Phase transitions and critical phenomena: the (micro...
Transcript of Phase transitions and critical phenomena: the (micro...
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Phase transitions and critical phenomena: the (micro)physicist
viewpoint
Francesca GulminelliLPC-Ensicaen et Université,
Caen, France
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Menu1. Phase transitions in physical systems
– Concepts and definitions– Classical theory: Landau
2. First order transitions– Phase coexistence– Extension to many densities
3. Second order and critical phenomena– Divergence of the correlation length– Scale invariance and renormalization
4. Extension to finite systems – Precursor of phase transitions– Thermodynamic anomalies– Frustration and ensemble inequivalence
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1 - Phase transitions in physical systems
• Concepts and definitions
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What is a phase transition?
• A small variation of a control parameterinduces a dramaticqualitative modification of the system properties.
SL
G
temperature
Pres
sure
Water phase diagram
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A paradigm in statistical mechanics: lattice models
• Site variables: ni, piconfiguration
or microstate
• solid state physics: ferro-para transition, spin glasses• molecular systems: liquid-gas transition• site and bond percolation: critical phenomena• Lattice QCD: deconfinement phase transition …..
Ising , Heisenberg, Lattice Gas, Potts, spherical…...
• Couplings among sites εij
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A small dictionary• Statistical ensemble (macrostate):
exhaustive ensemble of all microstatessatisfying some given conditions (constraints)
• Observable (extensive variable B) : expectation value of an operator on a microstate
• Conjugated field (intensive variable β) : what is shared among microstates
• Control parameter : constraintcharacterizing the ensemble (intensive variable or conservation law)
• Order parameter : average extensive variable changing among differentphases
H(k)=Σεijninj energyi,j∈k
N(k)=Σni particle numberi∈k
T = f (<H(k)>k) Temperatureμ = f (<M(k)>k) Chemical potential
‚ ‚ …{k}=
Control parameter
Ord
erpa
ram
eter
Phase I Phase II
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Information theory in a nutshell
• Controlled variables: (constraints)
• Statistical entropy:(lack of Information)
• Microstate probability: • Partition sum: (sum on the physical realizations of the system)
• Equations of state: (relations between extensive and intensive)
An equilibrium is the statistical ensemble of microstates whichmaximizes the statistical entropy under a given set of constraints
( ) ( )( )
logn nn
S p p I S= − = −∑
0: 1, , : 1, ,jAr j mβ = =… …
( )
( ) 0( : )( , )
n
nj j
Bn A A j
Z A e ββ −∑= ∀
= ∑( )( )( ) 1 ( ) 0nBn n
j jj
p Z e A Aβ δ−∑−= −∏
log ( )B Zβ∂ β< >= − A Sα = ∂
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What is a phase transition? • A small variation of a
control parameterinduces a dramaticqualitative modification of the system properties
⇒ « accident » in an Equation of State
⇒ Non-analitycity of the partition sum
( ) logB Zββ ∂< > = −
SL
G
temperature
Pres
sure
Water phase diagram(β)
(<Β>)
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Definition of a phase transition (thermo limit)• Non-analyticity of the
thermodynamical potentialG=-TlogZ(β)
• Order of the transition: discontinuity (or divergence) in
• First order:discontinuous
∞→N
logn
Zβ∂
logB Zβ∂< >= −βλ
βλ
lnZ(β,λ)
<Β>
( )
( )( )
nBn
Z e ββ −∑= ∑
SL
G
Temperature β−1
Pre
ssu
re λ
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T
Order of Phase
Transitions
Ex: Liquid-Gas
Β = Energy,Volume
cross-over
T
E
C
P
TPhase diagram
Β discontinuous
Order 1
T
E
δΒ divergent
Order 2
T
E
Β continuous
LT
G
G
LC
L
G
L
G1st order
2nd order
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Example: the ferro-para phase transition in solid-state physics
• Ising model
2nd order at T=Tc
1st order at h=0i
iM s= ∑
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1 - Phase transitions in physical systems
• The classical mean field Landau theory
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Landau theory: First order
• Constrained entropy or thermodynamic potential G/T
• Series developement around the transition point Β−ΒΙΙ => Β = 0
• two minima of equal depth => a first order transition
Lev Landau, 1936
2 3 41 1 1log2 3 4
Z B a B b B c Bβλ λ λ λβ= − + + + +…
( )log cZ S B L S Bβλβλ β λ− = − + + = − = min
0 100
I
II
Ba a a B
Bβ
λβ
⎧ <= + → = ⎨
>⎩
β,λ control variablesΒ order parameter
-Sc
β>0ΒΙ
Β
-Sc
β=0
Β
-Sc
β<0ΒΙΙ
Β
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Landau theory: second order
• Constrained entropy or thermodynamic potential G/T
• Series developement around the transition point Β−ΒΙΙ => Β = 0
• Symmetry Β −Β => a second ordertransition
Lev Landau, 1936
2 3 41 1 1log2 3 4
Z B a B b B c Bβλ λ λ λβ= − + + + +…
( )log cZ S B L S Bβλβλ β λ− = − + + = − = min
Β
-Sc
λ<λc
( ) ( )1/ 200 c c
aa a Bc
λ λ λ λ= − → = ± −
β,λ control variablesΒ order parameter
Β
-Sc
λ>λc
Β
-Sc
λ=λc
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Fluid phase diagram in the Landau picture
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2 – First order transitions
• Phase coexistence
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Equilibrium and instabilities
Β
-Sc
BSβ = ∂
Β
( )cS B S Bβ β− = − + Thermo potential G/T
( )0cS Bδ β β= ⇔ =
All Β are stationary points but a convex entropygives a multi-valued equation of state!=> Which one is the equilibrium state?
02 <cSδ Stability condition2
2 0cSB B
β∂ ∂= >
∂ ∂Instable region (spinodal)
2 cSB vt B
δδ
∂= − ∇
∂Diffusion equation(Ginzburg-Landau)
β
/ log logV cp T Z z s= −∂ = − =
phas
e I
phase II
instab
ility
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What happens in the spinodal region ?
Heat (Calories per gram)
Tem
pera
ture
(deg
rees
)
?
?
S
L
G
temperatureP
ress
ure
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What happens in the spinodal region ?
Heat (Calories per gram)
Tem
pera
ture
(deg
rees
)
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Β
-Sc
BSβ = ∂
ΒΒΙ ΒΙΙ21 VVV +=( ) ( )1 2
c c I c IIS V s b V s b= +
( )1 21cs s sα α= + − VV /1=α
tangent construction
= α +(1-α) Β ΒΙΙΒΙ
Phase Separation:Maxwell construction
β
/ log logV cp T Z z s= −∂ = − =
pt
βt
( ) 'I
II
bI II
tb
b b s dbβ − = ∫I II
tb b
s sb b
β∂ ∂= =
∂ ∂
same ordinate same derivative
(ΒΙ,ΒΙΙ) coexistence zone
BΙ BII
V1
V2
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2 – First order transitions
• extension to many densities b=Β/V
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Extension to many densities:from Maxwell to Gibbs
• Tangent hypersurface: equality of all fields
• Multi-dimensionalspinodal
I IIb b
s sb b
β∂ ∂= = ∀
∂ ∂
k
k k
s sb bb
Cs s
b b b
⎛ ⎞∂ ∂⎜ ⎟∂ ∂∂⎜ ⎟⎜ ⎟=⎜ ⎟
∂ ∂⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠
2 2
211
2 2
21
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Illustration in 2D
• the saddle: one single order parameter
⇒ direction of phase separation
(LG: B1=V Β2=E)
• The well: two orderparameters
(He3-He4: B1=Δ Β2=X3/X4)
Β2
Β1
EZ
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Example: nuclear matter
• Two possible phases L+G
• A region of first order phase transitions
• A line of second order phase transitions
Motivations 1/5ρp
ρn
β = 0.1 MeV −1
0.090.0850.080.0750.07
fm-3
fm-3
Sly230a NM phase diagram
C.Ducoin et al NPA 2006
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Phase coexistence in nuclearmatter
ρ p(fm
-3)
ρp
ρn0
0 0.05
0.05
0.1
0.1
0
-0.1
-0.2
s/β
(MeV
.fm-3
)
ρn (fm-3)
Β/V=(ρp ,ρn )
Gibbs construction
Sc= S-β E+βμtρ
,I II
s s n pρ ρ
βμρ ρ∂ ∂
= = =∂ ∂
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Spinodal of nuclear matterρ p
(fm-3
)
ρn (fm-3)
Sc= S-β E+βμtρ
• negative curvature => instability•1 eigen-value <0 => 1 spinodal
ρp
ρn0
0 0.05
0.05
0.1
0.1
0
-0.1
-0.2
s/β
(MeV
.fm-3
)
n pn
p n p
s s
Cs s
ρ ρρ
ρ ρ ρ
⎛ ⎞∂ ∂⎜ ⎟∂ ∂∂⎜ ⎟= ⎜ ⎟∂ ∂⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠
2 2
2
2 2
2
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Distillation• Distillation: generic
phenomenon of phase separation with >1 densities – ordered phase more
symmetric – desordered phase more
asymmetric
1
ρ p(fm
)
ρn (fm-3)
-3ρn /ρp =cteρ n
= ρ p
Sc= S-β E+βμtρ
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(3) Second order transitions and critical phenomena
•Divergence of the correlationlength
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Back to Landau : Φ2−Φ4
• Constrained entropy or thermo potential (Symmetry B -B )
• series developement around the transition point B = 0 λ=λc
• Second order transition => divergent susceptibility
2 41 1log min2 4cZ S a B cBβλ
λ− = − = + + =…
( )0 ca aλ λ λ= −
( )1/ 20c
aBc
λ λ= −
( )
( )
1
0
1
0
121
c c
c c
aB
a
λ λ λ λχ
β λ λ λ λ
−
−
− <∂
= =∂
− >
<B>
λc λ
χ
λλcb
-Sc
λ<λc
λ=λc
β,λ control variablesΒ order parameter
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Divergences and Fluctuations
• Distribution of the orderparameter Β
• Β Fluctuationdiverges at the critical point
( ) ( )( ) 1 ( );nBn nnn
p Z e p B p B Bββ β δ−−= = −∑
2
2
log ZB β
β β∂∂
= =∂ ∂
2Bσ=
12
2cS
Bχ
−⎛ ⎞∂
=⎜ ⎟∂⎝ ⎠
Ornstein-Zernike-Fisher 1964
mIsing h=0
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Fluctuations and Correlations• Distribution of the order
parameter BΒ Fluctuationdiverges at the critical point
• Correlation fonction ex: B = Ν => b=ρ(r)
• Correlation length ξcorrelations over all length scalesat the critical point
• Critical opalescence:Fourier response : radiatedintensity. q=2ksinθ/2 momentumtransfer
2
2
log ZB β
β β∂∂
= =∂ ∂
1 ;n nB Bn n
p Z e Z eβ ββ β
− −−= = ∑2Bσ=
( )( )' ( ) ( ')
' ( , ')
drdr b r b b r b
drdr G r r
= − < > − < >
=
∫∫
( )/( ) dsG s e s ηξ − − +−∝ 2
12
2cS
Bχ
−⎛ ⎞∂
=⎜ ⎟∂⎝ ⎠
( ) ( )∫ ⋅= sGesdqI sqi
Ornstein-Zernike-Fisher 1964
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Liquid sodium hexafluoride
T=42 oC T=44 oC
T=43.9 oC
p=37 atm
L Gtemperature
pres
sure
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Divergent fluctuations and spontaneous symmetry breaking
• Critical point: bifurcation• B=0 with β=0 (=> logZ
preserving B reflection symmetry):
⇒ spontaneous symmetrybreaking
Β
-Sc
λ<λc
λ>λc m
Distribution of the orderparameter for the Ising model
without external field
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Spontaneous symmetry breaking
• Critical point: bifurcation• B=0 with β=0 (=> logZ
preserving B reflection symmetry):
=> spontaneous symmetrybreaking
• ferro/para phase transition• superconductivity below Tc• Higgs mechanism in the Standard Model
Β
-Sc
λ<λc
λ>λc
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(3) Second order transitions and critical phenomena
•Scale invariance and renormalization
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Universality and criticalexponents
• 2 « universal » criticalexponents
γχ −−∝ TTc( )cB T T β∝ −
Only 2 exponents are independent
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Universality and criticalexponents
• 2 « universal » criticalexponents
γχ −−∝ TTc( )cB T T β∝ −
Ising n=1 XY n=2 Heisenberg n=3
Liquid/gas superfluidity ferromagnetismβ = 0.325
γ = 1.231
0.346 0.369
1.315 1.392
Only depend on the dimensionality of B
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Critical exponents and scaleinvariance
• x,y control parameters• scaling hypothesis (Widom 1965) ln Z(λ ax, λ by) = λ ln Z(x,y)• => only two independent critical exponents• definition of β,γ β=(1− b)/ a γ=( 2b-1)/ a
Scaling hypothesis scale invariance self-similarity
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Scale invariance and renormalization
Renormalization: at the critical point the configuration is self-similar: it will stay invariant. This is due to the divergence of the correlation length.
Wilson,1975
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• Observable setKi intensive variables (couplings) {s}=s1…sN degrees of freedom
• Scale change: re-iterated ad libitum
• Partition sum:preserved structure
• Fixed point and its attraction domain:
• Correlation length:infinite if at the fixed point
{ }( ) { }( )sfKsfKHH 2211 +=−= β
( ) ( )(1)log , log / , ( )dZ N K Z N x K Ng K= +
* * *x nx cK R K R K K⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦
( ) ( )( )*
nc nx c
n
K x R K
x K
ξ ξ
ξ
⎡ ⎤= ⎣ ⎦
= = ∞( ) 0* ≠Kξ
[ ](1) (1)( )xH R H H K= =
/ / dx N N x→ →
( 1) ( )i ixH R H+ ⎡ ⎤= ⎣ ⎦
( ) ( )i ixK R K+ ⎡ ⎤= ⎣ ⎦
1
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Scale invariance and renormalization
K1= βJ1
K*
line of criticalpoints
β<βc
K2=
βJ2
β>βc
line explored by a physical system varying β−1=Τ
If the flow of coupling constants converges to a non trivial fixedpoint:• the initial system is critical• the partition sum is scale
invariant on the critical line• all systems (observations)
converging towards the same fixed point have the same critical behavior => the same exponents
• this only depends on theobservable set => on the orderparameter
Kc
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(4) Extension to finite systems
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Phase transitions in finite systems
• Z analytic: transition rounded
• Ensemble inequivalence
<Β>
β
( )
1
( )nN
B
n
Z e ββ −∑
=
= ∑β
<Β>=>Intensive β
controlled
=>Extensive Βcontrolled
B Sβ = ∂
log ( )B Zβ∂ β< >= −
<Β>
β
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(4) Extension to finite systems
•Precursor of phase transition
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Finite size scaling
• Scale invariance at the critical point
• Scaling Hypothesis(Widom 1965)
⇒ Finite size scaling* Critical temperature* susceptibility
x c
c
Y λ λε ελ
∞−
∞
−∝ == ∞L
( )ξ λ λ −= −/ ( ) x
cY f L L
L f cteL Y cte
ξξ
⇒ =⎧⎨ ⇒ =⎩ ∼
1/ ( )c cT L T L ν−− ∝
ξ→ ∼const. Lβ ν χ2L ( ) γν ξ
−∝ >
1 , ( )cL t L T T L( ) βν ξ∝ <
21 , ( )cL t L T T L
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Finite size scalingF.Gulminelli, et al. PRL 2008
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The Yang-Lee theoremC.N. Yang, T.D.Lee Phys.Rev.1952
Phase transition: thermo potential non analytic for N → ∞
( )
1( )
nNB
nZ e ββ −∑
=
⎛ ⎞=⎜ ⎟⎝ ⎠
∑( )β− log Z
Origin of non-analyticities
γℜ ≠→ ∞
: ( ) 0log /
ZZ N N
analytic
β
( )i Zγ β η γ= + = 0 N
dn cNdη →∞⎯⎯⎯→
P.Borrmann, O.Mulken, J.Harting, Phys.Rev.Lett 84 (2000) 3511-3514A. Schiller et al Phys.Rev.C 66 (2002) 024322
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The pairing transition in finitenuclei
T.Sumaryada, A.Volya 2007
0 , 0
ˆ ˆ ˆ ˆ2 i i ij i ji i j
H n G p pε +
> >
= −∑ ∑
Second
order
Β=fraction of paired particles
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Yang-Lee theorem and Bimodalities
Partition sum and probability distribution
Normal distribution: no zeros
Bimodal distribution P = P1+P2:double saddle point approximation
πη +=
Δ(2 1)
ki k
B →∞Δ ⎯⎯⎯→ ΔNB N b
Β1Orderparameter
( )P Bβ
( ) ηγ β β
−= ∫ i BZ Z dBP B e
ß
η
ΔΒOrder parameter
K.C. Lee Phys Rev E 53 (1996) 6558Ph.Chomaz, F.G. Physica A (2002)
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The fragmentation transition of finite nuclei
, ,i j
i j in n i j c i
i j i j iij
q q pH n n n
r mε α= + +∑ ∑ ∑
2
2
F
L
G
G.Lehaut, F.G., O.Lopez,PRL 2009
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(4) Extension to finite systems
•Thermodynamic anomalies
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Illustration with the LG model
• Closest neighbors attractive coupling
• Belongs to the LG universality class
• Temperature (β) controlled: smooth equation of state β(Β)
216 particles
β=
Β= energy (MeV)
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216 particles
Ener
gy D
istr
ibut
ion
1
10
100
0.1
Liquid Gas
• Energy distribution at a fixed temperature
Most probable
Β= energy (MeV)
β=
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216 particles
Ener
gy D
istr
ibut
ion
d
1
10
100
0.1
Liquid Gas
• Energy distribution at a fixed temperature
• Discontinuity of the most probable
Most probableβ=
Β= energy (MeV)
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Energy distribution
1
10
100
0.1
Liquid Gas
• Bimodal distribution
• Energy distribution at a fixed temperature
• Discontinuity of the most probable
216 particles
Most probable
Β= energy (MeV)
β=
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Bimodality and ensemble inequivalence
1
10
100
0.1
Liquid Gas
216 particles
Β= energy (MeV)
β=
( ) ( )
( )( ) 1
1 ( )
nn B
n B
n
p Z e
p B Z B B e
ββ β
ββ β δ
− −
− −
=
= −∑
• β controlled
logB Zβ β∂< >= −
Energy distribution
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1
10
100
0.1
Liquid Gas
216 particles
Β= energy (MeV)
β=
( ) ( )
( )( ) 1
1 ( )
nn B
n B
n
p Z e
p B Z B B e
ββ β
ββ β δ
− −
− −
=
= −∑
( )( ) ( ) ( )log
log ( ) log
n n nB B B
n nS p p B B
p B Z Bβ β
δ
β
= − = −
= + +
∑ ∑( )( ) 1 ( )n n
B Bp S B Bδ−= −
( )logB B BS p Bββ β= ∂ = ∂ +
• Β controlled
• β controlled
logB Zβ β∂< >= −
Energy distribution
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Bimodality and ensemble inequivalence
1
10
100
0.1
Liquid Gas
216 particles
Β= energy (MeV)
β=
A first order phase transition occuring at the thermo limit:
⇒ the distribution of the order parameteris bimodal in the finite system
⇒ The underlying entropy has a convexintruder
=> The EoS β<−>Β is not the same if the controlled variable is extensive or intensive
⇒ The thermodynamics is not unique⇒ In particular thermodynamics
anomalies occur in the extensive ensemble: Negative heat capacity!
Energy distribution
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• Interfaces cannot be neglected in a finite system
• Phase separation has an extra energy and entropy cost
• Convexities persist• Local dishomogeneities
arise
entropy
How can a convex entropy be stable ???
ΒΙ Φmin ΒΙΙΦmax ΒEnergy or density
S(Β)βΒ
lnP(Β)
D.J.Wales R.S.Berry 1994K.Binder D.P.Landau 1984
( )( ) S B BP B e ββ
−∝
( )S B BsV V
⎛ ⎞≠ ⎜ ⎟⎝ ⎠
( ) 21 1 sss αα −+<
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Frustration and dishomogeneous phases
• Frustration is a genericphenomenon in physics
• It occurs whenever matter issubject to opposite interactions on comparable length scales
• Global variations of Β are replaced by local variations of Β
=>Phase coexistence is quenched=>dishomogeneous phases arise
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Frustration and dishomogeneous phases
• Example I: fluidtransition in a finitesystem
=>Phase coexistence is quenched=> dishomogeneous phases arise
β(Β)
ΒΙ ΒΙΙ
β(Β)
ΒΙ ΒΙΙ
2/3( ) vol surfE N e N e N= − +
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Frustration and dishomogeneous phases
• Example II: crust structure of a neutron star (matter of neutrons, protons in an homogeneous electron background)
A.Raduta, F.G. 2011
( ) ( )2 2/3( )coul p e nuclE r N rα ρ ρ α ρ= − −
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Frustration and dishomogeneous phases
• Example II: crust structure of a neutron star (matter of neutrons, protons in an homogeneous electron background)
Intensive controlled
Extensive controlled
A.Raduta, F.G. 2011
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Summary1. Phase transitions in physical systems
– Concepts and definitions– Classical theory: Landau
2. First order transitions– Phase coexistence– Extension to many densities
3. Second order and critical phenomena– Divergence of the correlation length– Scale invariance and renormalization
4. Extension to finite systems – Precursor of phase transitions– Thermodynamic anomalies– Frustration and ensemble inequivalence