fluctuation theorem and entropy production in statistical
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Fluctuation theorem and entropy production instatistical mechanics and turbulence
Michael Chertkov1, Vladimir Chernyak2, Christopher Jarzynski1
and Konstantin Turitsyn 1,3, Alberto Puliafito 1,4
1CNLS & Theory Division, LANL
2Wayne State, Detroit, 3Landau Institute, Moscow, 4INLN, Nice
Non-equilibrium Stat Mech & Turbulence, Warwick, July 2006
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Outline
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 3: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/3.jpg)
lamed
Outline
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 4: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/4.jpg)
lamed
Outline
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 5: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/5.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 6: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/6.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Equilibrium = Detailed Balance + No External FieldsFluctuation Dissipation Theoreme.g. molecule/polymer in a thermal bath
Non-Equilibrium #1Under Detailed Balance but in Time-Dependent External FieldWork (Jarzynski) Relatione.g. manipulations with bio-molecules
Non-Equilibrium #2Statistical Steady State with Broken Detailed BalanceFluctuation Theoreme.g. polymer in a steady flow; statistically steady turbulence
Generic Non-EquilibriumTemporarily Driven with Broken Detailed BalanceFluctuation Theoreme.g. Rayleigh-Taylor Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Equilibrium = Detailed Balance + No External FieldsFluctuation Dissipation Theoreme.g. molecule/polymer in a thermal bath
Non-Equilibrium #1Under Detailed Balance but in Time-Dependent External FieldWork (Jarzynski) Relatione.g. manipulations with bio-molecules
Non-Equilibrium #2Statistical Steady State with Broken Detailed BalanceFluctuation Theoreme.g. polymer in a steady flow; statistically steady turbulence
Generic Non-EquilibriumTemporarily Driven with Broken Detailed BalanceFluctuation Theoreme.g. Rayleigh-Taylor Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 8: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/8.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Equilibrium = Detailed Balance + No External FieldsFluctuation Dissipation Theoreme.g. molecule/polymer in a thermal bath
Non-Equilibrium #1Under Detailed Balance but in Time-Dependent External FieldWork (Jarzynski) Relatione.g. manipulations with bio-molecules
Non-Equilibrium #2Statistical Steady State with Broken Detailed BalanceFluctuation Theoreme.g. polymer in a steady flow; statistically steady turbulence
Generic Non-EquilibriumTemporarily Driven with Broken Detailed BalanceFluctuation Theoreme.g. Rayleigh-Taylor Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 9: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/9.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Equilibrium = Detailed Balance + No External FieldsFluctuation Dissipation Theoreme.g. molecule/polymer in a thermal bath
Non-Equilibrium #1Under Detailed Balance but in Time-Dependent External FieldWork (Jarzynski) Relatione.g. manipulations with bio-molecules
Non-Equilibrium #2Statistical Steady State with Broken Detailed BalanceFluctuation Theoreme.g. polymer in a steady flow; statistically steady turbulence
Generic Non-EquilibriumTemporarily Driven with Broken Detailed BalanceFluctuation Theoreme.g. Rayleigh-Taylor Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 10: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/10.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Fluctuation Theorem
Non-Conservative Forces but Autonomous Dynamics(Non-Equilibrium #2)
p(+Σ)
p(−Σ)= eΣ,
p(Σ) is the distribution of observed values of a quantity Σrepresenting dissipation or entropy production.
Bibliography:
Evans, Cohen, Morris (1993)Evans, Searles (1994)Gallavotti, Cohen (1995)Kurchan (1998)Lebowitz, Spohn (1999)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Work (Entropy Production) Relation
Non-Autonomous Dynamics (Non-Equilibrium #1): for any fixedvalue of an externally controlled, λ, system is in a Gibbs’ state,characterized by the free energy, F(λ). Executing a fixedtime-dependent protocol, λ(t), −τ < t < τ , repeatedly one getsthe following relation for average
Jarzynski equality:
〈e−Σ〉 = exp(Fλ(−τ) −Fλ(+τ)
)Bibliography:
Bochkov, Kuzovlev (1977)Jarzynski (1997)Crooks (1998,1999,2000)Hatano (1999)Hummer, Szabo (2001)Hatano, Sasa (2001)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Unified Framework:Generalized Fluctuation Theorems and Work Relations
Fluctuation Theorem:
p(+Σ′)
p(−Σ′)= eΣ′
,
Jarzynski Equality:
〈e−Σ′〉 = 1
Recent Progress:
Maes, Poincare (2003)Maes, Netocny (2003)Chernyak, Chertkov, Jarzynski (2005,2006)Seifert (2005)Kurchan (2005)Reid, Sevick, Evans (2005)Speck, Seifert (2005)Imparato, Peliti (2006)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Langevin Model
Over-damped classical system (e.g. polymer in a solution)
d
dtxi = Fi (x;λ) + ξi (t; x;λ), i = 1, · · · ,N
〈ξi 〉 = 0,⟨ξi (t) ξj(t
′)⟩
= Gij δ(t − t ′)
Fokker-Planck∂p
∂t= −∂ i (Fip) +
1
2∂i
(Gij(∂
jp))≡ Lλp = −∂ iJi
J(x, t) ≡ (1/2)G(v −∇)p, pS(x;λ) = exp (−ϕ(x;λ))
v i ≡ 2ΓijFj , Ai ≡ v i + ∂ iϕ, ΓijGjk = δik
Detailed Balance?
SATISFIED
v(x;λ) = −∇U(x;λ)pstat(x) ∝ e−U(x;λ)
BROKEN
v(x;λ) 6= −∇U(x;λ)A, Jstat 6= 0
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Langevin Model
Over-damped classical system (e.g. polymer in a solution)
d
dtxi = Fi (x;λ) + ξi (t; x;λ), i = 1, · · · ,N
〈ξi 〉 = 0,⟨ξi (t) ξj(t
′)⟩
= Gij δ(t − t ′)
Fokker-Planck∂p
∂t= −∂ i (Fip) +
1
2∂i
(Gij(∂
jp))≡ Lλp = −∂ iJi
J(x, t) ≡ (1/2)G(v −∇)p, pS(x;λ) = exp (−ϕ(x;λ))
v i ≡ 2ΓijFj , Ai ≡ v i + ∂ iϕ, ΓijGjk = δik
Detailed Balance?
SATISFIED
v(x;λ) = −∇U(x;λ)pstat(x) ∝ e−U(x;λ)
BROKEN
v(x;λ) 6= −∇U(x;λ)A, Jstat 6= 0
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Langevin Model
Over-damped classical system (e.g. polymer in a solution)
d
dtxi = Fi (x;λ) + ξi (t; x;λ), i = 1, · · · ,N
〈ξi 〉 = 0,⟨ξi (t) ξj(t
′)⟩
= Gij δ(t − t ′)
Fokker-Planck∂p
∂t= −∂ i (Fip) +
1
2∂i
(Gij(∂
jp))≡ Lλp = −∂ iJi
J(x, t) ≡ (1/2)G(v −∇)p, pS(x;λ) = exp (−ϕ(x;λ))
v i ≡ 2ΓijFj , Ai ≡ v i + ∂ iϕ, ΓijGjk = δik
Detailed Balance?
SATISFIED
v(x;λ) = −∇U(x;λ)pstat(x) ∝ e−U(x;λ)
BROKEN
v(x;λ) 6= −∇U(x;λ)A, Jstat 6= 0
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 17: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/17.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 18: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/18.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Path Integral
Forward/Reversed Protocol: λFt : [A,B]; λR
t = λF−t .
PF/R [X |x−τ ] = N exp
(−
∫ +τ
−τdt S+(xt , xt ;λ
F/Rt )
),
PF/R[X ] = pSA(x−τ )PF/R [X |x−τ ]
S+(x, x;λ) =1
2(xi − Fi )Γ
ij(xi − Fi ) +1
2∂iFi .
”Conjugated twin” of the trajectory: X † ≡ x†t+τ−τ , x†t = x−t
PR[X †|x†−τ
]= N exp
[−
∫ +τ
−τdt S+(x†t , x
†t ;λ
Rt )
]= N exp
[−
∫ +τ
−τdt S−(xt , xt ;λ
Ft )
], S−(x, x;λ) ≡ S+(x,−x;λ)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Path Integral
Forward/Reversed Protocol: λFt : [A,B]; λR
t = λF−t .
PF/R [X |x−τ ] = N exp
(−
∫ +τ
−τdt S+(xt , xt ;λ
F/Rt )
),
PF/R[X ] = pSA(x−τ )PF/R [X |x−τ ]
S+(x, x;λ) =1
2(xi − Fi )Γ
ij(xi − Fi ) +1
2∂iFi .
”Conjugated twin” of the trajectory: X † ≡ x†t+τ−τ , x†t = x−t
PR[X †|x†−τ
]= N exp
[−
∫ +τ
−τdt S+(x†t , x
†t ;λ
Rt )
]= N exp
[−
∫ +τ
−τdt S−(xt , xt ;λ
Ft )
], S−(x, x;λ) ≡ S+(x,−x;λ)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Configurational Entropy
Reversed Protocol, Forward Dynamics
PF[X ]
PR[X †]=
pSA(x−τ )PF[X |x−τ ]
pSB(x†−τ )PR[X †|x†−τ ]
= exp(RF[X ])
Entropy/Work
RF[X ] =
∫ F
dt λFt
∂ϕ
∂λ+
∫ F
dx · A
=
∫ F
dt
(λF
t
∂ϕ
∂λ+ 2xjΓ
ijF j + xj ∂jϕ
)
There exist other formulations,e.g. for Reversed Protocol and Reversed Dynamics
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Configurational Entropy
Reversed Protocol, Forward Dynamics
PF[X ]
PR[X †]=
pSA(x−τ )PF[X |x−τ ]
pSB(x†−τ )PR[X †|x†−τ ]
= exp(RF[X ])
Entropy/Work
RF[X ] =
∫ F
dt λFt
∂ϕ
∂λ+
∫ F
dx · A
=
∫ F
dt
(λF
t
∂ϕ
∂λ+ 2xjΓ
ijF j + xj ∂jϕ
)
There exist other formulations,e.g. for Reversed Protocol and Reversed Dynamics
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 22: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/22.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Fluctuation Theorem
ρF(R) ≡∫DX PF[X ] δ
(R − RF[X ]
)=
∫DX PR[X †] exp
(RF[X ]
)δ(R − RF[X ]
)= exp(R)
∫DX † PR[X †] δ
(R + RR[X †]
)= exp(R)ρF(−R)
Work (Jarzynski) Relation
⇒ 〈exp(−R)〉 =
∫dR exp(−R)ρF(R) =
∫dRρF(−R) = 1
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Equilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
Application in chem/bio physics
Jarzynski equality:
〈e−Σ〉 = exp(Fλ(−τ) −Fλ(+τ)
)Gain
Fast exploration of the phase space
But ...
The number of necessary observations grows as what is typical inequilibrium is atypical (rare) for a fast protocol, and vise versa.
Current Focus of Research
Design of efficient protocols for the free energy exploration
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Σ is the generalized work produced by an external field in time,or the energy (heat) dissipated by the system in time t,or the entropy generated in time t
Large Deviation Functional
P(Σ; t) ∝ exp (−tQ(Σ/t))
Fluctuation Theorem
Q(ω)− Q(ω) = ω
disclaimer
”complex” averaging (noise+disorder) may brake FT (see below)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Σ is the generalized work produced by an external field in time,or the energy (heat) dissipated by the system in time t,or the entropy generated in time t
Large Deviation Functional
P(Σ; t) ∝ exp (−tQ(Σ/t))
Fluctuation Theorem
Q(ω)− Q(ω) = ω
disclaimer
”complex” averaging (noise+disorder) may brake FT (see below)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 28: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/28.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Σ is the generalized work produced by an external field in time,or the energy (heat) dissipated by the system in time t,or the entropy generated in time t
Large Deviation Functional
P(Σ; t) ∝ exp (−tQ(Σ/t))
Fluctuation Theorem
Q(ω)− Q(ω) = ω
disclaimer
”complex” averaging (noise+disorder) may brake FT (see below)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 29: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/29.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Σ is the generalized work produced by an external field in time,or the energy (heat) dissipated by the system in time t,or the entropy generated in time t
Large Deviation Functional
P(Σ; t) ∝ exp (−tQ(Σ/t))
Fluctuation Theorem
Q(ω)− Q(ω) = ω
disclaimer
”complex” averaging (noise+disorder) may brake FT (see below)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 30: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/30.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 31: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/31.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Model
xi = σijxj − ∂xi U(x) + ξi
σij(t)xj is force exertedby flow on polymer
〈ξi (t)ξj(t′)〉 = 2T δ(t−t ′)δij
Steinberg et al.(2004)
Σ ≡∫ t
0dt ′xα(t ′)σαβxβ(t ′)
P± ∼ρ(t)=xt∫
ρ(0)=x0
Dρ exp[−S±]
S = S+ − S− =−Σ + U(x(0))− U(x(t))
T
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Model
xi = σijxj − ∂xi U(x) + ξi
σij(t)xj is force exertedby flow on polymer
〈ξi (t)ξj(t′)〉 = 2T δ(t−t ′)δij
Steinberg et al.(2004)
Σ ≡∫ t
0dt ′xα(t ′)σαβxβ(t ′)
P± ∼ρ(t)=xt∫
ρ(0)=x0
Dρ exp[−S±]
S = S+ − S− =−Σ + U(x(0))− U(x(t))
T
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 33: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/33.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Generating functional
Ψs ≡ exp
(−sΣ
T
), ∂tΨs = LsΨs
Ls = −∇α(σαβxβ − ∂xαU(x)
)+ T∇α∇α
∇α = ∂α +s
Tσαβxβ
Linear elasticity, U(x) = x2/(2τ) & σ = const
t →∞: looking for the “ground” state in a Gaussian form
Ψs = exp(−λst) exp(−xiB
ijs xj
)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 34: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/34.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Generating functional
Ψs ≡ exp
(−sΣ
T
), ∂tΨs = LsΨs
Ls = −∇α(σαβxβ − ∂xαU(x)
)+ T∇α∇α
∇α = ∂α +s
Tσαβxβ
Linear elasticity, U(x) = x2/(2τ) & σ = const
t →∞: looking for the “ground” state in a Gaussian form
Ψs = exp(−λst) exp(−xiB
ijs xj
)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 35: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/35.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Linear Elasticity, Constant Flow
d = 2
σ =
(a b + c/2
b − c/2 −a
)
Answer
P (Σ) ∼ exp (−tQ(Σ/(Tt)))
Q(w) =
√(4+c2τ2)(w2+c2τ2)
2cτ − 1− w2
20
15
10
5
0 20 10 0-10
Q(w
), L
arge
Dev
iatio
n F
unct
iona
l
w, Entropy production rate
0.04
0.02
0 10 5 0
Q(w
)
w
To notice
Q(w) depends only on rotation, c
FT is satisfied: Q(w)− Q(−w) = w
Linear asymptotics for large deviations, |ω| 1Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
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lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Chaotic flow case
Annealed vs Quenched
Fσ ≡ 〈Pσ(Ω)〉ξlog〈Fσ〉σ vs 〈logFσ〉σ
Standard FT for annealed: Q(w)− Q(−w) = w
Modified FT for quenched: 〈[Fσ]n〉 ∼ exp (−tQn(Ω/Tt))
Qn(w)− Qn(−w) = nw
Lack of FT for the Largish Deviations: w 〈w〉Gaussian σ: − log〈Fσ〉 ∝
√w
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 37: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/37.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Large Deviation FunctionalPolymer in a flow
Chaotic flow case
Annealed vs Quenched
Fσ ≡ 〈Pσ(Ω)〉ξlog〈Fσ〉σ vs 〈logFσ〉σ
Standard FT for annealed: Q(w)− Q(−w) = w
Modified FT for quenched: 〈[Fσ]n〉 ∼ exp (−tQn(Ω/Tt))
Qn(w)− Qn(−w) = nw
Lack of FT for the Largish Deviations: w 〈w〉Gaussian σ: − log〈Fσ〉 ∝
√w
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 38: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/38.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Work & Kolmogorov flux
Outline
1 Fluctuation Theorems & Work RelationsEquilibrium vs Non-EquilibriumLangevin ModelPath Integral & Configurational entropyDerivations & Applications
2 Statistics of Entropy ProductionLarge Deviation Functional and Fluctuation TheoremPolymer in a (gradient) flow, regular and chaotic
3 Entropy/Work Production in Turbulence: SpeculationsWork & Kolmogorov flux
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 39: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/39.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Work & Kolmogorov flux
∂tu + (u∇)u + ∇p = ν∆u + f
Work produced in time t
Σ1 = 1/2∫ t0 dt ′
∫dr(uf)
Energy dissipated in time t
Σ2 = ν/2∫ t0 dt ′
∫dr(∇u)2
Suggestions:
Test Large Deviations: at t τL, P(Σ) ∼ exp (−tQ(Σ/t))In case it works, Q(w) may serve gives an ultimate measureof possible ”universality” classes in turbulence
Check if Fluctuation Theorem applies: Q(w)− Q(−w) = w
Verify if Q1 = Q2 = · · ·Work Relations and Protocol Optimization may find practicalapplications in Non-Stationary Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 40: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/40.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Work & Kolmogorov flux
∂tu + (u∇)u + ∇p = ν∆u + f
Work produced in time t
Σ1 = 1/2∫ t0 dt ′
∫dr(uf)
Energy dissipated in time t
Σ2 = ν/2∫ t0 dt ′
∫dr(∇u)2
Suggestions:
Test Large Deviations: at t τL, P(Σ) ∼ exp (−tQ(Σ/t))In case it works, Q(w) may serve gives an ultimate measureof possible ”universality” classes in turbulence
Check if Fluctuation Theorem applies: Q(w)− Q(−w) = w
Verify if Q1 = Q2 = · · ·Work Relations and Protocol Optimization may find practicalapplications in Non-Stationary Turbulence
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production
![Page 41: Fluctuation theorem and entropy production in statistical](https://reader034.vdocuments.us/reader034/viewer/2022052514/628bb88fecc24b7452377834/html5/thumbnails/41.jpg)
lamed
Fluctuation Theorems & Work RelationsStatistics of Entropy Production
Entropy/Work Production in TurbulenceSummary
Summary
Summary
Production of entropy/work (in finite time) is a universalcharacteristic explaining breakdown of the detailed balance
Generalized Fluctuation Theorem gives a useful and nontrivialrelation
Large Deviation Functional is an ultimate and rich tester of anon-equilibrium stat mech system (e.g. turbulence)
Michael Chertkov, Los Alamos NL Fluctuation theorem and entropy production