bekenstein PRD(81); Schiffer & Bekenstein PRD(89); Zaslavskii CQG(96); Bousso JHEP(01)ht/00 [in de Sitter and Minkowski]; Bekenstein FP(05)qp/04 [how it works]; Marolf ht/04-GR17 [rev]; Rideout & Zohren CQG(06)gq [from causal sets].
@ Consequences: Karch ht/03 [fluid viscosity bound]; Medved MPLA(06)ht/05 [cosmology, dark energy and inflation].
@ Debate: Page gq/00, ht/00 [violations]; Bekenstein gq/00 [rebuttal]; Page ht/00 [violations and fix].
@ Quantum fields: Solodukhin PRD(01)gq/00 [scalar in a cavity]; Brevik et al AP(02) [R × S3 geometries]; Strominger & Thompson PRD(04)ht/03 [quantum correction to Bousso bound]; Page JHEP(08) [non-gravitational].
@ Charged system: Bekenstein & Mayo PRD(00)gq/99; Hod gq/99/PRD, PRD(00)gq/99; Mayo PRD(99)gq; Gour CQG(03)gq [rotating].
@ Causality-based: Brustein & Veneziano PRL(00)ht/99; Brustein et al PLB(01) [and conformal field theory], PRD(02) [non-singular cosmology]; Brustein ht/07-in [and cosmology, rev].
@ Related topics: Hod PRD(00)gq/99 [rotating system]; Birmingham & Sen PRD(01)ht/00 [black holes in conformal field theory]; Gour PRD(03)gq/02 [extensive]; Elizalde & Tort PRD(03) [massive scalar in S1 × S3]; Mignemi PRD(04)ht/03 [in 2D]; Berry & Sanders JPA(03)qp [and relationships]; Zachos JPA(07) [classical bound, including Rényi entropy].

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