Relativistic treatment of Verlinde's emergent force in Tsallis' statistics

TL;DR

This paper explores a relativistic classical approach within Tsallis' statistics to derive Newtonian gravity as an emergent entropic force, highlighting the limitations of using Renyi's distribution for this purpose.

Contribution

It introduces a relativistic Hamiltonian framework in Tsallis' statistics to derive gravitational force behavior, extending Verlinde's emergent gravity conjecture.

Findings

Tsallis' relativistic distribution reproduces Newton's inverse-square law.

Using Renyi's distribution does not successfully reproduce gravitational behavior.

The approach aligns with classical limits at small velocities.

Abstract

Following Chakrabarti,Chandrasekhar, and Naina [Physica A {\bf 389} (2010) 1571], we attempt a classical relativistic treatment of Verlinde's emergent entropic force conjecture by appealing to a relativistic Hamiltonian in the context of Tsalli's statistics. The ensuing partition function becomes the classical one for small velocities. We show that Tsallis' relativistic (classical) free particle distribution at temperature $T$ can generate Newton's gravitational force's $r^{-2}$ {\it distance's dependence}. If we want to repeat the concomitant argument by appealing to Renyi's distribution, the attempt fails and one needs to modify the conjecture.

Equations

$$\mathcal{Z}=\frac{V}{N!h^{3N}}\int {\left[1+(1-q)\beta (\sqrt{m^2{c}^4+p^2{c}^2}-m{c}^2)\right]}_{+}^{\frac{1}{q-1}}{d}^{4}p.$$

$$\mathcal{Z}=\frac{4\pi V}{N!h^{3N}}\underset{0}{\overset{\infty }{\int }}{\left[1+(1-q)\beta (\sqrt{m^2{c}^4+p^2{c}^2}-m{c}^2)\right]}^{\frac{1}{q-1}}{p}^{2}dp.$$

$$\mathcal{Z}=\frac{4\pi V}{N!h^{3N}}\underset{mc}{\overset{\infty }{\int }}y\sqrt{y^2-m^2{c}^2}{\left[1+(1-q)\beta c(y-mc)\right]}^{\frac{1}{q-1}}dy.$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}\underset{1}{\overset{\infty }{\int }}x\sqrt{x^2-1}{\left[1+(1-q)\beta m{c}^2(x-1)\right]}^{\frac{1}{q-1}}dx.$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}\underset{0}{\overset{\infty }{\int }}\left(s^{\frac{3}{2}}+s^{\frac{1}{2}}\right){\left(s+2\right)}^{\frac{1}{2}}{\left[1+(1-q)\beta m{c}^{2}s\right]}^{\frac{1}{q-1}}ds,$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}[(1-q)\beta m{c}^2{]}^{\frac{1}{q-1}}\underset{0}{\overset{\infty }{\int }}{s}^{\frac{3}{2}}{\left(s+2\right)}^{\frac{1}{2}}{\left[s+\frac{1}{(1-q)\beta m{c}^2}\right]}^{\frac{1}{q-1}}ds+$$

$$\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}[(1-q)\beta m{c}^2{]}^{\frac{1}{q-1}}\underset{0}{\overset{\infty }{\int }}{s}^{\frac{1}{2}}{\left(s+2\right)}^{\frac{1}{2}}{\left[s+\frac{1}{(1-q)\beta m{c}^2}\right]}^{\frac{1}{q-1}}ds.$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}[(1-q)\beta m{c}^2{]}^{-\frac{3}{2}}[\frac{B\left(\frac{5}{2},\frac{1}{1-q}-3\right)}{\beta m{c}^2(1-q)}\times$$

$$F\left(-\frac{1}{2},\frac{5}{2},\frac{1}{1-q}-\frac{1}{2};1-\frac{1}{2\beta m{c}^2(1-q)}\right)+$$

$$B\left(\frac{3}{2},\frac{1}{1-q}-2\right)F\left(-\frac{1}{2},\frac{3}{2},\frac{1}{1-q}-\frac{1}{2};1-\frac{1}{2\beta m{c}^2(1-q)}\right)].$$

$$\mathcal{Z}=\frac{2\pi V}{N!h^{3N}}{\left[\frac{2m}{\beta (1-q)}\right]}^{\frac{3}{2}}\frac{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{1}{1-q}-\frac{3}{2}\right)}{\Gamma \left(\frac{1}{1-q}\right)}.$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}\underset{0}{\overset{\infty }{\int }}\left(s^{\frac{3}{2}}+s^{\frac{1}{2}}\right){\left(s+2\right)}^{\frac{1}{2}}{\left[1-(q-1)\beta m{c}^{2}s\right]}_{+}^{\frac{1}{q-1}}ds.$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}\underset{0}{\overset{\frac{1}{\beta m{c}^2(q-1)}}{\int }}\left(s^{\frac{3}{2}}+s^{\frac{1}{2}}\right){\left(s+2\right)}^{\frac{1}{2}}{\left[1-(q-1)\beta m{c}^{2}s\right]}^{\frac{1}{q-1}}ds,$$

$$\mathcal{Z}=\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}[(q-1)\beta m{c}^2{]}^{\frac{1}{q-1}}\underset{0}{\overset{\frac{1}{\beta m{c}^2(q-1)}}{\int }}{s}^{\frac{3}{2}}{\left(s+2\right)}^{\frac{1}{2}}{\left[\frac{1}{(q-1)\beta m{c}^2}-s\right]}^{\frac{1}{q-1}}ds+$$

$$\frac{4\pi V{m}^3{c}^3}{N!h^{3N}}[(q-1)\beta m{c}^2{]}^{\frac{1}{q-1}}\underset{0}{\overset{\frac{1}{\beta m{c}^2(q-1)}}{\int }}{s}^{\frac{1}{2}}{\left(s+2\right)}^{\frac{1}{2}}{\left[\frac{1}{(q-1)\beta m{c}^2}-s\right]}^{\frac{1}{q-1}}ds.$$

$$\mathcal{Z}=\frac{2\pi V}{N!h^{3N}}{\left[\frac{2m}{\beta m(q-1)}\right]}^{\frac{3}{2}}[\frac{B\left(\frac{5}{2},\frac{1}{q-1}+1\right)}{\beta m{c}^2(q-1)}\times$$

$$F\left(-\frac{1}{2},\frac{5}{2},\frac{7}{2}+\frac{1}{q-1};-\frac{1}{2\beta m{c}^2(q-1)}\right)+$$

$$B\left(\frac{3}{2},\frac{1}{q-1}+1\right)F\left(-\frac{1}{2},\frac{3}{2},\frac{5}{2}+\frac{1}{q-1};-\frac{1}{2\beta m{c}^2(q-1)}\right)].$$

$$\mathcal{Z}=\frac{2\pi V}{N!h^{3N}}{\left[\frac{2m}{\beta (q-1)}\right]}^{\frac{3}{2}}\frac{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{1}{q-1}+1\right)}{\Gamma \left(\frac{1}{1-q}+\frac{5}{2}\right)},$$

$$\mathcal{Z}=\frac{2\pi V}{N!h^{3N}}{\left[\frac{2m}{\beta (q-1)}\right]}^{\frac{3}{2}}H(T),$$

$$\mathcal{Z}<\mathcal{U}>=\frac{V}{N!h^{3N}}\int [\sqrt{m^2{c}^4+p^2{c}^2}-m{c}^2]\times$$

$${\left[1+(1-q)\beta (\sqrt{m^2{c}^4+p^2{c}^2}-m{c}^2)\right]}_{+}^{\frac{1}{q-1}}{d}^{4}p,$$

$$\mathcal{Z}<\mathcal{U}>=\frac{V}{N!h^{3N}}\int [\sqrt{m^2{c}^4+p^2{c}^2}]\times$$

$${\left[1+(1-q)\beta (\sqrt{m^2{c}^4+p^2{c}^2}-m{c}^2)\right]}_{+}^{\frac{1}{q-1}}{d}^{4}p-m{c}^2\mathcal{Z}.$$

$$\mathcal{Z}<\mathcal{U}>=\frac{4\pi V{m}^4{C}^5}{N!h^{3N}}\underset{0}{\overset{\infty }{\int }}{x}^{\frac{3}{2}}(x+1)(\sqrt{x+2}\times$$

$${\left[1+(1-q)\beta m{c}^{2}x\right]}^{\frac{1}{q-1}}dx.$$

$$\mathcal{Z}<\mathcal{U}>=\frac{4\pi V{m}^4{C}^5}{N!h^{3N}}[\beta m{c}^2(1-q){]}^{\frac{1}{q-1}}\underset{0}{\overset{\infty }{\int }}{x}^{\frac{3}{2}}(x+1)(\sqrt{x+2}\times$$

$${\left[x+\frac{1}{(1-q)\beta m{c}^2}\right]}^{\frac{1}{q-1}}dx.$$

$$<\mathcal{U}>=\frac{\sqrt{2}4\pi V{m}^4{c}^5}{N!h^{3N}\mathcal{Z}}{\left[\frac{1}{\beta m{c}^2(1-q)}\right]}^{\frac{5}{2}+\frac{1}{q-1}}[\frac{B\left(\frac{7}{2},\frac{1}{1-q}-4\right)}{\beta m{c}^2(1-q)}\times$$

$$F\left(-\frac{1}{2},\frac{7}{2},\frac{1}{1-q}-\frac{1}{2};1-\frac{1}{2\beta m{c}^2(1-q)}\right)+$$

Full text

Keywords: Tsallis’ and Renyi’s relativistic distributions, classical partition function, entropic force.

PACS: 05.20.-y, 05.70.Ce, 05.90.+m

1. 1 Introduction
2. 2 Tsallis’ relativistic partition function for the free particle
3. 3 Tsallis’ relativistic mean energy of the free particle
4. 4 Specific heat in the linear constraints Tsallis’ scenario
5. 5 The relativistic, Tsallis entropic force
6. 6 The relativistic, Renyi’s entropic force
7. 7 Conclusions

1 Introduction

In 2011, Verlinde [1] put forward a conjecture that connects gravity to an entropic force. Gravity would then arise out of information regarding the positions of material bodies (it from bit). This idea links a thermal gravity-treatment to ’t Hooft’s holographic principle. As a consequence, gravitation ought to be be regarded as an emergent phenomenon. Verlinde’s conjecture attained considerable reception (just as an example, see [2]). For a superb overview on the statistical mechanics of gravitation, we recommend Padmanabhan’s work [3], and references therein.

Verlinde’s initiative originated works on cosmology, the dark energy hypothesis, cosmological acceleration, cosmological inflation, and loop quantum

gravity. The literature is immense [4]. A relevant contribution to information theory is that of Guseo [5], who proved that the local entropy function, related to a logistic distribution, is a catenary and vice versa. Such invariance may be explained, at a deeper level, through the Verlinde’s conjecture on the origin of gravity, as an effect of the entropic force. Guseo puts forward a new interpretation of the local entropy in a system, as quantifying a hypothetical attraction force that the system would exert [5].

The present effort does not deal with any of these issues. What we will do is to show that a simple classical reasoning centered on Tsallis’ relativistic probability distributions proves Varlinde’s conjecture. For Renyi’s relativistic instance, one needs to modify the conjecture to achieve a similar result.

Our point of departure is Ref. [6], in which their authors studied a canonical ensemble of $N$ particles for a classical relativistic ideal gas, and found its specific heat in the Tsallis-Mendes-Plastino (TMP) scenario [7]. We will not use here the TMP scenario. Inspired by [6], we appeal as well to our previous effort [8] for non-relativistic results and deal with Tsallis’ statistics with linear constraints as a priori information [7]. In addition to finding, for the first time ever, relativistic Verlinde-results in a Tsallis’context, we will, for the sake of completeness, register some advances regarding the relativistic Tsallis scenario with linear constraints for the ideal gas.

2 Tsallis’ relativistic partition function for the free particle

The celebrated and well-known Tsalis entropy is a generalization of Shanon’s one, that depends on a free real parameter $q$ [7]. The $q<1$ instance We consider first the case $q<1$. This case is not relevant to our Verlinde’s endeavor [8], but is a logical addition to the results of [6].

Tsallis’ relativistic q-partition function for $N-$free particles of mass $m$ reads [6]

[TABLE]

Using spherical coordinates and integrating over the angles the precedent integral we have

[TABLE]

With the change of variables $y^{2}=p^{2}+m^{2}c^{2}$ one now has

[TABLE]

Let $x$ be given by $y=mcx$. We have then

[TABLE]

With $s$ defined as $x=s+1$ we obtain:

[TABLE]

or

[TABLE]

[TABLE]

Appealing to reference [9] we have now a result in terms of Hyper-geometric functions $F$ and Beta functions $B$, namely,

[TABLE]

[TABLE]

[TABLE]

For $\beta mc^{2}>>1$, $mc^{2}>>k_{B}T$, we are in the non-relativistic case and have

[TABLE]

The case $q>1$ Let is now consider gravitationally relevant [8] case $q>1$ . We have for the partition function

[TABLE]

Integrating on the angles we have again

[TABLE]

or

[TABLE]

[TABLE]

By recourse to [9] we now obtain

[TABLE]

[TABLE]

[TABLE]

For $\beta mc^{2}>>1$, the classic case, the partition function reads

[TABLE]

which is the usual non relativistic Tsalli’s partition function for $q>1$ already obtained in [8]. Figure 1 displays the graph of the function $H(T)$ given by

[TABLE]

for $q=\frac{4}{3}$, the specific $q-$value needed for gravitaional considerations [8]. It tells us that ${\cal Z}$ is always positive, as it should be.

3 Tsallis’ relativistic mean energy of the free particle

Case $q<1$ Let us now calculate the average energy corresponding, firstly in the case $q<1$. For it we have

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

With changes in the variables similar to those made for the partition function, we obtain here

[TABLE]

[TABLE]

This last equation can be rewritten as

[TABLE]

[TABLE]

Returning again to reference [9], we obtain for $<{\cal U}>$

[TABLE]

[TABLE]

[TABLE]

From this last equation we obtain the mean energy expression for the non-relativistic case

[TABLE]

Case $q$ larger than one When $q>1$ we have

[TABLE]

[TABLE]

Making a similar reasoning as for the case $q<1$ we obtain

[TABLE]

[TABLE]

[TABLE]

For $\beta mc^{2}>>1$ (the non-relativistic case) we obtain the result of [8], i.e.,

[TABLE]

4 Specific heat in the linear constraints Tsallis’ scenario

Let is now calculate the specific heat for the case $q=\frac{4}{3}$, relevant for Verlinde-endeavors [8]. This was not done in [6]. We should first note, with respect to Hyper-geometric functions, that

[TABLE]

We now use the notation

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, we can write

[TABLE]

and, for the specific heat we have then

[TABLE]

[TABLE]

This expression is plotted in Figure 2. We see that the specific heat is always positive, as it happens in the non-relativistic case [8].

5 The relativistic, Tsallis entropic force

We arrive now at our main present goal. We specialize things now to $q=\frac{4}{3}$. Why do we select this special value $q=\frac{4}{3}$? There is a solid reason. This is because

[TABLE]

Since the entropic force is to be defined as proportional to the gradient of ${\cal S}$, there is a unique $q$-value for which the dependence on $r$ of the entropic force is $\sim r^{-2}$ when $\nu=3$. Thus we obtain, for $q=4/3$,

[TABLE]

From (2.12) we can write

[TABLE]

from which it is obtained that

[TABLE]

Following Verlinde [1] we define the entropic force as

[TABLE]

where $\vec{\nabla}$ indicates the four-gradient in Minkowskian space.

[TABLE]

where ${\vec{e}}_{r}$ is the radial unit vector. We see that $F_{e}$ acquires an appearance quite similar to that of Newton’s gravitational one, as conjectured by Verlinde en [1]. In Figures 3 and 4 the function $L=3-\beta<{\cal U}>$ is plotted. We see that $L$ is always positive. This entails that the relativistic entropic force is purely gravitational.

6 The relativistic, Renyi’s entropic force

In Renyi’s approach to our problem [8] the entropy is

[TABLE]

For $\alpha=\frac{4}{3}$, the expression for the entropy is

[TABLE]

The second term on the right hand of (6.2) is independent of $r$. Additionally, from (5.2) we obtain

[TABLE]

Here we need to derive the entropy with respect to the area, thus changing Verlinde´s conjecture. As in the non-relativistic case [8], we have then

[TABLE]

This is again a gravitational expression for the entropic force.

7 Conclusions

We obtained here the relativistic partition function ${\cal Z}$ of Tsalli’s theory with linear constraints, that adequately reduces itself to its non-relativistic counterpart for small velocities.

We do the same for the mean value of the energy$<{\cal U}>$ for the relativistic Hamiltonian of the ideal gas.

We obtain the associated specific heat that turns out to be positive, as befits an ideal gas.

From ${\cal Z}$ and $<{\cal U}>$ we obtained the relativistic entropy ${\cal S}$

We have presented two very simple relativistic classical realizations of Verlinde’s conjecture. The Tsallis treatment, for $q=4/3$, seems to be neater, as the entropic force is directly associated to the gradient of Tsallis’ entropy $S_{q}$, which acts as a ”potential”, as Verlinde prescribes. This is not so in the Renyi instance, in which one has to modify Verlinde’s $F_{e}$ definition and derive $S$ with respect to the area.

Strictly speaking, Verlinde’s conjecture can be unambiguously proved for the Tsallis entropy with $q=4/3$. The Renyi demonstration correspond to a modified version of Verlinde’s conjecture.

Of course, ours is a very preliminary, if significant, effort. A much more elaborate treatment would be desirable.