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O iginal A icle
THERMODYNAMIC STUDY OF SPECIFIC BLACK HOLE INSPIRED BY
GENERALIZED ENTROPY
D . Ranjan P asad
Assis an P o esso , Depa men o Physics,
S. N. S. R. K. S. College Saha sa. B. N. Mandal Uni e si y Madhepu a, Biha
Manusc ip ID:
IJAAR-130115
Abs ac :
The e m “black hole” is o a e y ecen o igin. I was coined in
1969 by he Ame ican scien is John Wheele , as a g aphic desc ip ion o
an idea ha da es o a leas wo hund ed yea s, o a ime when he e we e
wo heo ies abou ligh - he co puscula heo y and he wa e heo y. We
now ha e a mo e concise heo y in his espec - he wa e-pa icle duali y
heo y o quan um mechanics. John Michell, in 1783, was he i s o
p o ide a b eak h ough, h ough his pape , Philosophical ansac ions o
he oyal socie y o London. In his pape , he poin ed ou ha a s a ha
was su icien ly massi e and compac will ha e such a s ong g a i a ional
ield ha e en ligh canno escape-any ligh emi ed om he su ace o he
s a will be d agged back by he s a ’s g a i a ional a ac ion be o e i
could ge e y a . Michell sugges ed ha he e migh be la ge numbe o
s a s jus like his. Such en i ies which o m black oids in space came o
be known as black holes. Black hole he modynamics is a ich subjec ,
s addling bo h he classical and quan um aspec s o g a i y. The
he modynamic cha ges o a black hole such as en opy and empe a u e,
while in insically quan um in na u e, a e ela ed o classical a ibu es
such as ho izon a ea and su ace g a i y. Indeed, i was conside ing he
classical esponse o a black hole o in alling ma e ha led Ba deen,
Ca e , and Hawking o make he link be ween black hole a ia ions and
laws o he modynamics. Mo e ecen ly, ou unde s anding o black hole
he modynamics and he in e p e a ion o he a ious pa ame e s has also
been imp o ing. The ou line o his pape is as ollows: Fi s , we e iewed
he a ious he modynamic aspec s o non- o a ing and uncha ged black
holes like Reissne - No ds öm, BTZ and Ba deen black holes. The
indings we e hen inco po a ed o in es iga e, de i e and compa e
he modynamical pa ame e s o Schwa zschild and Schwa zschild-AdS
black holes. This s udy examines he he modynamic ea u es o he Van
de Waals black hole wi hin he con ex o h ee pa ame e s o en opy. I
is disco e ed ha he pa ame e s o black hole, including mass and
empe a u e, de ia e om hose ha a e de i ed wi h he Bol zmann-Gibbs
pa adigm.
ISSN: 2347-7075
Impac Fac o – 8.141
Volume - 13
Issue - 1
Sep embe - Oc obe 2025
Pp. 94-99
Submi ed:
26 Sep 2025
Re ised:
25 Oc 2025
Accep ed:
28 Oc 2025
Published:
31 Oc 2025
Co esponding Au ho :
D . Ranjan P asad
Quick Response Code:
Websi e: h ps://ijaa .co.in/
DOI:
10.5281/zenodo.17677782
DOI Link:
h ps://doi.o g/10.5281/zenod
o.17677782
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How o ci e his a icle:
D . Ranjan P asad. (2025). The modynamic S udy O Speci ic Black Hole Inspi ed By
Gene alized En opy. In e na ional Jou nal o Ad ance and Applied Resea ch, 13(1), 94–
99. h ps://doi.o g/10.5281/zenodo.17677782
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Fu he mo e, he en opy pa ame e in luences he beha iou o he Gibbs ee ene gy by
in oducing he modynamic ins abili ies, whe eas he emission a e is mos ly a ec ed by en opy
pa ame e s a low equencies. We p o ide exp essions o he modynamic quan i ies like p essu e,
empe a u e, hea capaci y, Gibbs ee ene gy, Helmhol z ee ene gy, and iso he mal comp essibili y.
We in es iga e he phase s uc u e o hese solu ions by analysing hei hea capaci y and Gibbs ee
ene gy. This in es iga ion also examines he c i ical beha io and phase ansi ions o black hole.
Addi ionally, we obse e bo h local and global s abili y o black hole in he canonical ensemble o a
a ie y o pa ame e alues. We also in es iga e he e ec o he en opy pa ame e and he spa si y
o Hawking adia ion on black hole e apo a ion.
Keywo ds: Black Hole, The Exp essions Fo Tempe a u e (T), Gibbs F ee Ene gy (G), Helmhol z
F ee Ene gy (F), Hea Capaci y (C).
In oduc ion:
Black holes (BHs), among he
g ea es mys e ious objec s in he cosmos,
a e egions o in ense space ime cu a u e
speci ied by an e en ho izon, a limi behind
which nei he ma e no ligh can escape.
BH he modynamics has ad anced
d ama ically since he 1970s, wi h impo an
inno a ions a ibu able o he e olu iona y
wo k o Bekens ein and Hawking.
Hawking's a ea heo em [1] asse s ha he
o al ho izon a ea o BH canno diminish
o e ime du ing any physical ac i i y ha
complies wi h he laws o classical physics.
This heo em posi s ha BHs e lec
he modynamic cha ac e is ics analogous o
en opy in he modynamic sys ems. Jacob
Bekens ein elabo a ed on his no ion by
sugges ing ha a BH's en opy is di ec ly
p opo ional o he a ea o i s e en ho izon.
The Bekens ein-Hawking en opy illus a es
a signi ican co ela ion be ween he
geome y o BHs and he modynamic
en opy [2]. S ephen Hawking's disco e y o
Hawking adia ion u he e i ied he
compa ison be ween BH he modynamics
and classical he modynamics. Quan um
physics a he e en ho izon in luences his
occu ence, leading o he emission o hea
adia ion om BHs. Thus, BHs possess a
empe a u e, e e ed o as Hawking
empe a u e, ha is in e sely p opo ional o
hei mass [3], [4], [5]. En opy has become
one o he g ea es uni e sal and key
concep s in physics since he de elopmen o
classical he modynamics. Non-ex ensi e
he modynamics sugges s ha en opy is
subjec o con ex and changes ac oss
heo ies, leading o a e-e alua ion o i s
de ini ion and pu pose [6]. Hawking's
adia ion e ealed ha BHs adia e a black
body spec um wi h a empe a u e,
es ablishing he Bekens ein-Hawking
en opy as a key componen o BH
he modynamics. The non-ex ensi eness o
Bekens ein-Hawking en opy emains
enigma ic, as classical he modynamic
en opy is addi i e and ex ensi e. Howe e ,
BH en opy de ies his p inciple, leading o
in e es in al e na i e o mula ions Renyi
and Tsallis en opies [3], [1], which a e non-
addi i e en opy models, help us unde s and
non-ex ensi e sys ems in new ways. Recen
imp o emen s, like Ba ow en opy [2],
Sha ma Mi al [3], and Kaniadakis en opy
concep s [4], [3], conside sol ing
cons ain s in he classical o mula ions.
Ano he ac o o conside is he spa si y o
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Hawking adia ion h oughou he
e apo a ion p ocess [3],. The mean ime
be ween he emission o successi e Hawking
quan a signi ican ly exceeds he na u al
imescale de e mined by he ene gy o he
emi ed quan a. This esea ch deepens ou
unde s anding o de ia ions om he usual
Bol zmann-Gibbs o malism by expanding
on ea lie wo k ha in es iga ed en opy
co ec ions and hei e ec s on BH
he modynamics. Co ec ions o he en opy-
a ea ela ion o quan um luc ua ions a e
discussed in [2]. This concep is ele an o
ou analysis o he e ec s o hese
co ec ions on he he modynamic p ope ies
o he an de Waals (VDW) BH, namely i s
mass- empe a u e connec ion and phase
diag am. As sugges ed in [3], he ac ha
en opy co ec ions could po en ially ha e a
signi ican e ec on BH s abili y and phase
ansi ions while s udying comp essibili y,
Gibbs and Helmhol z ee ene gy, and hea
capaci y. Addi ionally, he spa se Hawking
adia ion is explained by [3], which leads us
o wonde i he e is an e ec o en opy
changes on he emission spec um and he
dynamics o VDWBH e apo a ion. Ou
esul s con i m s onge he modynamics
compa ed o s anda d BH models such as
Schwa zschild and Ke -AdS. The VDWBH,
unlike he Schwa zschild BH, unde goes
phase ansi ions like luid sys ems, bu wi h
he ansi ions being induced by en opy
ins ead o g a i a ional o ces alone. Ou
analysis indica es ha he dominan
he modynamic pa ame e s, like en opy and
p essu e play an impo an ole in
es ablishing he phase s uc u e o a
VDWBH. Howe e , angula momen um
plays he cen al ole in cha ac e izing phase
ansi ions and he s abili y o Ke -AdS
BHs. The esul s con ibu e g ea ly o BH
he modynamics by illus a ing en opy-
based gene alized models enhance
heo e ical BH physics wi h co ec
modi ica ions in e ms o adia i e
p ope ies, mass, empe a u e, and ee
ene gy.
Law o he Black Hole:
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Along wi h Hawking's impo an
inding o BH adia ion, demons a ed a
undamen al link wi hin BHs and
he modynamic p inciples. The disco e y
ha BHs a e go e ned by he modynamic
p inciples ma ked a signi ican change in
heo e ical physics [4]. Nume ous s udies
ha e explo ed he complex ela ionship
be ween BHs and he modynamics,
expanding on he undamen al indings. BH
he modynamics ocuses on he s udy o
phase ansi ions. Da ies was he i s o
iden i y phase ansi ions by measu ing hea
capaci y a speci ic poin s [1]. The Hawking-
Page ansi ion ma ks a shi in he ee
ene gy o a BH, indica ing a change in
he modynamic s a es [2]. Ex ensi e
esea ch has been conduc ed on ansi ions
be ween non-ex emal and ex emal BH
s a es. Analogies wi h VDW sys ems ha e
p o ided de ails ega ding BH phase
ansi ions and c i ical conduc , leading o a
be e unde s anding o hei he modynamic
p ope ies The compa isons ha e enhanced
he analysis o BH he modynamics,
p o iding a mo e comp ehensi e iew o
c ucial p ocesses and phase ansi ions. The
cosmological cons an is ans o med in o a
dynamic a iable in he ex ended phase
space app oach o s udying BHs in AdS
space, simila ly o he way p essu e is
ea ed in classical he modynamics. This
me hod allows o he de ini ion o no el
he modynamic concep s, such as olume,
which leads o a mo e comp ehensi e and
exhaus i e analysis o BH he modynamics.
Resea che s can in es iga e c i ical
phenomena, phase ansi ions, and s abili y
condi ions in BHs by employing he
ex ended phase space amewo k. The
connec ion among he he modynamics o
cha ged AdS BHs ha a e o a ing and he
VDW luid is only quali a i e, e en hough
i is in e es ing. I 's no always easy o see
BH quan i ies like cha ge o angula
momen um a e linked oge he o luid
pa ame e s a and b because he
co esponding equa ions o s a e change.
Un o una ely, his limi a ion con inues o
exis in all o he challenging BHs,
po en ially in highe o lowe dimensions,
ha exhibi quali a i e VDW beha io . A
comp ehensi e summa y can be ob ained by
consul ing he mos ecen inspec ion [72]. I
is in e es ing o in es iga e he
he modynamic cha ac e is ics o VDWBH,
especially wi h ela ion o en opy, which
de i es om hei ascina ing simila i ies o
classical he modynamic sys ems. Hawking
adia ion and BH en opies a e among he
many ascina ing and exci ing aspec s o BH
he modynamics ha ha e gained a lo o
a en ion ecen ly. I may o e a pa h owa d
a mo e p o ound unde s anding o quan um
g a i y. Inco po a ing he cosmological
cons an Λ in o he undamen al law o BH
mechanics gi es us a consis en me hod o
analyse he he modynamics o he phase
spaces o he (an i)-de Si e spaces, o be
mo e p ecise. BHs in de Si e spaces a e
he modynamically uns able and p oduce
he mal adia ion a a ying empe a u es.
They can iden i y he ela ionship be ween
he he modynamic quan i ies on he wo
ho izons, which could lead o an e icien
me hod o s udy he he modynamic
quan i ies in de Si e space ime since
s udying he ho izon he modynamics o he
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de Si e BHs independen ly makes he
sys em he modynamically uns able. A lo o
esea ch has been done on he c i ical
beha iou o he e ec i e he modynamic
quan i ies in de Si e space ime because o
hese a gumen s. A s udy o he e ec i e
he modynamics o he o a ing de Si e
space ime has shown ha i beha es in a
way like ha o VDWs [3], [4]. The
e ec i e he modynamic quan i ies o he
BH space ime ha e ecen ly been he subjec
o an in es iga ion. We ea he
cosmological cons an in BH chemis y as a
he modynamic p essu e, es ablishing a
connec ion be ween BHs and
he modynamic sys ems. This amewo k
ex ends he in es iga ion o BH phase
ansi ions, aligning wi h he VDW equa ion
o s a e o eal gases. BH he modynamics
a e also changed by polyme iza ion in he
loop quan um g a i y domain, which
combines classical BH singula i ies wi h
quan um ones. The in e sec ion o hese wo
concep s enables he s udy o c i ical
phenomena and phase ansi ions o
quan um-co ec ed BHs in ela ion o egula
he modynamic sys ems. The e ec s o
polyme iza ion signi ican ly imp o e he
mic oscopic s uc u e o VDWBH, as hese
en i ies exhibi a phase ansi ion compa able
o ha o liquid-gas beha io . The
obse a ional cons ain s p o ided by BH
shadows, which ha e he po en ial o e eal
signa u es o quan um g a i y, keep
conside able impo ance. The inco po a ion
o BH chemis y wi h polyme iza ion yields
new insigh s in o he ealms o space- ime,
he modynamics, and quan um g a i y [2].
The cu en s udy highligh s signi ican
di e ences in he he modynamics o BHs
when compa ed o he Bol zmann-Gibbs
amewo k. The [3] s udy ha analyzes he
quan um Oppenheime -Snyde model
unde lines quan um “co ec ions” o he
mass ene gies o he BH which al e s hei
he modynamic p ope ies, including mass,
Hawking empe a u e, hea capaci y, and
s abili y. In he same way, he p esen
indings on he VDWBH show ha many o
he he modynamic a ibu es like p essu e,
empe a u e, Gibbs ee ene gy, Helmhol z
ee ene gy, and comp essibili y de ia e
om he classical he modynamic beha io .
Bo h s udies ocus on he ole o quan um
phenomena and some o he s a is ical
mechanics in he e olu ion and s abili y o a
BH. O he aspec s, like he adia ion o BHs,
phase ansi ions, and s abili y om he
he modynamics poin o iew, a e also
s udied ex ensi ely, suppo ing he heo y o
BHs and he applica ion o adi ional
he modynamic app oaches. This pape
concen a es on he he modynamic
p ope ies o VDWBH, inspi ed by he
connec ion be ween BH chemis y and he
h ee-pa ame e en opy s a is ics in BH
physics. This s udy is o ganized as ollows:
Sec ion 2 desc ibes he VDWBH and
essen ial he modynamic quan i ies in he
h ee-pa ame e en opy domain. In Sec ion
3, we add ess he modynamic s abili y,
including local s abili y, Gibbs ee ene gy,
Helmhol z ee ene gy, he equa ion o s a e
ac o , he comp essibili y ac o , emission
ene gy a e, and he spa si y o Hawking
adia ion du ing he BH e apo a ion p ocess.
Sec ion 4 summa izes and p o ides
conclusions om he s udy discussed he ein.
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Discussion:
Sec ion Snippe s:
The h ee-pa ame e en opy in he pe spec i e o Van de Waals black hole geome y
Conclusion:
The VDWBH's he modynamics a e
in es iga ed in his s udy h ough he use o
a h ee-pa ame e en opy amewo k. The
exp essions o empe a u e (T), Gibbs ee
ene gy (G), Helmhol z ee ene gy (F), hea
capaci y (C), and emission ene gy a e in
e ms o ho izon adius ha e been ob ained
by assigning he nega i e cosmological
cons an as a he modynamic p essu e.
Re e ences:
1. D. Sama e al.Nucl. Phys.
B(2023)S. Noji i e al.Phys. Le . B
(2022)
2. S. Noji i e al. Phys. Le . B (2023)
S. Noji i e al. Phys.
3. Le . B (2023) N. Kuma Gen.
Rela i . G a i . (2024)
4. X. Huang e al.J. High Ene gy
Phys.(2015) V.G. Czinne e al.
Phys. Le . B (2016)
5. J.D. Ba ow Phys. Le . B (2020) Q.
Huang e al. Ann. Phys. (2019) D.V.
ingh e al. Nucl. Phys. B (2022)
