Stochastic enhancement of absolute negative mobility Lukasz Machura - PowerPoint PPT Presentation

Stochastic enhancement of absolute negative mobility Lukasz Machura Jakub Spiechowicz, Jerzy � Luczka Department of Theoretical Physics Univeristy of Silesia, Katowice Barcelona, 2015 LM (IF U´ S) Noise induced ANM 7. UPON 2015 1 / 11

Motivation x ( t ) + dU ( x ) m ¨ x ( t ) + γ ˙ = F + A cos(Ω t + φ ) + ξ ( t ) dx 0.05 0.025 v 0 − 0.025 F − F + stall stall − 0.05 − 0.2 − 0.1 0 0.1 0.2 F LM, Kostur, Talkner, � Luczka, H¨ anggi Phys. Rev. Lett. 98, 040601 (2007) LM (IF U´ S) Noise induced ANM 7. UPON 2015 2 / 11

Motivation x ( t ) + dU ( x ) m ¨ x ( t ) + γ ˙ = F + A cos(Ω t + φ ) + ξ ( t ) dx 0.05 0.025 v 0 − 0.025 F − F + stall stall − 0.05 − 0.2 − 0.1 0 0.1 0.2 F Nagel, Speer, Gaber, Sterck, Eichhorn, LM, Kostur, Talkner, � Luczka, H¨ anggi Reimann, Ilin, Siegel, Koelle, Kleiner Phys. Rev. Lett. 98, 040601 (2007) Phys. Rev. Lett. 100, 217001 (2008) LM (IF U´ S) Noise induced ANM 7. UPON 2015 2 / 11

Main question Is it possible to enlarge the mean velocity reduce the fluctuations improve the quality of transport ? LM (IF U´ S) Noise induced ANM 7. UPON 2015 3 / 11

Main question Is it possible to enlarge the mean velocity reduce the fluctuations improve the quality of transport ? or in general Is it possible to improve the transport properties? LM (IF U´ S) Noise induced ANM 7. UPON 2015 3 / 11

Main question Is it possible to enlarge the mean velocity reduce the fluctuations improve the quality of transport ? or in general Is it possible to improve the transport properties? ”New operating principle: consider replacing the constant force by nonequilibrium noise!” ”Brownian motors in the micro-scale domain: Enhancement of efficiency by noise” → Monday talk by Jakub Spiechowicz J. Spiechowicz, P. H¨ anggi and J. � Luczka, Phys. Rev. E 90 , 032104 (2014) LM (IF U´ S) Noise induced ANM 7. UPON 2015 3 / 11

Main question or in general Is it possible to improve the transport properties? 0.035 λ = 4 0.03 λ = 16 0.025 λ = 64 λ = 512 0.02 deterministic 0.015 0.01 v 0.005 0 -0.005 -0.01 -0.015 0 0.05 0.1 0.15 0.2 f LM (IF U´ S) Noise induced ANM 7. UPON 2015 3 / 11

The idea: deterministic → dichotomic x ( t ) + dV ( x ) ¨ x ( t ) + γ ˙ = η ( t ) + a cos( ω t + φ ) + ξ ( t ) dx dichotomic noise η ( t ) = {− d , u } , d , u > 0 , probabilities of transition per unit time from one state to another Pr ( − d → u ) = µ d = 1 Pr ( u → − d ) = µ u = 1 , τ d τ u u η ( t ) − d start end t LM (IF U´ S) Noise induced ANM 7. UPON 2015 4 / 11

The idea: deterministic → dichotomic x ( t ) + dV ( x ) ¨ x ( t ) + γ ˙ = η ( t ) + a cos( ω t + φ ) + ξ ( t ) dx dichotomic noise η ( t ) = {− d , u } , d , u > 0 , probabilities of transition per unit time from one state to another Pr ( − d → u ) = µ d = 1 Pr ( u → − d ) = µ u = 1 , τ d τ u with one essential condition � η ( t ) � = F LM (IF U´ S) Noise induced ANM 7. UPON 2015 4 / 11

Let’s see... Comprehensive and fast numerical simulations (for poor) CUDA environment on a modern desktop GPU speed–up of order 3000x GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA J. Spiechowicz, M. Kostur and LM, Comput. Phys. Commun. 191, 140 (2015) LM (IF U´ S) Noise induced ANM 7. UPON 2015 5 / 11

simulations 0.20 deterministic d =0 . 010 ,µ d =0 . 100 ,µ u =0 . 010 d =0 . 010 ,µ d =0 . 010 ,µ u =1 . 000 0.15 d =0 . 010 ,µ d =10 . 000 ,µ u =1 . 000 d =0 . 001 ,µ d =10 . 000 ,µ u =0 . 001 d = − 0 . 010 ,µ d =0 . 010 ,µ u =0 . 010 0.10 v 0.05 0.00 0.05 0.0 0.1 0.2 0.3 0.4 0.5 � � f = η ( t ) LM (IF U´ S) Noise induced ANM 7. UPON 2015 6 / 11

a simple idea, T = 2 π/ω ≃ 1 . 28 x ( t ) + dV ( x ) ¨ x ( t ) + γ ˙ = u + a cos( ω t + φ ) + ξ ( t ) dx u LM (IF U´ S) Noise induced ANM 7. UPON 2015 7 / 11

a simple idea, T = 2 π/ω ≃ 1 . 28 x ( t ) + dV ( x ) ¨ x ( t ) + γ ˙ = − d + a cos( ω t + φ ) + ξ ( t ) dx -d u LM (IF U´ S) Noise induced ANM 7. UPON 2015 7 / 11

a simple idea, T = 2 π/ω ≃ 1 . 28 state u = 0 . 15: average residence time τ u = 1 ≃ 0 . 78 T state d = 0 . 3: average residence time τ d = 1 / 8 ≃ 0 . 1 T � � η ( t ) � = 0 . 1 0 . 15 η ( t ) = − 0 . 1 − 0 . 3 start end t -d u LM (IF U´ S) Noise induced ANM 7. UPON 2015 7 / 11

a simple idea d =0 . 3 ,µ d =8 . 00 ,u =0 . 15 ,µ u =1 . 00 d =0 . 3 ,µ d =0 . 80 ,u =0 . 15 ,µ u =0 . 10 0 . 15 0 . 15 − 0 . 1 − 0 . 1 − 0 . 30 − 0 . 30 t t d =0 . 3 ,µ d =0 . 08 ,u =0 . 15 ,µ u =0 . 01 0 . 15 0.005 dichotomic − 0 . 1 0 − 0 . 30 d =0 . 5 t d =0 . 3 µ b =12 d =0 . 3 ,µ d =80 . 00 ,u =0 . 15 ,µ u =10 . 00 d =0 . 3 d =0 . 3 0 . 15 d =0 . 3 µ b =80 u =0 . 15 -0.005 µ b =0 . 08 µ b =8 u =0 . 15 µ b =0 . 8 µ u =1 v − 0 . 1 u =0 . 15 u =0 . 15 µ u =10 u =0 . 15 − 0 . 30 µ u =0 . 01 -0.01 µ u =1 t µ u =0 . 1 d =0 . 5 ,µ d =12 . 00 ,u =0 . 15 ,µ u =1 . 00 0 . 15 -0.015 deterministic f =0 . 1 − 0 . 1 − 0 . 50 t LM (IF U´ S) Noise induced ANM 7. UPON 2015 8 / 11

Two state noise can also be better than deterministic force d = 0 . 85 , µ d = 8 . 7 , µ u = 0 . 708 , a = 8 . 95 , ω = 3 . 77 , γ = 1 . 546 , D = 0 . 001 LM (IF U´ S) Noise induced ANM 7. UPON 2015 9 / 11

Efficiency H. Linke, Appl. Phys. A 75, 167 (2002), Kostur, et. al, Physica A 371, 20, (2006) Stokes efficiency (against the friction force γ v ) γ � v � 2 ε S = P out = γ ( � v � 2 + σ 2 v − D ) P in d =0 . 3 ,µ d =0 . 08 ,u =0 . 15 ,µ u =0 . 01 Efficiency of the energy conversion 0 . 15 P out =? − 0 . 1 − 0 . 30 t LM (IF U´ S) Noise induced ANM 7. UPON 2015 10 / 11

Thank you for your attention! LM (IF U´ S) Noise induced ANM 7. UPON 2015 11 / 11