Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT,.. December, 2014
Contenido 1 Generalized information entropies depending on the probability
Contenido 1 Generalized information entropies depending on the probability 2 Superstatistics and Gravitation
Contenido 1 Generalized information entropies depending on the probability 2 Superstatistics and Gravitation 3 AdS-CFT, Ryu and Takayanagi proposal
Generalized information entropies depending on the probability Boltzman-Gibbs (BG) statistics works perfectly well for classical systems with short range forces and relatively simple dynamics in equilibrium. SUPERSTATISTICS: Beck and Cohen considered nonequilibrium systems with a long term stationary state that possess a spatio temporally fluctuating intensive quantity (temperature, chemical potential, energy dissipation). More general statistics were formulated.
The macroscopic system is made of many smaller cells that are temporarily in local equilibrium, is constant. Each cell is large enough to obey statistical mechanics. But has a different assigned to it, according to a general distribution f( ), from it one can get on effective Boltzmann factor B(E) = ˆ 1 0 d f( )e E, (1) where E is the energy of a microstate associated with each of the considered cells. The ordinary Boltzmann factor is recovered for f( )= ( 0). (2)
One can, however, consider other distributions. Assume a (or 2 ), distribution depending on a parameter p l, to be identified with the probability associated with the macroscopic configuration of the system. f pl ( )= Integrating over 0p l 1 1 p l 1 0 p l 1 p l p l e / 0 p l, (3) B pl (E) =(1+p l 0 E) 1 p l. (4)
By defining S = k P l=1 s(p l) where p l at this moment is an arbitrary parameter, it was shown that it is possible to express s(x) and a generic internal energy by s(x) = and ˆ x 0 dy + E(y) 1 E(y)/E, (5) ˆ x u(x) =(1+ /E dy ) 0 1 E(y)/E, (6) where E(y) is to be identified with the inverse function B pl (E) 1 0 de0 B pl (E 0 )
One selects f( )! B(E)! E(y)! s(x) and u(x) are then calculated. For the distribution ( 2 ), we have shown, Its expansion gives S k = X l=1 " S = k X l=1 p l ln p l + (p l ln p l ) 2 2! (1 p p l l ). (7) + (p l ln p l ) 3 3! + #, (8) where the first term corresponds to the usual Shannon entropy.
The same infinite expansion is obtained by making an extension of the Replica trick by including higher order derivative terms S k = X k 1 1 k! lim n!k @ k @n k Tr n. (9)
The corresponding functional including restrictions is given by = S k X l=1 p l X l=1 p p l+1 l E l, (10) where the first restriction corresponds to P l=1 p l =1and the second one concerns the average value of the energy and and are Lagrange parameters. By maximizing, p l is obtained as 1+lnp l + E l (1 + p l + p l ln p l )=p p l l, (11)
1.0 p l 0.8 0.6 0.4 0.2 1 2 3 4 5 be l Figure: Comparison of the two probabilities. Blue dotted line corresponds to p l = f( E l ), Eq. (10), and red dashed line to the E standard one p l = e l
Assume now the equiparable condition p l = 1, remember In our case S k = X p l ln p l k=1 apple S = k 1! S B k 1 1 in terms of S B, the Boltzmann entropy =ln. (12), (13) S k = S B k 1 2! e S B/k SB k 2 + 1 3! e 2SB SB k k 3. (14)
Entropy 1.5 1.0 0.5 1 2 3 4 5 6 7 W Figure: Entropies as function of (small). Blue dashed and red dotted lines correspond to S k, Eq.(12) and, S k, respectively (p l =1/ equipartition)
Entropy 7 6 5 4 3 200 400 600 800 1000 W Figure: Entropies as function of (large). Blue Dashed and red dotted lines correspond to S k, Eq. (12) and S k, respectively (p l =1/ equipartition)
The known generalized entropies depend on one or several parameters. The one proposed here depends only on p l. Other entropies depending only on p l can be proposed. Take, as an example, the Kaniadakis entropy S apple = k X l p 1+apple l p 1 apple l 2apple. (15) This entropy reduces to the Shannon entropy for apple =0. Inspired in this, we propose S = k X l p p l l p p l l, (16) 2 which expansion gives also a first term the Shannon entropy. There is more...pre 88, 062146 (2013).
Superstatistics and Gravitation According with Ted Jacobson s (and also E. Verlinde) proposal, we can reobtain gravitation from the entropy for a modified entropy S = A 4lp 2 + s, (17) one gets a modified Newton s law F = GMm apple R 2 1+4lp 2 @s. (18) @A A=4 R 2 Coming back to Eq.(14) and identifying S B = A F = GMm R 2 + GMm l 2 p apple 1 R 2 2l 2 p 4l 2 p e we get R 2 l 2 p. (19)
Superstatistics and Gravitation Generalized gravitation?, basically (Jacobson), Clausius relation Q T = 2 ˆ T ab k a k b ( )d d 2 A, (20) ~ and ˆ S B = one gets Einstein s Equations. In our case, even approximated, A generalization with integrals? Entropy 2010, 12, 2067. R ab k a k b ( )d d 2 A, (21) S = S B (1 S B ). (22)
AdS-CFT, Ryu and Takayanagi proposal Now, von-neumann entropy S A = tr A A log A, A =tr B ih. (23) For one-dimensional (1D) quantum many-body systems at criticality (i.e. 2D CFT) it is known S A = c L l 3 log sin, (24) L where l and L are the length of the subsystem A and the total system A [ B respectively is a UV cutoff (lattice spacing), c is the central charge of the CFT. For an infinite system S A = c 3 log l (25)
AdS-CFT, Ryu and Takayanagi proposal Ryu and Takayanagi proposed S A = Area of A 4G (d+2), (26) N where A is the d dimensional static minimal surface in AdS d+2 whose boundary is given by @A, and G (d+2) N is the d+2 dimensional Newton constant. Intuitively, this suggests that the minimal surface A plays the role of a holographic screen for an observer who is only accesible to the subsystem A. They show explicitly the relation (25) in the lowest dimensional case d =1, where A is given by a geodesic line in AdS 3.
AdS-CFT, Ryu and Takayanagi proposal For general d this also seems to work. A more general treatment and a proof of the conjecture were given by Maldacena and Lewkowycz. Which is the generalization of this conjecture if by means of S = T r [I ], (27) Tr n A = b 1 6 ( 1 n n) (Cardy-Calabrese), (28)
with c =1and b results to be b = e 3S 0, now utilizing the modified Replica trick one gets apple S = S 0 e 3 4 S 0 S 0 5 2 2! 2 6 S 0 + 1 apple 2 3 +e 4 3 S 0 S 0 5 3 3! 3 6 S2 0 + 5 3 4 S 0 + 1 3 2. (29) Collaborators: N. Cabo, R. Castañeda, J.L López, A. Martínez, J.Torres, S. Zacarías.