Thermodynamics of exponential kolmogorov-nagumo Averages



Jan Korbel
Pablo Morales
Fernando RoSAS
REFERENCE PAPER: 10.1088/1367-2630/ace4eb (Published in New JOurnal of Physics)
SLIDES AVAILABLE AT: SLIDES.COM/JANKORBEL
outline
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Motivation
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KOLMOGOROV-NAGUMO AVERAGES
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Rényi entropy
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EQUILIBRIUM THERMODYNAMICS
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APPLICATIONS TO MULTIFRACTALS
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NONEQUILIBRIUM THERMODYNAMICS
MOTIVATION
- Complexity science is a contemporary field that studies systems with many interacting subsystems and emergent behavior
- For many complex systems, the standard approaches known from thermodynamics and information theory do not apply
- One way to overcome these issues is to introduce the generalization of ordinary variables known from thermodynamics and information theory
- These "generalized measures" can provide a solution for the description of these systems
- Warning: There is no general measure that can be used for all types of complex systems
MOTIVATION
- Example: some generalized entropies are used because they are maximized by "non-Boltzmann distributions" (e.g., heavy tails)
- There are, however, many more thermodynamic and information properties that are connected to the given entropic functional
- Note: "Three faces of entropy" (a system can be described by more than one entropy)
- Shannon entropy can lead to non-Boltzmann distribution and generalized entropies can lead to Boltzmann distribution
- The role of constraints
- One has to start with the particular system, not with the particular entropy
KOLMOGOROV-NAGUMO AVERAGES
- In ancient Greece, three types of averages were studied:
- arithmetic mean n1∑ixi
- geometric mean n∏ixi
- harmonic mean n(∑ixi1)−1
- These three means have many applications in physics, geometry, and music.
- All three can be generalized into a Hölder mean (n1∑ixip)1/p
- The Hölder mean is used in many disciplines, including functional calculus, probability or signal processing
KOLMOGOROV-NAGUMO AVERAGES
- In 1930, Andrey Kolmogorov and Mitio Nagumo independently introduced a generalization called f-mean for an arbitrary continuous and injective function
⟨X⟩f:= f−1(n1i∑f(xi))
- The basic property is that the mean is invariant to the transformation f(x)↦af(x)+b
- The Hölder mean is obtained for f(x)=xp
KOLMOGOROV-NAGUMO AVERAGES
- The original arithmetic mean can be obtained for f(x)=x
- The arithmetic mean can be uniquely determined from the class of KN-mean by the following two properties:
- Homogeneity ⟨aX⟩=a⟨X⟩
- Translation invariance ⟨X+c⟩=⟨X⟩+c
- Note that the Hölder means satisfy Property 1 but not property 2
- What is the class of means that satisfies Property 2?
EXPONENTIAL KN AVERAGES
- The solution is given by the function f(x)=(eγx−1)/γ and the inverse function is f−1(x)=γ1ln(1+γx)
- This leads to the exponential KN mean⟨X⟩γ=γ1ln(i∑exp(γxi))
- We recover the arithmetic mean by γ→0
- Property 2 is important for thermodynamics since the average energy ⟨E⟩ does not depend on the energy of the ground state
- The additivity property is preserved for independent variables
⟨X+Y⟩γ=⟨X⟩γ+⟨Y⟩γ⇔X⊥⊥ Y
EXPONENTIAL KN AVERAGES
- A general additivity formula is given as ⟨X+Y⟩γ=⟨X+⟨Y∣X⟩γ⟩γ where the conditional mean is given as ⟨X∣Y⟩γ=γ1ln(i∑piexp(γ⟨Y∣X=xi⟩γ))
- The exponential KN mean is closely related to the cumulant generating function Mγ(X)=ln⟨exp(γX)⟩=γ⟨X⟩γ
RéNYI ENTROPY AS an EXPONENTIAL KN AVERAGE
- Rényi entropy can be naturally obtained as the exponential KN average of the Hartley information ln1/pk
- We define the Rényi entropy as Rγ(P)=1−γ1⟨ln1/pk⟩γ=γ(1−γ)1lnk∑pk1−γ
- The role of the prefactor will be clear later, but one of the reasons is that the limit γ→1 is the well-defined Burg entropy −∑klnpk
- We also naturally obtain the conditional entropy Rγ(X,Y)=Rγ(X)+Rγ(Y∣X)
EQUILIBRIUM THERMODYNAMICS OF EXPONENTIAL KN AVERAGES
- Let us now consider thermodynamics where all quantities are exponential KN averages
- We consider that the system entropy is described by Rényi entropy
- The internal energy is describe as the KN average of the energy spectrum ϵ Uγβ:=β1⟨βϵ⟩γ=βγ1lni∑piexp(βγϵi)
- The inverse temperature β=kBT1 ensures that the energy has correct units
MAXENT DISTRIBUTION
- Let us now calculate the MaxEnt distribution obtained from Rényi entropy with given constrained of internal energy Uγβ
- The Lagrange function is L=Rγ−α0i∑pi−α1β1⟨βϵ⟩γ
- The MaxEnt distribution πi can be obtained from
γ1∑kπk1−γπi−γ−α0−βγα1∑kπkeγβϵkeγβϵi=0
- By multiplying by πi and summing over i we obtain α0=γ1−α1 and therefore πi= (∑kπk1−γ)1/γ(∑kπkeγβϵk)1/γ exp(−βϵi)=Zβexp(−βϵi)
MAXENT DISTRIBUTION
- We obtained Boltzmann distribution from non-Shannonian entropy with non-arithmetic constraint
- We immediately obtain that lnπi=−β(ϵi−Uγβ(π))−(1−γ)Rγ(π) which leads to lnk∑e−βϵk=(1−γ)Rγ−βUγβ=Ψγ−γRγ where Ψγ=Rγ−βUγβ is the Massieu function
- The Helmholtz free energy can be expressed as Fγβ(π)=Uγβ(π)−β1Rγ(π)=(γ−1)β1lnk∑e(γ−1)βϵk.
- Thus, although we get the same distribution as for the ordinary thermodynamics, the thermodynamic relations are different!
THERMODYNAMIC INTERPRETATION
- What is the interpretation of equilibrium thermodynamics of exponential KN averages?
- Let us denote the equilibrium quantities as
Uγβ=Uγβ(π)
Rγβ=Rγ(π)
Fγβ=Uγβ−β1Rγβ
- We can now express each quantity to find its interpretation
THERMODYNAMIC INTERPRETATION
- We start with Rényi entropy Rγβ=γ(1−γ)1lni∑(Zβe−βϵi)1−γ=γ(1−γ)1lni∑e−(1−γ)β−γ1lnZβ
- Thus we get that Rényi entropy of a Boltzmann distribution is Rγβ =−γβ(F(1−γ)β− Fβ)
- By defining β′=(1−γ)β ⇒γ=1−ββ′ we can express it as
Rγβ= β2β′−βFβ′−Fβ
which is the β rescaling of the free energy difference.
- This can be interpreted as the maximum amount of work the system can perform by quenching the system from inverse temperature β to inverse temperature β′.
- This has been first discovered by John Baez.
THERMODYNAMIC INTERPRETATION
- By taking γ→0 we recover the relation between entropy and free energy Sβ=β2(∂β∂Fβ)
- As shown before, the free energy can be expressed as Fγβ=F(1−γ)β so it is clear that the temperature is rescaled from β to (1−γ)β
- Finally the internal energy can be expressed as Uγβ=β′−ββ′Fβ′−βFβ= −β′−βΨβ′−Ψβ
- Again, by taking γ→0 we recover
Uβ= −(∂β∂Ψβ)
APPLICATION TO MULTIFRACTALS
- One of the main applications of Rényi entropy is multifractals
- Let us first remind go through the fundamentals of multifractal analysis
- Let us consider a physical system whose state space is parcelled into distinct regions ki(s) indexed by i∈I
- Consider the probability of observing the system within region ki(s), which is denoted by pi(s).
- Let us assume that this probability observes a scaling property of the form
pi(s)=z(s)sαi
APPLICATION TO MULTIFRACTALS
- For small scales, i.e. s→0, let us assume that the frequency of the scaling exponent αi is given by a continuous probability distribution, whose density ρ has the form
ρ(α,s)dα=c(α)s−f(α)dα
where f(α) is called as multifractal spectrum
- Let us define the generalized dimension of Rényi entropy as s→0limlns(1−γ)Rγ(s)=Dγ
- We can express Rényi entropy as Rγ(s)=γ(1−γ)1ln(∫sαc(α)s−f(α) dα)1−γ∫s(1−γ)αc(α)s−f(α) dα
- The relation between the multifractal spectrum and generalized dimension is given by using the stepest descent approximation for s→0
APPLICATION TO MULTIFRACTALS
- By using the steepest descent, we define αγ=argαmin{(1−γ)α−f(α)}
- We introduce the Legendre transform of multifractal spectrum τγ=(1−γ)αγ−f(αγ)
- Thus, Rényi entropy can be expressed as Rγ=γ(1−γ)τγ−(1−γ)τ1lns+O(1)
- Therefore, the relation to generalized dimension is Dγ=γτγ−(1−γ)τ1
MULTIFRACTAL ENERGY CASCADES
- Let us now make a connection with turbulence cascades.
- We assume that the characteristic energy ϵ(s) scales as ⟨ϵ(s)γ⟩∼sMγ(ϵ)
- The cascade spectrum can be connected to the multifractal spectrum by using the fact that the MaxEnt distribution is Boltzmann, i.e., πi∝sαi∝e−βϵi which leads to αilns=−βϵi
- Thus, the scale is connected with the temperature and the scaling exponent with the energy
- Rescaling of the temperature β↦(1−γ)β corresponds to the change of scale s↦s1−γ
MULTIFRACTAL ENERGY CASCADES
Finally, the scaling exponents of thermodynamic potentials are
βUβγ(s) =γτγ−τ1lns+O(1)
βFβγ(s)=γ−1τγlns+O(1)
Mγ=γUγβ(s)=τ1−τγ+o(1)
LEGENDRE STRUCTURE
- We consider now a general non-equilibrium state and try to calculate the work done between that state and the equilibrium state
- The minimum work is given by the free energy difference Wrev=Fγ(p)−Fγ(π)=ΔUγβ−β1ΔRγ
- The difference between non-equilibrium and equilibrium internal energy can be expressed as
Uγβ(p)−Uγβ(π)=βγ1ln ∑iπieγβϵi∑ipieγβ0ϵi=β1γln(1+∑iπi1−γ∑i(pi−πi)πi−γ) = βγ1ln(1+γ ∇Rγ⋅(p−π))
LEGENDRE STRUCTURE
- Thus, the free energy difference can be expressed as β1(Fγ(p)−Fγ(π))=Rγ(π)−β1Rγ(p)+ βγ1ln(1+γ ∇Rγ⋅(p−π))
- This can be expressed as the Rényi-Bregmann divergence which is a generalized Bregmann divergence Dγ(p∣∣π)=Rγ(π)−Rγ(p)−C(∇Rγ,(p−π)) where C(x,y) is the link function, here C(x,y)=γ1 ln(1+γx⋅y)
- We can rewrite it as Dγ(p∣∣π)= − Rγ(p)−Rγ(p,q) where Rγ(p,q) is the Rényi cross entropy
- We can also express it as Dγ(p∣∣π)=Rγ(π)−Rγ(p)+γ1ln∑piΠi(γ) where Πi(γ)=∑iπi1−γπi−γ is the ratio of escort distribution and original distribution
NON-EQUILIBRIUM THERMODYNAMICS
- Let us now consider the case of arbitrary non-equilibrium state and its evolution in time
- We consider the standard master equation p˙i(t)=j∑(wij(t)pj(t)−wji(t)pi(t))
- Since the equilibrium distribution is Boltzmann, we obtain the standard detailed balance wji(t)wij(t)=πj(t)πi(t)=eβ(ϵj(t)−ϵi(t))
- It is possible to show that the second law of thermodynamics hold: Σ˙γ=R˙γ− βQ˙γβ≥0
- For relaxation processes (ϵi does not depend on time), we can show that the H-function Hγ(p):=Dγ(p∣∣π)=β(Fγ(p)−Fγ(π)) does not increase in time H˙γ ≤0
ENSEMBLE ENTROPY PRODUCTION
- Finally, we focus on trajectory thermodynamics
- Trajectory entropy production can be expressed as lnP~[x~]P[x]=Δs[x] +βq[x]=σ[x]
- Here x is a trajectory, x~ is its time reverse, P is trajectory probability, Δs=lnp(xi,ti)−lnp(xf,tf) and q is the trajectory heat
- The ensemble average can be obtained as ⟨⟨σ⟩⟩γ=γ1ln∫DxP[x]exp(γ lnP~[x~]P[x]) =γ1ln∫Dx(P[x])γ+1(P~[x~])−γ=Dγ(P[x]∣∣P~[x~])≥0
- Here Dγ is the Rényi-Csiszár divergence

Other topics in the paper:
- Trajectory thermodynamics and fluctuation theorems
- Information geometry and thermodynamic length
Thermodynamics of exponential kolmogorov-nagumo Averages Jan Korbel Pablo Morales Fernando RoSAS REFERENCE PAPER: 10.1088/1367-2630/ace4eb (Published in New JOurnal of Physics) SLIDES AVAILABLE AT: SLIDES.COM/JANKORBEL
Thermodynamics of exponential kolmogorov-nagumo means
By Jan Korbel
Thermodynamics of exponential kolmogorov-nagumo means
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