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Here equation(6.1-6.4) are known as the thermodynamic Maxwell’s relations which are universally satisfied by the charged Kalb Ramond black hole as introduced in this article. To establish the thermodynamic Maxwell’s relations here we remember that all the Kalb Ramond black hole thermodynamic potentials Gyr, Hxr, Fxr and Mgr are thermodynamic state functions and hence any infinitesimal reversible change in the thermodynamic potentials are exact differentials. Consequently we have: B. Verification of Ehrenfest’s Theorem and Prigogine-Defay ratio A. Thermodynamic Maxwell’s Relations

Figure 20 Here equation(6.1-6.4) are known as the thermodynamic Maxwell’s relations which are universally satisfied by the charged Kalb Ramond black hole as introduced in this article. To establish the thermodynamic Maxwell’s relations here we remember that all the Kalb Ramond black hole thermodynamic potentials Gyr, Hxr, Fxr and Mgr are thermodynamic state functions and hence any infinitesimal reversible change in the thermodynamic potentials are exact differentials. Consequently we have: B. Verification of Ehrenfest’s Theorem and Prigogine-Defay ratio A. Thermodynamic Maxwell’s Relations

Related Figures (22)

FIG. 1: Variation of the metric function h_(r) with r for Gauss-Bonnet coupling ais) = 0.001 > 0 and ADM mass parameter pt > 0 in presence of Kalb Ramond field. Here we use ps = 0.028, Ajs) = —1 in the Planckian unit.
FIG. 1: Variation of the metric function h_(r) with r for Gauss-Bonnet coupling ais) = 0.001 > 0 and ADM mass parameter pt > 0 in presence of Kalb Ramond field. Here we use ps = 0.028, Ajs) = —1 in the Planckian unit.
Now combining all of these allowed solution for the “branch singularity” in general we can write:
Now combining all of these allowed solution for the “branch singularity” in general we can write:
where C1 (t) be the Chebyshev polynomial with argument t [39, 40]. Let us now concentrate on the “killing horizon” which has important physical significances in the context of phase transition and critical phenomena in the black hole thermodynamics [46-54]. By setting h(rz) = 0 in equation(3.11) in k = 1 we get:
where C1 (t) be the Chebyshev polynomial with argument t [39, 40]. Let us now concentrate on the “killing horizon” which has important physical significances in the context of phase transition and critical phenomena in the black hole thermodynamics [46-54]. By setting h(rz) = 0 in equation(3.11) in k = 1 we get:
FIG. 2: Variation of the killing horizon ry vs five dimensional Gauss-Bonnet coupling and ADM mass parameter ju = 0.028 > | in presence of Kalb Ramond field with charge Qxr = 10~* Mpz.
FIG. 2: Variation of the killing horizon ry vs five dimensional Gauss-Bonnet coupling and ADM mass parameter ju = 0.028 > | in presence of Kalb Ramond field with charge Qxr = 10~* Mpz.
FIG. 3: Variation of the killing horizon ry with Kalb Ramond charge Qxr and ADM mass parameter pp = 0.028 > 0 in presence of Kalb Ramond field.
FIG. 3: Variation of the killing horizon ry with Kalb Ramond charge Qxr and ADM mass parameter pp = 0.028 > 0 in presence of Kalb Ramond field.
FIG. 4: Variation of the Hawking temperature Ty_ with five dimensional Gauss-Bonnet coupling a5) for Qkr = 10-* in presence of Kalb Ramond antisymmetric tensor field. Here we use the ADM mass parameter yw = 0.028 in the Planckian unit.
FIG. 4: Variation of the Hawking temperature Ty_ with five dimensional Gauss-Bonnet coupling a5) for Qkr = 10-* in presence of Kalb Ramond antisymmetric tensor field. Here we use the ADM mass parameter yw = 0.028 in the Planckian unit.
FIG. 5: Variation of the Hawking temperature Ty_ with Kalb Ramond charge @ xR in presence of Kalb Ramond antisymmetri tensor field. Here we use the ADM mass parameter ps = 0.028 with Gauss-Bonnet coupling ais) = at, :
FIG. 5: Variation of the Hawking temperature Ty_ with Kalb Ramond charge @ xR in presence of Kalb Ramond antisymmetri tensor field. Here we use the ADM mass parameter ps = 0.028 with Gauss-Bonnet coupling ais) = at, :
In presence of GB coupling (a(5)) the “Bekenstein Hawking entropy” in five dimension is defined as [55-69 B. Bekenstein Hawking entropy
In presence of GB coupling (a(5)) the “Bekenstein Hawking entropy” in five dimension is defined as [55-69 B. Bekenstein Hawking entropy
FIG. 6: Variation of Bekenstein Hawking entropy (S#) with five dimensional Gauss-Bonnet coupling a,s) in presence of charged Kalb Ramond antisymmetric tensor field with charge Qxr = 10-4 Mp7. Here we use js = 0.028 in the Planckian unit.
FIG. 6: Variation of Bekenstein Hawking entropy (S#) with five dimensional Gauss-Bonnet coupling a,s) in presence of charged Kalb Ramond antisymmetric tensor field with charge Qxr = 10-4 Mp7. Here we use js = 0.028 in the Planckian unit.
FIG. 7: Variation of Bekenstein Hawking entropy (S'7) with Kalb Ramond charge Qi r five dimensional Gauss-Bonnet coupling Q(5) = 0.004. in the Planckian unit.
FIG. 7: Variation of Bekenstein Hawking entropy (S'7) with Kalb Ramond charge Qi r five dimensional Gauss-Bonnet coupling Q(5) = 0.004. in the Planckian unit.
‘IG. 8: Variation of specific heat Cong C= Cexr) With five dimensional Gauss-Bonnet coupling a(5) Kalb Ramond charge Qir = 10~* Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
‘IG. 8: Variation of specific heat Cong C= Cexr) With five dimensional Gauss-Bonnet coupling a(5) Kalb Ramond charge Qir = 10~* Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
FIG. 9: Variation of specific heat Cong C= Caxr) With Kalb Ramond charge QR five dimensional Gauss-Bonnet coupling (5) = args. Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
FIG. 9: Variation of specific heat Cong C= Caxr) With Kalb Ramond charge QR five dimensional Gauss-Bonnet coupling (5) = args. Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
FIG. 10: Variation of inverse of isothermal compressibility K7* with five dimensional Gauss-Bonnet coupling a(5) Kalb Ramond charge Qkr = 10-*. Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
FIG. 10: Variation of inverse of isothermal compressibility K7* with five dimensional Gauss-Bonnet coupling a(5) Kalb Ramond charge Qkr = 10-*. Here we use ADM mass parameter ps = 0.028 in the Planckian unit.
FIG. 11: Variation of inverse of isothermal compressibility K-' with Kalb Ramond charge Qicr five dimensional Gauss-Bonnet coupling a/s5) = 0.00018. Here we use ADM mass parameter ys = 0.028 in the Planckian unit.
FIG. 11: Variation of inverse of isothermal compressibility K-' with Kalb Ramond charge Qicr five dimensional Gauss-Bonnet coupling a/s5) = 0.00018. Here we use ADM mass parameter ys = 0.028 in the Planckian unit.
FIG. 12: Variation of inverse of volume expansivity Born with five dimensional Gauss-Bonnet coupling a(5) Kalb Ramon« charge Qxr = 10~*. Here we use ADM mass parameter ps = 0.028 in the Planckian unit. Now using equation(5.16) the volume expansivity for the Kalb Ramond black hole can be computed as: where the expression for E(a(s), Q«r, Px, A(5), ) is mentioned in equation(5.18). Specifically in the asymptotic imit a5) + 0, the volume expansivity reduces to:
FIG. 12: Variation of inverse of volume expansivity Born with five dimensional Gauss-Bonnet coupling a(5) Kalb Ramon« charge Qxr = 10~*. Here we use ADM mass parameter ps = 0.028 in the Planckian unit. Now using equation(5.16) the volume expansivity for the Kalb Ramond black hole can be computed as: where the expression for E(a(s), Q«r, Px, A(5), ) is mentioned in equation(5.18). Specifically in the asymptotic imit a5) + 0, the volume expansivity reduces to:
FIG. 13: Variation of inverse of volume expansivity Bor with Kalb Ramond charge Qxr for five dimensional Gauss-Bonnet coupling a5) = 0.001. Here we use ADM mass parameter js = 0.028 in the Planckian unit system. In this context the specific heat at constant Kalb Ramond electric potential is defined as:
FIG. 13: Variation of inverse of volume expansivity Bor with Kalb Ramond charge Qxr for five dimensional Gauss-Bonnet coupling a5) = 0.001. Here we use ADM mass parameter js = 0.028 in the Planckian unit system. In this context the specific heat at constant Kalb Ramond electric potential is defined as:
IG. 14: Variation of specific heat Cog a C= Cop) with five dimensional Gauss-Bonnet coupling ais) for ®xr = 2 x 073, Acs) < 0. Here we use the ADM mass parameter pp = 0.028 in the Planckian unit.
IG. 14: Variation of specific heat Cog a C= Cop) with five dimensional Gauss-Bonnet coupling ais) for ®xr = 2 x 073, Acs) < 0. Here we use the ADM mass parameter pp = 0.028 in the Planckian unit.
FIG. 15: Variation of specific heat Cen p C= Caxp) with Kalb Ramond potential ®xr for five dimensional Gauss-Bonn¢ coupling a(5) = aig). Here we use the ADM mass parameter y = 0.028 in the Planckian unit.
FIG. 15: Variation of specific heat Cen p C= Caxp) with Kalb Ramond potential ®xr for five dimensional Gauss-Bonn¢ coupling a(5) = aig). Here we use the ADM mass parameter y = 0.028 in the Planckian unit.
After establishing the existence of the two Ehrenfests equations for known volume expansivity and isothermal com- pressibility here our prime objective is to determine the order of the phase transition. For this, we shall analytically check the applicability of the two Ehrenfest’s equations at the points of discontinuity $;,,Vi. In the context of equilibrium statistical mechanics such points are identified to be the “critical points” or “transition points” . Now taking infinitesimal reversible change in both sides of equation(6.9) we get:
After establishing the existence of the two Ehrenfests equations for known volume expansivity and isothermal com- pressibility here our prime objective is to determine the order of the phase transition. For this, we shall analytically check the applicability of the two Ehrenfest’s equations at the points of discontinuity $;,,Vi. In the context of equilibrium statistical mechanics such points are identified to be the “critical points” or “transition points” . Now taking infinitesimal reversible change in both sides of equation(6.9) we get:
To maintain the inherent assumptions of equilibrium statistical mechanics here we truncate the series at p = 1. Consequently we have: Now making use of equation(7.2) and equation(7.8) we get:
To maintain the inherent assumptions of equilibrium statistical mechanics here we truncate the series at p = 1. Consequently we have: Now making use of equation(7.2) and equation(7.8) we get:
which are usual consistency relations satisfied in the context of equilibrium statistical mechanics. These are commonly known as Widom scaling hypothesis. On the other hand for Enthalpy and Black hole mass the above mentioned consistency conditions are modified as: which are obviously a new results in the context of charged Kalb Ramond black hole induced by the 5D Gauss-Bonnet coupling (a(5)).
which are usual consistency relations satisfied in the context of equilibrium statistical mechanics. These are commonly known as Widom scaling hypothesis. On the other hand for Enthalpy and Black hole mass the above mentioned consistency conditions are modified as: which are obviously a new results in the context of charged Kalb Ramond black hole induced by the 5D Gauss-Bonnet coupling (a(5)).
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Condensed Matter PhysicsStatistical MechanicsThermodynamicsQuantum GravityPhase TransformationsSoft Condensed Matter PhysicsQuantum Field TheoryString TheoryHigh Energy PhysicsStatistical Thermodynamics
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