Is the entropy at the liquid-gas critical point of pure fluids proportional to a master dimensionless constant ?

0.41 MB·English

by  Yves Garrabos

Europhysi
s Letters PREPRINT
6
0 Is the entropy at the liquid-gas
riti
al point of pure (cid:29)uids
0
2 proportional to a master dimensionless
onstant?

n
a Yves Garrabos
J
Equipe du Super
ritique pour l'Environnement, les Matériaux et l'Espa
e - Institut de

5 Chimie de la Matière Condensée de Bordeaux - Centre National de la Re
her
he S
ien-

ti(cid:28)que - Université Bordeaux I - 87, avenue du Do
teur S
hweitzer, F 33608 PESSAC
]
h Cedex Fran
e.
c
e
m
PACS. 64.60.-i (cid:21) General studies of phase transitions.
- PACS. 05.70.Jk (cid:21) Criti
al pointphenomena.
t
a PACS. 64.70.Fx (cid:21) Liquid-vapor transitions.
t
s
.
t
a
m
Abstra
t. (cid:21) From a minimal set made of four s
ale fa
tors de(cid:28)ned at the liquid-gas
riti
al
- pointofapure(cid:29)uid,andoneadjustableparameterwhi
ha

ountsforparti
lequantume(cid:27)e
ts,
d
wedemonstratehereamastersingularbehaviorofthe
orrelationlengthfortheone-
omponent
n
o (cid:29)uidsub
lass,usinganasymptoti
s
aledilatationofthephysi
al(cid:28)elds. Su
hmasteΦr4behavior
c observed within the preasymtoti
domain is in
onformity with the renormalized d=3 Field
[ Theorypredi
tions at large
orrelation length s
ale of the(cid:29)u
tuatingorder parameter, forthe

omplete universality
lass of the symmetri
al uniaxial 3D-Ising-like systems. The following
1

onsequen
es are dis
ussed: (i) A
omparison between the
riti
al state of pure (cid:29)uids and the
v
zero-temperature state leads to an intuitive analogy with the (Nerst) third law of thermo-
8
dynami
s, whi
h authorizes spe
i(cid:28)
master form for hypers
aling within the sub
lass of pure
8
0 (cid:29)uids; (ii) A master
onstant value of the non-dimensional
riti
al entropy
an exist for all
1 the pure (cid:29)uids at the short-ranged lengths
ale of the mole
ular intera
tion. From this latter
0 hypothesis, we show that the needed four s
ale fa
tors are the four preferred dire
tions ex-
6 pressing
omplete thermodynami
(linear)
ontinuity
rossing the liquid-gas
riti
al point on
0 the(pressure, volume,temperature) phase surfa
e.
/
t
a
The liquid-gas
riti
al point (CP) of a pure (cid:29)uid is an unattainable single point on the
m p,v ,T
p
d- of freedo(pmresasruere
,ouppalretdi
.leAvosluamme,attteemrpoefraftau
rte,)tphhea
seritsiu
rafla
sNeta,ti=enawNthpi
ch, aTnc,ina(cid:28)nnditnecde=greNVVecs
c
n (the
riti
al density number) of a
riti
al amount of matter (cid:28)lling the volume ,
o is a thermodynami
ally unstable state, due to the diverging
hara
ter of the spontaneous
p V
c
(cid:29)u
tuations of extensive variables ( is dual to ). This
riti
al state appears then asa limit
:
v of∂2tUhermodynami
stability at whi
h all the stability determinants (the se
ond derivatives
Xi ∂Ωi∂Ωj), that were stri
tly positive for any point of the four-dimensional (4-D)
hara
teristi
φ (U,Ω )=0 U(Ω )
U i i
r surfa
e Ω =(S,,Vb,eN
o)me zero for CP (see for example [1℄). S is the total internal
a i
energy, while arethe threeasso
iatednaturalvariables. is thetotalentropy.
A
omprehensiveunderstandingofthediverging
hara
terofthespontaneous(cid:29)u
tuations
of extensive variables, whi
h
hara
terizes the
riti
al behavior of the one-
omponent (cid:29)uid

losetoitsunstable
riti
alstate,
omesfromthe(cid:28)eldtheory(FT)framework(seeforexample
[2℄). This theoreti
al s
heme a

ounts for the in(cid:28)nite degrees of freedom throughout the
(cid:13)

EDPS
ien
es
2 EUROPHYSICSLETTERS
Φ4 (n = 1)
Hamiltonian of the so-
alled d=3 -model of the symmetri
al uniaxial 3D-Ising-like
u > 0 Λ
4 0
systems, with asso
iated
oupling
onstant and (cid:28)nite
uto(cid:27) wave number . The
t
sele
ted pair of relevant s
aling (cid:28)elds, made of the thermal (cid:28)eld weakly (cid:29)u
tuating and the
h
ordering(cid:28)eld morestrongly(cid:29)u
tuating,takesexa
tzero-valueattheisolatednon-Gaussian
{t=0;h=0}
(Wilson-Fisher)(cid:28)xedpoint . Atthis (cid:28)xed point, the(cid:29)u
tuationsofthe1(s
alar)
m=hΦi h ξ ∼L=(V)d ∼∞
order-parameter(OP) ,
onjugatedto ,
anrea
hin(cid:28)nitesizen o.
T = T
c
This hypotheti
al situation at is then only expe
ted for an unphysi
al pure (cid:29)uid
V → ∞ N → ∞
of in(cid:28)nitne =sizeN(= (v )−)1and in(cid:28)nitennum=berNo→f∞parti
le≡s (1 ), but with (cid:28)nite number
density V p , su
h that c V→∞ CP vp,c. Asymptoti
ally
lose to this
t (cid:0) h (cid:1)

riti
al point, i. e. for small values of and , power law behaviors of universal features
are expe
ted whatever the sele
ted system. So that, the set of system-dependent parameters
whi
h
hara
terizesea
hone-
omponent(cid:29)uid ismade from i) the
riti
alparameters,su
has
T p n
c c c
, ,and ,ii)theinverse(
Λut)o−(cid:27)1wavenumberwhi
h
hara
terizesadis
retestrutureofthet
0
(cid:29)uid parti
les with spa
ing , and iii) the two-s
ale fa
tors whi
h relate analyti
ally
h
and totherespe
tivephysi
al(cid:28)eldspropertoea
hsystem[3℄. Startingfromthisasymptoti
des
ription of the two-s
aleuniversalityin the
lose vi
inity of CP, we havepostulated in [4℄,
that the
hara
teristi
set of (cid:29)uid-dependent parameters
orresponds to the minimal set of
p,v ,T
p
measured
riti
alparametersneeded to lo
alize CP on thenormalized (i.e. 3D) phase
surfa
e. Here normalizaNtionreferto parti
le properties forstandarvd t=herVmodynami
swritten
for a
onstant amount of matter. Our parti
le notation, i.e. p N, uses small letters
p Qmin = T ;v ;p ;γ′ = ∂p
withexpli
itsubs
ript . Theminimalsetreads c (cid:26) c p,c c c (cid:16)∂T(cid:17)vp,c(cid:27)CP [4,5℄.
We
an then make dimensionless the thermodynami
and
orrelation fun
tions of any one1-
(β )−1 = k T α = kBTc d

omponent (cid:29)uid, using the s
ale fa
tors, c B c for energy unit, and c (cid:16) pc (cid:17)
k α
B c
for lengt1h unit ( is the Boltzmann
onstant). is not dependent of the
ontainer size
L=(V)d αc
[6℄ and takes a
lear physi
al meaning: isvthe=spakBtiTacl extent of the short-ranged
(Lennard-Joneslike)mole
ularintera
tion. Therefore, c,I pc , isthemi
ros
opi
volume
oZft=hep
crvipt,ic
al inteYra
=tioγn′p
ce−ll 1(CIC).Introdu
ingthetwo
hara
teristi
dimensionlessnumber,
Qcmin =kB{Tβc ,;aαnd;Zc;Y }cTc ,(Zlea)d−s1t=o rnewvritethe minimal setin the more
onvenientformZ,
c c c c c . Now, c c c,I is the number of parti
lesthat (cid:28)ll the CIC. c
Y
c
and
anthentaketheirphysi
almeaningoftwo-s
alefa
torstoformulatethedimensionless
master behavior of all the one
omponent (cid:29)uids asymptoti
ally to their CP [5,7℄.
"Master" singular behavior of the one-
omponent (cid:29)uid sub
lass
As a matter of fa
t, asymptoti
master singular behavior of dimensionless potentials and
dimensionlesssizeofOP(cid:29)u
tuationsonlyo∆
τ
u∗rs=wkhenβa(nTa−ppTro)pri∆athe∗s
=aleβdi(lµata−tioµnm)ethod
B c c c p p,c
(SDM) is applied to the physi
al (cid:28)elds, , , and
∆m∗ = (α )d(n−n ) µ N
c c p
.∗ ,∗dual to ∗, is the
hemi
al potential per parti
le. In SDM, the
T H M
(cid:16)renormalized(cid:17) (cid:28)elds qf, qf, and qf, are proportional to the physi
al (cid:28)elds, in
omplete
analogy to the FT framework near the Wilson-Fisher (cid:28)xed point [3℄. For the
omplete one-

omponent (cid:29)uid sub
lass, these renorTm∗al≡izeTd∗(cid:28)=eldYs∆arτe∗now [8℄ de(cid:28)ned by
qf c
H∗ = Λ∗ 2H∗ = Λ∗ 2(Z )−d2 ∆h∗
qf qe qe c
(1)
M∗(cid:0)=Λ(cid:1)∗ M∗ =(cid:0)Λ∗ (cid:1)(Z )d2 ∆m∗
qf qe qe c
Λ∗ = 1 + λ
qe c
In Eqs. (1), a

ounts for quantum e(cid:27)e
ts on the
ut-o(cid:27) parameter for
T ∼= Tc [8℄. Writing λc = λq,f(cid:16)ΛαTc,c(cid:17), [ΛT,c = (2πmphkPBTc)12 is the thermal wavelength at
YvesGarrabos:Istheentropyattheliquid-gas
riti
alpointofpurefluidsproportionaltoamasterdimensionless
onstant?3
T =T h
c P
and isthe Plan
k
onstant℄, leadsto a relativequantum
orre
tionof the rangeof
ΛT,c λ
mole
ularintera
tionatCPproportionaltotheratio αc . Then q,f appearsasanadjustable
numeri
al prefa
tor whi
h in
orporates the quantum parti
le statisti
s.
When
riti
al properties of one standard (cid:29)uid (xenon [9℄) are known, Eqs. (1) permit
to de(cid:28)ne the
onstant amplitude values of the master
riti
al behavior [5℄ in the so-
alled
preasymptoti
domain (PAD) [10℄, where the singular powerlaws expressedat the (cid:28)rst-order
of the Wegner expansion [11℄, are expe
ted to be valid. Conversely, SDM is then able to
estimate all the
riti
al amplitudes appearing in the [two-terms℄ Wegner expansions for any
pure (cid:29)uid [5,9℄.
ξ+ nm T −Tc > 0 K
Figure 1 (cid:21) a)XLeogK-LrogArs
aCleOo2fSF6(inD2O ) as a3Hfuen
tion of (in ), along the
riti
al
iso
hore, for , , , , , αc ,and(βc)−1(seetherespe
tive
olorsintheinsertedTable);
b) dimensionless behaviors in units of and , showing failure of the
lassi
al
orrℓe∗s,p+onding
state s
heme;
) master dimensionleTss∗behavior of the renormalized
orrelation length qf , as a
fun
tion of the dilated thermal (cid:28)eld [see Eq. (2)℄. The arrow indi
ates the order of magnitude
α
c
of the expe
ted extension of the preasymptoti
domain. Ea
h (cid:29)uid length s
ale is given in the
inserted Table.
To demonstrate this impoQrtmanint featuΛre∗where any leading and (cid:28)rst
on(cid:29)uent amplitudes

an be estimated only using c and qe, we are here
on
erned by the
riti
al behavior
ξ+ T
c
of the a
tual
orrelaΛtion length ΛaloΛn∗g t=he1
riti
al iso
hore above [12℄. When the (cid:29)uid

uto(cid:27)wavenumber 0 issu
hthat 0 qe αc [8℄,therenormalized
orrelationlengthwrites
ℓ∗ =Λ ξ = ξ∗ = αξc ℓ∗,+
qf 0 Λ∗qe Λ∗qe. Within PAD, the two-terms master divergen
e of qf reads as
ℓ∗,+ =Z+(T∗)−ν 1+Z(1),+(T∗)∆
qf ℓ h ℓ i (2)
ν =0.6304±0.0013 ∆=0.502±0.002
where and are universal
riti
al exponents [13℄.
Z+ = 0.57 Z(1),+ = 0.385
The leading amplitude ℓ and the (cid:28)rst
on(cid:29)uent amplitude ℓ have

onstant values for the pure (cid:29)uid sub
lass (see [12℄ for detailed analysis and the Refs. [5,7,9℄
Z(1),+ = cte
for amplitude values of standard
riti
al xenon). Here, postulating ℓ for the pure
4 EUROPHYSICSLETTERS
Φ4 (n = 1)
(cid:29)uid sub
lass, leads to
onsider the simpler situation in the d=3 -theory, where the
u > 0
4
starting point for (in usual renormalized traje
tories), is
ertainly very
lose to the
ideal traje
tory between the Gaussian and the Wilson-Fisher (cid:28)xed points [14℄.
Xe Kr Ar CO
2
The published (cid:28)tting resultsof the
orrelationlength measurementsof , , , ,
SF D O 3He
6 2
, ,and ,[15℄havebeenreportedonFigure1ausingdimensionedquantities,whi
h
make
learly distinguishable ea
h spe
i(cid:28)
(cid:29)uid behavior. Figure 1b gives a representation of
t(βhe)d−i1mensioαnless quantities obtained from the
lassi
al theory of
orresponding states, using
c c
and units. The failure of the
lassi
al theory is eviden
ed from the importan
e
3He D O
2
of quantum e(cid:27)e
ts in and mole
ular intera
tion e(cid:27)e
ts in , when
omparison to
Xe
the standard monoatomi
is made. Final representation of the master behavior obtained
from SDM is given on Figure 1
. The s
atter between the data
orrespondsto the estimated
pre
ision (10%) on the determination of ea
h (cid:29)uid
orrelation length.
α
c
Sin
e is a measure of the mean range of intera
tion for
es, Eq. (2) also provides an
ℓe∗a,s+y≫
on1trol of the e(cid:27)e
tive extension of the
riti
al domain where the mandatory
ondition
qf is expe
ted to be validT. ∗T/hen0.0F1igure 1
indi
ates also the order of magnitude of
the PAD length (at least up to ) where Eq. (2) is valid (see also [7,12℄). Su
h
result
on(cid:28)rmsprevioussimilar
on
lusionsbasedon
arefulanalysesof
rossovermodels and
generalized
riti
al (cid:29)uid e.o.s. [5,9,16℄.
Chara
terization of the thermodynami
"CP vi
inity"
To rea
h CP from a thermodynami
approa
h needs to perform an in(cid:28)nite number of
transformationsbetweenin(cid:28)nite number ofnear-
riti
alequilibrium states,leadingto in(cid:28)nite
time to obtain the
riti
al state. By analogy with the Nernst prin
iple (the so-
alled third
law of thermodynami
s), we
an also reformulate the previous senten
e as follows. It is
impossible by any pro
edure, no matter how idealized, to rea
h exa
t
riti
al state of any pure
(cid:29)uid in a (cid:28)nite number of operations. In a (cid:28)nite number of operations (or (cid:28)nite time), it is
then only possible to border the
riti
al point as
lose as possible, and the (cid:28)nal near
riti
al
state at in
ipient equilibrium is imposed by the (cid:28)nite sized
ontainer. For pure (cid:29)uids in
absen
e of external (cid:28)eld, this (cid:28)nite size of the
ontainer is measured by i) (cid:28)nite extensive
V N n − n v − v
c p¯ p¯,c
values of and , (cid:28)xing the (small) mean value of (or , equivalently),
T
proportional to the physi
al OP; ii) the (cid:28)nite intensive value of , that (cid:28)xes the (small)
T −T
c
mean value of , proportional to the independent physi
al thermal (cid:28)eld. Therefore, the
µ −µ p−p
p¯ p¯,c c
(cid:28)nite intensive value of (or , equivalently), proportional to the independent
physi
al ordering (cid:28)eld, is also (cid:28)xed (from thermodynami
s prin
iples). This (cid:28)nite
riti
al
size of the
ontainer governs the natural way for the unstable
riti
al (cid:29)uid to rea
h in
ipient
stabilityonanequilibriumstatea
tedbythisnear
riti
al
ontainer,theso-
alledreservoir in
statisti
alme
hani
s. Forsu
hathermodynami
equilibriumvery
losetotheCP,allthetotal
Π≡{U,H,A,G,J}

hara
teristi
potentials [17℄arehomogeneousfun
tions ofthe (cid:28)rst order
Ω ≡ {S,V,N,T,p,µ }
i p¯
in terms of their three natural variables among , and Euler's thoerem
SdT − Vdp + Ndµ = 0
p¯
applies [1℄. Correlatively, the Gibbs-Duhem equation, , requires
a normalized des
ription leading to the 3-D representation of thermodynami
equilibrium
N V
states. As and are two independent extensive variables, in addition to the standard
normalization per parti
le mentionned above, another equivalent normalizeπd=s
Πhemeno=

uNrs
using densities for a system at
onstant volume. The density notation, i.e. V or V ,
uses small letters.
When the above (cid:28)rst-order (equilibrium) s
heme is used in the "CP vi
inity", the (cid:28)nite

riti
al parameters are basi
ally
onstitued from the non-zero values of appropriate "(cid:28)rst
order" derivatives. On this thermodynami
point of view, all the
riti
al parameters, the
so-
alledpreferred
riti
aldire
tionsin the following,re(cid:29)e
ttopologi
al
ontinuityofthermo-
dynami
sattheCP,dire
tlyasso
iatedtotheproperanalyti

ontinuityofthethreeintensive
YvesGarrabos:Istheentropyattheliquid-gas
riti
alpointofpurefluidsproportionaltoamasterdimensionless
onstant?5
T p µ
p¯
variables , , and when "
rossing CP". For example of our main present
on
ern, the
s = ∂gp¯ s = ∂g
normalizedvalueof p¯,c (cid:16)∂T (cid:17)p,CP [or c (cid:16)∂T(cid:17)p,CP℄,shouldbede(cid:28)nedasoneamongthe
preferreddire
tions. Obviously,the extendedsetmade with all the
riti
aldire
tions(see be-
low), in
ludes our minimal set hypothetized as
ontaining the needed information to des
ribe

riti
alphenomena(i.e. theneededinformationto
al
ulatealltheleadingand(cid:28)rst
on(cid:29)uent
amplitudes of the power law singularities of "se
ond order" derivatives). This raises the fol-
lowingquestionresultingfromthermodynami
equivalen
ebetweenunattainable
riti
alstate
and unattainable absolute zero state: Is the entropy at the exa
t liquid-gas
riti
al point of
any pure (cid:29)uid a
hara
teristi
value of ea
h (cid:29)uid parti
le, related to a master dimensionless

onstant re(cid:29)e
ting universal (cid:29)uid nature approa
hing asymptoti
ally CP?
N V
Close to the
riti
al point, we are mainly
on
erned by the (cid:29)u
tuations of (or ,
alternatively). Whentheenergyandthenumberofparti
lesofa(cid:29)uidin
onta
twithaparti
le
reservoir
an (cid:29)u
tuate, the basi
link between statisti
alme
hani
sand thermodynami
sof a
T µ
p¯
(cid:28)xedvolumeofmatter,where and aretheoperatingvariables
anbe
orre
tlyestablished
J(T,V,µ ) =
p¯
only from the Grand
anoni
al statisti
al distribution. The Grand potential
−pV
isthennaturallysele
tedto
hara
terizetheequilibriumstateofthesystemmaintained
ajt(T
,oµns)ta=ntJvolume. Its equilibriump(sTt,aµte)is
hara
terized by the Grand potential density
p¯ V, whose opposite, i.e. p¯ , gives the
hara
teristi
surfa
e
lose to the CP
s
hematized in (cid:28)gure 2 [18℄. This result
omplements the Canoni
al statisti
al des
ription
A(T,V,N)

onne
tead (toT,tvhe) =HeAlmholtz free energy , where the Helmholtz free energy per
parti
le p p N
hara
terizes the equilibrium state of the system of
onstant amount
of matter in
onta
t with an energy reservoir (a thermostat). To dis
uss di(cid:27)eren
es between
{p(T,v );s (T,v )}
p p p
minimalandextendedsets,needsto
onsiderthee.o.s. pairs, ,asso
iated
a {n(T,µ );s(T,µ )} j(T,µ ) µ ;T
p p p p p
to , or, , asso
iated to . However, only their
ommon
diagram
ontainsthreenewunmeasured
hara
teristi
parameters,inadditiontothemeasured
p;T
ones inµ∗the≡usgu∗al= β dµiagram (seae th=e rjespe+
tgive b=ina−rpyvdiag+ramµs
onstrau∗
ted=inj∗Fi+gugre∗ 2)=:
−Zi)+µp,∗c µp,c c p,c, with p,c p,c p,c c p,c (β )p−,c1and p,c p,c p,c
c p,c; p,c is another energy s
ale fa
tor (in addition to c );
′
x∗ = δc (β )−1 α
ii) p,c kB (obtained without use of c and c), where the preferred dire
tion,
δ′ = ∂gp
c (cid:20)(cid:16)∂T (cid:17)vp(cid:21)CP,
hara
terizesthermodynami

ontinuity in this diagram ;
s∗ = sp,c (β )−1 α
iii) p,c kB (alsoobtained withoutuseof c and c), wherethe
riti
alentropyper
s = − ∂µp > 0 γ′
parti
le p,c (cid:20)(cid:16)∂T (cid:17)p(cid:21)CP is a preferred dire
tion
onne
ted to c (measured on the
′
p;T δ µ ;T
c p
diagsram=, Fγi′gv. 2a−),δa′nd (unmesa∗sur+edx∗on=theY Z −jd∗iagram, F−ijg∗. 2=b)Z, throughout the
relation p,c c p,c c, leading to p,c p,c c c p,c, with p,c c.
v ∂sp
Y Z = (cid:20) (cid:16) ∂v (cid:17)h=hc(cid:21)CP
The produ
t c c kB is independent of the redu
tion pro
ess, then is a
parti
{lZe p,YropZer}ty,
hara
teristi
of ea
h on(eβ-
o)−m1ponenαt (cid:29)uid. De fa
to, the non-dimensional
c c c c c
pair Φ (obtain(jed with,oTu,tvus)e=of0 ajnd ) i(sTa,svso)
i=atedJto the two preferred
dire
tions of jp¯,V=cte p¯,V=cte p , where p¯,V=cte p N V=cte is the Grand
(cid:0) (cid:1)
potential per parti
le of a (cid:29)uidTm;apin;tZain;eYd;aµt∗
o;nss∗tant volume.
Finally, the extended set c c c c p,c p,c CP is the
omplete set from whi
h we
(cid:8) (cid:9)
are able to
al
ulate all the othUer=
hTaSra+
teGris+tiJ

riti
al paramuc,epte=rssusin+g lµinc,epa+riz−edpcvt0h,ecrmo-
dynamu∗i
s.=Fµo∗r e+xasm∗pl−e,Zfrom , we obtain Tc c,p Tc Tc and
then c,p c,p c,p c.
6 EUROPHYSICSLETTERS
µ ;p µ ;T p;T
p p
Figure2(cid:21)Preferreddire
tions
rossing
riti
alpointin (a), (b),and (
)diagramsob-
p(T,µp¯)
tainedfromproje
tionsofthe3D
hara
teristi
surfa
e and
orrespondingisothermal(blue),
isobari
(bla
k), iso
hori
(green), and iso
hemi
al potential (yellow) iso
lines. Non-homogenous
two-phasedomain
orresponds to theliquid-vapor equilibrium(LVE) (red)line. Seealso [18℄.
We
an now refovrmu=latkeB
Trciti
al thermod1ynami
s, at the s
ale of the mi
ros
opi
intera
-
tion
ell of volume c,I pc (cid:28)lled with Zc parti
les, where we expe
t that all the system
1
information is
ontained. Multiplying then the above parti
le relations by Zc, we obtain
j∗ = −1
c,I
s∗ −1 = Y −X
c,I c c
u∗ = s∗ +µ∗ −1 (3)
c,I c,I c,I
x∗
X = c,p Z X Y
where c Zc . As expe
ted, c disappears in the above equations. c, as well as c, are
two
hara
teristi
properties per CIC.
Sin
je∗th=e G−r1and potential (whi
h favorizes the lo
al ordering) takes master value at
rit-
i
ality, c,I , whatever the sele
ted CIC volume, it seems natural to postulate that the
esn∗tro=pycoants
triti
ality(whi
hfaµv∗ori−zeus∗thelo
aldisorder)
analsotakeamaster
onstantvalue,
c,I . From Eqs. (3), c,I c,I is tuh∗en agl∗so≡
onµs∗tanth.∗M=oreu∗gen−erja∗lly,=al1l t+heur∗emain-
ain∗g
=ritji∗
al+frge∗e en=er−g1ies+pµe∗r CIC volume, c,I, c,I c,I, c,I c,I c,I c,I, and
c,I c,I c,I c,I, are also master
onstants, ex
ept a possible
ommon
onstant
YvesGarrabos:Istheentropyattheliquid-gas
riti
alpointofpurefluidsproportionaltoamasterdimensionless
onstant?7
value
orresponding to an energy translation. Finally, the two dimensionless preferred dire
-
X Y
c c
tions and arerelated. TheremainingsetofthQem
hinara
teristi
parameterswhi
h
ontains
c
the
omplete information at the CIC s
ale is well . Our above suggestion
omplements
s =0 T =0
p
the third law of thermodynami
s ( at ), and de(cid:28)nes parti
le entropy and parti
le
Z
c
free energy from (unknown)
onstants of proport(iβon)a−li1ty to at exa
t CP, in
onformity
c
with basi
thermody(nαam)−i
1prin
ipes. Therefore is the unique (cid:29)uid-dependent s
ale
c
fa
tor for energy, as is the unique (cid:29)uid-dependent s
ale fa
tor for length, as initially
postulated in [4℄.
Asa
on
lusion,fromaformulationof
riti
al thermodynami
s inunitsofthepropertiesof
theCICvolume,theaboveanalysisshowsthattheappropriates
aledilatationofthephysi
al
(cid:28)eldsseemsadequatetoobserves
alingoftheirasymptoti

riti
alsingularities(withonlyone
adjustableparameterto a

ountforquantume(cid:27)e
ts). Todes
ribethe(cid:29)uid singularbehavior
withinthepreasymptoti
domain,s
aledilatationsoftwoindependent(cid:28)elds
anthenbeused
as
ontrolledsimpli(cid:28)
ationsoflinear
ombinationsofthree(cid:28)elds[19℄asso
iatedtothedensity
formulation of non-symmetrized thermodynami
potentials at (cid:28)nite distan
e to the CP.
Referen
es
2nd
[1℄ Modell M. and Reid R.C., Thermodynami
sand its Appli
ations, ed., (Prenti
e Hall, New
York) 1983. 4th
[2℄ Zinn-Justin J., Quantum Field Theory and Criti
al Phenomena, ed., (Oxford University
Press) 2002.
[3℄ Wilson K.G., Phys. Rev. B, 4 (1971) 3174; Wilson K.G. and Kogut J.K., Phys. Rep. C, 12
(1974) 75.
[4℄ Garrabos Y., Thesis, Paris (1982).
[5℄ Garrabos Y.,J.Phys.(Paris), 46 (1985)281 (see alsohttps://hal.

sd.
nrs.fr/

sd-00015956 or
http://fr.arxiv.org/abs/
ond-mat/0512347); J. Phys. (Paris), 47 (m19p8,c6) 197.
[6℄ This is notthe
ase of these
ond
riti
al length s
ale de(cid:28)nedby rhoc.
[7℄ Garrabos Y., Le Neindre B., Wunenburger R., Le
outre-Chabot C., and Beysens D., Int. J.
Thermophys., 23 (2002) 997.
[8℄ Garrabos Y., preprint, https://hal.

sd.
nrs.fr/

sd-00015988 or
http://fr.arxiv.org/abs/
ond-mat/0512408.
[9℄ Bagnuls C., Bervillier C., and Garrabos Y., J. Phys.-Lettres, 45 (1984) L-127.
[10℄ Bagnuls C. and Bervillier C., J. Phys.-Lettres, 45 (1984) L-95; Phys. Rev. B, 32 (1985) 7209;
Phys. Rev. E, 65 (2002) 066132; Bagnuls C., Bervillier C., Meiron D.I., and Ni
kel B.G., Phys.
Rev. B, 35 (1987) 3585.
[11℄ Wegner F.J., Phys.Rev. B, 5 (1972) 4529.
[12℄ Garrabos Y., Palen
ia F., Le
outre C., Erkey C. J., Le Neindre B., preprint,
https://hal.

sd.
nrs.fr/

sd-00016105 or http://fr.arxiv.org/abs/
ond-mat/0512456.
[13℄ Guida R. and Zinn-JustinJ., J. Phys.A: Math. Gen., 31 (1998) 8130.
[14℄ Bagnuls C. and Bervillier C., Cond. Matter Phys., 3 (2000) 559.
[15℄ Data and sour
es for (cid:29)uid-dependent
orrelation length measurementsare givenin [12℄.
[16℄ Kiselev S.B. and Ely J.F., J. Chem. Phys., 119 (2003) 8645; and referen
es therein.
H A G J
[17℄ , , , refer to (total) Enthalpy,Helmholtz and Gibbs free energies, and Grand potential.
[18℄ Figure 2 provides
omplete thermodynami
information for a (cid:29)uidmaintained at
onstantvol-
ufrmeeee
nloesregytoptehrepCaPrtia
nledg
pan=bNeGu≡sedµpin,c
(oT∼m,pp)lefmorenat(cid:29)tuoidthoefu
sounaslt3aDntraepmroeusennttaotfiomnaottfetrhe(sGeeibfbors
example, http://www.publi
.iastate.edu/ jolls/homepage.html).
[19℄ seeforexample,KimY.C.,FisherM.E.,andOrkoulasG.,Phys.Rev.E, 67(2003)061506; and
referen
es therein.