The pros and cons of swimming in a noisy environment

Transcription

1 The pros and cons of swimming in a noisy environment Piero Olla ISAC-CNR & INFN Sez. Cagliari I Monserrato (CA) ITALY December 16, 2014

2 Streamlined version of a microswimmer Main approximations and simplifications Smallness of the moving parts (the beads): a R. Small stroke amplitude: R R. Discrete design: three identical beads connected by immaterial links. Internal forces exerted along the links. Symmetric deformations (it turns out that two degrees of freedom are sufficient for propulsion). The swimmer is constrained to move horizontally (no rotations and transverse translations). No external forces Quiescent fluid P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 2/9

3 The scallop theorem Repeating backwards a set of deformations, will bring the microswimmer center of mass back to its original position, irrespective of how fast the swimming strokes have been carried out. x 2 3 x 2 z 2 3 x 1 x 1 z z 1 2 Basic deformations z 2 1 S The swimming cycle takes the form of a closed trajectory in deformation space. We need at least two degrees of freedom. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 3/9

4 Optimization in the noiseless case We want to minimize dissipation at fixed migration velocity. S z 2 z 1 Migration speed u migr S T stroke Dissipation in one cycle Q Tstroke 0 ż 2 dt We define efficiency as the ratio η = u migr /Q. (This allows to phase out the uninteresting dependence of Q (u migr /z) 2 on z). P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 4/9

5 Optimization in the noiseless case We want to minimize dissipation at fixed migration velocity. S z 2 z 1 Migration speed u migr S T stroke Dissipation in one cycle Q Tstroke 0 ż 2 dt We define efficiency as the ratio η = u migr /Q. (This allows to phase out the uninteresting dependence of Q (u migr /z) 2 on z). Both quantities Q and u migr scale as S/T stroke only geometry left to optimize. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 4/9

6 Optimization in the noiseless case We want to minimize dissipation at fixed migration velocity. S z 2 z 1 Migration speed u migr S T stroke Dissipation in one cycle Q Tstroke 0 ż 2 dt We define efficiency as the ratio η = u migr /Q. (This allows to phase out the uninteresting dependence of Q (u migr /z) 2 on z). Both quantities Q and u migr scale as S/T stroke only geometry left to optimize. The shape of S that minimizes the length of S for fixed area, is the circle. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 4/9

7 Optimization in the noiseless case We want to minimize dissipation at fixed migration velocity. S z 2 z 1 Migration speed u migr S T stroke Dissipation in one cycle Q Tstroke 0 ż 2 dt We define efficiency as the ratio η = u migr /Q. (This allows to phase out the uninteresting dependence of Q (u migr /z) 2 on z). Both quantities Q and u migr scale as S/T stroke only geometry left to optimize. The shape of S that minimizes the length of S for fixed area, is the circle. For fixed trajectory S, the minimal Q is achieved at constant ż. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 4/9

8 In the presence of noise Consider first the weak noise regime = expansion around optimized noiseless case r 0 s z 2 z 1 ω stroke t+ψ ṡ +g s = ξ s ; ψ +g ψ = ξ ψ ; r 0 ξ i (t)ξ j (0) = 2Kδ ij δ(t) P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 5/9

9 In the presence of noise Consider first the weak noise regime = expansion around optimized noiseless case r 0 s z 2 z 1 ω stroke t+ψ ṡ +g s = ξ s ; ψ +g ψ = ξ ψ ; r 0 ξ i (t)ξ j (0) = 2Kδ ij δ(t) We must minimize heat production Q [ ] 1+ g ż gs 2 +[(1+s)(1+g ψ )] 2 +K g = min For fixed migration velocity and stroke frequency u migr (z 1 ż 2 z 2 ż 1 ) (1+s) 2 [1+g ψ ] = fixed ω stroke = fixed g ψ = 0. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 5/9

10 Better efficiency Small deformations g linear in s. g s = α s (ψ)[s s s (ψ)] g ψ = α ψ (ψ)[s s ψ (ψ)] Notice that the form of α s,ψ is irrelevant in the noiseless case, as long as s s,ψ = 0. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 6/9

11 Better efficiency Small deformations g linear in s. g s = α s (ψ)[s s s (ψ)] g ψ = α ψ (ψ)[s s ψ (ψ)] Notice that the form of α s,ψ is irrelevant in the noiseless case, as long as s s,ψ = 0. Result in the tangentially uniform case, α s,ψ = constant (no localization tangentially): Q Q opt 0 1+ Kα ψ α s (α ψ 2) Dissipation for fixed u migr is reduced with respect to the noiseless opt optimal value Q 0. (Optimal noisy swimming tangential deceleration for larger deformations). P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 6/9

12 Strong noise We may try to overcome the constraints of the scallop theorem and exert control only on one of the degrees of freedom, the other being activated by the noise noise + control z x 2 z 1 noise + constant spring 1 x 1 ż 1 +z 1 = ξ 1 ; ż 2 g(z) = ξ 2 ; (1) ξ i (t)ξ j (0) = 2Kδ ij δ(t); (2) u migr [z 1 (z 2 +g)] ; Q [g 2 +K z2 g] ;. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 7/9

13 Strong noise We may try to overcome the constraints of the scallop theorem and exert control only on one of the degrees of freedom, the other being activated by the noise noise + control z x 2 z 1 noise + constant spring 1 x 1 ż 1 +z 1 = ξ 1 ; ż 2 g(z) = ξ 2 ; (1) ξ i (t)ξ j (0) = 2Kδ ij δ(t); (2) u migr [z 1 (z 2 +g)] ; Q [g 2 +K z2 g] ;. Optimization problem: Q = min, under the constraint that u migr = fixed (and fixed spring constant), and that the PDF for z, from which Q and u migr are calculated, satisfies the stationary Fokker-Planck equation associated with Eqs. (1) and (2). P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 7/9

14 Results Optimization tells us that g is linear in z: g = αz 1 βz 2. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 8/9

15 Results Optimization tells us that g is linear in z: g = αz 1 βz 2. We find u migr α/(1+β): migration is larger, when the drive αz 1 is larger, and the constant β of the spring that limits z 2, is smaller. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 8/9

16 Results Optimization tells us that g is linear in z: g = αz 1 βz 2. We find u migr α/(1+β): migration is larger, when the drive αz 1 is larger, and the constant β of the spring that limits z 2, is smaller. Difficult to compare efficiency of the strongly noisy swimmer with that of its optimal deterministic counterpart, as the distributions of the stroke time T stroke and deformation amplitude z are not peaked around well defined mean values. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 8/9

17 Conclusions It is possible in principle to exploit thermal noise to increase the efficiency of a microswimmer (minimize dissipation at fixed mean migration speed). It is possible to exploit noise also to overcome some of the limitations of the scallop theorem (we do not need to act on two separate degrees of freedom to produce migration). Of course, what one gains from the point of view of efficiency, will be lost in the form of fluctuations in the migration velocity. P. Olla, ISAC-CNR The pros and cons of swimming in a noisy environment 9/9