thermodynamics of abstract composition rules
DESCRIPTION
Thermodynamics of abstract composition rules. T.S.Biró, MTA KFKI RMKI Budapest. Product, addition, logarithm Abstract composition rules, entropy formulas and generalizations of the Boltzmann equation Application: Lattice SU2 with fluctuating temperature. - PowerPoint PPT PresentationTRANSCRIPT
Thermodynamics of abstract composition rules
• Product, addition, logarithm
• Abstract composition rules, entropy formulas and
generalizations of the Boltzmann equation
• Application: Lattice SU2 with fluctuating temperature
T.S.Biró, MTA KFKI RMKI Budapest
Talk given at Zimányi School, Nov. 30. – Dec. 4. 2009, Budapest, Hungary
Thanks to: G.Purcsel, K.Ürmössy, Zs.Schram, P.Ván
Non-extensive Thermodynamics
The goal is to describe:
• statistical
• macro-equilibrium
• irreversible
properties of long-range correlated (entangled) systems
Non-extensive Thermodynamics
The goal is to describe:
• statistical
• macro-equilibrium
• irreversible
properties of long-range correlated (entangled) systems
Non-extensive Thermodynamics
Generalizations done (more or less):
• entropy formulas
• kinetic eq.-s: Boltzmann, Fokker-Planck, Langevin
• composition rules
Most important: fat tail distributions canonically
Applications (fits)
• galaxies, galaxy clusters• anomalous diffusion (Lévy flight)• turbulence, granular matter, viscous fingering• solar neutrinos, cosmic rays• plasma, glass, spin-glass• superfluid He, BE-condenstaion• hadron spectra• liquid crystals, microemulsions• finance models• tomography• lingustics, hydrology, cognitive sciences
Logarithm: Product Sum
additive extensive
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
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
Entropy formulas, distributions
Boltzmann – GibbsRényiTsallisKaniadakis …
EPJ A 40: 325, 2009
Entropy formulas from composition rules
Joint probability = marginal prob. * conditional prob.
The last line is for a subset
Entropy formulas from composition rules
Equiprobability: p = 1 / N
Nontrivial composition rule at statistical independence
Entropy formulas from composition rules
ppLp
L
bLaLabL
def
1ln)ln()(
ln
))(())(())((
1
1. Thermodynamical limit: deformed log
Entropy maximum at fixed energy
)()(
)()(
)(
)(
fixed ))()((
max))()((
22
2
2
11
1
1
1221
1
1221
1
ESEX
SYES
EX
SY
EEXEXX
SSYSYY
Generalized kinetic theory
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
Generalized Stoßzahlansatz
)(ln)(ln
)(
0
234123412341
jaiaaijffeG
Fp
GGwfDF
General H theorem
function rising monotonica
)(G iff 0
))((4
1
)()(
))((
ij
1234341243211234
11
0
jiS
GGw
fDFFS
fFp
pS
General H theorem: entropy density formula
df)f(ln)f(F)f(
)f(ln))f(F(
2)G(ln)G(
a
a
a
Detailed balance: G = G 12 34
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
yxhyxc
zczL
c
xcxL
cxxh
cxy
yxyxh
c
c
Important example: product class
axyyxyxa
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
Important example: product class
axyyxyxa
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
QCD is
like
this
!
Relativistic energy composition
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:
q > 1
q < 1
Q²
α
Simulation using non-additive rule
Non-extensive Boltzmann Equation
(NEBE) :
• Rényi-Tsallis energy addition rule
• random momenta accordingly
• pairwise collisions repeated
• momentum distribution collected
with Gábor PurcselPRL 95: 162302, 2005
Evolution in NEBE phase space
Stationary energy distributions in NEBE program
x + y x + y + 2 x y
Thermal equilibration in NEBE program
Scaling variable E or X(E)?Károly Ürmössy
Scaling variable E or X(E)?Károly Ürmössy
Microscopic theory in non-extensive approach: questions, projects, ...
• Ideal gas with deformed exponentials
• Boltzmann and Bose distribution
• Fermi distribution: ptl – hole effect
• Thermal field theory with stohastic temperature
• Lattice SU(2) with Gamma * Metropolis method
As if temperature fluctuated…
• EulerGamma Boltzmann = Tsallis
• EulerGamma Poisson = Negative Binomial
max: 1 – 1/c, mean: 1, spread: 1 / √ c
Euler - Gamma distribution
Tsallis lattice EOS
Tamás S. Bíró (KFKI RMKI Budapest) and
Zsolt Schram (DTP ATOMKI Debrecen)
• Lattice action with
superstatistics
• Ideal gas with power-law tails
• Numerical results on EOS
Lattice theory
A =
DU dt w (t) e t A(U) -S(t,U)c
DU dt w (t) e -S(t,U)c
v
Expectation values of observables:
t = a / a asymmetry parametert s
Action: S(t,U) = a(U) t + b(U) / t
Su2 Yang-Mills eos on the lattice with Euler-Gamma distributed inverse temperature: Effective action method
preliminary
with Zsolt Schram (work in progress)
Method: EulerGamma * Metropolis
• asymmetry thrown from Euler-Gamma
• at each Monte Carlo step / only after a while
• at each link update / only for the whole lattice
• meaning local / global fluctuation in space
• c = 1024 for checking usual su2
• c = 5.5 for genuine quark matter
Ratio
e / T4
(e-3p) / T4
Ideal Tsallis-Bose gas
For c = 5.5 we have 1 / a = 4.5 and e ≈ 4 e_0
Summary
• Non-extensive thermodynamics is not only
derivable from composition rules, but it is
realized by QCD interactions in the high-
energy limit and can be seen in heavy-ion
collisions!
Topical Review Issue of EPJ A