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Unique additive information measures – Boltzmann-Gibbs-Shannon, Fisher and beyond
Peter VánBME, Department of Chemical Physics
Thermodynamic Research GroupHungary
1. Introduction – the observation of Jaynes2. Weakly nonlocal additive information measures3. Dynamics – quantum mechanics4. Power law tails
Microcanonical and canonical equilibrium distributions
5. Conclusions
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1 Entropy in local statistical physics(information theoretical, predictive, bayesian)
(Jaynes, 1957):
The measure of information is unique under general physical conditions.
(Shannon, 1948; Rényi, 1963)
– Extensivity (density)– Additivity
)()()( 2121 fsfsffs
fkfs ln)( (unique solution)
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The importance of being unique:
Robustness and stability: link to thermodynamics.
Why thermodynamics?
• thermodynamics is not the theory of temperature• thermodynamics is not a generalized energetics• thermodynamics is a theory of STABILITY
(in contemporary nonequilibrium thermodynamics)
•The most general well-posedness theorems use explicite thermodynamic methods (generalizations of Lax idea)
•The best numerical FEM programs offer an explicite thermodynamic structure (balances, etc…)
Link to statistical physics: universality (e.g. fluid dynamics)
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)()()( 2121 fsfsffs
)(')(')(
)(')(')(
2211212
1212211
fsffsfffsf
fsffsfffsf
1
2
2
121
)(')(')('
f
fs
f
fsffs
.)(' constfsf
Cfdff
fs ln)(
ffs ln)(
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2 Entropy in weakly nonlocal statistical physics (estimation theoretical, Fisher based)
(Fisher, …., Frieden, Plastino, Hall, Reginatto, Garbaczewski, …):
– Isotropy
))(,(),( 2DffsDffs
– Extensivity
– Additivity
),(~),(~))(,( 22112121 DffsDffsffDffs
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2
2
12 )(
ln))(,(f
DfkfkDffs
Fisher
Boltzmann-Gibbs-Shannon
(unique solution)
There is no estimation theory behind!
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2
2
12 )(
ln))(,(f
DfkfkDffs
– What about higher order derivatives?
– What is the physics behind the second term? What could be the value of k1?
– What is the physics behind the combination of the two terms?
Equilibrium distributions with power law tails
Dynamics - quantum systems
Dimension dependence
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Second order weakly nonlocal information measures
– Isotropy in 3D
))(,)(),(
,)(,,)(,(),,(32222
22223
23
fDTrfDTrfDTr
DffDDfDffDDfDffsfDDffs
– Extensivity
– Additivity
),,(~),,(~))(),(,(
22
222
11
212
2121
fDDffsfDDffs
ffDffDffs
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(unique solution)
))(1
)(1
)(1
)(1
)()32(
1
)()(
)()(
)(ln
),,(
3236
2225
24
2243
25
2
632
32
6
6
634
4
522
2
1
23
fDTrf
k
fDTrf
kfDTrf
kDffDDff
k
DffDDff
DfkkDffDDf
fk
f
Dfkk
f
Dfkk
f
Dfkfk
fDDffs
Depends on the dimension of the phase space.
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ln
)())(,(
2
2
12 kks
FisherBoltzmann-Gibbs-Shannon
– Mass-scale invariance (particle interpretation)
–
),(),( ss
f probability distribution
3 Quantum systems – the meaning of k1
21 k
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Why quantum?
048
ln2
''1
2
11
Rk
kRp
mkRR
k
kR
02
''2
22
ExD
m
Time independent Schrödinger equation
Probability conservationMomentum conservation+ Second Law (entropy inequality)
Equations of Korteweg fluidswith a potential
0 v0Pv )C(
2
2
1 mk
0)C()C(s s j
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Schrödinger-Madelung fluid
222),,(
22v
v
SchM
SchMs
2
8
1 2rSchM IP
(Fisher entropy)
Bernoulli equation
Schrödinger equation
v m
ie
21 k
2
)(
22),,(
22v
vm
s SchMSchM
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Logarithmic and Fisher together?
Nonlinear Schrödinger eqaution of Bialynicki-Birula and Mycielski (1976)(additivity is preserved):
Existence of non dispersive free solutions – solutions with finite support - Gaussons
Ubmt
i2
2
ln2
Stationary equation for the wave function:
02
ln''2
22
ExD
km
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12
ln)(
.
2
2
2
1
fdxEdxm
pf
dpfkf
Dfkfextr
4 MaxEnt calculations – the interaction of the two terms(1D ideal gas)
fR :
048
ln2
''1
2
11
Rk
kRp
mkRR
k
kR
boundarys RRJ |'
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048
ln2
''1
2
11
Rk
kRp
mkRR
k
kR
CR
R
)0(
0)0('
There is no analitic solution in general
There is no partition function formalism
symmetry
Numerical normalization
One free parameter: Js
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-2 -1 1 2x
0.2
0.4
0.6
0.8
1
1.2
R
18
,12
),5,2,5.1,3.1,2.1,1.1,1(4 111
mkk
k
k
k
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Conclusions
– Corrections to equilibrium (?) distributions
– Corrections to equations of motion
– Unique = universal• Independence of micro details• General principles in background
Thermodynamics Statistical physics
Information theory?
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Concavity properties and stability:
ff
Dff
Df
f
Df
fk
k
kfsff
4
1
4
44
)(1
)(ˆ
2
23
2
11
2
Positive definite
If k is zero (pure Fisher): positive semidefinite!
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One component weakly nonlocal fluid
),,,(C vv ),,,,(Cwnl vv
)C(),C(),C(s Pjs
Liu procedure (Farkas’s lemma):
constitutive state
constitutive functions
0 v
0)C()C(s s j0Pv )C(
... Pvjs2
)(s),(s2
e
vv
2),(s),,(s
2
e
vv
),( v basic state
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0:s2
ss2
1 22
s
vIP
rv PPP
reversible pressurerP
Potential form: Qr U P
)()( eeQ ssU Euler-Lagrange form
Variational origin
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Schrödinger-Madelung fluid
222),,(
22v
v
SchM
SchMs
2
8
1 2rSchM IP
(Fisher entropy)
Bernoulli equation
Schrödinger equation
v ie
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Alternate fluid
2)(),(
AltAlts
)(
42IP Altr
Alt
2Alt
AltU
Korteweg fluids:
22)( IP prKor