aa r X i v : . [ h e p - t h ] M a y Debye entropic force and modified Newtonian dynamics
Xin Li , ∗ and Zhe Chang , † Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences
Verlinde has suggested that the gravity has an entropic origin, and a gravitational system couldbe regarded as a thermodynamical system. It is well-known that the equipartition law of energy isinvalid at very low temperature. Therefore, entropic force should be modified while the temperatureof the holographic screen is very low. It is shown that the modified entropic force is proportional tothe square of the acceleration, while the temperature of the holographic screen is much lower thanthe Debye temperature T D . The modified entropic force returns to the Newton’s law of gravitationwhile the temperature of the holographic screen is much higher than the Debye temperature. Themodified entropic force is connected with modified Newtonian dynamics (MOND). The constant a involved in MOND is linear in the Debye frequency ω D , which can be regarded as the largestfrequency of the bits in screen. We find that there do have a strong connection between MONDand cosmology in the framework of Verlinde’s entropic force, if the holographic screen is taken tobe bound of the Universe. The Debye frequency is linear in the Hubble constant H . PACS numbers: 98.80.Cq,95.35.+d
I. INTRODUCTION
Gravity, the most universal force in nature, is stillunclear in quantum level. The profound relation be-tween black hole entropy and thermodynamics, whichproposed by Hawking et al. [1], throws new light to thegenius of gravity. This fact leads some physicists investi-gating gravity from a point of view of thermodynamics.Jacobson[2] derived Einstein’s gravitational field equa-tion from the first law of thermodynamics. The furtherinvestigation of Padmanabhan[3] reinterpreted entropyas the equipartition rule of energy. It also provided athermodynamic interpretation of gravity. These resultssupport that the gravity has an entropic origin[4]. Re-cently, Verlinde[5] proposed an interesting scenario thatinterpret gravity as entropic force caused by the changesin the information associated with the positions of mate-rial bodies. Verlinde’s scenario has two important conse-quences. One is that, by making use of the the entropicforce together with the Unruh temperature[6], one couldderive the second law of Newton. The other is that, bymaking use of the the entropic force together with theholographic principle[7] and the equipartition law of en-ergy, one could derive Newton’s law of gravitation andKomar mass[8] in relativistic case.In the framework of entropic force, various aspects ofgravitational physics have been discussed. A succinctlist includes: a Friedmann equation derived from entropicforce[9, 10], the implication for loop quantum gravity[11],the construction of holographic actions from black holeentropy[12, 13], the holographic dark energy derived fromthe entropic force[14], the correct entropy/area relation ∗ Electronic address: [email protected] † Electronic address: [email protected] in the relativistic case[15], and the modified entropicforce derived from the modified the equipartition law ofenergy[16]. More related works can be seen in [17–24]It should be noticed that Verlinde used the free parti-cle’s the equipartition law of energy to derive the New-ton’s law of gravitation. It is well known that theequipartition law of energy does not hold at very lowtemperatures, and the equipartition law of energy de-rived from Debye model is in good agreement with ex-perimental results for most of the solid objects in very lowtemperatures. Verlinde got the Newton’s law of gravita-tion with the assumption that each bit on holographicscreen is free of interaction. It should be more generalthat the bits on holographic screen interact each others.In such case, one could anticipate that the Newton’s lawof gravitation must be modified. For example, Gao[16]used the three dimensional Debye model, which modifiedthe equipartition law of energy, to modify the entropicforce. Such modification can interpret the current accel-eration of the Universe while without invoking any kindof dark energy.Since the equipartition law of energy in Verlinde’sscenario[5] is one dimensional free particle’s equipartitionlaw of energy, we use the one dimensional Debye modelto modified the entropic force. We find that this modi-fied entropic force is able to derive the famous modifiedNewtonian dynamics(MOND). MOND was constructedto explaining the flat rotational curves of spiral galaxies.There are a great variety of observations which showthat the rotational velocity curves of all spiral galaxiestend to some constant values[25]. These include the Oortdiscrepancy in the disk of the Milky Way[26], the veloc-ity dispersions of dwarf Spheroidal galaxies[27], and theflat rotation curves of spiral galaxies[28]. These facts vi-olate sharply the prediction of Newtonian dynamics orNewton’s gravity.The most widely adopted way to resolve these difficul-ties is the dark matter hypothesis. It is assumed thatall visible stars are surrounded by massive nonluminousmatters. Another approach is the famous MOND[29]. Itassumed that the Newtonian dynamics does not hold ingalactic scale. The particular form of MOND is given as mµ (cid:18) aa (cid:19) a = F , lim x ≫ µ ( x ) = 1 , lim x ≪ µ ( x ) = x, (1)where a is at the order of 10 − cm/s . At beginning,as a phenomenological model, MOND explains well theflat rotation curves with a simple formula and a new pa-rameter. In particular, it deduces naturally a well-knownglobal scaling relation for spiral galaxies, the Tully-Fisherrelation[30]. By introducing several scalar, vector andtensor fields, Bekenstein[31] rewrote the MOND into acovariant formulation. He showed that the MOND satis-fies all four classical tests on Einstein’s general relativityin Solar system.In this paper, we will derive the MOND in the frame-work of Verlinde’s entropic force. II. MOND FROM MODIFIED ENTROPICFORCE
In verlinde’s scenario, the change of a particle’s posi-tion ( △ x ) corresponds the change of entropy ( △ S ) asso-ciated with the information on the boundary. Motivatedby Bekenstein’s entropic bound, Verlinde postulated thatthe change of entropy is proportional to the change of aparticle’s position, △ S = 2 πk B mc ~ △ x. (2)Such particle will experience an effective force when theholographic screen carries a temperature T , together withthe first law of thermodynamics, the force equals to F △ x = T △ S. (3)According to the Unruh formula, the temperature in (3)corresponds the acceleration k B T = 12 π ~ ac (4)where a denotes the acceleration. Since the holographicscreen carries temperature, the equipartition law of en-ergy gives the relation between the total energy of thescreen and the temperature E = 12 N k B T, (5)where N denotes the number of degrees of freedom (bits)on the screen. The number of bits N on the screen isassumed to proportional to the area A of the screen N = Ac G ~ . (6) Verlinde assumed that the energy of the screen is pro-portional to the mass that would emerge in the part ofspace enclosed by the screen E = M c . (7)By making use of the Eq. (2), (3) and (4), one couldobtain the second law of Newton F = ma . By makinguse of the Eq. (2–7) together with the relation A = 4 πR ,one could obtain Newton’s law of gravitation F = G MmR .In a statistic system, there are interactions amongmolecules. Thus the equipartition law of energy for freemolecule only valid for the situation that the kinetic en-ergy of molecule is much larger than the effective po-tential of the interaction between each molecule. There-fore, the equipartition law of energy is invalid at very lowtemperatures. It is found that Debye model, which mod-ified the equipartition law of energy, is in good agreementwith experimental results for most solid objects. Follow-ing Verlinde’s scenario, we know that the gravity can beexplained as an entropic force, it means that the gravitymay have a statistical thermodynamics explanation. Ifso, the gravity should modified while the correspondedstatistics changes.Here, we modified the equipartition law of energy asone dimensional Debye model E = 12 N k B T D ( y ) , (8)where the one dimensional Debye function is defined as D ( y ) ≡ y Z y ze z − dz. (9) y is related to the Debye frequency ω D , and defined as y ≡ ~ ω D k B T = 2 πcω D a . (10)Following Verlinde’s method, we obtain the modificationof Newton’s law of gravitation GMR = a D ( 2 πcω D a ) . (11)There lies two limit case for the modification of Newton’slaw of gravitation (11). One is the high temperature limit y ≪
1, the Debye function D ( y ) reads D ( y ) ≈ y Z y dy = 1 . (12)Then, the modified equipartition law of energy (8) re-turns to (5). Therefore, at very high temperature, theNewtonian gravity is recovered. The other is the lowtemperature limit y ≫
1, the Debye function D ( y ) reads D ( y ) ≈ y Z ∞ ze z − dz = π y . (13)Then, the modification of Newton’s law of gravitation(11) reads GMR = π cω D a = a a , (14)where the constant a is defined as a = 12 cω D π . (15)Combining the Eq. (11) and (15), we find that D ( y ) = D ( π a a ) = 6 π aa Z π a a ze z − dz (16)Defining the function µ ( x ) = 6 π x Z π x ze z − dz, (17)where x = aa , one could find that this function has prop-erties lim x ≫ µ ( x ) = 1 , lim x ≪ µ ( x ) = x. (18)Therefore, the modification of Newton’s law of gravita-tion (11) reads GMR = a D ( 2 πcω D a ) = aµ ( x ) . (19)Equation (19) is just the combination of the MOND rela-tion (1) and Newton’s law of gravitation. Thus, the mod-ified entropic force (11) could account for the MOND. Tospecific this point, by making use of the formula (19) wegive the modified Poisson equation for the gravitationalpotential φ ∇ · ( µ ( |∇ φ | /a ) ∇ φ ) = 4 πGρ, (20)where ρ is the energy density of matter sources. Thisequation is just modified Poisson equation in MOND[32].Milgrom[29, 32] observed the relation between theMOND constant a and the Hubble constant H that2 πa ≈ cH . (21)In MOND[32], this relation (21) is just a coincidences,and it may point to a strong connection between MONDand cosmology. In Verlinde’s scenario, we show that thiscoincidence is just a consequence of modified entropicforce. By making use of the formula (15) and (21), wefind that the Debye frequency ω D is proportional to theHubble constant H H ≈ ω D . (22) Since the Debye frequency ω D is the largest oscillate fre-quency of the particle on holographic screen, its wavelength λ < c/ω D should be less than scale of the screen.On the other hand, the cosmological horizon L h ∼ c/H has the same scale as the holographic screen, for thescreen could be regarded as the bound of the Universe.The relation (22) tells us that λ < c/ω D = c/ H < c/H . (23)Thus, the relation (22) is reasonable. III. CONCLUSION
In this paper, we find that a modified entropic force isconnected with modified Newtonian dynamics. In Ver-linde’s scenario, gravity has an entropic origin and gravi-tational system could be regarded as a thermodynamicalsystem. However, it is well-known that the equipartitionlaw of energy is invalid at very low temperature. There-fore, entropic force should be modified while the temper-ature of the holographic screen is very low. It is shownthat the modified entropic force is proportional to thesquare of the acceleration, while the temperature of theholographic screen is much lower that the Debye temper-ature T D = ~ ω D k B . And the modified entropic force returnsto the Newton’s law of gravitation while the temperatureof the holographic screen is much larger that the Debyetemperature. This thermodynamical approach gives apossible origin of the famous MOND. Also, the constant a involved in MOND is linear in the Debye frequency ω D , which can be regarded as the largest frequency of thebits in screen. We find that there do have a strong con-nection between MOND and cosmology in the frameworkof Verlinde’s entropic force, if the holographic screen isthe bound of the Universe. It is shown that the Debyefrequency is linear in the Hubble constant H . A specificmicroscopic statistical thermodynamical model of space-time may give the origin of our modified entropic forcemodel and the microscopic origin for the MOND and thecosmology. We will study it in the future. Acknowledgments
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