On the entropic derivation of the r 2 Newtonian gravity force

Transcription

1 arxiv: v [gr-qc] 5 Apr 08 On the entropic derivation of the r Newtonian gravity force A. Plastino,3,4, M.C.Rocca,,3, Departamento de Física, Universidad Nacional de La Plata, Departamento de Matemática, Universidad Nacional de La Plata, 3 Consejo Nacional de Investigaciones Científicas y Tecnológicas (IFLP-CCT-CONICET)-C. C. 77, 900 La Plata - Argentina 4 SThAR - EPFL, Lausanne, Switzerland April 6, 08 Abstract Following Verlinde s conjecture, we show that Tsallis classical free particle distribution at temperature T can generate Newton s gravitational force s r distance s dependence. If we want to repeat the concomitant argument by appealing to either Boltzmann-Gibbs or Renyi s distributions, the attempt fails and one needs to modify the conjecture. Keywords: Tsallis, Boltzmann-Gibbs, and Renyi s distributions, classical partition function, entropic force.

2 Introduction Eight years ago, Verlinde [] advanced a conjecture that links gravity to an entropic force, so that gravity would result from information regarding the positions of material bodies. His model joins a thermal gravity-treatment to t Hooft s holographic principle. This would entail that gravitation should be viewed as an emergent phenomenon. Verlinde s notion received much attention, of course (just as an example, see []). For an excellent overview on the statistical mechanics of gravitation, the reader is directed to Padmanabhan s article [4], and references therein. Verlinde s work attracted efforts on cosmology, the dark energy hypothesis, cosmological acceleration, cosmological inflation, and loop quantum gravity. The literature is immense [3]. In particular, an important contribution to information theory is that of Guseo [5], who has proved that the local entropy function, related to a logistic distribution, is a catenary and vice versa. This special invariance may be explained, at a deeper level, through the Verlindes conjecture on the origin of gravity, as an effect of the entropic force. Guseo advances a novel interpretation of the local entropy in a system, as quantifying a hypothetical attraction force that the system would exert [5]. This paper deals with none of these issues, though. We just show that extremely simple classical reasoning based on the Tsallis, probability distributions straightforwardly proves the conjecture. In Boltzmann-Gibbs and Renyi s instance, one needs to modify the conjecture to achieve a similar result. Tsallis q-entropy of the free particle Tsallis q-partition function for a free particle of mass m in dimensions reads [6] [ ] Z = V +( q) p q d p, (.) m + with the particle probability distribution ξ(p) being ξ = [ ] +( q) p q, (.) Z m +

3 where V is the volume of an hypersphere in dimensions and we assume q >. (.) can be recast as Z = π Γ ( )V With the change of variables x = p m Z = (mπ) Γ ( ) V that after integration becomes The mean energy is or 0 ] [+( q) p q p dp. (.3) m + (q ) [ ] (mπ) Z = V (q ) 0 one has [+( q)x] q x dx, (.4) Γ ( Γ ( ) q q q q + < U >= V ] [+( q) p q Z m + < U >= V (mπ) Z Γ ( ) so that after integration we find and finally < U > = For the entropy one has [6] < U > = (q ) 0 (q ) ). (.5) p m d p, (.6) [+( q)x] q x dx, (.7) ( ) Γ + + q ( ), (.8) Γ + + q [q +(q )]. (.9) S = ln q Z +Z q < U >. (.0) 3

4 3 The Tsallis entropic force We specialize things now to = 3 and q = 4. Why do we select this special 3 value q = 4? There is a solid reason. This is because 3 S = ln q Z +Z q < U >. Since the entropic force is to be defined as proportional to the gradient of S, there is a unique q-value for which the dependence on r of the entropic force is r when = 3. Thus we obtain, for q = 4/3, ( 6mπ Z = )3 8π Γ ( Following Verlinde [] we define the entropic force as )r 3, (3.) < U >= 9. (3.) F e = λ(m,m) S, (3.3) where λ is a numerical parameter depending on the masses involved, m and a new one M that we place at the center of the sphere. Thus, F e = 4 [ ( Γ 8π ) ] 3 ( ) kb T λ(m,m) e 6mπ r r, (3.4) where e r is the radial unit vector. We see that F e acquires an appearance quite similar to that of Newton s gravitation, as conjectured by Verlinde en []. Note that entropic force vanishes at zero temperature, in agreement with Thermodynamics third law [7]. 4 An illustrative example Assume that we deal with a large mass M and a very small one m. One has F e = 4 [ ( Γ ) 8π ] 3 ( ) kb T 6mπ 4 λ(m,m) r e r = GmM r e r. (4.)

5 We obtain for λ(m,m) λ (m,m) = π 3G 5 m 3 M [ (. (4.) k B T 4 4 )] 4 Γ 3 Ifweselect M=Sunmassm=Jupiter mass, T=3 Kthenλ(m,M) = Kgmeters. When m=earth mass, then λ(m,m) = 3. 0 Kgmeters s 4. Energies involved s. In [8], different q-values have been associated to energies of CERN experiments [9, 0]. q-statistics is seen to be meaningful at very high energies (TeVs) for q =.5, high ones (GeVs) for q =.00, and at low energies (MeVs) for q = Then we see that q= 4/3 should be associated with n energy of (TeVs), an energy that can be expected to arise shortly after the Big Bang, where quantum gravity effects should be apparent. 5 The Boltzmann-Gibbs entropy of the free particle Now the classical partition function Z is p Z = V e m d p, (5.) with V V = π Γ ( ) r. (5.) Since ( ) p πm e m d p =, (5.3) we have ( πm Z = so that the mean energy < U > is < U > = V Z ) π p 5 Γ ( +)r, (5.4) p m e m d p. (5.5)

6 We appeal now to the well known relation: ( ) p p πm m e m d p =, (5.6) so that which leads to an entropy: < U > =, (5.7) S = lnz +. (5.8) 6 The Boltzmann-Gibbs entropic force Our hyper-sphere s area A is A = π Γ ( ).r (6.) The hyper-sphere s volume, as a function of its area reads [ ( Γ )] V = π ( ) The derivative of S with respect to A is S = A A. (6.). (6.3) A Specialize things now to = 3. Following Verlinde [], with a slight modification, we define the entropic force that arises out of forcing the particle of mass m to remain enclosed in a given volume as F e = λ(m,m) Replacing A 3 s value in (6.4) we find or S 3 A 3 = λ F e = λ(m,m)k B T 3 3, (6.4) A 3 Γ ( ) 3 r, (6.5) F e = 3λ(m,M)k BT. (6.6) 8π r WeseeagainthatF e acquiresanappearancequitesimilartothatofnewton s gravitation, as conjectured by Verline in []. 6 π 3

7 7 A second illustrative example Let us replace the enclosing effect of a spherical cavity by the gravitational one of a large mass M on a very small one m, that is, F e = 3λ(m,M)k bt 8π r = GmM, (7.) r and deduce λ(m,m) as λ(m,m) = 8πGmM 3Tk b. (7.) If we select M=Sun mass m=jupiter mass, T=3 K then λ(m,m) = 4,6 0 7 meters. When m=earth mass, then λ(m,m) =, meters. 8 The Renyi entropic force In Renyi s approach to our problem the entropy is []-[] [ ] (mπ) Γ ( ) α α Z = V (α ) Γ ( α + ) α >, (8.) α [ ] (mπ) Γ ( ) α Z = V ( α) Γ ( + ) α <, (8.) α that for = 3 becomes Z 3 = γ(α,m,)a 3 3 A 3 = 4πr, (8.3) while for the mean energy one has and for the entropy < U > = < U > = [α+(α )] [ ( +)( α)] α >, (8.4) α <, (8.5) S = lnz +ln[+( α) < U >] α +. (8.6) 7

8 The second term on the right hand of (8.6) is independent of r. Additionally, lnz 3 = 3 lna 3 +ln[γ(m,)]+ln(3 4π). (8.7) Slightly modifying, as in the BG case, Verlinde s entropic form we have F e = λ(m,m) S 3 A 3 = λ 3 8πr. (8.8) We see that(8.8) coincides with(6.6). Renyi s entropic force is just Boltzmann- Gibbs one. 9 Conclusions We have presented three very simple classical realizations of Verlinde s conjecture. The Tsallis one, for q = 4/3 seems to be cleaner, as the entropic force is directly associated to the gradient of Tsallis entropy S q, which acts asa potential, asverlinde prescribes. This isnot so intheclassical BGand Renyi instances, in which one has to modify Verlinde s F e definition. The Tsallis case also gives interesting indications regarding the energies involved. Remarkably enough, Boltzmann-Gibbs and Renyi s entropic forces coincide. Strictly speaking, Verlinde s conjecture can be unambiguously proved for the Tsallis entropy with q = 4/3. The Boltzmann-Gibbs and Renyi demonstrations correspond to a modified version of Verlinde s conjecture. Of course, ours is a very preliminary, if significant, effort. A much more elaborate model would be desired. 8

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