An Entropy depending only on the Probability (or the Density Matrix) December, 2016
The Entropy and Superstatistics The Entropy and Superstatistics 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 possesses a spatio-temporally fluctuating intensive quantity (temperature, chemical potential, energy dissipation). More general statistics were formulated.
The Entropy and Superstatistics The macroscopic system is made of many smaller cells that are temporarily in local equilibrium, β is constant. Each cell is large enough to obey statistical mechanics. But has a different β assigned to it, according to a general distribution f(β), from it one can get on effective Boltzmann factor B(E) = ˆ 0 dβf(β)e βe, (1) where E is the energy of a microstate associated with each of the considered cells. The ordinary Boltzmann factor is recovered for f(β) = δ(β β 0 ). (2)
The Entropy and Superstatistics One can, however, consider other distributions. Assume a Γ (or χ 2 ), distribution depending on a parameter p l, to be identified with the probability associated with the macroscopic configuration of the system. f pl (β) = 1 ( ) β 0 p l Γ 1 pl ( ) β 1 1 p l p l e β/β 0p l, (3) β 0 p l Integrating over β B pl (E) = (1 + p l β 0 E) 1 p l. (4)
The Entropy and Superstatistics By defining S = k Ω 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) by s(x) = ˆ x 0 dy α + E(y) 1 E(y)/E, (5) where E(y) is to be identified with the inverse function B pl (E) 0 de B pl (E ).
The Entropy and Superstatistics One selects f(β) B(E) E(y), S(x) is then calculated. For our distribution Γ(χ 2 ), Its expansion gives l=1 S = k Ω l=1 S Ω [ k = p l ln p l + (p l ln p l ) 2 2! (1 p p l l ). (6) + (p l ln p l ) 3 3! + ], (7) where the first term corresponds to the usual Boltzmann-Gibbs (Shannon) entropy.
The Entropy and Superstatistics The corresponding functional including restrictions is given by Φ = S Ω k γ p l β l=1 Ω l=1 p p l+1 l E l, (8) where the first restriction corresponds to Ω l=1 p l = 1 and the second one concerns the average value of the energy and γ and β are Lagrange parameters. By maximizing Φ, p l is obtained as 1 + ln p l + βe l (1 + p l + p l ln p l ) = p p l l. (9)
The Entropy and Superstatistics Assume now the equipartition condition p l = 1 Ω, remember S k = Ω k=1 p l ln p l S B k = ln Ω. (10) In our case [ S = kω 1 1 ], (11) Ω 1 Ω in terms of S B (the Boltzmann entropy), S reads S k = S B k 1 2! e S B/k ( SB k ) 2 + 1 ( 2S B SB 3! e k k ) 3. (12)
The Entropy and Superstatistics Entropy 1.5 1.0 0.5 Figure 1: Entropies as function of Ω (small). Blue dashed and red dotted lines correspond to S k, and S B k, respectively (p l = 1/Ω equipartition).
H-Theorem H-Theorem The usual H-theorem is established for the H function defined as ˆ H = d 3 pf ln f. (13) The new H function can be written as ˆ H = d 3 p(f f 1). (14) Considering the partial time derivative, we have ˆ H t = d 3 f ln f f r[ln f + 1]e t. (15)
H-Theorem Using the mean value theorem for integrals, and realizing that the factor e f ln f is always positive, it follows from the conventional H-theorem that the variation of the new H-function with time satisfies H 0. (16) t
Distribution and their Associated Boltzmann Factors Distribution and their Associated Boltzmann Factors For the f pl (β), Γ(χ 2 ) distribution we have shown that the Boltzmann factor can be expanded for small p l β 0 E, to get B pl (E) = e β 0E [ 1 + 1 2 p lβ 2 0E 2 1 3 p2 l β3 0E 3 +... ]. (17) We follow now the same procedure for the log-normal distribution, f pl (β) = 1 2πβ[ln(pl + 1)] 1 2 exp { [ln β(p 1 l+1) 2 β 0 ] 2 2 ln(p l + 1) }. (18)
Distribution and their Associated Boltzmann Factors The Boltzmann factor can be obtained at leading order, for small variance of the inverse temperature fluctuations, B pl (E) = e β 0E [ 1 + 1 2 p lβ 2 0E 2 1 6 p2 l (p l + 3)β 3 0E 3 +... ]. (19) In general, the F -distribution has two free constant parameters. We consider, particulary the case in which one of these parameters is chosen as v = 4. For this value we define a F -distribution as f pl (β) = Γ( 8p l 1) 2p l 1 Γ ( 4p l +1 2p l 1) 1 β0 2 ( ) 2pl 1 2 β p l + 1 (1 + β 2p l 1 β 0 p l +1 ) ( 8p ). (20) l 1 2p 1 l
Distribution and their Associated Boltzmann Factors Once more the associated Boltzmann factor can not be evaluated in a closed form, but for small variance of the fluctuations we obtain B pl (E) = e β 0E [ 1 + 1 2 p lβ0e 2 2 + 1 3 p 5p l 1 l p l 2 β3 0E 3 +... ]. (21) As can be observed the first correction term to all these Boltzmann factor is the same. This will imply that their associated entropies also coincide up to the first correction term.
The Generalized Replica Trick The Generalized Replica Trick The corresponding generalized entanglement entropy to the entropy (6) is given by S k = T r(i ρρ ), (22) with ρ the density matrix, but this exactly corresponds to a natural generalization of the Replica trick namely S k = k 1 1 k! lim k n k n k T rρn. (23)
The Generalized Replica Trick As shown, by example, by C. Pasquale and J. Cardy for several examples of 2dCFT s T rρ n A = c n b c( 1 n n) 6 (24) where ρ A = T r B ρ for a system composed of the subsystems A and B, with c n a constant, b a parameter depending on the model and c the central charge.
The Generalized Replica Trick Then the usual Von Neumann entropy (k = 1) S A n T rρn n=1 = c 3 ln b = S A. (25) Our generalized Replica trick taking the second and third derivation will give S = S A + e 3SA ( 4 16 S A 1+ 25 8 S A ) e 4S A 3 The correction terms are exponentially suppressed. 6 ( 1 S A 27 + 5 181 S A+ 125 ) 729 S2 A +.... (26)
Attempts to relate this Entropy to Gravitation and Holography According to Ted Jacobson s (and also E. Verlinde) proposal, we can reobtain gravitation from the entropy, for a modified entropy S = A 4lp 2 + s, (27) one gets a modified Newton s law F = GMm [ ] R 2 1 + 4lp 2 s. (28) A A=4πR 2
Coming back to our entropy and identifying S B = A F = GMm R 2 + GMmπ l 2 p [1 π R2 2l 2 p 4l 2 p we get ] e π R2 l 2 p. (29) Generalized gravitation? From Clausius relation (Jacobson) δq T = 2π ˆ T ab k a k b ( λ)dλd 2 A, (30) and ˆ δs B = η R ab k a k b ( λ)dλd 2 A, (31) one gets Einstein s Equations. In our case, approximately one gets A nonlocal gravity? δs = δs B (1 S B ). (32)
Now, von-neumann entropy S A = tr A ρ A logρ A, ρ A = tr B Ψ Ψ. (33) For a 2dCFT with periodic boundary conditions S A = c 3 log ( L πα sin ( π l L )), (34) where l and L are the length of the subsystem A and the total system A B respectively α is a UV cutoff (lattice spacing), c is the central charge of the CFT. For an infinite system S A = c 3 log l α. (35)
Ryu and Takayanagi proposed S A = Area ofγ A 4G (d+2), (36) N where γ 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 γ A plays the role of a holographic screen for an observer who is only accesible to the subsystem A. They show explicitly the relation (35) in the lowest dimensional case d = 1, where γ A is given by a geodesic line in AdS 3.
Let us explore the generalization of this conjecture if S = T r [I ρ ρ ]. (37) As we have shown the entropy we propose gives further terms also functions of S A which are polynomials exponentially suppressed. So on the gravitational side (the AdS 3 spacetime) is not clear how to generalize it, i.e. which modified theory of gravitation would provide a spacetime solution for which one could construct a minimal surface area corresponding to the 2dCFT generalized entropy. Moreover, recently Xi Dong found that a derivative of holographic Rényi entropy S n = 1 1 n ln T rρn with respect to n, gives the area S A of a bulk codimensional-2 cosmic brane.
The classical action I( B n ) on the replicated bulk B n is related to the area of the brane as n I( B n ) n ( n 1 n )S n = An 1 4G N, for A n 2 2, A 2 4G n = 2S 2 + 2S 2 S 2. (38) Our entropy corresponds to S = S 0 + 1 [ 2 e S 2 S 2 2S 2 + (S 2 + S 2 ) 2], (39) where the quantities S 2, S 2 its derivatives: S 2 = 2(I( B 2 ) I( B 1 )), and S 2 can be written vs. I( B n ) and S 2 = 2 n I( B n ) n=2 I( B 2 ) + I( B 1 ), (40) S 2 = n I( B n ) n=2 + 2(I( B n ) I( B 1 )) + 2 2 ni( B n ) n=2.
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