The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H...
-
Upload
simon-barnett -
Category
Documents
-
view
220 -
download
0
Transcript of The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H...
![Page 1: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/1.jpg)
The thermodynamical limit of abstract composition rules
T. S. Bíró, KFKI RMKI Budapest, H
• Non-extensive thermodynamics
• Composition rules and formal log-s
• Repeated rules are associative
• Examples
Talk by T. S. Biro at Varoš Rab, Dalmatia, Croatia, Sept. 1. 2008.
![Page 2: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/2.jpg)
From physics to composition rules
Entropy is not a sum: correlations in the common probability
Energy is not a sum: (long range) interaction inside the system
Thermodynamical limit: extensive but not additive?
![Page 3: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/3.jpg)
Short / long range interaction vs. extensivity
dN
dN
dconst
N
E
rrrg
rdrrgnEN
E
d
N
dN
/1
1
ln
.
)()(
)()(
short range
long range
![Page 4: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/4.jpg)
Short / long range correlation vs. extensivity
dN
dN
dconst
N
S
rrgrg
rdrgrgnSN
S
d
N
dN
/1
1
ln
.
)(ln)(
)(ln)(
short range
long range
![Page 5: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/5.jpg)
Typical g( r ) functions
ga
scrystal strin
gy
liqu
id
![Page 6: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/6.jpg)
From physics to composition rules
Abstract composition rule h(x,y)
anomalous diffusion
multiplicative noise
coupled stochastic equations superstatistics
fractal phase space filling
chaotic dynamics
power-law tailed distributions extended logarithm and exponential
Lévy distributions
![Page 7: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/7.jpg)
From composition rules to physics
Abstract composition rule h(x,y)
h(x,0) = x, general rules
associative (commutative) rules
Formal logarithm L(x)
equilibrium distribution: exp ּס L
generalized entropy: L̄ ¹ ּס ln
![Page 8: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/8.jpg)
Thermodynamical limit:
repeated rules
![Page 9: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/9.jpg)
Thermodynamical limit: repeated rules
N-fold composition
![Page 10: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/10.jpg)
Thermodynamical limit: repeated rules
recursion
00x
![Page 11: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/11.jpg)
Thermodynamical limit: repeated rules
use the ‘ zero property ’ h(x,0) = x
![Page 12: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/12.jpg)
Thermodynamical limit: repeated rules
The N limit: scaling differential equation
![Page 13: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/13.jpg)
Thermodynamical limit: repeated rules
solution: asymptotic formal logarithm
Note: t / t_f = n / N finite ratio of infinite system sizes (parts’ numbers)
![Page 14: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/14.jpg)
Thermodynamical limit: repeated rules
The asymptotic rule is given by
Proof of associativity:
![Page 15: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/15.jpg)
Thermodynamical limit: associative rules are attractors
If we began with h(x,y) associative, then it has a formal logarithm, F(x).
Proportional formal logarithm same composition rule!
![Page 16: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/16.jpg)
Boltzmann algorithm: pairwise combination + separation
With additive composition rule at independence:
Such rules generate exponential distribution
![Page 17: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/17.jpg)
Boltzmann algorithm: pairwise combination + separation
With associative composition rule at independence:
Such rules generate ‘exponential of the formal logarithm’ distribution
![Page 18: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/18.jpg)
Entropy formulae from canonical equilibrium
Equilibrium: q – exponential, entropy: q - logarithm
All composition rules generate a non-extensive entropy formula in the th. limit
![Page 19: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/19.jpg)
Entropy formulae from canonical equilibrium
Dual views: either additive or physical quantities
Associative composition rules can be viewed as a canonical equilibrium
![Page 20: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/20.jpg)
Rules and entropies
h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln
composition rule formal logarithmformal
exponentialequilibrium distribution
entropy formula
ffLS
eeZf
ttL
xxL
yxyxh
f
EEL
eq
ln)(ln
)(
)(
),(
11
)(
1
Gibbs, Boltzmann
![Page 21: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/21.jpg)
Rules and entropies
h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln
composition rule formal logarithmformal
exponentialequilibrium distribution
entropy formula
a
ffLS
aEeZf
etL
axxL
axyyxyxh
a
f
aEL
eq
ata
a
1
11
/)(
11
1
)(ln
1
1)(
1ln)(
),(
Pareto, Tsallis
![Page 22: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/22.jpg)
Rules and entropies
h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln
composition rule formal logarithmformal
exponentialequilibrium distribution
entropy formula
a
f
EEL
eq
a
a
aaa
fLS
eeZf
ttL
xxL
yxyxh
a
/111
)(
/11
/1
ln)(ln
)(
)(
),(
Lévy
![Page 23: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/23.jpg)
Rules and entropies
h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln
composition rule formal logarithmformal
exponentialequilibrium distribution
entropy formula
c
c
f
ff
c
cvcvvL
eq
xcxcc
cLS
eZf
ctctL
xL
cxy
yxyxh
/2
/2
1
111
2/
/1/1)(
1
2
2
)(ln
)/tanh()(
ln)(
1),(
Einstein
![Page 24: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/24.jpg)
Classification based on h’(x,0)
• constant addition Gibbs distribution
• linear Tsallis rule Pareto
• pure quadratic Einstein rule rapidity
• quadratic combined Einstein-Tsallis
• polynomial multinomial rule rational
function of power laws
![Page 25: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/25.jpg)
Interaction and kinematics
• Assume that the interaction energy can be
expressed via the asymptotic, free individual
energies. This gives an energy composition
law as:
),(212112EEUEEE
![Page 26: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/26.jpg)
Interaction and kinematics
Let U depend on the relative momentum
squared:
)cos(2
)cos(2
)(
2
21
2
21
2
2
1112
baQ
ppmEEQ
QUEEE
![Page 27: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/27.jpg)
Interaction and kinematics
Average over the directions gives for the kinetic
energy composition rule (with F’ = U)
2/122/1221
2121
21
41
2112
2111
)(
)0()2()2(
)22()22(
Km
Km
b
KKb
KKKKma
UmKUmKU
baFbaFKKK
![Page 28: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/28.jpg)
Interaction and kinematics
In the extreme relativistic limit (K >> m) it gives
)0(4
)0()4(
21
21
2112U
KK
FKKFKKK
![Page 29: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/29.jpg)
Rule and asymptotic rule
h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln
Original composition rule
Asymptotic formal logarithm
formal exponential
equilibrium distribution
entropy formula
1)(ln
1
1)(
)0(1ln)(
0)0()(),(
111
/)(
11
1
aaf
aEL
eq
ata
a
fLS
aEeZf
etL
GaaxxL
GxyGyxyxh
Pareto - Tsallis
![Page 30: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/30.jpg)
Interaction and kinematics
In the non-relativistic limit (K << m) the angle
averaged composition rule has the form:
)()()0(1)0,(
)0()()(
4
)2()2(),(
32
2xUxUUxh
UyUxU
xy
xyyxFxyyxFyxyxh
x
non-trivial formal logarithm non-additive entropy formula
![Page 31: The thermodynamical limit of abstract composition rules T. S. Bíró, KFKI RMKI Budapest, H Non-extensive thermodynamics Composition rules and formal log-s.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bf8a1a28abf838c8a8a7/html5/thumbnails/31.jpg)
Summary
• Extensive composition rules in the
thermodynamical limit are associative and
symmetric, they define a formal logarithm, L
• The stationary distribution is exp o L, the
entropy is related to L inverse o ln.
• Extreme relativistic kinematics leads to the
Pareto-Tsallis distribution.