A simple phenomenologic model for particle transport in spaceperiodic potentials in underdamped systems

Transcription

1 A siple phenoenologic odel for particle transport in spaceperiodic potentials in underdaped systes IG MARCHENKO 1,(a,b), II MARCHENKO 3, A ZHIGLO 1 1 NSC Kharov Institute of Physics and Technology, Aadeichesaya 1, 6118 Kharov, Uraine Kharov National University, Svobody Sq 4, 6177 Kharov, Uraine 3 NTU Kharov Polytechnic Institute, Frunze 1, Kharov, Uraine PACS 54-a Fluctuation phenoena, rando processes, noise, and Brownian otion PACS 56- Transport processes PACS 6Cb Nuerical siulation; solution of equations Abstract We consider the otion of an underdaped Brownian particle in a tilted periodic potential in a wide teperature range Based on the previous data [1] and the new siulation results we show that the underdaped otion of particles in space-periodic potentials can be considered as the overdaped otion in the velocity space in the effective double-well potential Siple analytic ressions for the particle obility and diffusion coefficient have been derived with the use of the presented odel The results of analytical coputations atch well with nuerical siulation data Despite particle diffusion and transport in washboard-type potentials have been subject of active research for decades [,3] investigations in this field eep surprising us In particular, it was shown in [1] that the particle diffusion coefficient in low friction systes can onentially rise with a teperature drop This research is iportant as it deals with both acadeic and practical issues This proble is involved in physical processes that occur in Josephson tunneling junctions, superionic conductors, phase-loced-loop frequency control systes, etc [] The research of the so-called Brownian otors is of special interest [3,4] Studies of the features of the transport of atos and point and linear defects in the ystal lattice in external fields are of extree iportance for the developent of new technologies for aterial science Thorough discussion of applications, references, and ethods of studying Brownian diffusion in washboard potentials can be found in [] Successful analytical ethods were developed for desibing the particle diffusion and transport in overdaped probles [5]; however, these ethods are of liited usability for systes with low energy dissipation In this paper we construct a siple phenoenological odel of the particle transport in underdaped systes, aiing it in particular at erientalists Following our previous wor [1] we analyze the particle diffusion and obility in a wide teperature range by eans of nuerical siulation We then propose an analytical odel and copare its results with those of coputer siulation The particle otion is desibed by Langevin equation We consider the case of one-diensional diffusion in a spatially periodic potential U (x) d x U ( x) x F ( t), (1) dx under the action of additional constant force F Here x is the particle coordinate, is its ass, is the friction coefficient (t) desibes theral fluctuations, assued Gaussian white noise, ( t) ( t') T ( t t'); () here is the Boltzann constant, and T is the teperature Overdot stands for tie differentiation The potential energy of the particle is U U( x) cos x (3) a where a is the constant of the one-diensional lattice The spatially periodic force exerted upon the particle by the ystal 1 1 lattice is F lat F sin( a x), where F a U The paraeters of the potential are the sae as in [1, 6] The siulation schee and statistical analysis used here are desibed in those papers To copare our results with those obtained by other authors we use diensionless teperature 1/ T T /U and friction coefficient ( ) a U [,7] In the previous paper [1] the teperature dependence of the diffusion coefficient of an enseble of particles oving in the tilted periodic potential was studied In this paper we are ore interested in directional transport The average velocity of the particles is used as a characteristic of this otion, N d ; (4) (a) E-ail: arch@iptharovua (b) Present address: NSC Kharov Institute of Physics and Technology, Aadeichesaya 1, 6118 Kharov, Uraine

2 I G Marcheno, I I Marcheno, A Zhiglo here N ( ) is the particle distribution function in velocity space Particle diffusion and transport in the underdaped case was first studied by coputer siulation in [8] Papers [9 1] present further studies of the (F) relation for different s The teperature dependence of the obility was not investigated properly We study the (T ) dependence in detail in this paper Fig 1 shows the dependence of obility / F ties the on the force F for different teperatures It is nown that is asyptotically independent of F at low force values; it satisfies the Einstein relation D/(T ) D here is the particle diffusion coefficient In spatially periodic structures D satisfies the Arrhenius relation D D ( U / T ) This linear dependence F is shown as the lower dashed line in the inset in Fig 1 An upper dashed line in the inset corresponds to F with 1/ ; such a regie (of the particle otion in viscous ediu) is realized at large F Fig1 shows that the obility at an arbitrary force value is between the two asyptotes corresponding to the Einstein relationship for the periodic lattice and for viscous drag As the teperature drops a ore abrupt transition fro one functional relationship to another is observed zero obility occurs This pattern of a hysteresis is often used for the qualitative interpretation of siulation results Fig 1 shows the siulation results for the obility as a function of the driving force F in the presence of theral fluctuations It is seen that an inease in teperature results in curves becoing flatter All these curves intersect at one point, at F 95 F (when ) This force value corresponds to F according to classification in [] To lain the features of ( F, ) behavior we consider how the distribution function N ( ) changes with an inease in force It has been nown for a while that N ( ) is biodal; we need to understand its shape better, at different F In our previous paper [1] Fig 3 showed the change in N () with an inease in F One axiu corresponds to zero velocity value and the other one to F / Albeit substantial redistribution of the particles between the loced and running solutions taes place with an inease in force the qualitative shape of the curves reains unchanged As deonstrated in Fig (in current paper), showing fitting of the data obtained in [1], N ( ) is accurately represented as a su of two Gaussians with the sae widths but different aplitudes The fitting parabolas in Fig have axia at velocity values of and F / Fig 1: The dependence of particle velocity on the force at different teperatures Stc F / / Stc The friction coefficient T =97, T =19, 3 T =194, 4 T =388, 5 T =58, 6 T =776 Inset: (F) at T =388 The Einstein relation for periodic lattice and the stationary velocity of the particle otion in viscous ediu are shown by the dashed lines A pattern of a particle otion at low friction is desibed in [] When the driving force F adiabatically ineases fro zero in the absence of theral fluctuations the particle obility is equal to zero up to certain force value, F 3This is the so-called loced solution At F F3 the obility is changed abruptly until the value of 1/ is reached At F F 3 the particle oentu acquired while driven through the lattice period is sufficient to overcoe the bonding force at the lattice sites; the particle then oves in the direction of F, with oscillating velocity The particle velocity averaged over the oscillation period is Stc F / This is the running solution [] If the force is adiabatically reduced fro large values, the obility reains constant, 1/, down to the value of F F1, at which the reverse jup to Fig : The velocity distribution function for F 9F T =19, 141 Approxiation with two Gaussians 1/ with dispersions ( T / ) is shown as a dashed line Least-square fitting shows that in the entire investigated range of actuating forces F the teperature dependence of N() can be rather accurately desibed by N( ) A T F / B T with F -dependent A and B To lain the teperature behavior of the particle transport the qualitative pattern desibed in [] requires further developent Let us consider first the case of deterinistic particle otion (with theral noise absent) The particles ay have different initial conditions, in particular velocities At fixed values of F and different classes of solutions of the equation of otion are realized depending on the value of initial velocity Fig 3 shows nuerical solutions of the equations of otion for different initial conditions All the particles were put into a local iniu of the total potential energy U( x) Fx at t Depending on its initial velocity the particle is either in a loced state (curve 1), or (5)

3 evolves towards the running state, oving with the average speed of F F / (curves and 3) It is seen in Fig 3 that there exists certain itical value of the initial velocity, At the particle is in a loced state, and at a running solution ensues arrows in Fig 3 The presented curves correspond to different values of the actuating force Each plot was constructed using the data obtained for different values of the initial velocities of the particle Fig 3: Tie dependence of the particle velocity for different initial velocities Stc F / The value of depends on F and Fig 4 shows the diagra of existence of loced and running solutions depending on the value of the actuating force F at a fixed value of ' 141 If F is lower than certain itical value F 1 the syste allows only one, loced, solution At F F1 two solutions appear, a loced and a running one Which solution is realized depends on the initial velocity At the particle dynaics evolves towards a loced solution and at towards a running one The dashed line in Fig 4, which separates the solutions, is the relation (F) for the given friction coefficient Fig 5 shows (F) for different values of It is seen that (F) varies approxiately linearly with the force near its itical value, for all values of studied Analysis of the data of coputer siulation also shows that the value of F is a linear function of at low friction The siulation data for ( F, ) near F F1 are accurately approxiated as F ( F, ) (6) F with diensionless fitting paraeters 88, 6, 15 (7) is the particle velocity required to overcoe the potential 1/ barrier when friction is absent: ( U / ) Let us study the transition to the stationary solution in the absence of theral noise Fig 6 shows a ind of soothed over oscillations particle acceleration, as a function of the particle velocity Naely, disete particle acceleration values x n1/ were calculated based on the difference between consecutive axiu velocity values, d dt x x n1/ t x n1 n1 t x n axia One such pair of consecutive n ; the lower index nuerates the x n is pointed with Fig 4: A phase diagra of loced and running solutions in (,F) plane At F F1 only loced solutions exist (the loced phase is colored blue) At F F3 only running solutions are realized (colored red) At interediate F both solutions coexist: at initial below the itical velocity (shown as the dashed line) the particle is eventually stopped at the full potential iniu; at larger the particle tends to the running solution, drifting with average velocity F / (solid red line) ' 141 Fig 5: Dependence of the itical velocity on the applied force in the absence of external noise for different friction coefficients Linear approxiation (6,7) is shown by dashed lines Curve 1 corresponds to 141, 354, 3 77 Dotted line - Fig 6 shows that the acceleration changes linearly with the velocity, away fro the region near the itical velocity It pushes the particle to the stationary solution At the x x slope d / d is /( ) whereas at this slope is / Let us consider the consequences of the contact of the enseble of particles with a theral reservoir While studying the otion of such an enseble osed to the action of external force the ain attention is usually paid to theral fluctuations of the force Classical consideration of this situation is given in Ch 11 of [] Under a contact with the theral reservoir while the force is changing adiabatically hysteresis loop is fored and one ore itical value of force F F eerges At F F when the teperature tends to zero a jup fro zero obility to a finite value occurs 3

4 I G Marcheno, I I Marcheno, A Zhiglo F c d (11) and ) we cast the N ( ) in the following for Fig 6: Particle acceleration as a function of velocity for different values of effective force The points with zero acceleration are pointed with arrows No theral force 1 F 4F, F 98F, 3 F F, 4 F 3F, 5 F 5F For desibing average (over oscillation period) dynaics of the particles in low dissipation systes we put forward the following heuristic odel, suggested by the ( ) relation noticed above We assue that particles are in certain effective velocity potential W ( ) [11], dependent on the values and F On the surface this approach is siilar to the ethod in [] However, the potential introduced here has a different diensionality (of [ ] /[ t] ), as the dynaics is desibed in velocity space, vs energy space in [] This odel we present in effect is ore ain the odel for the otion of active particles for soe special type of the potential [11] The otion of the particles can be desibed by the following equations: where the noise force satisfies x W (, F) ( t) (8) T ( t) ( t') Q ( t t' ) ( t t') (9) According to the siulations, the noralized velocity distribution function N ( ) is closely approxiated by (5) We thus put forward the following for of the effective potential: c W ) F ( at d at (1) where c and d are constants, with appropriate choice of which N ( ) ust closely atch the true distribution function in a wide region in -space around the axia of N () As it is shown below, an accurate fit for the particle obility and diffusion coefficients is achieved ( F, ) when c and d are fixed based on the following recipe 1) We require the potential continuity at, c at T T N( ) F d at T T (1) as if there is a twofold degeneracy in running states, where / T The nd relation between c and d follows fro the noralization N d T F T c d T T By perforing integration this is rewritten as c T 1 erf 1 erf T T F 1 d d 1 d T (13) (14) Understanding the for of N ( ) fro the first principles is clearly desirable This is wor in progress The average velocities can be found both analytically, through integration (eqs (4) and (13)), and nuerically The plots of functions are given in the lower part of Fig 7 as a function of the actuating force F Sybols represent the results of nuerical integration of Langevin equation (1 ), and solid curves correspond to calculations based on forulae (1 14) The results are shown for two values of the teperature, T =19 and T =388 It is seen that the siulated results for the average velocity of the underdaped otion of particles in the tilted space-periodic potential agree well with those for an overdaped otion in the double-well velocity potential (1) If overlapping of the two Gaussians is neglected (by replacing with in Eq (13)) the coputation of the average velocity is considerably siplified We thus obtain for F F F T (15) Approxiate solutions according to this siplification are shown in Fig 7 as dashed lines These also atch well the 4

5 siulation data This siplification gets less accurate, ectedly, at sall F, when F / (and ) is no longer uch saller than the Gaussian widths, ( T / ) 1/ Let us consider the way the proposed odel desibes the teperature behavior of the diffusion According to the Kubo relation, the diffusion coefficient can be found fro the integral of the velocity autocorrelation function, which can be written as the product of the velocity variance v ( v v) and its correlation tie corl [11]: D d v t vt v corl T corl (16) Here the inetic teperature introduced: T v has been Fig 7: Dependence of the obility (lower plot) and the inetic teperature (upper plot) on the actuating force Sybols show the siulation results, solid curves denote calculations according to forulae (8 9), the dashed line is approxiation (1 11) 1 T =19, T =388 ' 141 Fig 8 shows the agreeent between (F) curves found in siulations and through solving the odel (Eqs (1 14)) at different values of T and Curves 1 and correspond to ' 141, T =776 and T =194 respectively Curves 3 and 4 are for T =776, ' 8, ' 71 respectively The figure deonstrates that the proposed analytical odel accurately desibes the behavior of (F) for different values of in the underdaped case Fig 8: Average velocity as a function of applied force for different friction coefficients Curves depict calculations according to Eq (8 9), sybols show the siulation results The upper plot T =194, the lower plot T =776 Curve 1 8, and 3 ' 141, 4 ' 71 T N d (17) By nuerical integration with the distribution function (1) we find the dependence of inetic teperature on the acting force It is plotted in the upper part of Fig 7, together with the results based on siplified N ( ) (that assues non-overlapping Gaussians for loced and running states; dashed line) and the results for T found through direct Monte-Carlo siulation The siplified ression reads T T F T T (18) The Figure deonstrates good agreeent of the odel results, based on (1), and the siulation results Analysis of the obtained results shows that the axiu value of excess in inetic teperature T x is approxiately equal to U / and is actually independent of the value of T Maxial T is observed at the sae fixed value of F / F, which is also independent of T Thus, at low T values ( T U / ) if the correlation tie were fixed the diffusivity should have only changed slightly with the teperature At the sae tie paper [1] revealed an onential inease in D with a teperature drop This ust thus be due to the features of corl behaviour at low teperatures ( Q W ); we estiate this below For this purpose we use a siple two-state theory in which the velocity perfors transitions between two disete states [1], fro to and bac, with Kraers rates r Here, W Q аnd and W r Q are the absolute values of the curvatures of the potential W ( ) at, F / and respectively 5

6 I G Marcheno, I I Marcheno, A Zhiglo nuerical siulation results Proposed distribution function is obtained fro the picture of averaged particle dynaics in velocity space (8), with licitly presented dependence of the velocity potential W ( ) on F and Thus, we showed that the transport of particles in space-periodic potentials in the underdaped case can be treated as overdaped otion of active Brownian particles in velocity space with a double-well potential This allows one to efficiently use analytical ethods developed for the overdaped case Siple analytic approxiations for, T, corl can be used in designing, planning and analyzing erients Fig 9: corl as a function of actuating force F for different teperatures 1 T = 9 K ( T =97), T = 1 K ( T =19), 3 T = 18 K ( T =194) ' 141 Inset: teperature dependence of the axiu value corl Approxiation (1) for the effective potential is accurate near its axia, this aes the predictions based on (1) for D and correct (as the presented results show) However it is not reliable near, as can be seen fro the absence of continuous derivative of (1) at, whereas in part the siulations for N ( ) show that the latter is sooth at Thus cannot be reliably inferred fro our odel (1) So we only estiate corl (F ) up to an -dependent factor (not affecting the ain teperature dependence, as is clear below) According to [1] the correlation tie is 1/( r r corl ) Estiating for potential (1) we obtain diensionless T corl corl / F F 1 T, (19) REFERENCES [1] MARCHENKO IG, MARCHENKO II, Europhys Lett 1 (1) 55 [] RISKEN H, The Foer-Planc equation Methods of solution and applications (Springer) 1989 [3] HÄNGGI P, MARCHESONI F Rev Mod Phys 81 (9) 337 [4] MACHURA L, KOSTUR M, TALKNER P, et al Phys Rev- E7 (4) 6115 [5] REIMANN P, AN DEN BROECK C, LINKE H, et al, Phys Rev E 65 () 3114 [6] MARCHENKO IG, MARCHENKO II, JETP Letters 95 (3) (1), 137 [7] SANCHO JM, LACASTA AM Eur Phys J Special Topics 187 (1) 49 [8] COSTANTINI G, MARCHESONI F, Europhys Lett, 48 (5) (1999) 491 [9]LINDENBERG K, LACASTA AM, SANCHO JM, ROMERO AH New Journal of Physics 7 (5) 9 [1]LINDENBERG K, SANCHO JM, LACASTA AM, SOKOLO IM, Phys Rev Lett 98 (7) 6 [11]LINDNER B, NICOLA EM Phys Rev Lett 11 (8) 1963 [1]CW GARDINER Handboo of stochastic ethods for Physics, cheistry and natural sciences (Springer-erlag), 1981 where Fig 9 shows dependence corl corl ( F) / for different teperatures It follows fro (19) that at the axiu the value of corl really behaves as ( ~ corl T / T) with certain T~ The relation (19) obtained in this paper fully coincides with the results of coputer siulation in [1], where it was also found that the correlation tie ineases onentially with the deease in T The sae functional dependence is observed in results of direct nueric siulations of (8 1) Suing up, we showed that the velocity distribution function for underdaped particles perforing Brownian otion in a tilted periodic potential is closely approxiated by the su of two Gaussians (5), and presented licit ression (1) for that function valid for a wide range of the teperatures and friction coefficients Quantities lie obility, diffusion coefficient, inetic teperature of the particles found in the proposed odel agree well with direct 6