Optimal driving waveform for the overdamped, rocking ratchets Maria Laura Olivera Instituto de Ciencias de la Universidad Nacional de General Sarmiento. Buenos Aires Argentina Directors: R. ´ Alvarez-Nodarse & N.R. Quintero IMUS Doc-course May 20, 2010 Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 1 / 21
Plan 1 Introduction to the Ratchet effect 2 Motivation 3 Working Model 4 Objetives and methodology of the project 5 References Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 2 / 21
Introduction to the Ratchet effect Introduction Ratchet transport phenomena means a net directed motion of particles or solitons induced by zero − average forces . i.e. presence of the thermal noise and some prothotipical perturbation can drives the system out of the equilibrium (directed transport). The ratchet effect usually occurs when there is a break of at less one of main symmetries of the system: temporal symmetries: shift-symmetry (caused by time dependent forces) spatial symmetries (caused by ratchet potential ) We focus in the breaking of the temporal symetries Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 3 / 21
Introduction to the Ratchet effect Introduction Elements present in the physical system: Potential U(x) Friction force η ˙ x Force f(t) if any Noise ξ ( t ) The quantity of foremost in the context of transport in periodic systems is the particle current v: v = ˙ x The ratchet optimization is given by the maximun of average current of the particle. We study ratchet current and its relationship with the kind of driving force f(t): the driving force shape that maximizes � ˙ x � . Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 4 / 21
Introduction to the Ratchet effect Introduction The simplest system is a particle or soliton moving in a potential under influence of a friction force and noise. According to Newton’s second law: x + U ′ ( x ) + η ˙ m ¨ x − f ( t ) = ξ ( t ) , where the left-hand represent the deterministic, conservative part of the particles dynamics, while the right-hand side account for the effects of the thermal environment. Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 5 / 21
Introduction to the Ratchet effect The system is assumed to be overdamped, that is, the inertial term ¨ x is negligible and we arrive to: x + U ′ ( x ) = f ( t ) + ξ ( t ) . η ˙ Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 6 / 21
Motivation A few works have investigated the dependence of the ratchet current on amplitudes and phases of forces. We focus in two of them. 1. The first one: conjectures there exist a particular force waveform that optimally enhances directed transported. proposes a measure of ’degree of symmetry breaking’ (DSB). He defines the DSB of the symmetries of the force f(t) by the expresion: D s ( f ) ≡ �− f ( t + T / 2) � T ≡ (1) f ( t ) uses the biharmonic force to prove that (1) exhibits tha same quantitaive behaviour as a funtion of the symmetry-breaking parameters Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 7 / 21
Motivation consideres tha case of the elliptic force: f ellip ( t ) = ǫ f ( t ; T , m , θ ) ≡ ǫ sn (Ω t + Θ; m ) cn (Ω t + Θ; m ) , and put a biharmonic aproximation of it in (1). recovers the same optimal values obtained for the biharmonic force Concludes that the biharmonic force optimally enhances directed transport. Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 8 / 21
Motivation 2. The second one: Claims that the optimal driving waveform among a wide class of admissible funtions for an overdamped, rocking ratchet is dichotomous, with a form: � L , 0 ≤ t ≤ d f ∗ ( t ) = N ∗ , d ≤ t ≤ T where N ∗ ≤ 0 is a solution of ν ′ ( N ∗ ) = ν ( L ) − ν ( N ∗ ) L − N ∗ uses: � d � � ˙ x � = ν ( M ) d + ν − M (1 − d ) 1 − d put the dichotomous force in it and maximizes it with respect to M and to N, finding the dichotomic optimal force Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 9 / 21
Motivation 2. The second one: Then fixes the parameters ǫ 1 =L/3 and ǫ 2 =2L/3 valid for for the optimal biharmonic force in the limit of small force amplitud in the dichotomous force. concludes that the dichotomous responses is better than the biharmonic responses for larges L Question: The dichotomous responses is better than the biharmonic responses for larges L but the values of parameters fixed are valided only in limit of small force amplitud for the biharmonic force. Then we suspect that the main conclusion: ”the optimal force is dichotomous”, is not correct. Moreover, we think that there is not a general optimal force. Probably there is an optimal force for every family of forces. Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 10 / 21
Working Model Working Model We study the ratchet dynamic of an overdamped particle with position x(t) moving in a potential, suject to a force and noise. x − U ′ ( x ) + f ( t ) + ξ ( t ) = 0 − η ˙ where: η ˙ x represent the friction, with η the damping coeficient and ˙ x the derivate on time of the position, the velocity U ′ ( x ) is the derivate on position of the symmetric periodic potential ξ ( t ) is an independent Gaussian white noise Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 11 / 21
Working Model Working Model We study the ratchet dynamic of an overdamped particle with position x(t) moving in a potential, suject to a force and noise. x − U ′ ( x ) + f ( t ) + ξ ( t ) = 0 − η ˙ where: η ˙ x represent the friction, with η the damping coeficient and ˙ x the derivate on time of the position, the velocity U ′ ( x ) is the derivate on position of the symmetric periodic potential ξ ( t ) is an independent Gaussian white noise Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 11 / 21
Working Model Working Model We study the ratchet dynamic of an overdamped particle with position x(t) moving in a potential, suject to a force and noise. x − U ′ ( x ) + f ( t ) + ξ ( t ) = 0 − η ˙ where: η ˙ x represent the friction, with η the damping coeficient and ˙ x the derivate on time of the position, the velocity U ′ ( x ) is the derivate on position of the symmetric periodic potential ξ ( t ) is an independent Gaussian white noise Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 11 / 21
Working Model Working Model We study the ratchet dynamic of an overdamped particle with position x(t) moving in a potential, suject to a force and noise. x − U ′ ( x ) + f ( t ) + ξ ( t ) = 0 − η ˙ where: η ˙ x represent the friction, with η the damping coeficient and ˙ x the derivate on time of the position, the velocity U ′ ( x ) is the derivate on position of the symmetric periodic potential ξ ( t ) is an independent Gaussian white noise Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 11 / 21
Working Model Hypothesis: ξ ( t ) is independent Gaussian withe noise of zero mean, � ξ ( t ) � = 0 , with � ξ ( t ) ξ ( t ′ ) � = D δ ( t − t ′ ) , where D is noise coeficient and δ is dirac’s delta f(t) breaks the time-shift symmetry − f ( t ) = f ( t + T / 2) Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 12 / 21
Working Model We take: f(t) biharmonic force: f ( t ) = ǫ 1 cos ( q ω t + φ 1 ) + ǫ 2 cos ( p ω t + φ 2 ) where T=2 π / ω , φ 1 and φ 2 are the phases, p and q are coprimes with p+q odd and the amplitudes ǫ 1 and ǫ 2 are small U(x) periodic potential: U ( x ) = U 0 (1 − cos ( x )) We fix: q=1 p=2 ǫ 1 = ǫ f 0 ǫ 2 = (1 − ǫ ) f 0 Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 13 / 21
Objetives and methodology of the project Goals We will: study the average ratchet velocity dependence on the force strength check numerically that φ opt = 0 check if there is average velocity ratchet dependence on the frequency ω Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 14 / 21
Objetives and methodology of the project Goals We will: check the theorem [Lade, 2008]: If force(s) exist that optimize the current : � T x � = 1 � ˙ v ( f ( t )) dt , T 0 then one such optimal force is dichotomous, that is, for all t, f(t)=M or N, for some M and N. Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 15 / 21
Objetives and methodology of the project Numerical work Methodology to solve the problem: Numerical simulations Proofs of the analytic results Numerical simulations: We solve the equation: x ( t ) = − η f ( t ) − U ′ ( x ) + ξ ( t ) ˙ using f ( t ) = ǫ 1 cos ( q ω t + φ 1 ) + ǫ 2 cos ( p ω t + φ 2 ) U ( x ) = U 0 (1 − cos ( x )) with U 0 = 1, ω = 0 . 1, φ 1 = 0, φ 2 = π , p=2, q=1, η = 1, ǫ = 1 / 3,and √ D= 0 . 32 we obtain x(t) averaging 1000 realizations. Maria Laura Olivera (ICI.UNGS) Optimal driving waveform rocking ratchets May 20, 2010 16 / 21
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