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.
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
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.
GENERALIZATIONS
o Composition Rules
o Associative Limit
o Zeroth-Law Compatibility
o Universal Thermostat Independence
4
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 !
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!
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!
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
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…
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
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
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
Generalized absolute temperature
dE)E(B
)E(A)E(L
dS)S(G
)S(F)S(L
)E(L
)S(L
T
1
13
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
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
Example: Tsallis entropy
Sa1lna
1)S(L
)Sa1()Sa1()Sa1(
SSaSSS
2112
212112
16
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
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
Non-additive: Tsallis - Entropy
a/1
i
eq
i
Rényii
a1
i
ii
a1
iTsallis
)E(a1Z
1w
Swlna
1)S(L
wwa
1S
Power law factorizes Energy is non-additive
212112
a/1
2
a/1
1
a/1
12
a/1eq
EEˆaEEE
Eˆa1Eˆa1Eˆa1
Eˆa1Z
1w
Abstract Composition Rules
)y,x(hyx
EPL 84: 56003, 2008
Repeated Composition, large-N
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
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
Asymptotic rules are associative and attractors among all rules…
Asymptotic rules are associative
).),,((
))()()((
))()(()(
)))()((,()),(,(
1
11
1
zyx
zLyLxLL
zLyLLLxLL
zLyLLxzyx
Associative rules are asymptotic
),(),(
)0(
)(
)0(
)()(
)(
)0(
))0,((
)0()0,(
)()(
)()()),((
0
2
yxhyx
xdz
zxL
xxhxh
yyhh
yxyxh
x
Scaled Formal Logarithm
xxL
axLa
xL
axLa
xL
LL
a
a
)(
)(1
)(
)(1
)(
0)0(,1)0(
0
11
Deformed logarithm
)(ln)/1(ln
))(ln()(ln 1
xx
xLx
aa
aa
Deformed exponential
)()(/1
))(exp()(
xexe
xLxe
aa
aa
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
Examples for composition rules
Example: Gibbs-Boltzmann
WlnkSW/1ffor
flnfS
)E(eZ
1f
x)x(L
1)0,x(h,yx)y,x(h
eq
2
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
Example: Einstein
),(),(
)tanh()(
)tanh(Ar)(
1)0,(
1),(
1
22
2
2
yxhyx
c
zczL
c
xcxL
cxxh
cxy
yxyxh
c
c
Example: Non associative
yxyx
zazL
a
xxL
axh
yx
xyayxyxh
c
c
),(
)1()(
1)(
1)0,(
),(
1
2
Important example: product class
axyyxyx
a
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
Important example: product class
axyyxyx
a
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
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 )
Asymptotic rule for m=0
)0(U2/
eq
2
E)0(U21Z
1f
xy)0(U2yx)y,x(
)0(Ux21)0,x(h
Physics background:
rdQ
d
r
1
dQ
d)0(U
0Q
2
0Q
2
2
2
q > 1
q < 1
Q²
α
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
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 +⋯
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!
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
Derivation: formulas
The solution is:
𝐿 𝑆 =𝑒𝑎𝑆 − 1
𝑎
This generates
𝑳 −𝒍𝒏 𝑷𝒊 = 𝟏
𝒂 𝑷𝒊−𝒂 − 𝟏
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 )
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
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!
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!
Infinite heat capacity limit
• 𝑃𝑖 → 1
𝑍 𝑒−𝐸𝑖/𝑇𝑓𝑖𝑡 with
• 𝑻𝒇𝒊𝒕 = 𝟏
𝜷 = 𝑻 𝐥𝐢𝐦
𝑪→∞ 𝒆−𝑺/𝑪
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 !
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
Gaussian approximation
• After Legendre transformation also fluctuates as Gaussian:
• P ∆𝛽 ∝ 𝑒
− 𝐶𝑇2
2∆𝛽2 + ⋯
• Thermodynamic ”uncertainty” minimal
Gaussian approximation and beyond
Beta fluctuation Particle spectra :
log lin
1 / T
𝒆−𝜷𝝎
C T
Boltzmann-Gibbs
Gauss
Euler
Euler
Gauss
Boltzmann-Gibbs
Summary figure
1 / E
1 / C
BG
Summary figure
1 / E
1 / C
BG
Physical point, found
Linear scaling: extensive
Summary figure
1 / E
1 / C
1-q
BG
Physical point, found by fitting q
to the best averages
Linear scaling: extensive
Tsallis formula
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
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
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
