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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.
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?
Short / long range interaction vs. extensivity
dN
dN
dconst
N
E
rrrg
rdrrgnEN
E
d
N
dN
/1
1
ln
.
)()(
)()(
short range
long range
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
Typical g( r ) functions
ga
scrystal strin
gy
liqu
id
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
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
Thermodynamical limit:
repeated rules
Thermodynamical limit: repeated rules
N-fold composition
Thermodynamical limit: repeated rules
recursion
00x
Thermodynamical limit: repeated rules
use the ‘ zero property ’ h(x,0) = x
Thermodynamical limit: repeated rules
The N limit: scaling differential equation
Thermodynamical limit: repeated rules
solution: asymptotic formal logarithm
Note: t / t_f = n / N finite ratio of infinite system sizes (parts’ numbers)
Thermodynamical limit: repeated rules
The asymptotic rule is given by
Proof of associativity:
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!
Boltzmann algorithm: pairwise combination + separation
With additive composition rule at independence:
Such rules generate exponential distribution
Boltzmann algorithm: pairwise combination + separation
With associative composition rule at independence:
Such rules generate ‘exponential of the formal logarithm’ distribution
Entropy formulae from canonical equilibrium
Equilibrium: q – exponential, entropy: q - logarithm
All composition rules generate a non-extensive entropy formula in the th. limit
Entropy formulae from canonical equilibrium
Dual views: either additive or physical quantities
Associative composition rules can be viewed as a canonical equilibrium
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
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
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
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
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
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
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
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
Interaction and kinematics
In the extreme relativistic limit (K >> m) it gives
)0(4
)0()4(
21
21
2112U
KK
FKKFKKK
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
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
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.
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