six lectures about (advanced) statistical...
TRANSCRIPT
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LECTURES ABOUT
(ADVANCED) STATISTICAL
PHYSICS
T.S.Biró, MTA Wigner Research Centre for Physics, Budapest
Lectures given at: University of Johannesburg, South-Africa,
November 26 – November 29, 2012.
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1. Ancient Thermodynamics (… - 1870)
2. The Rise of Statistical Physics (1890 – 1920)
3. Modern (postwar) Problems (1940 – 1980)
4. Corrections (1950 – 2005)
5. Generalizations (1960 – 2010)
6. High Energy Physics (1950 – 2010)
2
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LECTURE THREE ABOUT
(ADVANCED) STATISTICAL
PHYSICS
T.S.Biró, MTA Wigner Research Centre for Physics, Budapest
Lectures given at: University of Johannesburg, South-Africa,
November 28, 2012.
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GENERALIZATIONS
o Composition Rules
o Associative Limit
o Zeroth-Law Compatibility
o Universal Thermostat Independence
4
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Entropy formulas
• 𝑆 = 𝑙𝑛 𝑁!
𝑁𝑖!𝑖 Boltzmann (permutation)
• 𝑆 = − 𝑃𝑖 𝑙𝑛 𝑃𝑖 Gibbs (Planck)
• 𝑆 = 11−𝑞ln 𝑃𝑖
𝑞 Rényi
• 𝑆 = 1𝑞−1 ( 𝑃𝑖 − 𝑃𝑖
𝑞) Tsallis (Chravda, Aczél, Daróczy,…)
There are (much) more !
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Canonical distribution with Rényi entropy
1q
1
i
)S(Li
iq
i
1q
i
iii
q
i
q
)EE()q1(1
e
1p
Ep
pq
q1
1
maxEppplnq1
1
This cut power-law distribution is an excellent fit to particle spectra in high-energy experiments!
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Canonical distribution with Tsallis entropy
1
1
1
1
)1(
1
1
1
1
max)(1
1
qiq
i
i
q
i
iiii
q
i
q
EqZp
Eq
pqq
Eppppq
This cut power-law distribution is an excellent fit to particle spectra in high-energy experiments!
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Why to use the Tsallis / Rényi entropy formulas?
• It generalizes the Boltzmann-Gibbs-Shannon formula
• It treats statistical entanglement between subsystem and reservoir (due to conservation)
• It claims to be universal (applicable for whatever material quality of the reservoir)
• It leads to a cut power-law energy distribution in the canonical treatment
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Why not to use the Tsallis / Rényi entropy formulas?
• They lack 300 years of classical thermodynamic foundation
• Tsallis is not additive, Rényi is not linear
• There is an extra parameter q (mysterious?)
• How do different q systems equilibrate ?
• Why this and not any other ?
• It looks pretty much formal…
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Again the Zeroth Law: (E1,…)=(E2,…)
2
2
12
1
12
1
1
12
2
12
2
2
12
1
1
12
12
2
2
12
1
1
12
12
SS
S
E
ES
S
S
E
E
0dEE
EdE
E
EdE
0dEE
SdE
E
SdS
Factorization = ? 10
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The temperature for non-additive composition rules
const.)E,E(C
)E,E(C
)S,S(H
)S,S(H
SCBAHGFSCBAHGF
)E(SE
E
S
S)E(S
E
E
S
S
2121
2112
2121
2112
212212112121121221
22
1
12
2
12
11
2
12
1
12
11
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The temperature for non-additive composition rules
222
22
11
11
1
22
22
22
22
22
11
11
11
11
11
2121
2112
2121
2112
T
1
)E(L
)S(L
)E(L
)S(L
T
1
)E(S)E(A
)E(B
)S(G
)S(F)E(S
)E(A
)E(B
)S(G
)S(F
1const.)E,E(C
)E,E(C
)S,S(H
)S,S(H
12
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Generalized absolute temperature
dE)E(B
)E(A)E(L
dS)S(G
)S(F)S(L
)E(L
)S(L
T
1
13
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Admissible composition rules
)S(L)S(L)S(L
)LL(S
SL
SL
SSF
GS
SF
G
HSSFG
1S
SFG
1H
22111212
2112
12
2
12
1
12
22
2
12
11
1
2112
221
12
112
12
14
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Admissible composition rules
)E(L)E(L)E(L
)LL(E
EL
EL
EEA
BE
EA
B
CEEAB
1E
SAB
1C
22111212
2112
12
2
12
1
12
22
2
12
11
1
2112
221
12
112
12
15
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Example: Tsallis entropy
Sa1lna
1)S(L
)Sa1()Sa1()Sa1(
SSaSSS
2112
212112
16
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Heterogeneous equilibrium
1Sa1Sa1a
1S
Sa1lna
1Sa1ln
a
1Sa1ln
a
1
212112 aa
22
aa
11
12
12
22
2
11
1
1212
12
17
Tsallis - Nauenberg dispute
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Nonextensive thermodynamics: a summary
i iiiii
21122112
maxw)E(Lww)S(L
)E(L
)S(L
T
1
)E(L)E(L)E(L)S(L)S(L)S(L
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Non-additive: Tsallis - Entropy
a/1
i
eq
i
Rényii
a1
i
ii
a1
iTsallis
)E(a1Z
1w
Swlna
1)S(L
wwa
1S
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Power law factorizes Energy is non-additive
212112
a/1
2
a/1
1
a/1
12
a/1eq
EEˆaEEE
Eˆa1Eˆa1Eˆa1
Eˆa1Z
1w
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Abstract Composition Rules
)y,x(hyx
EPL 84: 56003, 2008
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Repeated Composition, large-N
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Scaling law for large-N
)0,x(hdy
dx :N
)0,x(hyxx
)0,x(h)y,x(hxx
yy,0x),y,x(hx
2
1n2n1nn
1nn1n1nn
N
1nn0n1nn
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Formal Logarithm
)yy(x
),y(xx),y(xx
)x(L)x(LL)x,x(
)x(Ly
21
2211
21
1
21
x
0 )0,z(h
dz
2
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Asymptotic rules are associative and attractors among all rules…
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Asymptotic rules are associative
).),,((
))()()((
))()(()(
)))()((,()),(,(
1
11
1
zyx
zLyLxLL
zLyLLLxLL
zLyLLxzyx
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Associative rules are asymptotic
),(),(
)0(
)(
)0(
)()(
)(
)0(
))0,((
)0()0,(
)()(
)()()),((
0
2
yxhyx
xdz
zxL
xxhxh
yyhh
yxyxh
x
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Scaled Formal Logarithm
xxL
axLa
xL
axLa
xL
LL
a
a
)(
)(1
)(
)(1
)(
0)0(,1)0(
0
11
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Deformed logarithm
)(ln)/1(ln
))(ln()(ln 1
xx
xLx
aa
aa
Deformed exponential
)()(/1
))(exp()(
xexe
xLxe
aa
aa
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Formal composition
rules
Differentiable rules
Asymptotic rules
Associative rules
Formal Logarithm
1. General rules repeated infinitely asymptotic rules
2. Asymptotic rules are associative
3. Associative rules are self-asymptotic
4. For all associative rules there is a formal logarithm mapping it onto
the simple addition
5. It can be obtained by scaling the general rule applied for small
amounts
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Examples for composition rules
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Example: Gibbs-Boltzmann
WlnkSW/1ffor
flnfS
)E(eZ
1f
x)x(L
1)0,x(h,yx)y,x(h
eq
2
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Example: Rényi, Tsallis
ényi Rln1
1)(
Tsallis )(1
)1(1
),1ln(1
)(
1)0,(,),(
11
/
2
q
nona
aqa
non
a
eqa
fq
SL
ffa
S
aEZ
faxa
xL
axxhaxyyxyxh
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Example: Einstein
),(),(
)tanh()(
)tanh(Ar)(
1)0,(
1),(
1
22
2
2
yxhyx
c
zczL
c
xcxL
cxxh
cxy
yxyxh
c
c
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Example: Non associative
yxyx
zazL
a
xxL
axh
yx
xyayxyxh
c
c
),(
)1()(
1)(
1)0,(
),(
1
2
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Important example: product class
axyyxyx
a
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
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Important example: product class
axyyxyx
a
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
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Relativistic energy composition
)cos1(EE2Q
)EE()pp(Q
)Q(UEE)E,E(h
21
2
2
21
2
21
2
2
2121
( high-energy limit: mass ≈ 0 )
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Asymptotic rule for m=0
)0(U2/
eq
2
E)0(U21Z
1f
xy)0(U2yx)y,x(
)0(Ux21)0,x(h
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Physics background:
rdQ
d
r
1
dQ
d)0(U
0Q
2
0Q
2
2
2
q > 1
q < 1
Q²
α
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Derivation as improved canonical
• Derivation:
– Microcanonical entropy maximum for two
– Reservoir-independent temperature: the best one can
– Which composition rule leads to higher order agreement (cannot be the simple addition)
– Make the choice of the additive L(S) universal separation constant = 1 / heat capacity
– Result: L(S) is Tsallis entropy, S is Rényi entropy
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Derivation: formulas
• 𝑆 = − 𝑃𝑖 ln 𝑃𝑖 →𝑖 𝐿 𝑆 = 𝑃𝑖 L(−ln𝑃𝑖)𝑖
• 𝐿 𝑆 𝐸1 + 𝐿 𝑆 𝐸 − 𝐸1 = 𝑚𝑎𝑥.
• 𝛽1 = 𝐿′ 𝑆(𝐸1) ∙ 𝑆
′ 𝐸1
= 𝐿′ 𝑆 𝐸 − 𝐸1 ∙ 𝑆′ 𝐸 − 𝐸1
Taylor: 𝑆 𝐸 − 𝐸1 = 𝑆 𝐸 − 𝐸1𝑆′ 𝐸 +⋯
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Derivation: formulas
𝛽1 = 𝐿′ 𝑆(𝐸) ∙ 𝑆′ 𝐸
− 𝐸1 𝑆′(𝐸)2𝐿′′ 𝑆 𝐸 + 𝑆′′ 𝐸 𝐿′(𝑆 𝐸 )
The content of the square bracket be zero!
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Derivation: formulas
𝛽 = 𝐿′ 𝑆(𝐸) ∙ 𝑆′ 𝐸
and the content of the bracket [ ] is zero:
𝐿′′(𝑆)
𝐿′(𝑆)= −𝑆′′ 𝐸
𝑆′ 𝐸 2 = 1
𝐶(𝐸)
Universal Thermostat Independence:
𝑳′′(𝑺)
𝑳′(𝑺)= 𝒂
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Derivation: formulas
The solution is:
𝐿 𝑆 =𝑒𝑎𝑆 − 1
𝑎
This generates
𝑳 −𝒍𝒏 𝑷𝒊 = 𝟏
𝒂 𝑷𝒊−𝒂 − 𝟏
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Derivation: Tsallis entropy
The canonical principle becomes:
𝟏𝒂 𝑷𝒊𝟏−𝒂 −𝑷𝒊 − 𝜷 𝑷𝒊 𝑬𝒊 − 𝜶 𝑷𝒊 = 𝒎𝒂𝒙.
The entropy with q = 1-a
𝑺𝑻𝒔𝒂𝒍𝒍𝒊𝒔 = 𝟏
𝒒 − 𝟏 (𝑷𝒊 − 𝑷𝒊
𝒒 )
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Derivation: Rényi entropy
The Rényi entropy is the original one,
but the Tsallis entropy is to be maximized canonically
𝑺𝑹é𝒏𝒚𝒊 = 𝑳−𝟏( 𝑺𝑻𝒔𝒂𝒍𝒍𝒊𝒔 ) =
𝟏
𝟏 − 𝒒 𝒍𝒏 𝑷𝒊
𝒒
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Improved Canonical Distribution
• 𝑃𝑖 = 𝑍1−𝑞 + 1 − 𝑞
𝛽
𝑞 𝐸𝑖
1
𝑞−1
• Expressed by the reservoir’s physical parameters via using our results:
• 𝑃𝑖 = 1
𝑍1 +
𝑍−1/𝐶
𝐶−1 𝑒𝑆/𝐶 1
𝑇 𝐸𝑖
−𝐶
Check infinite C limit!
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Improved Logarithmic Slope
•1
𝜏= −
𝑑
𝑑𝐸𝑖 𝑙𝑛 𝑃𝑖 = 𝑇0 +
1
𝐶 𝐸𝑖
• Quark coalescence:
𝐶𝑚𝑒𝑠𝑜𝑛= 2 𝐶𝑞𝑢𝑎𝑟𝑘
𝐶𝑏𝑎𝑟𝑦𝑜𝑛= 3 𝐶𝑞𝑢𝑎𝑟𝑘
• 𝑇0 = 𝑇𝑒−𝑆/𝐶 𝑍1/𝐶 1 − 1 𝐶
Check infinite C limit!
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Infinite heat capacity limit
• 𝑃𝑖 → 1
𝑍 𝑒−𝐸𝑖/𝑇𝑓𝑖𝑡 with
• 𝑻𝒇𝒊𝒕 = 𝟏
𝜷 = 𝑻 𝐥𝐢𝐦
𝑪→∞ 𝒆−𝑺/𝑪
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Finite subsystem corrections to infinite heat capacity limit
• 𝑇1 = 𝑇 1
1+1 ∙ 𝐸1𝐶𝑇 + ⋯
traditional S-expansion
• 𝑇1 = 𝑇𝑒−𝑆/𝐶
𝑒𝑆(𝐸1)/𝐶
1+0 ∙ 𝐸1𝐶𝑇 +𝛼 ∙
𝐸12
𝐶2𝑇2 + ⋯ Our expression
Traditional: T1 < T, falling in E1; Ours: T1 < T, but rising in E1 !
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Gaussian approximation
• Deviations from S=max equilibrium are traditionally considered as Gaussian:
• P ∆𝐸 = 𝑒𝑆 𝐸1 +𝑆 𝐸−𝐸1−∆𝐸 ≈
𝑒−𝑆′ 𝐸−𝐸1 ∆𝐸+
1
2 𝑆"(𝐸−𝐸1) ∆𝐸
2 ≈
∝ 𝑒−1
𝑇 ∆𝐸−
1
2𝐶𝑇2 ∆𝐸2
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Gaussian approximation
• After Legendre transformation also fluctuates as Gaussian:
• P ∆𝛽 ∝ 𝑒
− 𝐶𝑇2
2∆𝛽2 + ⋯
• Thermodynamic ”uncertainty” minimal
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Gaussian approximation and beyond
Beta fluctuation Particle spectra :
log lin
1 / T
𝒆−𝜷𝝎
C T
Boltzmann-Gibbs
Gauss
Euler
Euler
Gauss
Boltzmann-Gibbs
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Summary figure
1 / E
1 / C
BG
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Summary figure
1 / E
1 / C
BG
Physical point, found
Linear scaling: extensive
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Summary figure
1 / E
1 / C
1-q
BG
Physical point, found by fitting q
to the best averages
Linear scaling: extensive
Tsallis formula
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Summary figure
1 / E
1 / C
1-q
BG
Physical point, found by fitting q
to the best averages
Linear scaling: extensive
Anomalous scaling: non-extensive
Tsallis formula
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Summary figure
1 / E
1 / C
1-q
BG
Physical point, found by fitting q
to the best averages
Linear scaling: extensive
Anomalous scaling: non-extensive
A realistic reservoir model
Tsallis formula
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Summary figure
1 / E
1 / C
1-q
BG
Physical point, found by fitting q
to the best averages
Linear scaling: extensive
Anomalous scaling: non-extensive
Black hole
A realistic reservoir model
Tsallis formula
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