globular proteins that are found in nature are special sequences of amino acids that fold rapidly into their native states under physiological conditions . the folding process in such proteins is considered to take place through motion in a mildly rugged free energy landscape with a prominent native basin .such a landscape forms through the principles of minimal frustration and maximal compatibility  . on the other hand ,random sequences are expected to form rough landscapes in which many local minima compete for occupancy , as illustrated in a model , for instance , in ref . .the question : what sequences make good folders , has been addressed mostly in the context of lattice models and various criteria have been proposed .one of these is that the lowest energy state the native state should have large thermodynamic stability  .another is that the location of the specific - heat maximum should coincide with that of the structural susceptibility  .it should be noted that the specific heat is a measure of fluctuations in the energy whereas the susceptibility is a measure of fluctuations in the number of pairs of `` amino acids '' which stay at their native spatial separation .a more intuitional criterion has been proposed by socci and onuchic .it is formulated in terms of two temperatures ( ) : and .the former is the folding ( or melting ) temperature that depends on the energy spectrum and defines a temperature below which the probability of occupancy of the native state , , is substantial .the latter relates to the dynamics and is the of the glass transition below which the protein gets trapped in a non - native state and folding times , , become very long .bad folders are then those sequences for which . the larger the ratio of to , the better the folding properties of the sequence .the concepts behind this picture appear to be satisfactory in lattice models where the conformations , including the native one , are defined in a clear - cut manner and are structurally separated .small lattice systems are simple enough to even allow for master - equation based exact analysis of the folding process  . here, we show that conceptual problems arise in off - lattice models and that they can be remedied by slightly relaxing the criteria for what it means for the system to stay in its native state .we focus on the highly studied structure - based , or go - like , coarse - grained model  which , by its construction , leads to folding with a minimal obstruction .this model has various variants but , invariably , it is defined primarily in terms of the contact map which specifies which pairs of amino acids form a non - bonded interaction ( most often through a hydrogen bond ) in the native state , which itself may be taken from the protein data bank ( pdb )  .the contact maps can be used in all - atom simulations for descriptive purposes , but in the structure - based models they also define the dynamics of the system because the contacts are endowed with effective potentials .the minima of the potentials are located at the native separations .  in molecular dynamics studies of folding ,one starts from an extended conformation and evolves the system until the native state is reached .time corresponds to the median first passage time . in the simplest approach ,the native state is declared to be reached when all of the native contacts become established .similarly , is calculated as the probability of all contacts being established simultaneously in a long equilibrated run and corresponds to a at which crosses .the plot of as a function of is typically u - shaped and the at the center of the u will be denoted as . the corresponding optimal will be denoted as .we define operationally as the at which on the low- side of the u.  the puzzling observation is that with these simple definitions , in off - lattice models is often found in the region in which is very small some previous examples are in refs . . furthermore , there are many go - model - based proteins for which the supposedly well folding chains are technically bad folders as they have a that is smaller or nearly equal to the corresponding value of . here, we examine 21 proteins within six go - like models and find that a minor change in the definition of what it means to be in the native state makes to coincide with in all cases .the change involves requiring that a fraction , , or more of the native contacts is established simultaneously , where , instead of being equal to 1 as in the simple approach , is changed to .the precise value of depends on the model and , to a lesser extent , on the protein .this result indicates that a situation with a few missing contacts makes the system practically reside within the very bottom of the native basin .we also find that the small reduction in leaves the nearly intact while boosting the apparent thermodynamic stability in a substantial manner .relaxing the definition involved in calculating is expected to enhance the apparent stability .what is surprising , however , is that making the model system technically a good folder requires to shift away from 1 by only 3 percentage points or less .it should be noted that the concept of the effective , the fraction of time in which all contacts are made simultaneously , is quite distinct from that of the average fraction , , of the native contacts that are present . it is often assumed that the temperature corresponding to crossing of through signals a transition to folding .this assumption does not seem to have been tested by the actual determination of the corresponding .an argument in favor of this idea can be derived from the behavior of the -dependent free energy , , which develops two minima which are of equal depths at a . however , at of , various subdomains may be set correctly without a proper establishment of the overall shape of the protein and hence the relevance of instead of .furthermore , an exact analysis  of a discretized -hairpin , performed within the so called munoz - eaton model  , indicates that does not constitute a reaction coordinate even for such a simple system ( see also refs .this is related to the fact that go - like models are defined in terms of contacts , but not in terms of .it should be noted , however , that all - atom simulations of a 3-helix bundle interpreted in terms of a bayesian framework suggest that using is a reasonable approximation in this case  . in each of the models studied here , is found to be close to the location of a maximum in the specific heat .the corresponding transition , however , should be associated with an onset of globular conformations since is significantly higher than and well above the whole region of in which folding to one of the globular states the native state is appreciable .we find that at the mean radius of gyration decreases rapidly on cooling .we also show that the -range of proper folding and other characteristics , such as mechanostability , depend on the description of the backbone stiffness .there are two ways of looking at the folding process .the perspective taken here is that it is a kinetic phenomenon .another is that it is a smoothed out equilibrium phase transition that should be characterized through trajectories that last much longer than typical folding times .studying is an approximate way to describe this transition . in this paper , we show that the two perspective can be made compatible if the equilibrium description is focused on , _i.e. _ on the fraction of time in which nearly all contacts are present instead of on average number of contacts. fluctuations in the total energy do not indicate folding .there is some analogy to spin glasses here .phase transitions in ferromagnets show , in the thermodynamic limit , as a divergence in the magnetic susceptibility , which is a measure of fluctuations in magnetization and is related to spin - spin correlation functions .however , in spin glasses , it is the nonlinear susceptibility , and not susceptibility , that is sensitive to the transition from the paramagnetic phase . at the transition, the susceptibility displays merely a cusp .this nonlinear susceptibility is related to different spin correlations .the six models we discuss here differ by the selection of the contact potential and the description of the local backbone stiffness , but the contact map is common : it is the map denoted by ov+rcsu in ref . , which is an extension of the ov map .the ov map is derived by considering overlaps between effective spheres assigned to heavy atoms in the native state .the radii are equal to the van der waals radii multiplied by 1.24  .the rcsu part adds to ov those contacts which are identified by considerations of a chemical nature . in the case of the highly mechanostable i27 domain of titin ( pdb:1tit ) ,the most important role of rcsu is to add an ionic bridge which contributes to the strength of a mechanical clamp in this protein .generally , ov+rcsu leads to similar and sometimes superior folding properties compared to ov .the improvements show as the broader outline of the u - shaped dependence of on and stronger thermal stability .the potentials assigned to the contacts between residues and are given by \ ] ] where is either 6 ( the 12 - 6 potential ) or 10 ( the 12 - 10 potential ) .the parameters are derived pair - by - pair from the native distances between the residues the minimum of the potential must coincide with the -c-c distance .consecutive -c atoms are tethered by the harmonic potential , where is the native distance between and .note that the stiffness coefficient used here incorporates the customary factor of in the hookean energy term .we shall adopt this convention in the definition of other harmonic terms discussed below .we consider three versions of the local backbone stiffness . in many of our previous works , we have used the chirality ( ch ) potential where is the number of residues and the coeffcient is set to , as ceases to be -dependent at larger values of  .the chirality of residue is defined as where and is the average . is the native value of .a more common way to account for the stiffness is to combine the bond - angle term with the dihedral term + k^3_{\phi}[1- cos(3(\phi - \phi_n))]$ ] where the subscript indicates the native values . for small departures of from , , where ) . in this limit, the dihedral term is equivalent to the chirality potential  .the values of the coefficients can be derived through all - atom simulations by matching average conformation - dependent energies to the coarse - grained expressions .poma et al . have studied sugar - protein complexes at room in this way and , as a byproduct , derived the following effective values for the parameters pertaining to proteins : =1.5 kcal / mol ,=100 kcal/(mol ) , =45 kcal/(mol rad ) , and =5 kcal/(mol rad ) .the contacts considered were defined by using the ov approach and the contact potential was 12 - 6 . in units of ,these parameters are : =66.66 / , = 30 /rad , and = 3.33 /rad . assuming , for simplicity , the equality of the two dihedral coeffcients , we get =0.66 /rad .this model of the backbone stiffness will be denoted as an ( for angular ) .a similar set of values for the backbone stiffness has been used by clementi et al . ( with a different contact map ) .it is : =100 / , =20 /rad , =1 /rad , and =0.5 /rad .this model will be denoted as an.  finally , we consider models with no backbone stiffness at all , such as the self organized polymer ( sop )  . in the original sop model , the harmonic representation of the peptide bondis expressed as a combination of the 12 - 6 and fene potentials so that they act partially as the 12 - 6 contact energies to accelerate the numerics . here, we use the sop idea of not including any backbone stiffness also to the systems with the 12 - 10 contacts .we provide comparisons of protein stretching in the sop models relative to the models with finite backbone stiffness .note that the contact map in the original sop model is defined through a cutoff distance whereas we use ov+rcsu .the models we study here can be characterized by pairs of symbol such that the first symbol denotes the contact potential and the second the model of the backbone stiffness .these are : i ( 12 - 6 , ch ) , ii ( 12 - 6 , an ) , iii ( 12 - 10 , an ) , iv ( 12 - 10 , an ) , v ( 12 - 6 , sop ) , vi ( 12 - 10 , sop ) .each of these is studied as described in refs . using our own code .in particular , we use the overdamped langevin thermostat and the characteristic time scale in the simulations , , is of order 1 ns .the equations of motion were solved by the 5th order predictor - corrector method . due to overdamping, our code is equivalent to the brownian dynamics approach .a contact is considered to be established if its length is within 1.5 or 1.2  for the 12 - 6 and 12 - 10 potentials respectively  .the equilibrium parameters are determined on at least 3 long trajectories that last for 100 000 .it should be noted that the all - atom derived coarse - grained parameters generally depend on the location in the sequence and on the protein so they should be considered more as an order of magnitude estimates than precise values . in particular , the effective calculated within the sheets has been found to be kcal / mol and within the helices :  .the uniform value of 1.5 kcal / mol used above has been selected for two reasons : 1 ) it is characteristic of the turn regions and 2 ) it coincides with the strength of a better - binding hydrogen bond .it corresponds to 755 k so the room temperature ought to be about 0.4 .however , due to the considerable uncertainty in this assessment , our previous practice has been to assume , when using models with a uniform , that the room situation occurs when folding is optimal as this is when the model protein behaves as the real one . of 1.5 kcal / mol is consistent with 1.55 kcal / mol ( or 106 16 pn , which itself is consistent with 110 pn of ref . ) that was obtained by matching characteristic values of the force , needed to unravel proteins by constant - speed stretching within the coarse - grained model , to the experimental results  .this estimate involves making extrapolations to the experimental speeds and adopting the ov - based contact map .when using the ov+rcsu map and 38 proteins the effective becomes 1.35 kcal / mol ( or 93 pn )  .the results pertaining to the thermodynamic and folding properties of the six models are illustrated in figures [ twelve ] , [ ten ] , and [ sop ] for the i27 domain of titin .mechanical stability of this protein has been assessed experimentally in many studies . the ovrcsu contact map for 1tit consists of 198 contacts. table i summarizes the results for all of the 21 proteins studied .the bottom panels show the -dependence of and  the specific heat normalized to its maximal value .these two quantities do not depend on the choice of .the top panels show the median and the middle panels show the effective .the general convention is that the solid data points connected by the solid lines correspond to of 1 ( the undoctored value ) whereas the open symbols connected by the dotted lines relate to the value of at which is in the very middle of the u - shaped folding curve it is written in the right - hand corner of the middle panels .this value of is indicated by the vertical line .the folding curve is obtained by fitting the results based on at least 105 trajectories to the function described by .the horizontal dashed lines indicate the level of the optimal multiplied by three from which an estimate of can be deduced .the horizontal lines in the middle and bottom panels indicate the level corresponding to ( either for or for ) .the shift in the apparent varies between 0.04 for model i and 0.25 for model iii .it is 0.13 , 0.22 , 0.06 , and 0.11 for models ii , iv , v , and vi respectively .the 12 - 10 potentials are generally effectively more contained , or narrower , than the 12 - 6 ones and the effects of being smaller than 1 are stronger .even the shifted effective values of are much lower by at least 50% than . on the other hand ,the separation between and does not exceed 0.07 in any model .the typical value of is of order 1 ( though the smallest is 0.7 model vi and then it is 0.85 model i ) which is puzzlingly high .the sop model v has properties which are very similar to that of model i : the kinetics are close but at =1 the stability of model i is higher .we find that when we remove the dihedral term in model ii , while keeping the bond - angle term , shifts to a half - way position between models i and ii .we now consider the mechanical stability .figure [ stretch ] illustrates stretching of 1tit for and .the terminal residues of the protein are attached to two harmonic springs  .one of the springs is anchored at one end and another is moving at a speed of 0.005 / .we monitor the force , , exerted by the protein on the moving spring as a function of the moving end displacement , . in the initial state , the protein is set in its native state .when pulling is implemented at , the patterns obtained by using the six models are seen to be remarkably similar though the heights of the force peaks are distinct . on the other hand , at , the force maximaare within the noise level .notice that one can not model experiments on stretching by simulating the corresponding kinetics at ( or which is very close to ) .when comparing mechanostability of many proteins , as in ref. , it is convenient to choose one fixed that , for most proteins , would be within the basin of good foldability , though not necessarily equal to . for modelsi , v , and vi such a temperature is 0.3  . for 1tit at this , we get of 2.04.14 , 2.05.12 , and 2.44.09 / respectively when extrapolating to the typical experimental speed of 600 nm / s .these estimates were obtained based on 100 trajectories .it should be noted , however , that each model has its own calibration of the value of  . on considering 46 cases of the theory - experiment comparison ( in addition to 38 of ref . , also 1qys  , 2 directions of stretching for 1a1 m  and 5 experimental pulling speeds for 1g6p  ) , we get the following calibrations for : 93.1.1 , 119.7.0 and 92.2.9 pn  for models i , v , and vi respectively. the corresponding values of the coefficient of determination , , are 0.846 , 0.622 and 0.661 , indicating that model i correlates with the experiment much better than the simpler sop models .similar estimates apply when stretching at the individually determined .in summary , the simple criterion of folding that all contacts are established may often lead to a perception that the structure - based model is unphysical because its thermodynamic stability is substantial in a low - temperature region in which folding is difficult or absent .this may happen especially when the contact map has too few elements to be appropriate dynamically , as is the case of , for instance , the csu - based contact map  used in ref .one needs to have a sufficient inter - residue connectivity for a model protein stay three - dimensional on its own  .there are many definitions of the contact map and none of them is perfect .the ov+rcsu map selected here is particularly good , as tested in ref . but even this one does not remove the unphysicality . we have argued here , that allowing for a small percentage of the native contacts not being established boosts the substantially but affects the time scales of folding only in a minor way ( with the exception of model vi ) .determination of the effective thermodynamic stability should then be less strict ( which also helps in hiding imperfections in the definition of the contact map ) for the model to work well . the simple strict criterion , however , is simpler to use because it does not require having the knowledge of the exact value of the parameter . it should be quite adequate when comparing relative thermodynamic stabilities of proteins .the good performance of the sop models testifies to the fact that the properties of the structure - based models depends primarily on the contact map , though the particular choices for the backbone stiffness affect the optimal folding temperatures .model polypeptide chains can be used to describe globular proteins if folding takes place in the temperature range in which getting to the native state is associated with a substantial probability of staying in the vicinity of this state .we have introduced parameter that defines what should be meant by the effective native vicinity : the temperature at which is at the center of the u - shaped plot of _ vs. _ . on decreasing beyond 0.97 ,this plot becomes increasingly broadened , optimal times get shorter , but the center of the optimal folding kinetics approximately stays unchanged . another way to definethe relevant vicinity of the native state is to determine a at which the optimal is shorter by , say , 10% compared to the =1 situation .this approach is harder to implement numerically because determining a precise value of would require generating a large number of trajectories .however , qualitatively , it would yield similar results to what we proposed here .we appreciate stimulating discussions with a. b. poma and b. rycki .the project has been supported by the national science centre , grant no .2014/15/b / st3/01905 and by the eu joint programme in neurodegenerative diseases project ( jpnd cd fp-688 - 059 ) through the national science centre ( 2014/15/z / nz1/00037 ) in poland .d. k. west , p. d. olmsted , and e. paci , j. chem . phys . * 124 * , 154909 ( 2006 ) . h. m. berman , j. westbrook , z. feng , g. gilliland , t. n. bhat , h. weissig , i. n. shindyalov , and p. e. bourne , nucleic acids res .* 28 * , 235 - 242 ( 2000 ) ; www.rcsb.org ..summary of the results obtained for models i - vi .the protein pdb structure codes are in the first column . in the case of 1cfc and 2p6j, we consider the first of the nmr - derived structures .the original structure file of cohesin 1aoh has missing several side groups and we use the repaired structure as described in ref . . in the case on 1emb , we consider the chromophore molecule to be represented by three mutually connected effective amino acids ( threonine , glycine , and tyrosine ) at locations 65 - 67 .for each of the models , there are three columns : the column with heading gives the values parameter at which the kinetic optimum coincides with , the next column lists the values of at this value of , the last column lists the temperatures corresponding to the maximum in the specific heat .the temperatures are in the units of and the error bars in each entry are smaller than 0.01 .the two bottom lines provide the average over the proteins and the standard deviation for a given model . [cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ]

Abstract: in structure - based models of proteins , one often assumes that folding is accomplished when all contacts are established . this assumption may frequently lead to a conceptual problem that folding takes place in a temperature region of very low thermodynamic stability , especially when the contact map used is too sparse . we consider six different structure - based models and show that allowing for a small , but model - dependent , percentage of the native contacts not being established boosts the folding temperature substantially while affecting the time scales of folding only in a minor way . we also compare other properties of the six models . we show that the choice of the description of the backbone stiffness has a substantial effect on the values of characteristic temperatures that relate both to equilibrium and kinetic properties . models without any backbone stiffness ( like the self - organized polymer ) are found to perform similar to those with the stiffness , including in the studies of stretching .


recent developments of new experimental techniques have allowed significant progress in the lagrangian description of turbulence  . although this aspect has been investigated theoretically since the beginning of the 20th century , the extreme difficulty of recording particle trajectories in high reynolds number turbulence restricted experimental investigations .the availability of very high speed imaging sensors gives access to the lagrangian acceleration in fully developed turbulence .it has been shown that the probability density function ( pdf ) of a lagrangian acceleration component is strongly non - gaussian , _i.e. _ , the pdf exhibits extremely large tails  .this has been also observed in direct numerical simulations of fully developed turbulence   these observations resemble features obtained in the field of non - extensive statistical mechanics  .this opens two promising perspectives : first toward a statistical theory of turbulence and second toward a dynamical modeling of the motion of fluid particles .the latter comes from the theoretical link first established by c. beck  between the tsallis entropy and a stochastic equation for the lagrangian acceleration .these developments led to a series of models for the pdf of lagrangian accelerations .the multitude of models needs to be tested by an accurate comparison with experimental data , especially in the low probability tails of the acceleration pdf .  herewe present new experimental results that have allowed us to improve the estimation of the acceleration pdf .first we describe the modifications that have been implemented on the previous experiment , then we show the acceleration pdf and compare it to the different classes of models .finally we show and discuss some pdf s of acceleration conditioned on velocity which are improvements over results presented previously in sawford et al .  .the experimental set - up has been described in detail in previous publications  .the reader should refer to these for detailed information on the experiment .let us recall briefly the main features .the flow under consideration is of the von krmn kind : the water is driven by two counter - rotating disks , 20 cm in diameter , 33 cm apart , mounted in a cylindrical container .the rotation frequency of the disks is regulated at 3.5 hz for the experiment under consideration here . the integral scale of the flow has been shown to be 7 cm and the energy dissipation rate to be 1.14 m  .the transverse and axial spatial coordinates of small tracer particles ( diameter 46  m , density ) are recorded by two silicon strip detectors .each detector provides a 1d recording of the particle trajectories at a sampling rate up to 70 khz .the root - mean - square velocity is 0.43 ms , so that the taylor based reynolds number is 690 .the kolmogorov time is  ms , the sampling frequency is  khz so that corresponds to 65 samples .we study here only one transverse component of the acceleration .even though the velocity components have been shown to be anisotropic , the acceleration components are close to isotropic at that reynolds number  .  compared to the experiment reported earlier , we are now able to record data three time faster ( in terms of global efficiency , _i.e. _ , including transfer rate from the digitizer to the computer , repetition rate , etc . ) because of a hardware upgrade of the data - acquisition computers .this allowed us to obtain more data in a reasonable time .the laser used for illumination has also been upgraded to a frequency doubled pulsed yag laser with mean output power as high as 35 watts , a typical pulse duration of 200 ns and a pulse repetition rate of up to 70 khz .this pulsed laser improves the efficiency of the silicon strip detectors due to the short duration of the pulses .another difference lies in the signal processing .as stressed in voth et al . , one has to low - pass filter the trajectories to filter the noise in order to have access to the acceleration . in our previous work ,the filtering and differentiation were achieved by fitting a parabola to the data , in a time window whose duration had to be appropriately chosen . here, this procedure has been modified to improve the performance of the data processing algorithm .we now use a differentiating and filtering kernel so that the acceleration is obtained by convolution , where is the convolution product and is the spatial coordinate of the particle .we have chosen a gaussian kernel   to achieve both a low pass filter and the second order differentiation of trajectories , one simply has to differentiate twice the gaussian kernel in practice , however , time is discrete and one has to use a finite time window , therefore the kernel is truncated .this requires a modification of the normalization .the following kernel was used in the data analysis ( for discrete times ) : where and are normalization constants chosen to ensure that and , _i.e. _ , the second derivative of a constant is 0 and the second derivative of a parabola is 2 .note that , if equals one time sample , one recovers the simple 3 points discrete second derivative ( independently of the value of ) .if one converges to the continuous kernel when goes to infinity .the results obtained from the filtering operation are found to be in agreement with those obtained by the parabolic fitting procedure .note that due to the truncation , the actual width of the filter is a slightly lower that .the kernel parameters are samples , .the solid line is the stretched exponential fit .insert : linear plot . ]the results presented here required continuous acquisition of data for two weeks .the data set consists of points , which include the new measurements and the data reported in the previous publications .the experimental probability density function ( pdf ) is presented in fig .[ pdf ] as dots .the new data allows the estimation of the pdf to reach probabilities more than one order of magnitude lower than the results in  , _ i.e. _ , less than .[ pdf ] also shows the stretched exponential fit : with , , and is a normalization constant ( ) .this expression is the same as the one proposed earlier  , with slightly different values due to improved accuracy of the fit and perhaps also a difference in the reynolds number .the initial values were , , at .this stretched exponential fit is shown to follow the experimental data very well .  .the _ rms _ error is estimated by splitting the data set in 6 subsets . ]  in fig .[ error ] , an estimation of the statistical dispersion of the pdf is shown .it has been obtained by splitting the whole data set into 6 subsets .the relative _ rms _ uncertainty is displayed .one observes a minimum of 3% for small values of the acceleration , , for which the number of samples is maximum .the uncertainty then increases as increases .it remains lower than 40% for .  .the kernel parameters are samples , .the solid line is the stretched exponential fit . ]one consequence of our better estimation of the acceleration pdf is that we can conclude that the fourth moment converges .[ flatness ] displays the contribution to the fourth moment : .the first observation is that this function decreases as increases . from our fit ,the decay is consistent with a stretched exponential : this means that the integral converges .the data gives a flatness of and the stretched exponential fit gives a flatness value equal to .the possible influence of the particle response time on this value is expected to be low as the characteristic time of the particle is .the filtering procedure has a limited impact because of the low value of the filter width . changing the width from to decreases the variance by 10 and the flatness by 15 .the shape of the pdf is unchanged for .thus we expect the flatness to be slightly underestimated but do not expect any qualitative change in the shape of the pdf .   among the large class of superstatistics  ,four specific examples have been proposed to model the observed acceleration pdf .the first one , proposed by c. beck  ( and referred to as beck- here after ) , is based on the -distribution .one very interesting feature of this analysis is the possibility of making a connection between a statistical approach and a dynamical approach .it allows us to build a stochastic equation for the lagrangian time dynamics of acceleration . in the same spirit , a. m. reynolds  and then c. beck  proposed a second model ( referred as reynolds and beck - log resp . )based on log - normal statistics .the approach of reynolds is much more ambitious .he tries to build a full stochastic equation that models the acceleration .this model takes into account not only the small time scales dynamics but also the large scales .he takes all numerical parameters from experiments or direct numerical simulations reported in the literature .therefore there are no free parameters in his model , once the reynolds number is chosen .thus the result of the model is not a fit to the experimental data .another more classical approach has been proposed by t. arimitsu and n. arimitsu  ( referred as arimitsu ) .they use the properties of scale invariance of the navier - stokes equation to build a multifractal description of the acceleration pdf .these four models are displayed in figures [ model ] and [ modelflat ] together with the experimental data .the first obvious observation is that the -distribution model ( with ) does not fit the data .the decay of the tails displays qualitatively incorrect behavior .indeed , for any value of the parameter , the model predicts a power - law decay which is not observed . for close to 3/2 , which gives the best fit , the fourth moment diverges which is not the case for the experimental data .  ,  .thick dashed line : beck - log ,  . solid line : arimitsu , and  .dot - dashed : reynolds  .insert : linear plot . ]  . left : linear plot , right : loglog plot .same symbols as previous figure . ]the modification of the distribution to log - normal , introduced later by c. beck , makes the fit qualitatively much better .in particular the flatness , for takes a finite value .the reynolds model does not agree as well with the data as the others .one reason is that there are no free parameters in this model .second , we low - pass filter the acceleration at the scale .this scale is small but not zero .if the temporal position signals from the reynolds model are low - pass filtered at the same scale then the overlap with the experimental data is highly improved  .the multifractal model by arimitsu also provides a good agreement with the experimental results . concerning the region of low acceleration region, the arimitsu model captures the correct behavior , contrary to the other models .reynolds and beck s log - normal models overestimate the zero acceleration probability whereas the tsallis statistics model underestimates it .  for all these models the departure from the experimental curve for large accelerations is best viewed in fig .[ modelflat ] .most of the proposed models do not peak at the same value of acceleration as the experimental curve .nevertheless most of them are very close to the experiment .   .the curves have been shifted for clarity by a factor 2 , the top one being unchanged .( b ) the conditional mean acceleration .( c ) the conditional acceleration variance .dots are the experiment , the solid line is a 6th order polynomial fit . ]conditional acceleration statistics are important in many second order stochastic models  .a very large number of statistics is required to obtain acceleration pdf s conditional on large velocities since large velocities are rare events .the new large data set allowed us to calculate distributions conditional on larger velocities than those shown in sawford et .figure [ cond ] shows pdf s of the transverse component of acceleration conditional on the transverse component of velocity .the conditional probability distributions are highly non - gaussian .they have almost the same shape as the unconditional distribution .the conditional acceleration variance increases with increasing velocity which seems to be due mainly to a widening of the tip of the distribution as can be see in in figure [ cond ] . for homogeneous isotropic turbulencethe conditional mean acceleration should be zero .departures from zero reflect the anisotropy of our flow although dns of homogeneous isotropic turbulence has also shown slight departures from zero  .sawford et .al . present a scaling argument that indicates that the conditional acceleration component variance to leading order .a fit to our data in fig .. shows good agreement with this .the dependence of acceleration distributions on the velocity violates local homogeneity  which is an assumption of kolmogorov 41 theory .our new set of data helps to discriminate among different classes of models .in particular , the model based on the tsallis entropy is shown to disagree with the experimental observations : the acceleration pdf does not display any power law decrease and is well fitted by a stretched exponential decay .one consequence is that the fourth moment of the acceleration component is finite and we observe a value of the flatness close to 55 .the models based on log - normal statistics proposed by beck and reynolds or multifractal analysis ( arimitsu ) are indeed close to our experimental observations .since the only difference between beck s two models is the underlying statistics ( tsallis or log - normal ) , the hypothesis of tsallis statistics must be discarded in order to reproduce the observed behavior of the acceleration pdf .the use of a log - normal statistics and the physical link of the parameter in beck s models to the dissipation rate are in agreement with the refined 1962 kolmogorov - obukhov theory of turbulence which is known to describe well the features of intermittency .we also show that the joint pdf of acceleration and velocity is not gaussian .the acceleration variance conditioned on velocity is also not constant .these results have important consequences on the structure of the stochastic equation required to model dispersion by turbulent flows .g. a. voth , a. la porta , a. m. crawford , j. alexander and e. bodenschatz , measurement of particle accelerations in fully developed turbulence " , _ j. fluid mech ._ , * 469 * pp . 121 - 160 ( 2002 ) . g. a. voth , a. la porta , a. m. crawford , e. bodenschatz c. ward and j. alexander , a silicon strip detector system for high resolution particle tracking in turbulence " , _ rev ._ , * 72*(12 ) , pp .43484353 ( 2001 ) .a. la porta , g. a. voth , a. m. crawford , j. alexander and e. bodenschatz , fluid particle accelerations in fully developed turbulence " , _ nature _ , * 409 * pp . 10171019 ( 2001 ) .n. mordant , p. metz , o. michel and j .- f .pinton , measurement of lagrangian velocity in fully developed turbulence " , _ physical review letters _ * 87*(21 ) 214501 ( 2001 ) . n. mordant , j. delour , e. lvque , a. arnodo and j .- f .pinton , long time correlations in lagrangian dynamics : a key to intermittency in turbulence " , _ physical review letters _ * 89 * ( 25 ) 254502 ( 2002 ) .s. ott and j. mann , an experimental investigation of the relative diffusion of particle pairs in three - dimensional turbulent flow " , _ j. fluid mech . _ * 422 * pp .207 - 223 ( 2000 ) .p. vedula & p. k. yeung similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence " , _ phys .fluids _ , * 11 * , 1208 ( 1999 ) .t. gotoh and d. fukayama , `` pressure spectrum in homogeneous turbulence '' , _ phys ._ , * 86 * , 3775 ( 2001 ) .c. beck , dynamical foundations of nonextensive statistical mechanics " , _ phys .lett . _ * 87 * , 180601 ( 2001 ) .t. aritmitsu and n. arimitsu , multifractal analysis of fluid particle accelerations in turbulence " , arxiv : cond - mat/0210274 ( 2002 ) .a. m. reynolds , on the application of nonextensive statistics to lagrangian turbulence " , _ phys . fluids _ ,* 15*(1 ) l1 ( 2003 ) . c. beck , lagrangian acceleration statistics in turbulent flows " , arxiv : cond - mat/0212566 ( 2002 ) . b. l. sawford , p. k. yeung , m. s. borgas , p. vedula , a. la porta , a. m. crawford , e. bodenschatz , acceleration variance and conditional variance statistics in turbulence , to be submitted .c. beck and e. g. d. cohen , superstatistics " , arxiv : cond - mat/0205097 ( 2002 ) .a. m. reynolds , private communication . b. l. sawford , reynolds number effects in lagrangian stochastic models of turbulent dispersion " , _ phy .fluids _ , * a3 * , p. 1577b. l. sawford , p. k. yeung , eulerian acceleration statistics as a discriminator between lagrangian stochastic models in uniform shear flow " , _ phys . fluids _ , * 12 * , pp .2033 ( 2000 ) .

Abstract: we report experimental results on the acceleration component probability distribution function at to probabilities of less than . this is an improvement of more than an order of magnitude over past measurements and allows us to conclude that the fourth moment converges and the flatness is approximately 55 . we compare our probability distribution to those predicted by several models inspired by non - extensive statistical mechanics . we also look at acceleration component probability distributions conditioned on a velocity component for conditioning velocities as high as 3 times the standard deviation and find them to be highly non - gaussian .


remarkable detection capabilities appearing in animal hearing are essentially governed by both a passive mechanical and an active biophysical procedures in the cochlea  .the active amplification of the living cochlea was conjectured by gold in 1948  and is now qualitatively studied widely through the hopf equation in mathematical models  . in order to give a better understanding to the experimental observations ,a more accurate model is required .furthermore the physical origin of the active oscillation remains still unclear . in mammals , the outer hair cells ( ohcs )are known to be force generators for auditory sensitivity and frequency selectivity . proposed mechanisms of the force generationare contraction of the ohc itself  and an active motion of the hair bundle  .energy depot model can describe the active phenomena  , because the energy supplied by the depot can induce a negative dissipation . a similar mechanism for a negative stiffness in the bullfrog s hair bundlehas been reported  .it is reasonable to consider the ohc as an energy depot since the ohcs play an important role in the active amplification of the cochlea even though we do not know the exact mechanism of the energy supply . in natureall living things have some mechanisms of protection for survival .we expect that there might be a similar mechanism to protect cochlea from damage by any external factors such as loud stimulation .hence , we propose a braking mechanism in the cochlea to prevent the basilar membrane ( bm ) from an excessive oscillation which could damage the cochlea .for this purpose , a critical velocity of the ohc oscillation will be introduced in our model .  in the following sections , we describe our model and reproduce the essential phenomena of the active cochlea observed in recent experiments such as the compressive nonlinearity , the dynamics of the bm response , nonlinear distortion and the spontaneous bm oscillation ( sbmo ) .it will also be pointed ut that our model naturally includes the hopf bifurcation model  in the case of weak stimulation . however , in the regime of strong stimulation , the present model is essentially different from the hopf bifurcation model .it shows that the response of the bm is non - linear and compressive at the cf .( note that the slope of the solid line is less than one . ) otherwise , the response of the bm is a linear function of the stimulation yielding that the slopes are .these results agree with the experimental observations  . ]it is well known that a change of transmembrane potential in the ohc induces a somatic vibration .this implies that a part of the electric energy is converted into the mechanical energy . in this sense , we believe that the ohc functions as an energy depot for the bm .the ohc as the energy depot stores energy supplied and converts it into kinetic energy overcoming dissipation .this energetics can be described by the following energy balance equation  . where is the energy density of the energy depot , the rate of the energy pumping into the energy depot per unit area , the rate of the internal energy dissipation , the rate of the energy converting into the kinetic energy of the bm , and the velocity of the bm . considering that constant contribution of can be incooperated into and that conversion rate of the energy into the kinetic energy of the bm will not be sensitive to the sign of , we assume that is an even function of without a constant term . the energy depot model in which has been extensively discussed  . in this paper , we include the next higher contribution to and introduce a critical velocity of the bm to describe a braking mechanism  .hence the conversion rate can be written as where is the critical velocity .for a finite the oscillation of the bm can be braked by the negative conversion rate into the kinetic motion of the bm .note that corresponds to a system without the braking mechanism .the critical velocity will be roughly estimated from the contraction of the ohc later for our numerical calculation .khz and the stimulation is .the onset time is much shorter than the offset time which is not shown in full scale .this result is qualitatively consistent with the experimental observations  . , , and .the stimulation is unloaded at .the lower panel shows the response represents the case without the braking mechanism , showing a continuous increase as long as the stimulation is on . ]kgm , , m/s ( solid thin line ) ; , , m/s ( solid thick line ) ; , , ( dashed thin line ) , and , , ( dashed thick line ) .the velocity of the bm is ( not ) saturated when the stimulation is sufficiently large with ( without ) the braking mechanism . ]now it is worthwhile to compare our newly proposed model with the one previously introduced in ref .  . whereas we treat the energy density ( ) of the energy depot(ohc ) , in ref . the energy density ( ) of the propagating hydrodynamic wave is considered .the model in ref . describes the coupling of the active elements to the propagating wave , while the present model describes the coupling between the energy depot(ohc ) and the bm oscillation .the emphasis in this paper is on showing that the active and compressive response arises naturally from the generalized energy depot concept .an adiabatic approximation in which the adaptation of the energy depot is very fast  yields that it should bear in mind that is not constant .the oscillation of the bm is then governed by where is the density , the physical damping coefficient , the stiffness , and the displacement of the bm . includes the contributions of a sound wave ( external stimulation ) and a noise .the noise is assumed to be a gaussian white noise . is the active force density acting on the bm by the ohc , note that the langevin equation , eq .( [ full ] ) , describes the passive and the active response in a unified and natural way .it should be noted that the active force , eq .( [ active ] ) , can be either positive or negative depending on the value of . when is smaller than ( weak stimulation ) , a positive active forceis provided on the bm ( active mechanism ) . on the other hand , when is larger than ( strong stimulation ) , a negative active force , hence an extra drag force , is exerted on the bm ( braking mechanism ) .the mass density of the bm is known to be about / m  and the reported damping coefficient , , of the bm has a large variation from to kg m  depending on the models . for a numerical calculation, we set the damping in the range of kg m .it has been reported that the guinea pig ohcs are able to change its length up to when the transmembrane potential is varied .this corresponds to a displacement of m for the ohcs of m at the high - frequency end  .hence , in the regime of high frequency of , the adjustable parameter is roughly estimated about m/s . by considering the ohc as a capacitor and using the observed membrane potential , membrane capacitance  , and the scale of the ohc  , we estimate the value of to be about kg s .the response of the bm to the sound wave can be then obtained straightforwardly using eqs .( [ full ] ) and ( [ active ] ) . in the following calculations , we set and the characteristic frequency ( cf ) , . it will be shown later that these relations arise from the conditions of the hopf bifurcation . in the present calculations ,the noise width of the distribution is chosen to be zero db .the compressive and dynamical responses of the bm are obtained as shown in fig .[ compressive ] and fig .[ dyn ] , respectively .[ compressive ] demonstrates that the response of the bm is non - linear and compressive at the cf , while the response is linear at other frequencies .this compressive response of the bm agrees well with the observations  as expected .[ dyn ] shows the time course of the bm response , reproducing the experimental observation that the onset time of the response is shorter than the offset time , although the onset time is somewhat larger than the experimental values  .it is important to notice that the velocity of the bm increases continuously as long as the stimulation is being turned on if the braking mechanism is not included .hence , the braking mechanism introduced in our model is essential to explain the experimental observation at a sufficiently strong stimulation .a dependence of the bm velocity on is plotted in fig .the results clearly show that lower critical velocity produces smaller response at the same stimulation , thus producing a supercompressive behavior as observed in experiment  .this is the effect of the braking mechanism , which is significantly different from the hopf bifurcation model .note that the saturation of the bm response disappears if the braking mechanism is not considered ., the brownian fluctuation is suppressed , while in the region of , the brownian fluctuation is amplified . for the plots ,the intensity of the thermal noise is , and the strength of the fluctuation is .( a ) , m/s .( b ) , m/s . ]  khz , , both strengths of the stimulation are , the response is measured at the place where the , /s and . ]the gain of the oscillator is defined as the ratio of the displacement of the bm to the stimulation , using eq .( [ full ] ) , we obtain the gain , which is a scaling function of the stimulation when the stimulation is not too large , .this result agrees well with the experiment  and the result of the hopf bifurcation model  , .indeed our model includes the hopf model naturally . to show this , we expand the active force , eq .( [ active ] ) , up to the lowest nonlinear term when is small .( [ full ] ) is then approximately written as in the fourier spectra space , this directly corresponds to the hopf equation  with the following bifurcation conditions , hence our energy depot model with a braking mechanism reduces to the hopf bifurcation model in the limit of weak stimulation .one of the most important results of our model is an amplification of the thermal noise itself in the absence of an external stimulation . at the bifurcation point or when , the thermal noise is well suppressed .however , when , so that too much of the internal energy is converted into the kinetic energy , the thermal noise can be largely amplified as shown in fig . [ vc ] . here , we set the mean average of the noise at and the width at .since it can be generally assumed that the oscillators are distributed in the vicinity of the bifurcation point , the noise may be amplified incoherently in the absence of the input signal . however , when a weak input signal with the same cf is introduced , it may phase - lock to the already existing amplified noise , thus enhancing the selectivity as discussed in ref .  . because it is believed that the sbmo is crucial for understanding the spontaneous otoacoustic emission  , our model may provide a clue to explore this interesting and important phenomenon .[ dist ] shows the nonlinear distortion of the response of the bm at the position where the cf is .the stimulation contains two tunes , and .the strengths both are .our model shows clearly the distortions at , in the response of the bm .in this paper , we propose a theoretical scheme for the active and passive response of the bm based on a concept of the energy depot model with a braking mechanism . according to the experimental observations ,the ohc is assumed to play a role of the energy depot by supplying an active force to the bm and also by reabsorbing an excessive kinetic energy like the regenerative - brake system in automobiles and electric vehicles  .it is shown that a single equation of motion , eq ( [ full ] ) , can produce all the essential passive and active behaviors . moreover, our model shows that the brownian noise can be sufficiently amplified under a certain condition , thus leading to the sbmo .although we applied our model only to the mammalian cochlea , it is equally possible to be applied to the nonmammalian vertebrates  because the only required physical characteristics are a supply of the active force and a self - adaptation . in this sense, we believe that the concept of the energy depot model with a braking mechanism is quite universal and can be applied to any underdamped biological systems .this work is supported by the korea science and engineering foundation ( kosef ) ( r01 - 2006 - 000 - 10083 - 0 ) .furness and c. m. hackney , _ the structure and composition of the stereociliary bundle of vertebrate hair cells _ , in _ vertebrate hair cells _ ,r. a. eatock , r. r. fay and a. n. popper ( springer , new york , 2006 ) , chapter . 3 , pp .

Abstract: high auditory sensitivity , sharp frequency selectivity , and otoacoustic emissions are signatures of active amplification of the cochlea . the human ear can also detect very large amplitude sound without being damaged as long as the exposed time is not too long . the outer hair cells are believed as the best candidate for the active force generator of the mammalian cochlea . in this paper , we propose a new model for the basilar membrane oscillation which successfully describes both the active and the protective mechanisms by employing an energy depot concept and a critical velocity of the basilar membrane . one of the main results is that thermal noise in the absence of external stimulation can be amplified leading to the spontaneous basilar membrane oscillation . the compressive response of the basilar membrane at the characteristic frequency and the dynamic response to the stimulation are consistent with the experimental results as expected . our model also shows the nonlinear distortion of the response of the basilar membrane .


parametrically excited surface waves , or faraday waves , are generated when a container of fluid is vertically vibrated . above an acceleration threshold , waves appear on the fluid surface .the waves vibrate at half the driving frequency , a feature associated with the parametric resonance that is responsible for pumping energy into the surface - wave modes .  in most `` faraday experiments , ''the container is driven sinusoidally .that is , its vertical position may be described by , and its acceleration by . in this paper, we explore the consequences of driving the container with a triangle - wave forcing .although the difference between the two waveforms matters little physically , we shall show that it simplifies the analysis remarkably .there are two broad motivations for our study .first , although the theory of hydrodynamic stability occupies an important conceptual place in current courses in hydrodynamics , standard examples such as thermal convection ( rayleigh - bnard convection , or rbc ) and shear flow between rotating concentric cylinders ( taylor - vortex flow , or tvf ) are too complicated to present in standard undergraduate courses .textbooks for such courses ( e.g. , ) usually limit themselves to heuristic arguments .by contrast , our analysis is simple enough to present in a single class to third- or fourth - year students .( the calculation s difficulty is equivalent to solving the kronig - penney model for energy bands in a one - dimensional solid .this example is commonly presented in undergraduate solid - state - physics courses . )a second motivation comes from the increasingly prominent role that the faraday experiment occupies as an example of a non - equilibrium pattern - forming system .most of the work on pattern formation done to date has focused on convection ( rbc ) and shear flow ( tvf ) . however , a recent series of papers has established a number of advantages for the faraday experiment .very large cells ( wavelengths wide ) may be created , and the basic ( fast " ) time scale can be very short ( sec . )the short time scale is particularly important because the dynamics of pattern - forming instabilities is often slow .current work looks at features that occur on time scales of to .convection , by contrast , usually has sec . , or slower .as we shall show below , our variant of the faraday experiment clarifies our understanding of the instability itself .the analysis separates into two pieces : one is the derivation of the surface - wave dispersion relation ( section iii ) ; the other is the description of how parametric pumping injects energy into the fluid ( section ii ) . by choosing triangle - wave forcing, we greatly simplify the latter piece .we hope that this will ultimately allow the nonlinear analysis ( particularly of high - viscosity fluids ) to be pursued further than it has been hitherto . for this reason, we make no restriction on the fluid viscosity ; the only approximation made is in linearizing the navier - stokes equations .the body of this paper is organized as follows : in section [ sec : mathieu ] , we study the related problem of the mathieu equation with delta - function forcing .solving this problem turns out to be mathematically equivalent to solving for the acceleration thresholds of faraday waves . in section [ sec : faraday ] , we apply this analysis to the faraday experiment with triangle - wave forcing . in section [ sec:2freq ], we consider a simple extension , driving by an asymmetric triangle wave , which models the recent two - frequency " experiments that have attracted much attention . section [ sec : conclusion ] contains a brief conclusion .an infinitely extended continuous system such as a fluid can be thought of as containing a continuum number of modes .to linear order , the modes are independent and the system s behavior can be determined from the study of each mode separately .nonlinear effects imply mode coupling . in this section, we examine the effects of parametric forcing on a single mode . in section [ sec : faraday ] , we apply the results derived here to the continuum of modes present in the faraday experiment .there are two ways to pump energy into an oscillator .the first is by direct forcing , where the prototypical equation is where is the driving force and is often chosen to be . for small damping , a directly forced oscillator has a resonant response ( of amplitude ) at .the second way to pump energy into an oscillator is by parametric forcing , whose prototypical equation is the ( damped ) mathieu equation : = 0 .\label{eq : mathieu}\ ] ] when is sinuosoidal , there is an infinite series of resonances centered on , , , , , for integer .the strongest of these resonances is the first ( ) , and a response at half the driving frequency is good evidence that a system is being driven parametrically .below , we shall find it convenient to rephrase the resonance conditions in terms of periods : , where and .although may be sinusoidal , the solutions are not .let and assume the term in eq .( [ eq : mathieu ] ) is proportional to and can be rewritten as , implying that has a component . adding this to generates a term , and so on .( [ eq : mathieu ] ) must in general be solved numerically . for special choices of , however , eq . ( [ eq : mathieu ] ) may be solved analytically . the simplest case is equal to a sum of delta functions . a slightly more complicated case , equal to a square wave , was discussed in these pages some years ago . in section [ sec : faraday ] , we show that analyzing a fluid container driven by a triangle wave is equivalent to studying the mathieu equation driven by a series of equal - amplitude delta functions , with alternating signs .( see fig .[ fig : driving]b . ) here , we review the analysis of for the particular case of delta - function forcing with equal and then with alternating signs .   in unitswhere time , damping rate , and forcing are all scaled by , we wish to solve = 0 .\label{eq : delta1}\ ] ] the forcing function is shown in fig .[ fig : driving]a . in between delta - function kicks , " the system is just a free , damped oscillator .let be the solution valid between the kicks at and .one finds with and .the kick imposes two conditions linking and :   is continuous .   velocity jumps discontinuously .this condition can be derived by integrating eq .( [ eq : delta1 ] ) over a small interval of time centered on .the conditions may be expressed as :  [ eq : conditions ]   imposing the continuity and velocity - jump conditions allows one to relate to : where and are the real and imaginary parts of , respectively , , , and is the scaled forcing amplitude .( [ eq : solution ] ) and ( [ eq : an ] ) constitute a solution to eq .( [ eq : delta1 ] ). given initial conditions , we can find by first iterating the map to find the proper .here , is the matrix in eq .( [ eq : an ] ) . then use eq . ([ eq : solution ] ) to find .( [ eq : an ] ) can also be used to find the acceleration threshold for resonant response .parametric resonance differs from ordinary resonance in that it is a true instability . if is below a threshold , at long times . for , grows without bounds .( adding a nonlinear term in to eq .( [ eq : delta1 ] ) will limit the oscillator s amplitude . in practice, nonlinearities are always present in the physical situation being modeled . )our goal now is to find .the key observation is that the long - time amplitude of the motion is controlled by the eigenvalues of the matrix : if their magnitude is greater than , the motion s amplitude will grow each period .if the magnitude of both eigenvalues is less than , damping will reduce the motion each cycle and the system will tend towards . loosely speaking , accounts for the energy pumped into the system during each period , while represents the energy lost during .the matrix has unit determinant .if we write in the general form the eigenvalues are given by where .instability arises when both eigenvalues are real and the magnitude of the largest exceeds .simple algebraic manipulations then give for the threshold condition in the present case , the matrix elements are and . inserting these into the threshold condition eq .( [ eq : threshold ] ) , we find this threshold condition is plotted in fig .[ fig : delta_tongues]a for different choices of the damping .the curves generated by choosing the plus sign are centered on and are known as subharmonic tongues , " while the curves generated by choosing the minus sign are centered on , and are known as harmonic tongues .these curves are qualitatively similar to those of the mathieu eq . with sinusoidal forcing .= 6 cm   in our version of the faraday experiment , the container s vertical position is a triangle wave . taking two time derivatives of , we see that the acceleration is a periodic sequence of delta functions of equal magnitude but alternating sign . for this reason, we generalize the analysis of section [ sec : equalsigns ] to a driving term of the form \right ] \right\}x(t ) \label{eq : doubledelta}\ ] ] let be the solution valid over  ] . except for the kicks at and  ] , we find    where and . combining these , we can write .since det ( det )(det , we have the same threshold condition , , leading to the resonance tongues are shown in fig . [fig : delta_tongues]b .there are only subharmonic tongues , a feature that turns out to be special to the use of our particular driving function eq .( [ eq : doubledelta ] ) .  to understand more intuitively why there is resonance for the subharmonic case but not for the harmonic case , consider the two time series shown in fig .[ fig : timeseries ] .fig [ fig : timeseries]a shows a time series for , ) , where is the natural period of the oscillator .[ fig : timeseries]b shows the case , .the delta - function impulses are indicated by up and down triangles on each curve while the force vector for each kick is indicated by an arrow above or below the triangle . because the delta functions are multiplied by the coordinate , the sign of the velocity increment depends on the position of the particle relative to the origin . in the subharmonic case, the impulse always increases the speed of the motion . in the harmonic case , the up triangles decrease the speed , while the down triangles add to it .because the overall motion is damped , it is clear that the net effect of the kicks is to decrease the energy of the system .one can understand this better by considering the matrices and in the limit of zero damping ( harmonic case ) .the cos and sin terms are then and , respectively , and the product is just the identity matrix , _ independent of the value of the forcing _ .thus , no energy is added to the system even as it is forced arbitrarily hard .( one kick adds energy while the next one removes it . ) when damping is added , as in fig .[ fig : timeseries]b , the system is always stable about .the harmonic case differs from the harmonic case of sinusoidal driving , where there is still a resonance because the forcing is distributed over the whole period of the oscillator .note that when , higher order tongues have higher thresholds than the tongue .the bottom of this first tongue defines an overall threshold that represents the global - minimum amplitude required to excite oscillations .we can find by differentiating eq .( [ eq : doubledelta_thresh ] ) for .this yields for .this transcendental equation can be solved numerically or perturbatively for small , which gives  ] .to lowest order in , the resonance conditions are just the zero - viscosity ones : and . at this order ,the overall forcing threshold is given by .in this section , we shall apply the above results to the linear - stability analysis of the flat surface of a viscous fluid held in a laterally infinite container of fluid of infinite depth .( the finite - depth case is straightforward , but the algebra is longer . ) already in 1883 , rayleigh had suggested that the instability studied fifty years earlier by faraday might be due to the phenomenon of parametric resonance , and he wrote down and explored the properties of eq .( [ eq : mathieu ] ) independently of mathieu .seventy years later , benjamin and ursell showed that the normal modes of the linearized , inviscid navier stokes equations each obeyed a mathieu equation . in the benjamin - ursell calculation, the effects of damping must be added perturbatively ( using as the bulk dissipation coefficient , where is the kinematic viscosity and the magnitude of the wavenumber of the disturbance ) . recently ( 1994 ) kumar and tuckerman performed a complete linear analysis that is valid for all viscosities . since the analysis for arbitrary viscosity is only slightly more complicated than that for small viscosity , we present the more general result .we begin by deriving the equations of motion , following kumar and tuckerman .the navier - stokes equations of motion for the fluid layer are   \vec{u } & = & - { 1\over\rho }     \vec{\nabla } p + \nu \nabla^2 \vec{u } - g(t ) \hat{z } \\\vec{\nabla } \cdot \vec{u } & = & 0 .\end{aligned}\ ] ]  here , is the fluid velocity field in the lab frame of reference and is the pressure field .there is a simple solution , corresponding to a fluid at rest in its own frame of reference : and .linearizing about the at - rest solution , we find for the perturbing fields and     ( in the new frame of reference , gravity does not appear in the bulk equations of motion .it will resurface in the boundary conditions . ) applying the operator ( and using the incompressibility condition , we eliminate the pressure and obtain where is the -component of the velocity field .  at the bottom of the cell , no - slipboundary conditions imply that . using the incompressibility condition ,we rewrite this as   at the air - fluid interface , there are three conditions .the first is a kinematic one that relates the fluid velocity to the position of the interface : \zeta =  u |_{z=\zeta},\ ] ] where is the location of the interface .to linearize , we drop the term and evaluate the right - hand side at , giving .the second condition asserts that there are no tangential stresses at the interface .since the interface normal is just to first order , one has that at , where the stress tensor is defined as \,\delta_{ij } + \eta\,(\partial_i u_j +  \partial_j u_i ) , \label{e : stresstensor}\ ] ] with and ranging over , , and , and where is the ordinary fluid viscosity . taking the horizontal divergence of eq .( [ e : stresstensor ] ) , we have where is the horizontal velocity field at and the incompressibility condition was used to write the last identity .the third condition relates the normal stress to the laplace pressure caused by surface tension acting on a curved interface : where is the mean curvature of the interface and the second term comes from writing the curvature to first - order in .we have , at , taking the horizontal divergence of the navier stokes equation , one can obtain a second relation involving the pressure field and use this to rewrite eq .( [ e : pressure ] ) in terms of and alone :   because the horizontal eigenfunctions of a laterally infinite container are , we can replace by , and by .we then have the following linearized equations of motion for :  ' '' ''  as with the mathieu eq ., one can easily solve the unforced equations between kicks , where surface waves propagate and are damped by viscosity .let be the -component of the velocity valid for .let be the amplitude of surface displacement over the same time period . from eqs .( [ eq : eqsofmotion ] ) , we find  [ eq : un_zetan ]   2  eqs .( [ eq : un_zetan ] ) give and .the normal - stress condition ( eq .[ eq : eqsofmotiond ] ) gives the dispersion relation for the surface waves : where .note that this expression for resembles the one found for the mathieu equation .[ eq : solution ] and the line following . )the laterally infinite container supports a continuum of modes indexed by the 2d wavenumber .a mode of wavenumber has a damping coefficient .damping gives a ( second - order ) decrease in the frequency , so that one complication , relevant only for higher viscosities , is that eq .( [ eq : dispersion ] ) is an implicit equation for , since . one can write ( and solve ) an explicit , fourth - order equation for , but not much insight is gained .the important point is that is the dispersion relation for free surface waves propagating on a deep , viscous fluid .such dispersion relations have been catalogued for a variety of physical situations ( small or large depth , small or large viscosity , viscoelastic effects , surfactant layers on the surface , etc . ) .one can look up or calculate the appropriate for each physical situation one wishes to study .the dispersion relation is completely independent of the way the waves are excited ( parametric or direct forcing ) . fornow , we leave as arbitrary .the next step is to derive the matrices relating the complex amplitudes to to . at , continuity of yields re re .the velocity - jump condition is derived by integrating the normal - stress condition eq .( [ eq : eqsofmotiond ] ) over a small interval of time centered on .we find  ' '' ''   \label{eq : fluidjump}\ ] ]  where \zeta_{n+1/2 } + \rm{c.c.}\ ] ] inserting this into eq .( [ eq : fluidjump ] ) , we find   ' '' ''  2  define now and . then and . here ,the matrices and have exactly the same form as before , except that now $ ] .the tongues then are given by .one difference is that in a laterally infinite container , all modes are available .thus , whatever the driving frequency ( or whatever the ) , the response of all modes must be considered .this implies that the overall threshold discussed above is the relevant concept .since the tongue is in general the easiest to excite , one expects waves at just half the driving frequency .  for small damping , one has , , , giving a threshold of which is slightly higher than the corresponding condition for sine - wave forcing , , edwards and fauve have explored the effects of exciting faraday waves with a driving signal that is the sum of two sine waves of rationally related frequency .they showed that for fixed frequency ratio ( 5:4 in their original work ) , the relative amplitude and phase of the two sine waves could be considered as independent control parameters .varying these new parameters dramatically affects the pattern - formation selection .for example , they produced hexagons and quasi - periodic patterns ( quasi - patterns " ) under conditions where single - frequency forcing would lead to a stripe pattern .  here , we look at a simple model of two - frequency forcing , an asymmetric triangle - wave forcing ( fig . [fig : driving]c ) .this waveform has the crucial feature that it is asymmetric under time reversal , , in contrast with the single - frequency " driving waveform ( fig . [fig : driving]b ) . as edwards and fauve have argued , time - reversal invariance of implies that the nonlinear amplitude equations for the parametrically excited waves must be invariant under ; however , this symmetry is removed for asymmetric .  in the delta - function model ,asymmetric triangle - wave forcing is easily solved by replacing the in the matrices and ( cf .[ eq : m1m2 ] ) with and , where .the tongue boundaries for the mathieu eq . are then given by the tongues are plotted for several choices of and in fig .[ fig : delta_tongues]c .note the reappearance of the harmonic tongues for . in this simple model, there is no analog of the bicritical point " discussed by edwards and fauve , where two different wavenumbers go unstable simultaneously .nonetheless , the symmetry argument given above suggests that asymmetric - triangle - wave forcing will alter the nonlinear pattern selection in much the same way as two frequencies do .in this paper , we have shown how the linear - stability analysis of faraday waves is greatly simplified for triangle - wave forcing .our analysis is simple enough to include in an undergraduate hydrodynamics course .one could also combine the analysis with a simple experiment to make a good senior thesis project .references and give careful discussions of the required experimental apparatus .our calculation may also serve as the starting point for a more extensive nonlinear analysis . to date , nonlinear amplitude equations for faraday waves have been derived only in the limit of small damping  ; however , many interesting phenomena are seen when to  . with delta - function driving , the effects of the parametric pumping are straightforward , even in the nonlinear regime .the hard part of the calculation becomes the determination of nonlinear terms in the dispersion relation for surface waves .however , this is a problem that has received much attention over the years , and a number of results are available .this work was supported by nserc ( canada ) .k. kumar k. l. s. tuckerman , parametric instability of the interface between two fluids , " _ j. fluid mech . _ * 279 * 49 - 68 ( 1994 ) ; cf .j. beyer and r. friedrich , faraday instability : linear analysis for viscous fluids , " phys .e * 51 * , 1162 - 1168 ( 1995 ) .lord rayleigh , on maintained vibrations , " phil . mag . * 15 * 229 - 235 ( 1883 ) . on the crispations of fluid resting upon a vibrating support , " phil. mag . * 16 * 50 - 58 ( 1883 ) . reprinted in _scientific papers by lord rayleigh , _ vol .ii , pp . 188 - 193 and pp . 212 - 219 .dover , 1964 .the electromagnetic shaker tables that are usually used to generate faraday waves will not give an accurate output if a function generator is used to create the triangle wave . because of the inductance of the vibration coils , the different frequency components of the input waveform will have different phase shifts and the result will be far from a triangle wave .a straightforward work - around is to use the feedback system of , which requires a modest amount of programming .alternatively , one could design a vibrator specifically for triangle waveforms .

Abstract: we show that the linear - stability analysis of the birth of faraday waves on the surface of a fluid is simplified considerably when the fluid container is driven by a triangle waveform rather than by a sine wave . the calculation is simple enough to use in an undergraduate course on fluid dynamics or nonlinear dynamics . it is also an attractive starting point for a nonlinear analysis .  2