α = 0.9
yields
q = 1.58
α =2
yields
q =1
L.J.L. Cirto, V.R.V. Assis and C. T., Physica A 393, 286 (2014)
FERMI-PASTA-ULAM MODEL
WITH LONG-RANGE INTERACTIONS
H. Christodoulidi, C. T. and T. Bountis (2014), 1405.3528 [nlin.CD]
H. Christodoulidi, C. T. and T. Bountis (2014)
α = 0.7
yields
q = 1.25
α = 1.4
yields
q =1
Tackled by
Jacob D. Bekenstein
Stephen W. Hawking
Gary W. Gibbons
Gerard ‘t Hooft
Leonard Susskind
Michael J. Duff
Juan M. Maldacena
Thanu Padmanabhan
Robert M. Wald
and many others
SINCE THE PIONEERING BEKENSTEIN-HAWKING RESULTS,
PHYSICALLY MEANINGFUL EVIDENCE HAS ACCUMULATED
(e.g., HOLOGRAPHIC PRINCIPLE) WHICH MANDATES THAT
lnWblack hole ∝ AREA
THIS IS PERFECTLY ADMISSIBLE AND MOST PROBABLY CORRECT.
HOWEVER,
IS THIS QUANTITY THE THERMODYNAMICAL ENTROPY???
ENTROPIES
W
S BG
1
= k B ∑ pi ln
pi
i=1
W
Sq = k B ∑ pi ln q
i=1
1
pi
⎛ 1⎞
Sδ = k B ∑ pi ⎜ ln ⎟
⎝ pi ⎠
i=1
W
(S1 = S BG ) → nonadditive if q ≠ 1
C. T. (1988)
(S1 = S BG ) → nonadditive if δ ≠ 1
C. T. (2009)
δ
⎛
1⎞
= k B ∑ pi ⎜ ln q ⎟
pi ⎠
⎝
i=1
W
Sq,δ
→ additive
δ
(Sq,1 = Sq ; S1,δ = Sδ ; S1,1 = S BG )
C. T. (2011)
→ nonadditive if (q,δ ) ≠ (1,1)
C. T. and L.J.L. Cirto, Eur Phys J C 73, 2487 (2013)
See also: R. Hanel and S. Thurner, EPL 93, 20006 (2011) and EPL 96, 50003 (2011)
R. Hanel, S. Thurner and M. Gell-Mann, PNAS 111, 6905 (2014)
Various arguments (phenomenological, holographic principle,
string theory, area law, etc) yield
SBG (L) ≡ k B lnW (L) ∝ Ld−1 (d > 1)
hence
W (L) ∝ Φ(L) ν
Ld−1
ln Φ(L)
⎛
ρ⎞
= 0; e.g., Φ(L) ∝ L ⎟
⎜⎝ with lim L→∞
d−1
⎠
L
d
hence, for d > 1, the entropy which is extensive is Sδ with δ =
d −1
W (L)
i.e.,
Sδ =d/(d−1) (L) = k B ∑
i=1
Consequently
black hole
δ =3 2
S
(L) = k B
⎛ 1⎞
pi ⎜ ln ⎟
⎝ pi ⎠
W (N )
∑
i=1
d
d−1
∝ Ld (d > 1)
3
2
⎛ 1⎞
pi ⎜ ln ⎟ ∝ L3 !!!
⎝ pi ⎠
C.T. and L.J.L. Cirto, Eur. Phys. J. C 73, 2487 (2013)
SYSTEMS ENTROPY SBG ENTROPY Sq
ENTROPY Sδ
W (N )
(δ ≠ 1)
(q ≠ 1)
(ADDITIVE) (NONADDITIVE) (NONADDITIVE)
 µN
( µ > 1)
 Nρ
( ρ > 0)
EXTENSIVE
NONEXTENSIVE
NONEXTENSIVE
NONEXTENSIVE
EXTENSIVE
(q = 1 −1 / ρ )
NONEXTENSIVE
Nγ
ν
(ν > 1;
0 < γ < 1)
NONEXTENSIVE NONEXTENSIVE
EXTENSIVE
(δ = 1 / γ )
(q = 1+1/n)
SIMPLE APROACH: TWO-DIMENSIONAL RELATIVISTIC FREE PARTICLE
L.J.L. Cirto, C. T., C.Y. Wong and G. Wilk, 1409.3278 [hep-ph]
Ep = 108 TeV (EECR)
Extreme Energy Cosmic Rays
Newton
E = mc2 + p2/2m
Ee = 7 TeV
Ep = 7 TeV
Einstein (1905)
5
ton
pro
ctr
on
E = (m2c4 + p2c4)1/2
ele
|
|
E
2
mc
1012
1011
1010
9
10
8
10
1 7
10
106
105
104
103
102
101
100
10 1
10 2
10 3
10 4
10
mp
me
938.3 MeV/c2
0.511 MeV/c2
p [GeV/c]
10 410 310 210 1100 101 102 103 104 105 106 107 108 10910101011
thanq !