A Brownian ratchet driven by Coulomb friction

Transcription

1 XCIX Congresso Nazionale SIF Trieste, 23 September 2013 A Brownian ratchet driven by Coulomb friction Alberto Petri CNR - ISC Roma Tor Vergata and Roma Sapienza Andrea Gnoli Fergal Dalton Giorgio Pontuale Andrea Puglisi Giacomo Gradenigo Alessandro Sarracino PRIN Project: Friction laws for granular media: ageing, memory and microscopic dynamics FIRB-IDEAS Project: Granular Chaos

2 The ratchet: how to extract work from molecular chaos An old problem (Smoluchowski 1912, Feynman 1963) Solution: rectification of thermal fluctuations becomes possible when operating under nonequilibrium conditions, as is the case in living organisms (2nd principle, Maxwell demon) The Feynman ratchet often referred to as a Brownian motor; could drive and control the activity at small scale: - rapidly increasing nanotechnology - small scale biological systems

3 Breaking equilibrium T 1 molecular gas granular gas T 2 ratchet effect with elastic collisions ratchet effect with inelastic collisions C. Van den Broeck, R. Kawai, P. Meurs, Phys. Rev. Lett. 93, (2004) Costantini G, Marconi Marini Bettolo U, Puglisi A, Phys. Rev. E 75, 4 (2006) geometric asymmetry temperature difference breaking of spatial symmetry breaking of temporal symmetry geometric asymmetry dissipative collisions Coulomb friction can break equilibrium and drive the motor even with elastic collisions and single temperature

4 A possible realization of a granular ratchet a usually neglected factor with asymmetric geometry The motion of the rotator can be modelled by a stochastic equation I = inertia, σ = sign function, Г = viscous drag

5 Stochastic dynamics It can be rewritten two relevant time scales inter-collisional time stopping time n = grain number density Σ = total rotator cross-section (perimeter x height) control parameter β β >> 1: friction dominated dynamics, Rare Collisions Limit RLC << 1: collision dominated dynamics, Frequent Collisions Limit FLC

6 Boltzmann equation Under the assumption of Molecular Chaos, from the stochastic equation one can derive the Boltzmann equation for the rotator velocity distribution function p t The collisional term contains - the rates for the transitions that include the Heaviside step function and the particles velocity pdf - the velocity dependent collision frequency We assume Gaussian particle velocity pdf

7 FCL ( ) The two limits The Boltzmann equation can be expanded in the small probe form factor = 0 for symmetric probes α = restitution coefficient This predicts: - zero drift for symmetric probe (A FCL =0) - ω v 0 for an asymmetric shape - zero drift for elastic collisions (α=1)

8 The two limits RCL ( ) Each collision produces an independent increment probe form factor = 0 for symmetric probes α = restitution coefficient This predicts But also - zero drift for symmetric probe (A RCL =0) - ω v 0 3 for an asymmetric shape - the rotation direction can change (with a zero crossing at some value of β) - the effect should survive also for α=1 (elastic collisions) - since in RCL ω v 0 3 while ω v 0, one expects a maximum drift at intermediate β

9 Monte Carlo simulations RCL dashed lines = theory FCL

10 Experimental Setup S L R high-speed camera for particle tracking and statistics diffusive lighting dome ball bearings angular encoder for probe monitoring friction source probe (not connected to the shaker granular gas plexiglass container accelerometer electrodynamic shaker

11 Experimental observation of net drift effect slow drift due to mechanical imprecisions

12 Experimental measurement of the drift effect in the two limits acceleration = 14 g FCL Frequent Collision Limit Rare Collision Limit RCL acceleration = 4 g grains pdf

13 Experimental measurement of the drift effect in the two limits stochastic resonances inversion of rotation

14 Elastic collisions: just simulations GR probe

15 Coulomb friction alone produces a ratchet effect even in the ideal (elastic) case. Being energy injected through an equilibrium bath, a possible miniaturized device acting as a micromotor driven by friction can be envisaged Conclusions Friction can be enough to supply the dissipation needed to drive the system out-of equilibrium The ratchet effect is computed, simulated and experimentally observed It is predicted that it can be observed even with non-dissipative collisions It must be observable at the micro and nanoscale not with macro grains (inelastic) Stochastic resonance that depends on the competition between friction and collisions It can separate rotations in opposite directions

16 Some useful references R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics I (Addison-Wesley, Reading, MA, 1963) p. Chap. 46 C. Van den Broeck, R. Kawai, P. Meurs, "Microscopic Analysis of a Thermal Brownian Motor", Phys. Rev. Lett. 93, (2004) G. Costantini, U.M.M. Bettolo, A. Puglisi, Granular Brownian ratchet model, Phys. Rev. E 75, 4 (2006) P. G. de Gennes, Brownian motion with dry friction, J. Stat. Phys. 119, 953 (2005) J. Talbot, R.D. Wildman, and P. Viot, Kinetics of a Frictional Granular Motor Phys. Rev. Lett. 107, (2011) J. Talbot, A. Burdeau, and P. Viot, Kinetic analysis of a chiral granular motor J. Stat. Mech. P03009 (2011) B. Cleuren, R. Eichhorn, Dynamical properties of granular rotors, J. Stat. Mech. P10011 (2008) Our work A. Gnoli, A. Petri, F. Dalton, G. Pontuale, G. Gradenigo, A. Sarracino and A. Puglisi, Brownian Ratchet in a Thermal Bath Driven by Coulomb Friction, Phys. Rev. Lett. 110, (2013) A. Gnoli, A. Sarracino, A. Puglisi, and A. Petri, Nonequilibrium fluctuations in a frictional granular motor: Experiments and kinetic theory, Phys. Rev.E 87, (2013) R. Balzan, F. Dalton, V. Loreto, A. Petri, and G. Pontuale, Brownian motor in a granular medium, Phys. Rev. E 83, (2010) Thank you for your attention!