Brownian motors in the micro-scale domain: Enhancement of efficiency - - PowerPoint PPT Presentation

Brownian motors in the micro-scale domain: Enhancement of efficiency by noise

Part of Phys. Rev. E 90, 032104 (2014) Jakub Spiechowicz1, Peter H¨ anggi2 and Jerzy Luczka1

1Institute of Physics, University of Silesia, 40-007 Katowice, Poland 2Institute of Physics, University of Augsburg, 86135 Augsburg, Germany JS, PH and J L Brownian motors efficiency 1 / 10

Le Chat` elier-Braun principle

v = µF. Typical response Absolute negative mobility

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Driven Brownian motor

Minimal model exhibiting ANM expressed in the dimensionless variables ¨ x + γ ˙ x = −V ′(x) + a cos(ωt) + F +

• 2γDG ξ(t).

We replace the constant static load F with the random force η(t) F → η(t), assuming η(t) = F.

Fundamental question

Can noise η(t) induce more effective transport than the constant force F?

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Nonequilibrium noise

Generalized white Poissonian noise η(t) =

n(t)

• i=1

ziδ(t − ti), where n(t) is Poissonian counting process Pr{n(t) = k} = (λt)k k! e−λt. The process η(t) presents white noise of finite mean and a covariance given by η(t) = λzi, η(t)η(s) − η(t)η(s) = 2DPδ(t − s). Its intensity reads DP = λz2

i

2 .

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Distribution of the amplitudes {zi} of the δ-kicks ρ(z) = ζ−1θ(z)e−z/ζ, zk

i = k!ζk,

k = 1, 2, ... The mean value and the intensity of white Poissonian shot noise η(t) = λζ =

• DPλ ≥ 0,

DP = λζ2.

2 4 6 8 10 20 40 60 80 100 η(t) t (b) η(t) 2 4 6 8 10 20 40 60 80 100 η(t) t (c) η(t)

Figure : In (b): λ = 2, DP = 0.5; in (c): λ = 0.5, DP = 2.

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Quantities of interest

Average velocity lim

t→∞˙

x(t) = v + vω(t) + v2ω(t) + ... v = lim

t→∞

ω 2π t+2π/ω

t

ds ≺ v(s) ≻ . Velocity fluctuations σ2

v = v2 − v2,

v(t) ∈ [v − σv, v + σv]. Stokes efficiency, Pout = γv2, Pin = γ[v2 + σ2

v − DG]

εS = Pout Pin = v2 v2 + σ2

v − DG

.

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Average velocity

• 0.2
• 0.1

0.1 0.2 0.2 0.4 0.6 0.8 v η(t) = F (b)

F λ = 151

• 0.2

0.2 0.4 10−1 100 101 102 103 104 v λ (c)

F = 0.58 η(t) = 0.58

Figure : Anomalous transport regime. Parameters are a = 8.95, ω = 3.77, γ = 1.546, DG = 0.001, λ = 151.

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Velocity fluctuations

σ2

v = v2 − v2.

1.8 1.9 2 2.1 0.2 0.4 0.6 0.8 σv η(t) = F (e)

F λ = 151

1.8 2 2.2 2.4 10−1 100 101 102 103 104 σv λ (f)

F = 0.58 η(t) = 0.58

Figure : Anomalous transport regime. Parameters are a = 8.95, ω = 3.77, γ = 1.546, DG = 0.001, λ = 151.

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Stokes efficiency

εS = v2 v2 + σ2

v − DG

.

0.01 0.02 0.2 0.4 0.6 0.8 εS η(t) = F (h)

F λ = 151

0.01 0.02 0.03 10−1 100 101 102 103 104 εS λ (i)

F = 0.58 η(t) = 0.58

Figure : Anomalous transport regime. Parameters are a = 8.95, ω = 3.77, γ = 1.546, DG = 0.001, λ = 151.

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Take home message

Fundamental question

Can noise η(t) induce more effective transport than the constant force F?

Answer

Yes, the Brownian motor can move much faster, its velocity fluctuations are much smaller and the motor efficiency increases several times in both normal and absolute negative mobility regimes. New operating principle: consider replacing the constant force by nonequilibrium noise!

• J. Spiechowicz, P. H¨

anggi and J. Luczka, Phys. Rev. E 90, 032104 (2014)

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