Heat conduction induced by non-Gaussian athermal fluctuations

Kiyoshi Kanazawa (YITP)Takahiro Sagawa (Tokyo

Univ.)Hisao Hayakawa (YITP)

2013/07/03 Physics of Granular Flows @YITP

Difference between thermal & athermal fluctuation

Introduction:Thermodynamics for small systems

To understand small systems• small systems (μm ～ nm)

(Ex. Colloidal particles)→manipulation of small systems (Ex. Optical tweezers)

• Role of fluctuations→ Feynman Ratchet (heat engine / heat conduction)

• Framework for small system(Theoretical limit of manipulation)

Thermal noise

laser

Analogy between Macro & MicroMacroscopic

thermodynamicsMicroscopic

thermodynamics• macro bath & macro

system

• Work & Heat (The 1st law)• Irreversibility (The 2nd

law)

• Efficiency (Carnot efficiency)

• macro bath & micro system (Ex) water & colloidal particle

• Manipulation (Optical tweezers)

• The 1st & 2nd laws?

• Efficiency of small systems

cm order order

Heat Bath(Macroscopic

)

Macroscopic System

Heat Bath(Macroscopic

)

Microscopic

Brownian particle trapped by optical tweezers(“Single particle gas”)Langevin Eq. with a potential

　 ( width of the potential)

white Gaussian noise

Can we definethermodynamic quantities

(work) & (heat)?(for this manipulation process) small ⇔

compressionbig ⇔ expansion

Colloidal

particle

laser

𝑉 (𝑥 ,𝑎)=12 ( 𝑥

𝑎 )2

water

The 1st law for small systems(Stochastic energetics)

• (work) = macro degrees of freedom

• (heat) = micro degrees of freedom𝑚 𝑑2 �̂�

𝑑𝑡2=−

𝜕𝑉 ( �̂� ,𝑎 )𝜕 �̂�

−𝛾 𝑑�̂�𝑑𝑡

+𝜉Environmental

effect(micro degrees of

freedom)

𝑑 �̂�=𝑑 �̂�+𝑑�̂�𝑑�̂�=𝜕𝑉𝜕 𝑎

𝑑𝑎K. Sekimoto, Prog. Theor. Phys. Supp. 130, 17 (1998).K. Sekimoto, Stochastic Energetics (Springer).K. Kanazawa et. al., Phys. Rev. Lett. 108, 210601 (2012).

The 2nd law for small systems• Average of work obeys the 2nd law!

• Maximum efficiency is achievedfor quasi-static processes!

Equality holds for quasi-static processes

Colloidal

particle

laser

𝑉 (𝑥 ,𝑎)=12 ( 𝑥

𝑎 )2

water𝑑𝑎𝑑𝑡

=( finite )⟹ ⟨ Δ �̂� ⟩>Δ𝐹

Heat conduction for small systems

Non-equilibrium equalities

Heat

Average

Fluctuation

Fourier lawFluctuation relation

μm

• Small systems ()

Ex.) Biological motors Nanotubes

• Characterized byNon-equilibrium equalities

Brownian motor• Vanes (Angles ) driven by

noises () & viscosity • Spring synchronizes the

angles

𝑑�̂�𝑑𝑡

=(−𝑅𝑑 �̂�1𝑑𝑡

+𝜉 ) 𝑑�̂�1

𝑑𝑡

Heat

(spring)

Thermal Thermal

�̂�1 �̂�2

𝑉=12(𝑞1−𝑞2)

2

Gaussian

noise

⟨ 𝜉 1 �̂�2 ⟩=2𝑅𝑘𝐵 𝑇 1𝛿(𝑡1−𝑡 2)⟨ �̂�1 �̂�2 ⟩=2𝑅𝑘𝐵𝑇2 𝛿(𝑡1− 𝑡2)

Detail

Electrical circuits (LRC)

Experimental realization(S. Ciliberto et al. (2013))

𝑇 1 𝑇 2

Summary of introduction• The 1st law

→• The 2nd law

→

• Fourier law

• Fluctuation relation

Thermodynamics

for small systems

Non-equilibriumequalities

Main: Heat conduction induced by non-Gaussian fluctuations

Thermal & athermal fluctuations

Fluctuation

Thermal noise

(Gauss noise)• Thermal fluctuation→from eq. environment　

Ex.)Nyquist noise Brownian noise

• Athermal fluctuation→from noneq. environment 　

Ex.)Electrical shot noise Biological fluctuation Granular noise

Athermal noise(Non-

Gaussian)

• Weak electrical current→particle property→”come” or “do not come” (spike noise ）

• A typical of non-Gaussian noise

• Noises happen times per unit time.Intensity (fight distanse) =

Poisson noise (shot noise)

E

Athermal env. & non-Gaussian noise

Fluctuations in athermal env. ⇒ non-Gaussian noise

(ii) Membrane of Red Blood Cell with ATP receptions

(i) Shot & burst noise in electrical circuitsEx.

ATP

Athermal

Env.

Abstraction

water

Non-Gaussia

n

Thermal Env.

Apply shot noise(zero-mean)

Gaussian

What corrections appear in the Fourier law?Between thermal systems Between athermal systems

• Fourier law (FL)• Fluctuation theorem

(FT)

• Extension of FL & FT?• Correction terms?

Thermal()

Thermal()

𝐽

+

Correction?

Correction?

conducting

wire

Athermal Athermal

𝐽

conducting

wire

• Non-Gaussian athermal fluctuations

• A conducting wire synchronizing the angles

𝑞1 𝑞2

𝑉 (𝑞1 ,𝑞2)

Non-Gaussiannoise （）

Non-Gaussian noise （）

conducting

wire

Heat ()Athermal Athermal

Non-equilibrium Brownian motor

𝑑�̂�1

𝑑𝑡=−

𝜕𝑉 (�̂�1 , �̂�2 )𝜕𝑞1

+𝜉

𝑑�̂�2

𝑑𝑡=−

𝜕𝑉 ( �̂�1 ,�̂�2 )𝜕𝑞2

+ �̂�

𝑑�̂�𝑑𝑡

=(− 𝑑 �̂�1𝑑𝑡

+𝜉 )∗ 𝑑�̂�1𝑑𝑡

Langevin

Eq.

HeatK.

Kanazawa et. al., PRL,

108, 210601(2012)

Characterizationof non-

Gaussianity

Non-Gaussia

n

(i) Generalized Fourier Law

• Perturbation in terms of • Not only but also contribute to .

𝐽=−∑𝑛=2

∞

𝜅𝑛 Δ 𝐾𝑛 ,𝜅𝑛=12 ∙𝑛 ! ⟨ 𝑑𝑛𝑉 ( �̂�)

𝑑 �̂�𝑛 ⟩eq

⟨ 𝑑𝑛𝑉𝑑 �̂�𝑛 ⟩

eq

Δ 𝐾𝑛couplin

g

Harmonic potential Quartic potential

𝑉 (𝑧 )= 𝜅2

𝑧 2

⟨ 𝑑𝑛𝑉𝑑 �̂�𝑛 ⟩

eq

Δ 𝐾𝑛couplin

g

𝐽=𝜅 Δ𝑇 𝐽=𝜅 ′ Δ𝑇+𝜒8Δ𝐾 4

The ordinary Fourier law

Corresponding correction

𝑉 (𝑧 )= 𝜅2

𝑧 2+⋯+𝜒2𝑛

𝑧 2𝑛⟹ 𝐽=𝜅 ′ Δ𝑇+⋯+𝜒4𝑛

Δ 𝐾2𝑛

Correspondence

Numerical check of GFL(Setup)

Gaussian noise (thermal) Two-sided Poisson noise (athermal)

• Variance = 2T• High order cumulants

= 0

• Flight distance = • Transition rate =

Gaussian vs. Two-sided Poisson

Numerical check of GFL(Results)

• Quartic potential

• Changing the parameter • The direction of heat

current depends on

We can change the direction of heat current by choosing an appropriate conducting

wire

B

Ａ Ｃ

eq. eq.

eq.

(ii)Absence of the 0th law

• Does the 0th law exists?（ Equilibrium between A and B, B and C → A and C ）

　　　　　• The direction of heat depends on the device.

←Violation of the 0th law• But, the 0th law recovers if we fix the device.

(+we can define effective temperature.)

Absent for athermal systems

(iii) Generalized Fluctuation theorem

•Perturbation in terms of •Harmonic Potential → Ordinary Fourier law

•But, the fluctuation theorem is modified.

In a case of the Gaussian & two-sided Poisson noise

We can further sum up the expansion!

lim𝑡→∞

𝑃 (+𝑞 , 𝑡)𝑃 (−𝑞 , 𝑡)

=∆ 𝛽𝑞+ 𝑞𝑇 ′−2 𝜆′exp [ 2𝑞2−𝑇 ′ ∆ 𝑇

4𝑇 ′ 𝜆 ′√𝑞2+𝑇𝑇 ′ ]sinh 𝑞2𝑇 ′ 𝜆 ′

A special case of the Gaussian and two-sided Poisson

noise

Numerical check of the linear part ofthe generalized fluctuation relation

•The Gaussian vs. two-sided Poisson case

•Consistent with our generalized FTnot with the conventional FT

Conventional

Modified

lim𝑡→∞

1𝑡log

𝑃 (+𝑞)𝑃 (−𝑞)

Conclusion•Generalized Fourier law

•Generalized fluctuation theorem

•Violation of the 0st lawThe direction of heat depends on the contact device(If we fix the contact device, the 0th law recovers)

Non-Gaussian Brownian motor

lim𝑡→∞

𝑃 (+𝑞 , 𝑡)𝑃 (−𝑞 , 𝑡)

=∆ 𝛽𝑞+∑𝑛=3

∞

[ 𝐾𝑛Ξ𝑛(𝑞 ;𝑇 ,𝑇 ′)+𝐾 ′𝑛Ξ𝑛(−𝑞 ;𝑇 ′ ,𝑇 )]

𝐽=−∑𝑛=2

∞

𝜅𝑛 Δ 𝐾𝑛 ,𝜅𝑛=12 ∙𝑛 ! ⟨ 𝑑𝑛𝑉 ( �̂�)

𝑑 �̂�𝑛 ⟩𝑒𝑞

K. Kanazawa, T. Sagawa, and H. Hayakawa, Phys. Rev. E 87, 052124 (2013)