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) anggi 2 and Jerzy � Jakub Spiechowicz 1 , Peter H¨ Luczka 1 1 Institute of Physics, University of Silesia, 40-007 Katowice, Poland 2 Institute of Physics, University of Augsburg, 86135 Augsburg, Germany JS, PH and J� L Brownian motors efficiency 1 / 10

Le Chat` elier-Braun principle � v � = µ F . Typical response Absolute negative mobility JS, PH and J� L Brownian motors efficiency 2 / 10

Driven Brownian motor Minimal model exhibiting ANM expressed in the dimensionless variables x = − V ′ ( x ) + a cos( ω t ) + F + � ¨ x + γ ˙ 2 γ D G 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 ? JS, PH and J� L Brownian motors efficiency 3 / 10

Nonequilibrium noise Generalized white Poissonian noise n ( t ) � η ( t ) = z i δ ( t − t i ) , i =1 where n ( t ) is Poissonian counting process Pr { n ( t ) = k } = ( λ t ) k e − λ t . k ! The process η ( t ) presents white noise of finite mean and a covariance given by � η ( t ) � = λ � z i � , � η ( t ) η ( s ) � − � η ( t ) �� η ( s ) � = 2 D P δ ( t − s ) . Its intensity reads D P = λ � z 2 i � . 2 JS, PH and J� L Brownian motors efficiency 4 / 10

Distribution of the amplitudes { z i } of the δ -kicks ρ ( z ) = ζ − 1 θ ( z ) e − z /ζ , � z k i � = k ! ζ k , k = 1 , 2 , ... The mean value and the intensity of white Poissonian shot noise � D P = λζ 2 . � η ( t ) � = λζ = D P λ ≥ 0 , 10 10 � η ( t ) � � η ( t ) � (b) (c) 8 8 6 6 η ( t ) η ( t ) 4 4 2 2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 t t Figure : In (b): λ = 2, D P = 0 . 5; in (c): λ = 0 . 5, D P = 2. JS, PH and J� L Brownian motors efficiency 5 / 10

Quantities of interest Average velocity t →∞ � ˙ lim x ( t ) � = � v � + v ω ( t ) + v 2 ω ( t ) + ... � t +2 π/ω ω � v � = lim ds ≺ v ( s ) ≻ . 2 π t →∞ t Velocity fluctuations σ 2 v = � v 2 � − � v � 2 , v ( t ) ∈ [ � v � − σ v , � v � + σ v ] . Stokes efficiency, P out = γ � v � 2 , P in = γ [ � v � 2 + σ 2 v − D G ] � v � 2 ε S = P out = . � v � 2 + σ 2 P in v − D G JS, PH and J� L Brownian motors efficiency 6 / 10

Average velocity 0.4 0.2 F F = 0 . 58 0.1 λ = 151 � η ( t ) � = 0 . 58 0.2 0 (c) � v � � v � (b) 0 -0.1 -0.2 -0.2 10 − 1 10 0 10 1 10 2 10 3 10 4 0 0.2 0.4 0.6 0.8 � η ( t ) � = F λ Figure : Anomalous transport regime. Parameters are a = 8 . 95, ω = 3 . 77, γ = 1 . 546, D G = 0 . 001, λ = 151. JS, PH and J� L Brownian motors efficiency 7 / 10

Velocity fluctuations σ 2 v = � v 2 � − � v � 2 . 2.4 2.1 F F = 0 . 58 λ = 151 � η ( t ) � = 0 . 58 2.2 2 (f) σ v (e) σ v 1.9 2 1.8 1.8 10 − 1 10 0 10 1 10 2 10 3 10 4 0 0.2 0.4 0.6 0.8 � η ( t ) � = F λ Figure : Anomalous transport regime. Parameters are a = 8 . 95, ω = 3 . 77, γ = 1 . 546, D G = 0 . 001, λ = 151. JS, PH and J� L Brownian motors efficiency 8 / 10

Stokes efficiency � v � 2 ε S = . � v � 2 + σ 2 v − D G F F = 0 . 58 0.02 0.03 λ = 151 � η ( t ) � = 0 . 58 (h) 0.02 ε S ε S 0.01 (i) 0.01 0 0 10 − 1 10 0 10 1 10 2 10 3 10 4 0 0.2 0.4 0.6 0.8 � η ( t ) � = F λ Figure : Anomalous transport regime. Parameters are a = 8 . 95, ω = 3 . 77, γ = 1 . 546, D G = 0 . 001, λ = 151. JS, PH and J� L Brownian motors efficiency 9 / 10

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) JS, PH and J� L Brownian motors efficiency 10 / 10