# A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas

aa r X i v : . [ g r- q c ] J a n A Note on Eﬀects of Generalized and Extended Uncertainty Principles on J¨uttner Gas

Hooman Moradpour ∗ , Sara Aghababaei † , Amir Hadi Ziaie ‡ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),University of Maragheh, Maragheh P.O. Box 55136-553, Iran Department of Physics, Faculty of Sciences, Yasouj University, Yasouj 75918-74934 Iran

In recent years, the implications of the generalized (GUP) and extended (EUP) uncertainty prin-ciples on Maxwell-Boltzmann distribution have been widely investigated. However, at high energyregimes, the validity of Maxwell-Boltzmann statistics is under debate and instead, the J¨uttnerdistribution is proposed as the distribution function in relativistic limit. Motivated by these con-siderations, in the present work, our aim is to study the eﬀects of GUP and EUP on a system thatobeys the J¨uttner distribution. To achieve this goal, we address a method to get the distributionfunction by starting from the partition function and its relation with thermal energy which ﬁnallyhelps us in ﬁnding the corresponding energy density states.

I. INTRODUCTION

A general prediction of any quantum gravity theory isthe possibility of the existence of a minimal length in na-ture, known as the Planck length, below which no otherlength can be observed. It is commonly believed that inthe vicinity of the Planck length, the smooth structure ofspacetime is replaced by a foamy structure due to quan-tum gravity eﬀects [1]. Therefore, the Planck scale canbe regarded as a separation line between classical andquantum gravity regimes. There is a general consensusthat in the scale of this minimal size, the characteris-tics of diﬀerent physical systems would be altered. Forinstance, the introduction of a minimal length scale re-sults in a generalization of the Heisenberg un-certaintyprinciple (HUP) in such a way that it incorporates gravi-tationally induced un-certainty, postulated as the gener-alized uncertainty principle (GUP) [2]. In fact, the HUPbreaks down for energies near the Planck scale, i.e., whenthe Schwarz-schild radius is comparable to the Comptonwavelength and both are close to the Planck length. Thisdeﬁciency is removed by revising the characteristic scalethrough the modiﬁ-cation of HUP to GUP.In recent decades, numerous studies on the eﬀectsof GUP in various classical and quantum mechanicalsystems have been performed (see e.g., [3–6]). Uncer-tainty in momentum is also bounded from below andit is proposed that its minimum is non-zero, a proposalwhich modiﬁes HUP to the extended uncertainty prin-ciple (EUP) [7]. In the presence of EUP and GUP, thegeneral form of modiﬁed HUP is proposed as∆ x ∆ p ≥ ~ (cid:0) α (∆ x ) + η (∆ p ) + γ (cid:1) , (1)in which α , η and γ are positive deformation parame-ters [8]. It should be noted that there is another formu-lation of GUP and EUP [9], and also the extended forms ∗ [email protected]r † [email protected] ‡ [email protected] of HUP, like GUP, may break the fundamental symme-tries such as Lorentz invariance and CPT [10].On the other hand, it is known that heavy ions can beaccelerated to very high kinetic energies constituting anensemble of ideal gas with relativistic velocities in largeparticle accelerators [11]. In such high energy regimes,minimal length eﬀects may appear and could have theirown inﬂuences on the statistics of ideal gases. Therefore,particle accelerators could provide a setting to exam-ine the phenomena related to short-distance physics [12].Based on minimum observable length, the quantum grav-ity implications on the statistical properties of ideal gaseshave been investigated in many studies, see, e.g., [13–16]and references therein. In the framework of GUP, i ) de-formed density of states and an improved deﬁnition ofthe statistical entropy have been introduced in [17], ii )Maxwell-Boltzmann statistics is investigated in [18], and iii ) employing Maxwell-Boltzmann statistics, the ther-modynamics of relativistic ideal gas has also analyzedin [19]. In the same manner, there have been some workson the deformation of statistical concepts in the EUPframework [20].J¨uttner distribution is a generalization of Maxwell-Boltzmann statistics to the relativistic regimes, whichappears in high energy physics. Since quantum gravity isa high energy physics scenario, its statistical eﬀects maybe more meaningful in the framework of J¨uttner distribu-tion function compared to the Maxwell-Boltzmann dis-tribution [21]. Here, our main aim is to study the eﬀectsof GUP and EUP on Maxwell-Boltzmann and J¨uttnerdistributions and density of states in energy space. Toachieve this goal, we begin by providing an introductorynote on Maxwell-Boltzmann and J¨uttner distributions.We then address a way to ﬁnd these functions by start-ing from the partition function of the system in the nextsection. The eﬀects of GUP and EUP on these statisticsare also studied in the subsequent sections, respectively.The last section is devoted to a summary on the work. II. THE MAXWELL-BOLTZMANN ANDJ ¨UTTNER DISTRIBUTION FUNCTIONS

We begin by considering an ideal gas composed of non-interacting particles and set the units so that ω = 2 π ~ =1, where ω denotes the fundamental volume of each cellin the two-dimensional phase-space. This value of ω originates from the well-known commutation relation be-tween the canonical coordinates x and p , and indeed, itis the direct result of HUP [22]. Therefore, any changesin HUP can aﬀect ω . Non-relativistic gas

Let us consider a 3-dimensional classical gas consistingof N identical non-interacting particles of mass m with E = m v , where E and v denote the energy and veloc-ity of each particle, respectively. At temperature T , theMaxwell-Boltzmann (MB) distribution function is givenby f MB ( v, β ) = Z MB exp (cid:18) − βmv (cid:19) , (2)where Z MB is a normalization constant, and β ≡ /K B T with K B being the Boltzmann constant. In terms of E we have f MB ( E, β ) = 4 πv ( E ) f MB ( v ( E ) , β ) dvdE = Z MBE E exp( − βE ) , (3)in which Z MBE is a new normalization constant and E denotes the density of states with energy E . The nor-malization constants can be calculated using the normal-ization constraint Z ∞ f MB ( v, β ) d v = Z ∞ f MB ( E, β ) dE = 1 (4)The extremum of f MB ( E, β ) is located at E = β ≡ E extMB or equally v = √ βm ≡ v extMB . One can also evaluate thepartition function of the mentioned gas (with Hamilto-nian H = p m ) as Q N = 1 N ! Z ... Z| {z } N exp( − βH ) d N x d N p = (cid:0) Q NR1 (cid:1) N N ! , (5)where Q NR1 = Z exp( − βH ) d x d p = V (cid:18) πmβ (cid:19) , (6) denotes the single particle partition function of a non-relativistic gas and V refers to the total volume of thesystem. In this manner, the corresponding thermal en-ergy per particle ( U ) takes the form U NR = Z ∞ E f MB ( E, β ) dE = − ∂ ln Q NR1 ∂β = 32 β = 3 E extMB . (7)Although the use of Eq. (7) dates back to before the dis-covery of the special relativity theory by Einstein, theultra-relativistic expression of E produces interesting re-sults in this framework [22]. Relativistic gas

In the relativistic situations, where E = p p c + m c in which c denotes the light velocity and m is the restmass, one can employ Eq. (5) to get Q R1 = Q NR1 Ψ( σ ) , Ψ( σ ) = i m c p π H (1)2 ( iσ )( iσ ) , (8)as the partition function of single particle [21, 23, 24].Finally, we obtain the thermal energy per particle as U R = 1 β " − iσ H ′ (1)2 ( iσ ) H (1)2 ( iσ ) = − ∂∂β ln Q R1 . (9)In the above equations, σ = βmc , H ( j ) n ( iσ ) is the n -thorder Hankel function of the j -th kind, and prime denotesderivative with respect to the argument of function. Theabove results were ﬁrst reported by in 1911 J¨uttner [21],who attempt to calculate the energy of a relativistic idealgas using the conventional theory of relativistic statisticalmechanics. According to J¨uttner’s results, a comprehen-sive study of a 3-dimensional relativistic system requiresthe J¨uttner distribution ( f J ) f J ( γ, β ) = Z J ( γ − γ exp (cid:0) − βmγ (cid:1) , (10)instead of MB distribution ( f MB ) [25–30]. J¨uttner distri-bution is indeed the relativistic extension of generalizedisotropic MB distribution when E ( p ) = mγ ( p ) c . Here, Z J is the normalization constant and γ = √ − v refers tothe Lorentz factor, where the units have been set so that c = 1. In terms of E , simple calculations give us J¨uttnerdistribution as f J ( E, β ) = Z JE a J ( E ) exp (cid:0) − βE (cid:1) , (11)where a J ( E ) = E √ E − m denotes the density of statesin energy representation, and in terms of v one ﬁnds f J ( v, β ) = Z JV (cid:18) √ − v (cid:19) exp (cid:18) − β m √ − v (cid:19) , (12)in which Z JE and Z JV are new normalization con-stants [26]. These constants can be evaluated using thenormalization condition Z ∞ f J ( γ, β ) dγ = Z ∞ m f J ( E, β ) dE = 1= Z f J ( γ ( v ) , β ) dγdv dv = Z f J ( v, β ) d v, (13)which clearly states that f J ( v, β ) = 14 πv f J ( γ ( v ) , β ) dγdv . (14)It is ﬁnally useful to note that the extremum of f J ( v, β ) islocated at v = q − ( βm ) ≡ v extJ leading to E extJ = β ,a solution which is valid only when βm <

5. There arealso other proposals for J¨uttner function ( f J ( γ, β )) [25–30], but the standard form Eq. (11) is considered in thispaper is conﬁrmed by some previous studies [27–29]. Thecorresponding thermal energy per particle (i.e., h γ i ) (orequally, the ratio UN in Eq. (12)) can also be obtained byusing f J ( γ, β ), as U R ≡ m h γ i = Z ∞ m E f J ( E, β ) dE = − ∂∂β ln Q R1 . (15)Although Eqs. (7) and (15) are simple examples,they conﬁrm that the mean value of energy (or equally,thermal energy) can be calculated by using eitherthe partition function or the distribution function.Moreover, employing these equations, one can ﬁnd thedistribution functions whenever the partition function isknown. Indeed, if the phase-space geometry is deformed,then the partition function will also be modiﬁed. There-fore, one can ﬁnd the corresponding modiﬁed MB andJ¨uttner distributions by directly using Eqs. (7) and (15)for the non-relativistic cases, repetitively. III. GENERALIZED UNCERTAINTYPRINCIPLE, PARTITION AND DISTRIBUTIONFUNCTIONS

In the units of ~ = c = 1, [ x k , p l ] = iδ kl is the standardcommutation relation between the canonical coordinates x and p . This relation leads to HUP in the frameworkof quantum mechanics and is the backbone of calculating ω [18, 22]. Thus, the volume element d x d p changeswhenever diﬀerent coordinates (commutation relations)are used [13–15, 18]. For GUP, we have [14, 31]∆ X ∆ P ≥

12 [1 + η (∆ P ) + ... ] , (16)where η denotes the GUP parameter and is based onmodiﬁed commutation relations (cid:2) X k , P l (cid:3) = i (cid:18) δ kl (1 + ηP ) + η ′ P k P l (cid:19) , (cid:2) P k , P l (cid:3) = 0 , (cid:2) X k , X l (cid:3) = i η − η ′ + (2 η + η ′ ) ηP ηP ( P k X l − P l X k ) , (17)where k, l = 1 , , P and X are generalized coordinates which are not neces-sarily equal to the canonical coordinates p and x . In thismanner, assuming η ′ = 0 and η is independent of ~ , oneﬁnds d x d p → d X d P (1 + ηP ) , (18)which must be considered as the volume element in X − P space instead of d x d p [13–15, 18]. This means that thedensity of states in the X − P phase space is aﬀected bythe factor of (1 + ηP ) [13]. In this situation, the singleparticle partition function can also be found as Q GUP1 = Z exp( − βH ( P, X )) d X d P (1 + ηP ) , (19)where H ( P, X ) denotes Hamiltonian in generalized co-ordinates [13, 14, 19, 20]. The corresponding thermalenergy ( U GUP ) can be calculated using the relation U GUP = − ∂∂β ln Q GUP1 , (20)along with Eq. (19), which ﬁnally gives U GUP = R H ( P ) exp( − βH ( P )) d P (1+ ηP ) R exp( − βH ( P )) d P (1+ ηP ) . (21)In obtaining this equation, the fact that H ( ≡ E ) is in-dependent of X has been used which cancels integrationover d X . Indeed, the density of states in phase-space ischanged under the shadow of GUP [13, 14, 18], a resultwhich aﬀects the distribution function.For a single free particle with H = P m , the ideal gaslaw is still valid, and therefore Q NR , GUP1 = Q NR1 I (cid:18) mηβ , (cid:19) , (22)while the explicit form of the function I (cid:0) mηβ , (cid:1) can befollowed in [15] and Q NR1 is introduced in Eq. (6). Theeﬀects of GUP are stored in I (cid:0) mηβ , (cid:1) , and at the limitof η →

0, one gets the ordinary single partition functionof a free particle.Correspondingly, the partition function of a single freerelativistic particle can also be evaluated using H = P + m in Eq. (19). By doing so one ﬁnds Q R , GUP1 = Z exp (cid:16) − β p P + m (cid:17) d X d P (1 + ηP ) , (23)for which the solution reads Q R , GUP1 = Q R1 (cid:18) − η

152 1 βm (cid:19) , (24)when m ≫ β [19, 20]. Maxwell-Boltzmann statistics

Bearing the recipe which led to the expression for f MB ( E, β ) in mind, one can get the modiﬁed MB dis-tribution in the X − P space as f GUPMB ( E, β ) = 4 πP ( E ) exp( − βE )(1 + ηP ( E )) dPdE = Z GUPMBE E exp( − βE )(1 + 2 ηmE ) , (25)where Z GUPMBE denotes the normalization constant in thepresence of GUP. The thermal energy then reads U GUP = Z ∞ E f

GUPMB ( E, β ) dE. (26)One can also ﬁnd the normalization constant Z GUPMBE as Z GUPMBE = (cid:18) Z ∞ E exp( − βE )(1 + 2 ηmE ) dE (cid:19) − , (27)which is equal to Z MBE in the limit where η → f MB ( E, β ) is recoveredthrough Eq. (25) at the appropriate limit of η = 0.For the density of states in HUP framework we have a MB ( E ) = √ E . This relation is modiﬁed in the pres-ence of GUP eﬀects and thus, the density of states willtake the following form a GUPMB ( E ) = √ E (1 + 2 ηmE ) , (28) M a x w e ll - B o l t z m annd i s t r i bu t i on (cid:1) = (cid:0) = (cid:2) = (cid:3) = FIG. 1: MB distribution versus energy for η = 0 . , , .

5. Theordinary MB distribution is denoted by the solid curve. Here,we have set the units so that ~ = c = k B = 1. which is in agreement with the results of [18]. The ex-tremum of f GUPMB ( E, β ) is also located at E extMB = (29)1 + 10 mηE extMB mη s mηE extMB (1 + 10 mηE exMB ) − ! , which clearly indicates E extMB → E extMB whenever η → β = 1). J¨uttner statistics

In the relativistic situation, where H = √ P + m ( ≡ E ), following the above recipe, we get the modiﬁedJ¨uttner distribution as f GUPJ ( E, β ) = Z GUPJE E √ E − m exp (cid:0) − βE (cid:1)(cid:0) η [ E − m ] (cid:1) , (30)which recovers f J ( E, β ) in the limit where η →

0. Here, Z GUPJE is also the normalization constant evaluated by uti-lizing the normalization constraint R ∞ m f GUP J ( E, β ) dE =1. We also ﬁnd a GUPJ ( E ) = E √ E − m (cid:0) η [ E − m ] (cid:1) , (31)as the density of states in J¨uttner statistics in thepresence of GUP. Figure (2) shows the behavior of f GUPJ ( E, β ) for some positive values of η parameter. IV. EXTENDED UNCERTAINTY PRINCIPLE,PARTITION AND DISTRIBUTION FUNCTIONS

The modiﬁed Heisenberg algebra in the EUP frame-work can be recast into the following form J un tt e r d i s t r i bu t i on η = η = η = η = FIG. 2: Behavior of J¨uttner distribution against energy for η = 0 . , , .

5. The solid curve belongs to the ordinary J¨uttnerdistribution and we have set the units so that ~ = c = k B = 1. [ X i , P j ] = i ( δ ij + αX i X j ) , (32)where α is a small positive parameter known as the EUPparameter. In the limit of α →

0, the canonical com-mutation relation of the standard quantum mechanics isrecovered. Based on the commutation relation (32), theHUP is modiﬁed by(∆ X i )(∆ P i ) ≥

12 [1 + α (∆ X i ) + ... ] , (33)which leads to a non-zero minimum uncertainty in mo-mentum as (∆ P i ) min = √ α . Here, we apply the coordi-nate representation of the operators X i and P i expressedas X i = x i ,P i = ( δ ij + αx i x j ) p j , (34)where x i and p j satisfy the standard commutation rela-tion of ordinary quantum mechanics. This representationyields the following commutation relation for the momen-tum operator [ P i , P j ] = iα ( x i p j − p i x j ) . (35)In the X − P space, the modiﬁed volume element d Xd P (1 + αX ) , (36)should be considered instead of d xd p [20]. We then pro-ceed to consider the consequences of such a modiﬁcationin calculating the partition and distribution functions.For a single particle, the partition function in X − P space can be found as Q EUP1 = Z exp( − βH ( P, X )) d X d P (1 + αX ) , (37) whence we get the corresponding thermal energy as U EUP = − ∂∂β ln Q EUP1 , (38)The above expression can also be combined with Eq. (37)to give Eq.(21). For the free non-relativistic and relativis-tic particles, one ﬁnds Q NR , EUP1 = V eﬀ ( α, r ) (cid:18) πmβ (cid:19) = V eﬀ ( α, r ) V Q

NR1 , (39)and Q R , EUP1 = V eﬀ ( α, r ) V Q R1 . (40)respectively, where we have deﬁned V eﬀ ( α, r ) = R r d X (1+ αX ) as the eﬀective volume, and in the limit of α →

0, the usual volume V is recovered. Since V eﬀ ( α, r )is independent of β , the thermal energy related to EUP isthe same as what we obtained in Eqs. (7) and (9), respec-tively. Consequently, EUP does not aﬀect the Maxwell-Boltzmann and J¨uttner distribution functions, becausethe corresponding eﬀective volume has no dependenceon β . V. CONCLUSION

The J¨uttner function is the relativistic version of MBdistribution and is proper for studying relativistic (highenergy) systems. On the other hand, the minimal lengthcomes into play in the realms of high energy physics.Hence, compared with MB distribution, the study of itsef-fects on J¨uttner distribution would be more meaning-ful. Thus, our attempt in the present work was to addressan algorithm with the help of which, one can get the dis-tribution function, starting from the partition function.Motivated then by the abovementioned arguments, westudied the eﬀects of GUP and EUP (two aspects of quan-tum gravity) on J¨uttner distribution and the correspond-ing density of states in energy space. We also addressedthe consequence of applying our approach to the MB dis-tribution in order to ﬁnd the density of states Eq. (28)which is in agreement with previous reports [13, 18], aresult which conﬁrms our approach. The results of ourstudy are summarized in Tables (I) and (II) for the non-relativistic and relativistic regimes, respectively.It is obvious from Figs. (1) and (2), that the eﬀects ofthe existence of a non-zero minimal length ( η = 0) ondistribution functions become more sensible as energyincreases. This means that the probability of achiev-ing high energy states when η = 0 is smaller than the η = 0 case. It is also worth mentioning that thoughthere exist some proposals to test observable eﬀects ofthe minimal length [32], the Planck scale is currentlyfar beyond our reach. Since, by comparing the Planckenergy ( ≈ TeV) [33] to the energy achieved in theLarge Hadron Collider ( ≈ ≈ − m) to the uncertainty within the posi-tion of the LIGO mirrors ( ≈ − m) [35] or the Plancktime ( ≈ − s) to the shortest light pulse produced inlaboratory ( ≈ − s) [36], we observe that we are at best15 orders of magnitude away from achieving the Planckscale. In this regard, more future developments withinthese experimental setups are expected in order to seekfor the footprints of GUP eﬀects in nature.Finally, regarding the results reported in [37] and [38]the usefulness of Tsallis distribution function at high en-ergy physics is expected. In line with these results, someresearchers study the possibility of describing the distri-bution of transverse momentum in Large Hadron Collider and Relativistic Heavy Ion Collider employing the Tsallisdistribution, expressed as [39–42] f T ( q, β ) = Z T [1 − (1 − q ) βE ] − q . (41)Here Z T and q denote the normalization constant andnon-extensivity parameter, respectively. Although, uti-lizing our approach to investigate the eﬀects of GUP andEUP on (41) is straightforward, such a study needs morecareful analysis owing to the issues raised by [37] whichstates a criterion on the domains of validity of Maxwell-Blotzmann, J¨uttner and Tsallis distributions as a specialhigh energy phenomenon. 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