aa r X i v : . [ h e p - t h ] J u l Thermodynamics on Fuzzy Spacetime
Wung-Hong HuangDepartment of PhysicsNational Cheng Kung UniversityTainan, Taiwan
ABSTRACT
We investigate the thermodynamics of non-relativistic and relativistic ideal gases on thespacetime with noncommutative fuzzy geometry. We first find that the heat capacities ofthe non-relativistic ideal boson and fermion on the fuzzy two-sphere have different values,contrast to that on the commutative geometry. We calculate the “statistical interparticlepotential” therein and interprete this property as a result that the non-commutativity ofthe fuzzy sphere has an inclination to enhance the statistical “attraction (repulsion) inter-particle potential” between boson (fermion). We also see that at high temperature the heatcapacity approaches to zero. We next evaluate the heat capacities of the non-relativisticideal boson and fermion on the product of the 1+D (with D=2,3) Minkowski spacetime bya fuzzy two-sphere and see that the fermion capacity could be a decreasing function of tem-perature in high-temperature limit, contrast to that always being an increasing function onthe commutative geometry. Also, the boson and fermion heat capacities both approach tothat on the 1+D Minkowski spacetime in high-temperature limit. We discuss these resultsand mention that the properties may be traced to the mechanism of “thermal reduction ofthe fuzzy space”. We also investigate the same problems in the relativistic system with freeKlein-Gordon field and Dirac field and find the similar properties.*E-mail: [email protected] 1
Introduction
Physics on the noncommutative spacetime had been received a great deal of attention [1-8].Historically, it is a hope that the deformed geometry in the small spacetime would be possibleto cure the quantum-field divergences, especially in the gravity theory. The renovation of theinteresting in noncommutative field theories is that it have proved to arise naturally in thestring/M theories [3,4]. In the noncommutative geometric approach [5] to the unification ofall fundamental interactions, including gravity, the space-time is the product of an ordinaryRiemannian manifold M by a finite noncommutative space F. The need for F is to avoidthe fermion doubling problem [5]. An advantage of this approach over the traditional grandunification approach is that the reduction to the Standard Model gauge group is not due toplethora of Higgs fields, but is naturally obtained from the order one condition, which is oneof the axioms of noncommutative geometry [5].Motivated by the physical interesting of noncommutative spacetime we will in this paperstudy the thermodynamics of ideal gas on the spacetime which is the product of a 1+DMinkowski manifold by a noncommutative fuzzy geometry. Note that the noncommutativefuzzy sphere can appear naturally in the string/M theory [6,7]. It is known to correspond tothe sphere D2-branes in string theory with background linear B-field [8]. Also, in the presenceof constant RR three-form potential, the D0-branes can expand into a noncommutative fuzzysphere configuration [9].In section II we first review the mathematical property of fuzzy sphere [10-11]. Then,to get some feelings about the thermal property on the fuzzy geometry we first evaluatethe heat capacity of the non-relativistic ideal boson and fermion on the fuzzy two-sphere.We see that they have different values, contrast to that on the commutative geometry [12].In section III we calculate the “statistical interparticle potential” [13] and see that thenoncommutativity of the fuzzy sphere has an inclination to enhance the statistical “attraction(repulsion) interparticle potential” between boson (fermion). This statistical property maybe used to explain why the ideal boson and fermion on the fuzzy two-sphere have differentvalue of heat capacity.In section IV we evaluate the heat capacities of the non-relativistic ideal boson andfermion on the product of 1 + D Minkowski spacetime by a fuzzy two-sphere, with D=2,3.We find that the heat capacities therein approach to those on the 1+D Minkowski spacetimein the high-temperature limit. Also, the boson and fermion heat capacities become decreas-ing function of temperature in high-temperature limit, contrast to the property that fermionheat capacity is always an increasing function on the 1+D commutative geometry. Note thatin the 1+2 commutative spacetime the boson and fermion have same heat capacity valuewhich is an increasing function of temperature. The calculations are performed in section 4and summarized in table 1. We discuss these results and mention that the properties maybe traced to the mechanism of “thermal reduction of the fuzzy space”.2 able 1: The high-temperature heat capacities of non-relativistic ideal boson and fermionon the Minkowski spacetime + fuzzy two-sphere and that on the flat Minkowski spacetime. C ≈ k n (cid:16) J mR kT ) (cid:17) ± O ( n ) C ≈ k n − k kT ) n C ≈ k (cid:16) n + J mR kT ) (cid:17) n ± O ( n ) C ≈ k n ∓ k n
20 3 √ π mkT ) In section V we present a simple toy model to see such an interesting property. In sectionVI we investigate the relativistic system and evaluate the heat capacity of the free Klein-Gordon and Dirac field on the product of 1 + 2 Minkowski spacetime by a fuzzy two-sphere.We also find that the heat capacity therein approaches to that on the 1+ 2 Minkowskispacetime in high-temperature limit. Last section is devoted to a discussion. Note that theproperties of Casimir effect and effective potential on the noncommutative fuzzy space hadbeen studied by us in [14].
The noncommutative fuzzy two-sphere geometry is described by a finite dimensional algebragenerated by 3 matrices X i which satisfies the commutator [10][ X a , X a ] = i R q J ( J + 1) ǫ abc X c , a, b, c = 1 , , and J ∈ N. (2 . X a are (2 J + 1) × (2 J + 1) matrices proportional to the (2 J + 1)-dimensionalrepresent of the generators J of SU (2) algebra. It is known that in the limit J → ∞ at fixed R we get the ordinary sphere with radius R . The non-relativistic free particle with mass m on the fuzzy sphere has the spectrum [10] E ℓ = ℓ ( ℓ + 1)2 mR , ℓ = 0 · · · J, (2 . ℓ + 1, which will be used in the following calculations. To proceed, let us first investigate the classical statistics in which the partition function isdefined by Z = J X ℓ =0 (2 ℓ + 1) e − ℓ ( ℓ +1)2 mR kT . (2 . a X n = b f ( n ) = Z ba dxf ( x ) + 12 [ f ( a ) + f ( b )] + 112 [ f ′ ( a ) − f ′ ( b )] + · · · . (2 . Z ≈ mR kT (cid:20) − e − J ( J +1)2 mR kT (cid:21) + ( J + 23 ) e − J ( J +1)2 mR kT + · · · . (2 . E and heat capacity C are E ( T ) = − ∂ℓnZ∂β ≈ J ( J + 1)4 mR − J ( J + 1) m R kT . (2 . a ) C ( T ) = ∂E∂T ≈ J ( J + 1) m R kT . (2 . b )The asymptotic value of energy E ( T → ∞ ) is just the algebra mean value of (2.2) calculatedby P Jℓ =0 (2 ℓ + 1) E ℓ P Jℓ =0 (2 ℓ + 1) = J ( J + 1)4 mR = E ( T → ∞ ) . (2 . E J , in which J is a finite value. The finite asymptotic value E ( ∞ ) implies that the heat capacity becomeszero asymptotically.Note that we could not use the approximation result (2.6) to find the quantity in theordinary sphere by taking the limit of J → ∞ . This is because that the approximationadopted in there is suitable only under the condition J ( J +1)2 mR kT ≪
1. This property will befound in the following section.
The thermodynamics of ideal boson and fermion could be calculated from the following tworelations N = X p z − e βǫ ± E = X p ǫz − e βǫ ± , (2 . z is fugacity of the ideal gas, which is related to the chemical potential µ through theformula z ≡ exp ( µ/kT ) [12]. Using the spectrum (2.2) the numerical results of the energyfor ideal boson and fermion are plotted in figure 1.4 E FermionBoson
Figure 1: The energy of ideal boson and fermion on the noncommutative fuzzy sphere.
Figure 1 shows that the heat capacities of the ideal boson and fermion on the fuzzy two-sphere have different values, contrast to that on the commutative geometry. Also, as the gashave a finite maximum energy the associated heat capacity becomes zero asymptotically.
To see the above property we can perform the high-temperature expansion (i.e. J ( J +1)2 mR ≪ kT )to (2.8) with a help of Euler-Maclaurin summation formula (2.4). The results are N ≈ z (cid:20) mkT R (cid:18) − e − J ( J +1)2 mR kT (cid:19) + (cid:18) J + 23 (cid:19) e − J ( J +1)2 mR kT + 13 (cid:21) ∓ z (cid:20) mkT R (cid:18) − e − J ( J +1) mR kT (cid:19) + (cid:18) J + 23 (cid:19) e − J ( J +1) mR kT + 13 (cid:21) + O ( z ) . (2 . E ≈ mR (cid:20) z (cid:18) (2 mR kT ) (cid:18) − e − J ( J +1)2 mR kT (cid:19) − mkT R J ( J + 1) e − J ( J +1)2 mR kT (cid:19) ∓ z (cid:18) ( mR kT ) (cid:18) − e − J ( J +1) mR kT (cid:19) − mkT R J ( J + 1) e − J ( J +1) mR kT (cid:19)(cid:21) + O ( z ) . (2 . ≪ J ( J + 1) ≪ mR kT we can use (2.9) to express the fugacity z as afunction of N . After substituting this relation into (2.10) we find that the relation betweenthe energy density and number density becomes ε ≈ J mR − J mR mR kT ! n ± J m mR kT n , (2 . J ≫
1. 5 .2.2 Low-temperature Expansion
In the low temperature the fugacity does not approach to zero and we need to adopt anotherapproach. For the case of fermion gas we can first use the Euler-Maclaurin summationformula (2.4) to express the particle number as N = J X ℓ =0 ℓ + 1 z − e − J ( J +1)2 mR kT + 1 ≈ mR β (cid:18) − ℓn (1 + ze − J ( J +1)2 mR kT ) + ℓn (1 + z ) (cid:19) + 12 11 + z − + 12 2 J + 11 + z − e J ( J +1)2 mR kT . (2 . N ≈ mR β (cid:18) − ze − J ( J +1)2 mR kT + ℓnz (cid:19) + 12 (cid:16) − z − (cid:17) + 12 (2 J + 1) ze − J ( J +1)2 mR kT , (2 . z ≈ e N mR kT − J e − J ( J +1)2 mR kT . (2 . ze − J ( J +1)2 mR kT ≈ e N − J ( J +1)2 mR kT − J e − J ( J +1)2 mR kT → , (2 . N shall be less then the total acceptablestate N max ≡ P Jℓ =0 ℓ + 1. Thus the low temperature expansion in (2.13) is a consistentmethod.Next, we also use the Euler-Maclaurin summation formula (2.4) to express the particleenergy as E = J X ℓ =0 ℓ + 1 z − e − J ( J +1)2 mR kT + 1 ℓ ( ℓ + 1)2 mR ≈ mR ( kT ) (cid:18) − Li ( − z − e J ( J +1)2 mR kT ) + Li ( − z − ) (cid:19) − kT ℓn (1 + z − e J ( J +1)2 mR kT ) + 12 mR J J z − e J ( J +1)2 mR kT ! , (2 . Li ( ∓ y ) is the polylogarithm function which has a series expansion formula [15] Li ( y ) = ∞ X k =1 y k . (2 . z → ∞ the term Li ( − z − ) in (2.17) could beexpressed a series expansion. However, as z − e J ( J +1)2 mR kT → ∞ (as explained in (2.15)) we haveto use the “inversion formula” [15] Li ( y ) + Li ( y − ) = − π −
12 ( ℓn ( − y )) , (2 . E πR = 12 m n π kT ) ! + O ( e − J /R kT ) , (2 . n is the particle number density. Above result is just the relation in the commu-tative system with small correction O ( e − J /R kT ).Note that at zero temperature the particle will filled from the state ℓ = 0 to ℓ = J . Inthe case of J ≫ N = J X ℓ =0 ≈ J , E = J X ℓ =0 ℓ ( ℓ + 1)2 m ≈ m J , ⇒ E = 12 m N . E J = J ( J +1)2 mR thesystem therefore has a finite limiting energy as shown in (2.6a) and (2.11). This implies thatthe heat capacity becomes zero asymptotically.2. At low temperature, as the particles are at low energy level they does not feel theconstraint property of ℓ ≤ J , the system will behave as that on the commutative space, asshown in (2.19).3. It is well known that, in comparison with the normal statistical behavior, bosonsexhibit a larger tendency of bunching together, i.e., a positive statistical correlation. Incontrast, fermions exhibit a negative statistical correlation. Uhlenbeck [13] interpreted thisproperty by the “statistical interparticle potential”. In our model, particle on the fuzzysphere will be constrained between the state with quantum number ℓ = 0 and ℓ = J .Therefore the fermion will feel more statistic repulsive effect and the boson will feel morestatistic attractive effect, as shown in the next section. The extra statistical effect, whichis induced by the fuzzy property, will render the the heat capacities of the ideal boson andfermion on the fuzzy two-sphere to have different values, contrast to that on the commutativegeometry, as shown in (2.11). We now following the Uhlenbeck [12,13] to evaluate the “statistical interparticle potential”for the non-relativistic ideal boson and fermion on the fuzzy two-sphere.Define the one particle matrix element of the Boltzmann factor by F ij = < X i | e − βH | X j >, (3 . < X , X | e − βH | X , X > = F F ± F F , (3 . F = F and F = F and density matrix element becomes [12] < X , X | ˜ ρ | X , X > = 1 V " ± F F , (3 . ρ ≡ e − βH Tr e − β H [12] and V is the system volume.The “statistical interparticle potential” U is defined to be such that the Boltzmann factorexp( − βv ) is precisely equal to the correlation factor (bracket term) in the above equation,i.e., U = − kT ℓn " ± F F . (3 . Y mℓ ( θ, φ )[10] with ℓ ≤ J , we find that F = T r < X | e − βH | X > = X p < X | p >< p | e − βH | p >< p | X > = X ℓ X m X m Z dφdθ sin θ Y ∗ m ℓ ( θ, φ ) Y m ℓ ( θ, φ ) e − ℓ ( ℓ +1)2 mR kT = J X ℓ =0 (2 ℓ + 1) ℓ e − ℓ ( ℓ +1)2 mR kT . (3 . F = < X | e − βH | X > = X p < X | p >< p | e − βH | p >< p | X > = X ℓ X m X m Y ∗ m ℓ ( θ , φ ) Y m ℓ ( θ , φ ) e − ℓ ( ℓ +1)2 mR kT . (3 . J = 5 while solid line is that with J = 12. RU FermionBoson
Figure 2: “Statistical interparticle potential” U(R) on the fuzzy geometry. Dashed linesrepresents that with J = 5 while solid line is that with J = 12 . J will enhance the neg-ative statistical correlation between fermion and enhances the positive statistical correlationbetween bosons. Thus the thermal property of the boson and fermion gas on fuzzy spherewill have different heat capacity, contrast to that on the commutative geometry. We now investigate the thermodynamics of boson and fermion on 1+D Minkowski spacetimewith extra fuzzy sphere. We will see that at high temperature the thermodynamics of idealgas will behave as that on the 1+D Minkowski spacetime without the extra fuzzy sphere.This “mechanism of thermal reduction of fuzzy geometry” will qualitatively modify the heatcapacity of the gas.
In the high-temperature limit ( i.e J ( J +1)2 mR ≪ kT ) the thermodynamics of ideal boson andfermion on the 1+2 Minkowski spacetime with extra fuzzy sphere could be studied from thefollowing analysis. N = 2 πSh J X ℓ =0 (2 ℓ +1) Z dp p z − e ℓ ( ℓ +1)2 mR kT + p mkT ± ≈ z (cid:20) πSh Z dp p e − p mkT (cid:21) J X ℓ =0 (2 ℓ +1) e − ℓ ( ℓ +1)2 mR kT ∓ z (cid:20) πSh Z dp p e − p mkT (cid:21) J X ℓ =0 (2 ℓ + 1) e − ℓ ( ℓ +1) mkT ≈ πSh mkT h z · W ∓ z · W i . (4 . E = 2 πSh J X ℓ =0 (2 ℓ +1) Z dp p p m + ℓ ( ℓ +1)2 mR z − e ℓ ( ℓ +1)2 mR kT + p mkT ± ≈ πSh h mkT ( z · W ∓ z · W mkT ) ( z · W ∓ z · W i , (4 . S is the area of Minkowski space and we have defined W ≡ mkT R (cid:18) − e − J ( J +1)2 mR kT (cid:19) + ( J + 2 / e − J ( J +1)2 mR kT + 1 / . (4 . W ≡ mkT R (cid:18) − e − J ( J +1) mR kT (cid:19) + ( J + 2 / e − J ( J +1) mR kT + 1 / . (4 . W ≡ (2 mkT R ) (cid:18) − e − J ( J +1)2 mR kT (cid:19) − mkT R J ( J + 1) e − J ( J +1)2 mR kT . (4 . ≡ ( mkT R ) (cid:18) − e − J ( J +1) mR kT (cid:19) − mkT R J ( J + 1) e − J ( J +1) mR kT . (4 . z as a function of N . After substituting this relationinto (4.2) we can find the energy. The associated heat capacity is C ≈ k n J mR kT ) ! ± O ( n ) , ( gas on f lat space + f uzzy sphere ) , (4 . n is the particle number density. Above result shows that the heat capacity is adecreasing function of temperature. Note that in the 1+2 commutative spacetime the heatcapacity of boson and fermion has a same value C ≈ k n − k kT ) n , ( gas on f lat space ) , (4 . ℓ ≤ J the system will behaveas that on the 1+2+2 commutative space. However, at high temperature, as the system hasa maximum value in the spectrum ℓ ≤ J the heat capacity of the system therefore behave asthat on the 1+2 commutative space asymptotically. This “mechanism of thermal reductionof fuzzy geometry” will render the heat capacity of boson and fermion to be a decreasingfunction at high temperature. The high-temperature thermodynamics of ideal boson and fermion on the 1+3 Minkowskispacetime with extra fuzzy sphere could be studied from the following analysis. N = 4 πVh J X ℓ =0 (2 ℓ +1) Z dp p z − e ℓ ( ℓ +1)2 mR kT + p mkT ± ≈ z (cid:20) πVh Z dp p e − p mkT (cid:21) J X ℓ =0 (2 ℓ +1) e − ℓ ( ℓ +1)2 mR kT ∓ z (cid:20) πVh Z dp p e − p mkT (cid:21) J X ℓ =0 (2 ℓ + 1) e − ℓ ( ℓ +1) mR kT = 4 πVh √ π (2 mkT ) / h z · W ∓ z · W i . (4 . E = 4 πVh J X ℓ =0 (2 ℓ +1) Z dp p p m + ℓ ( ℓ +1)2 mR z − e ℓ ( ℓ +1)2 mR kT + p mkT ± ≈ πVh " √ π (2 mkT ) / z · W ∓ z · W
2) + √ π (2 mkT ) / z · W ∓ z · W , (4 . V is the volume of Minkowski space. W1, W2, W3 and W4 are defined in (4.3)-(4.6). Now, using (4.9) we can express the fugacity z as a function of N . After substituting10his relation into (4.10) we can find the energy. The associated heat capacity is C ≈ k n + J mR kT ) ! n ± O ( n ) , ( gas on f lat space + f uzzy sphere ) , (4 . n is the particle number density. Above result shows that the heat capacity is adecreasing function of temperature. Note that in the 1+3 commutative spacetime the heatcapacity of boson and fermion has a same value C ≈ k n ∓ k n √ π mkT ) , ( gas on f lat space ) , (4 . T c . Therefore, increasing the temperature beyond the T c it will be adecreasing function.More precisely, at low temperature the system does not feel the finite value property of J and the particle will have the thermal property like as that on the commutative 1+3+2commutative spacetime. In this case the heat capacity is an increasing function of temper-ature. However, at high temperature the quantum level of extra fuzzy space is all occupiedand the particle will behave as that on lower space. Thus it will have less heat capacity. This“mechanism of thermal reduction of the extra fuzzy space” could lead the heat capacity tobe a decreasing function of temperature and the heat capacity has a “peak value” near thetemperature of “ thermal reduction”. The interesting property could be seen in the followingtoy model. For example, let us consider a simplest toy model of classical particle which has spectrum E = a · n + b · ℓ, ≤ n ≤ ∞ , ≤ ℓ ≤ J. (5 . ≤ n ≤ ∞ is used to describe a simple harmonic oscillatorand the mode with finite quantum number 0 ≤ ℓ ≤ J is used to simulate that on the fuzzygeometry. In this model the partition function and the heat capacity could be evaluatedexactly. We plot the heat capacity in figure 3.11 C J = = Figure 3: The heat capacity of the model with spectrum (5.1) when a=b=1.
We have explicity seen that if J = 0 then there is a peak in the system capacity. Theassociated heat capacity at low temperature is C ≈ kT h a e − a/kT + b e − b/kT i , low temperature, (5 . n and ℓ contribute the similar behavior in heat capacity, whichis an increasing function at low temperature. On the other hand, at high temperature theassociated heat capacity becomes C ≈ k + 112 kT h b J ( J + 2) − a i , high temperature. (5 . n contributes heat capacity value k − a kT , which is anincreasing function, and quantum mode ℓ contributes heat capacity value kT [ b J ( J + 2)],which is a decreasing function at high temperature. Therefore, in the case of large J withrelation b J ( J + 2) > a the model will show a “peak value” in heat capacity, as shown infigure 3. We now turn to the problems with relativistic gas. The Klein-Gordon field equation on theproduct of Minkowski spacetime (with coordinate ~x ) by a fuzzy two-sphere (with coordinate J i ) is h ∂ t − ~ ∇ ~x − J − J − J i Φ + m Φ = 0 . (6 . e − ip · x and spherical harmonic function Y ℓm ( θ, φ ) then we see that the spectrum of scalar field is [10] E ℓ = ~p + ℓ ( ℓ + 1) R + m , ℓ = 0 · · · J, (6 . ℓ + 1. The Dirac field has a similar relation. Note that the finite value ofquantum number ℓ characterizes the fuzzy property of the fuzzy sphere. We will in followinganalyze the system will massless field for simplicity. Let us first investigate the classical statistics for the relativistic gas on fuzzy two-sphere. Inthis case the partition function is defined by Z = J X ℓ =0 (2 ℓ + 1) e − √ ℓ ( ℓ +1) /R kT . (6 . q J ( J + 1) /R ≪ kT ) has an approximation value Z ≈ R ( kT ) " − √ JRkT ! e − √ ℓ ( ℓ +1) /R kT + 12 (cid:20) J ) e − √ ℓ ( ℓ +1) /R kT (cid:21) + · · · . (6 . E and heat capacity C are E ( T ) = − ∂ℓnZ∂β ≈ J R − J R kT . (6 . a ) C ( T ) = ∂E∂T ≈ J R kT . (6 . b )The asymptotic value of energy E ( T → ∞ ) is just the algebra mean value of (6.2) calculatedby P Jℓ =0 (2 ℓ + 1) E ℓ P Jℓ =0 (2 ℓ + 1) = J R = E ( T → ∞ ) . (6 . E J , in which J is a finite value. The finite asymptotic value E ( ∞ ) implies that the heat capacity becomeszero asymptotically, as that in the non-relativistic system analyzed in section II. The thermodynamics of relativistic ideal boson and fermion field could be studied fromthe standard analysis [12] and we can perform the calculation from the (2.8). Using thespectrum (6.2) and with a help of Euler-Maclaurin summation formula (2.4) the results inhigh temperature are N ≈ z J − J / RkT + q J ( J + 1)(1 + 2 J )2 RkT ∓ z J + J / RkT + q J ( J + 1)(1 + 2 J ) RkT , (6 . ≈ z J / R + q J ( J + 1)(1 + 2 J )2 R + J R kT − J ( J + 1)(1 + 2 J )2 R kT ! ∓ z J / R + q J ( J + 1)(1 + 2 J )2 R + J R kT − J ( J + 1)(1 + 2 J ) R kT ! . (6 . z as a function of N . After substituting this relationinto (6.8) we find that the relation between the energy density ε and number density n becomes ε ≈ k J R − J R kT ! n ± J R kT k n , (6 . To proceed, let us first remark that in order to have the analytic result we will investigatethe system of massless free field on the 1+2 Minkowski spacetime with Extra Fuzzy Sphere.Note that, as the property we attempt to see is shown at high temperature, in which thequantum field will become asymptotic free and mass of the field is irrelevant. Also, theproperty we find will be shown in 1+3 Minkowski spacetime, after numerical analysis.Then, the thermodynamics of quantum Klein-Gordon field and Dirac field on the Kaluza-Klein spacetime of “1+2 + fuzzy sphere” could be studied from the standard analysis [12].The total particle number N and energy E could therefore be evaluated from the two relationsin (2.8). Using the spectrum (6.2) we find the following expressions in the high-temperatureapproximation (i.e. q J ( J +1) R ≪ kT ) N ≡ πSh J X ℓ =0 (2 ℓ + 1) Z dp p z − e β q p + ℓ ( ℓ +1) R ± ≈ πSh z J X ℓ =0 (2 ℓ + 1) Z dp p e − β q p + ℓ ( ℓ +1) R ∓ z J X ℓ =0 (2 ℓ + 1) Z dp p e − β q p + ℓ ( ℓ +1) R ≈ πSh z J X ℓ =0 (2 ℓ + 1) e − β q ℓ ( ℓ +1) R β β s ℓ ( ℓ + 1) R z J X ℓ =0 (2 ℓ + 1) e − β q ℓ ( ℓ +1) R β β s ℓ ( ℓ + 1) R . (6 . E ≡ πSh J X ℓ =0 (2 ℓ + 1) Z dp p q p + ℓ ( ℓ +1) R z − e β q p + ℓ ( ℓ +1) R ± ≈ πSh z J X ℓ =0 (2 ℓ + 1) Z dp p s p + ℓ ( ℓ + 1) R e − β q p + ℓ ( ℓ +1) R ∓ z J X ℓ =0 (2 ℓ + 1) Z dp p s p + ℓ ( ℓ + 1) R e − β q p + ℓ ( ℓ +1) R ≈ πSh z J X ℓ =0 (2 ℓ + 1) e − β q ℓ ( ℓ +1) R β R β ℓ ( ℓ + 1) + 2 s ℓ ( ℓ + 1) R R ∓ z J X ℓ =0 (2 ℓ + 1) e − β q ℓ ( ℓ +1) R β R β ℓ ( ℓ + 1) + 2 s ℓ ( ℓ + 1) R R , (6 . S is the area of Minkowski space. Using the Euler-Maclaurin formula in (2.4) toperform the summation in above the total particle number becomes N ≈ πSh (cid:20) z (cid:18) R ( kT ) − J ( kT ) e − J ( J +1) RkT + 12 ( kT ) (cid:19) ∓ z (cid:18) R ( kT / − J ( kT / e − J ( J +1) R kT/ + 12 ( kT / (cid:19)(cid:21) , (6 . E ≈ πSh (cid:20) z (cid:18) R ( kT ) − J ( kT ) e − J ( J +1) RkT + ( kT ) (cid:19) ∓ z (cid:18) R ( kT / − J ( kT / e − J ( J +1) RkT/ + ( kT / (cid:19)(cid:21) , (6 . z as a function of N . After substituting this relationinto (6.13) we can find the energy and the associated heat capacity at high temperaturebecomes C ≈ k J R kT ) ! n ± k
116 + 13192 J R kT ) ! n , (6 . n is the particle number density. Above result shows that the heat capacity offermion is a decreasing function of temperature, in high-temperature limit, contrast to that15lways being an increasing function on the commutative geometry [12]. Above property alsoshows in 1+3 Minkowski spacetime with extra fuzzy sphere, after numerical analysis.According to the equipartition theorem [12] the particle on the 1+2 Minkowski spacetimewill have two degrees of freedom. As each degree of freedom of the relativistic particle willcontribute the heat capacity k n in high-temperature limit. Result in (6.14) tells us that therelativistic particle in 1+2 Minkowski spacetime with extra fuzzy sphere behaves as that on1+2 Minkowski spacetime in high-temperature limit. This shows explicity the “mechanismof thermal reduction of the fuzzy space”. In this paper we have studied the thermodynamics of ideal gas on the spacetime with extrafuzzy geometry. We first evaluate the heat capacities of the non-relativistic ideal boson andfermion on the fuzzy two-sphere. We see that they have different values, contrast to that onthe commutative geometry [12]. We have calculated the “statistical interparticle potential”[13] and see that the noncommutativity of the fuzzy sphere has an inclination to enhancethe statistical “attraction (repulsion) interparticle potential” between boson (fermion). Thisstatistical property may be used to explain why the ideal boson and fermion on the fuzzytwo-sphere have different value of heat capacity. We also see that, at high temperature theheat capacity approaches to zero as the all quantum levels on fuzzy two-sphere are occupied.We next evaluate the heat capacity of the non-relativistic ideal boson and fermion onthe product of 1+D (with D=2,3) Minkowski spacetime by a fuzzy two-sphere and see thatthe heat capacity is a decreasing function of temperature in high-temperature limit. Weargue that at high temperature the quantum level of extra fuzzy space is all occupied andthe particle will behave as that on the a reduced space of 1+D Minkowski spacetime . This“mechanism of thermal reduction of the fuzzy space” could lead the heat capacity of bosonand fermion to be a decreasing function of temperature.We finally investigate the relativistic system and evaluate the heat capacity of the freescalar and Dirac field on the product of 1 + 2 Minkowski spacetime by a fuzzy two-sphere.We also find the similar properties in the relativistic system.16
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