Microscopic derivation of non-Gaussian Langevin equations for athermal

systemsADVANCED STATISTICAL DYNAMICS GROUP

KIYOSHI KANAZAWA

Jan 28, 2015Lunch seminar@YITP

Fluctuation in small systems

Experimental development (e.g., optical tweezers)　 → Manipulation & Observation of small systemsSingle particle “ideal gas”Thermodynamics for a single particle

bead

laser

Gaussian Langevin Equation (GL)

Motion of a fluctuating bead in waterThe GL Eq. is universal and simpleA foundation for thermodynamics

White Gaussian noise𝑎

My interest: Athermal fluctuationNon-Gaussian Langevin Eq. (NGL)

Athermal fluctuationOriginating from non-eq. environmentsCharacterized by non-GaussianityElectrical, biological, granular systems𝑑𝑉

𝑑 𝑡=−𝛾𝑉+√2𝛾𝑇 𝜉𝐺+ �̂�𝑁𝐺

White non-Gaussian noise

EX1) Avalanche noise EX2) Active noise

• Reverse voltage on diodes• Chain-reaction

• Biological motor• Fluctuation induced by ATP

Goal of this talk: Microscopic Derivation of NGL Eq.

Review of a derivation of GL Eq.

The central limit theorem (CLT)→Emergence of Gaussian noise

Why is the CLT violated for non-equilibrium systems?

1. Microscopic derivation of NGL Eq.2. Application of a granular example

Review on a derivation of GL Eq.

Our study on a derivation of NGL Eq.

KK, T.G. Sano, T. Sagawa, H. Hayakawa, to appear in PRL.

Derivation of GL Eq.:Example （ Rayleigh Piston ）

Piston in rarefied gas　（ linearized Boltzmann Eq. ）Markov jump process• Collision→discontinuous velocity jump of Piston

Strong environmental correlation（ non white noise ）• large → energy outflow to gas• small → energy inflow from gas

Heavy piston limit （）1. Correlation is renormalized only into viscosity2. Fluctuation is reduced to white noise3. Fluctuation is reduced to Gaussian (the CLT)

𝜀→0

General derivation of GL Eq.:the system size expansion

A single stochastic bath described by Markov jump noise→Weak coupling (e.g., mass ratio ）The following simplification is performed

1. Correlation is renormalized into viscosity2. Fluctuation is reduced to white noise3. Fluctuation is reduced to Gaussian (the CLT)

Universality in the weak coupling limit

Weak coupling

・・・ Markov jump noise　　 (ε-independent)

Emergence of the NGL Eq.＝ Simplification 1 & 2 are applicable, but simplification 3 is not applicable.

・・・ small parameter

𝜀→0

Where is the CLT applied in the system size expansion?

CLT→Simple sum of independent and identically distributed (i.i.d.) random variables 　　　→ Can we regard collisions as many i.i.d. variables?　→ Yes, we can if (:relaxation time, ： collision interval)The CLT in the expansion　＝ weak frictionWeak-coupling for a single bath　→ Weak friction

1√𝑛∑𝑖=1

𝑛

𝜉 𝑖→Gaussianrandomvariable（𝑛→∞）

This condition can be violated for systems with multiple baths

An asymptotic derivation of the NGL Eq.

System associated with multiple baths

Assumption1. Weak coupling ：2. Coexistence of both noise ：3. Large thermal viscosity ：

is -independent

Assumption 1→Environmental correlationdisappear (Reduction to white noise)

Assumption 3→the CLT is violated

・・・ Athermal viscous coefficient・・・ thermal viscous coefficient

The NGL Eq. is derived

ε-independent

Violation of the CLT

(a) A single bath→Automatically weak friction　　　 Sufficiently frequent fluc. during relaxation　　　＝ the CLT is applicable

(b) Multiple baths→Origins of fluc. and diss. are separeted　　　 Not sufficiently frequent fluc. during relaxation　　　＝ the CLT is not applicable

Granular motor: Modeling

A non-eq. Rayleigh Piston

Linearized Boltzmann Eq.

Assumption1. Weak coupling ：2. Coexistence of both noise ：3. Strong thermal viscosity ：

,

ε-independent

Granular motor: Results

When the granular velocity dist. is The exact distribution of the rotor’s angular velocity

Numerical data are consistent with our result but not with the Gaussian approximation

ConclusionA derivation of the GL. Eq.

A derivation of the NGL. Eq.1. Weak coupling2. Coexistence of both fluctuations3. Strong thermal friction

KK, T.G. Sano, T. Sagawa, H. Hayakawa, to appear in PRL (arXiv: 1407.5267).

Application to a granular1. An exactly solvable model2. Agreement with simulation

Strong dissipation