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additive models @xcite provide an important family of models for semiparametric regression or classification . some reasons for the success of additive models are their increased flexibility when compared to linear or generalized linear models and their increased interpretability when compared to fully nonparametric models .
it is well - known that good estimators in additive models are in general less prone to the curse of high dimensionality than good estimators in fully nonparametric models .
many examples of such estimators belong to the large class of regularized kernel based methods over a reproducing kernel hilbert space @xmath0 , see e.g. @xcite . in the last years
many interesting results on learning rates of regularized kernel based models for additive models have been published when the focus is on sparsity and when the classical least squares loss function is used , see e.g. @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and the references therein . of course , the least squares loss function is differentiable and has many nice mathematical properties , but it is only locally lipschitz continuous and therefore regularized kernel based methods based on this loss function typically suffer on bad statistical robustness properties , even if the kernel is bounded .
this is in sharp contrast to kernel methods based on a lipschitz continuous loss function and on a bounded loss function , where results on upper bounds for the maxbias bias and on a bounded influence function are known , see e.g. @xcite for the general case and @xcite for additive models .
therefore , we will here consider the case of regularized kernel based methods based on a general convex and lipschitz continuous loss function , on a general kernel , and on the classical regularizing term @xmath1 for some @xmath2 which is a smoothness penalty but not a sparsity penalty , see e.g. @xcite .
such regularized kernel based methods are now often called support vector machines ( svms ) , although the notation was historically used for such methods based on the special hinge loss function and for special kernels only , we refer to @xcite . in this paper we address the open question , whether an svm with an additive kernel can provide a substantially better learning rate in high dimensions than an svm with a general kernel , say a classical gaussian rbf kernel , if the assumption of an additive model is satisfied .
our leading example covers learning rates for quantile regression based on the lipschitz continuous but non - differentiable pinball loss function , which is also called check function in the literature , see e.g. @xcite and @xcite for parametric quantile regression and @xcite , @xcite , and @xcite for kernel based quantile regression .
we will not address the question how to check whether the assumption of an additive model is satisfied because this would be a topic of a paper of its own .
of course , a practical approach might be to fit both models and compare their risks evaluated for test data .
for the same reason we will also not cover sparsity .
consistency of support vector machines generated by additive kernels for additive models was considered in @xcite . in this paper
we establish learning rates for these algorithms .
let us recall the framework with a complete separable metric space @xmath3 as the input space and a closed subset @xmath4 of @xmath5 as the output space .
a borel probability measure @xmath6 on @xmath7 is used to model the learning problem and an independent and identically distributed sample @xmath8 is drawn according to @xmath6 for learning .
a loss function @xmath9 is used to measure the quality of a prediction function @xmath10 by the local error @xmath11 .
_ throughout the paper we assume that @xmath12 is measurable , @xmath13 , convex with respect to the third variable , and uniformly lipschitz continuous satisfying @xmath14 with a finite constant @xmath15 .
_ support vector machines ( svms ) considered here are kernel - based regularization schemes in a reproducing kernel hilbert space ( rkhs ) @xmath0 generated by a mercer kernel @xmath16 . with a shifted loss function @xmath17 introduced for dealing
even with heavy - tailed distributions as @xmath18 , they take the form @xmath19 where for a general borel measure @xmath20 on @xmath21 , the function @xmath22 is defined by @xmath23 where @xmath24 is a regularization parameter .
the idea to shift a loss function has a long history , see e.g. @xcite in the context of m - estimators .
it was shown in @xcite that @xmath22 is also a minimizer of the following optimization problem involving the original loss function @xmath12 if a minimizer exists : @xmath25 the additive model we consider consists of the _ input space decomposition _
@xmath26 with each @xmath27 a complete separable metric space and a _ hypothesis space _
@xmath28 where @xmath29 is a set of functions @xmath30 each of which is also identified as a map @xmath31 from @xmath3 to @xmath5 .
hence the functions from @xmath32 take the additive form @xmath33 .
we mention , that there is strictly speaking a notational problem here , because in the previous formula each quantity @xmath34 is an element of the set @xmath35 which is a subset of the full input space @xmath36 , @xmath37 , whereas in the definition of sample @xmath8 each quantity @xmath38 is an element of the full input space @xmath36 , where @xmath39 .
because these notations will only be used in different places and because we do not expect any misunderstandings , we think this notation is easier and more intuitive than specifying these quantities with different symbols .
the additive kernel @xmath40 is defined in terms of mercer kernels @xmath41 on @xmath27 as @xmath42 it generates an rkhs @xmath0 which can be written in terms of the rkhs @xmath43 generated by @xmath41 on @xmath27 corresponding to the form ( [ additive ] ) as @xmath44 with norm given by @xmath45 the norm of @xmath46 satisfies @xmath47 to illustrate advantages of additive models , we provide two examples of comparing additive with product kernels .
the first example deals with gaussian rbf kernels .
all proofs will be given in section [ proofsection ] .
[ gaussadd ] let @xmath48 , @xmath49 $ ] and @xmath50 ^ 2.$ ] let @xmath51 and @xmath52.\ ] ] the additive kernel @xmath53 is given by @xmath54 furthermore , the product kernel @xmath55 is the standard gaussian kernel given by @xmath56 define a gaussian function @xmath57 on @xmath58 ^ 2 $ ] depending only on one variable by @xmath59 then @xmath60 but @xmath61 where @xmath62 denotes the rkhs generated by the standard gaussian rbf kernel @xmath63 .
the second example is about sobolev kernels .
[ sobolvadd ] let @xmath64 , @xmath65 $ ] and @xmath58^s.$ ] let @xmath66 : = \bigl\{u\in l_2([0,1 ] ) ; d^\alpha u \in l_2([0,1 ] ) \mbox{~for~all~}|\alpha|\le 1\bigr\}\ ] ] be the sobolev space consisting of all square integrable univariate functions whose derivative is also square integrable .
it is an rkhs with a mercer kernel @xmath67 defined on @xmath68 ^ 2 $ ] .
if we take all the mercer kernels @xmath69 to be @xmath67 , then @xmath70 $ ] for each @xmath71 .
the additive kernel @xmath72 is also a mercer kernel and defines an rkhs @xmath73\right\}.\ ] ] however , the multivariate sobolev space @xmath74^s)$ ] , consisting of all square integrable functions whose partial derivatives are all square integrable , contains discontinuous functions and is not an rkhs .
denote the marginal distribution of @xmath6 on @xmath27 as @xmath75 . under the assumption that @xmath76 for each @xmath71 and that @xmath43 is dense in @xmath29 in the @xmath77-metric , it was proved in @xcite that @xmath78 in probability as long as @xmath79 satisfies @xmath80 and @xmath81 .
the rest of the paper has the following structure .
section [ ratessection ] contains our main results on learning rates for svms based on additive kernels . learning rates for quantile regression
are treated as important special cases .
section [ comparisonsection ] contains a comparison of our results with other learning rates published recently .
section [ proofsection ] contains all the proofs and some results which can be interesting in their own .
in this paper we provide some learning rates for the support vector machines generated by additive kernels for additive models which helps improve the quantitative understanding presented in @xcite .
the rates are about asymptotic behaviors of the excess risk @xmath82 and take the form @xmath83 with @xmath84 .
they will be stated under three kinds of conditions involving the hypothesis space @xmath0 , the measure @xmath6 , the loss @xmath12 , and the choice of the regularization parameter @xmath85 .
the first condition is about the approximation ability of the hypothesis space @xmath0 .
since the output function @xmath19 is from the hypothesis space , the learning rates of the learning algorithm depend on the approximation ability of the hypothesis space @xmath0 with respect to the optimal risk @xmath86 measured by the following approximation error .
[ defapprox ] the approximation error of the triple @xmath87 is defined as @xmath88 to estimate the approximation error , we make an assumption about the minimizer of the risk @xmath89 for each @xmath90 , define the integral operator @xmath91 associated with the kernel @xmath41 by @xmath92 we mention that @xmath93 is a compact and positive operator on @xmath94 . hence we can find its normalized eigenpairs @xmath95 such that @xmath96 is an orthonormal basis of @xmath94 and @xmath97 as @xmath98 . fix @xmath99 .
then we can define the @xmath100-th power @xmath101 of @xmath93 by @xmath102 this is a positive and bounded operator and its range is well - defined .
the assumption @xmath103 means @xmath104 lies in this range .
[ assumption1 ] we assume @xmath105 and @xmath106 where for some @xmath107 and each @xmath108 , @xmath109 is a function of the form @xmath110 with some @xmath111 .
the case @xmath112 of assumption [ assumption1 ] means each @xmath113 lies in the rkhs @xmath43 .
a standard condition in the literature ( e.g. , @xcite ) for achieving decays of the form @xmath114 for the approximation error ( [ approxerrordef ] ) is @xmath115 with some @xmath116 . here
the operator @xmath117 is defined by @xmath118 in general , this can not be written in an additive form .
however , the hypothesis space ( [ additive ] ) takes an additive form @xmath119 .
so it is natural for us to impose an additive expression @xmath120 for the target function @xmath121 with the component functions @xmath113 satisfying the power condition @xmath110 .
the above natural assumption leads to a technical difficulty in estimating the approximation error : the function @xmath113 has no direct connection to the marginal distribution @xmath122 projected onto @xmath27 , hence existing methods in the literature ( e.g. , @xcite ) can not be applied directly .
note that on the product space @xmath123 , there is no natural probability measure projected from @xmath6 , and the risk on @xmath124 is not defined . our idea to overcome the difficulty is to introduce an intermediate function @xmath125 .
it may not minimize a risk ( which is not even defined ) .
however , it approximates the component function @xmath113 well .
when we add up such functions @xmath126 , we get a good approximation of the target function @xmath121 , and thereby a good estimate of the approximation error .
this is the first novelty of the paper .
[ approxerrorthm ] under assumption [ assumption1 ] , we have @xmath127 where @xmath128 is the constant given by @xmath129 the second condition for our learning rates is about the capacity of the hypothesis space measured by @xmath130-empirical covering numbers . let @xmath131 be a set of functions on @xmath21 and @xmath132 for every @xmath133 the * covering number of @xmath131 * with respect to the empirical metric @xmath134 , given by @xmath135 is defined as @xmath136 and the * @xmath130-empirical covering number * of @xmath137 is defined as @xmath138 [ assumption2 ] we assume @xmath139 and that for some @xmath140 , @xmath141 and every @xmath142 , the @xmath130-empirical covering number of the unit ball of @xmath43 satisfies @xmath143 the second novelty of this paper is to observe that the additive nature of the hypothesis space yields the following nice bound with a dimension - independent power exponent for the covering numbers of the balls of the hypothesis space @xmath0 , to be proved in section [ samplesection ] .
[ capacitythm ] under assumption [ assumption2 ] , for any @xmath144 and @xmath145 , we have @xmath146 the bound for the covering numbers stated in theorem [ capacitythm ] is special : the power @xmath147 is independent of the number @xmath148 of the components in the additive model .
it is well - known @xcite in the literature of function spaces that the covering numbers of balls of the sobolev space @xmath149 on the cube @xmath150^s$ ] of the euclidean space @xmath151 with regularity index @xmath152 has the following asymptotic behavior with @xmath153 : @xmath154 here the power @xmath155 depends linearly on the dimension @xmath148 .
similar dimension - dependent bounds for the covering numbers of the rkhss associated with gaussian rbf - kernels can be found in @xcite .
the special bound in theorem [ capacitythm ] demonstrates an advantage of the additive model in terms of capacity of the additive hypothesis space .
the third condition for our learning rates is about the noise level in the measure @xmath6 with respect to the hypothesis space . before stating the general condition
, we consider a special case for quantile regression , to illustrate our general results .
let @xmath156 be a quantile parameter .
the quantile regression function @xmath157 is defined by its value @xmath158 to be a @xmath159-quantile of @xmath160 , i.e. , a value @xmath161 satisfying @xmath162 the regularization scheme for quantile regression considered here takes the form ( [ algor ] ) with the loss function @xmath12 given by the pinball loss as @xmath163 a noise condition on @xmath6 for quantile regression is defined in @xcite as follows . to this end , let @xmath164 be a probability measure on @xmath165 and @xmath166 . then a real number @xmath167 is called @xmath159-quantile of @xmath164 , if and only if @xmath167 belongs to the set @xmath168\bigr ) \ge
\tau \mbox{~~and~~ } q\bigl([t , \infty)\bigr ) \ge 1-\tau\bigr\}\,.\ ] ] it is well - known that @xmath169 is a compact interval .
[ noisecond ] let @xmath166 . 1 .
a probability measure @xmath164 on @xmath165 is said to have a * @xmath159-quantile of type @xmath170 * , if there exist a @xmath159-quantile @xmath171 and a constant @xmath172 such that , for all @xmath173 $ ] , we have @xmath174 2 .
let @xmath175 $ ] .
we say that a probability measure @xmath20 on @xmath176 has a * @xmath159-quantile of @xmath177-average type @xmath170 * if the conditional probability measure @xmath178 has @xmath179-almost surely a @xmath159-quantile of type @xmath170 and the function @xmath180 where @xmath181 is the constant defined in part ( 1 ) , satisfies @xmath182 .
one can show that a distribution @xmath164 having a @xmath159-quantile of type @xmath170 has a unique @xmath159-quantile @xmath183 .
moreover , if @xmath164 has a lebesgue density @xmath184 then @xmath164 has a @xmath159-quantile of type @xmath170 if @xmath184 is bounded away from zero on @xmath185 $ ] since we can use @xmath186\}$ ] in ( [ tauquantileoftype2formula ] ) .
this assumption is general enough to cover many distributions used in parametric statistics such as gaussian , student s @xmath187 , and logistic distributions ( with @xmath188 ) , gamma and log - normal distributions ( with @xmath189 ) , and uniform and beta distributions ( with @xmath190 $ ] ) .
the following theorem , to be proved in section [ proofsection ] , gives a learning rate for the regularization scheme ( [ algor ] ) in the special case of quantile regression .
[ quantilethm ] suppose that @xmath191 almost surely for some constant @xmath192 , and that each kernel @xmath41 is @xmath193 with @xmath194 for some @xmath195 .
if assumption [ assumption1 ] holds with @xmath112 and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 for some @xmath196 $ ] , then by taking @xmath197 , for any @xmath198 and @xmath199 , with confidence at least @xmath200 we have @xmath201 where @xmath202 is a constant independent of @xmath203 and @xmath204 and @xmath205 please note that the exponent @xmath206 given by ( [ quantilerates2 ] ) for the learning rate in ( [ quantilerates ] ) is independent of the quantile level @xmath159 , of the number @xmath148 of additive components in @xmath207 , and of the dimensions @xmath208 and @xmath209 further note that @xmath210 , if @xmath211 , and @xmath212 if @xmath213 . because @xmath214 can be arbitrarily close to @xmath215 , the learning rate , which is independent of the dimension @xmath216 and given by theorem [ quantilethm ] , is close to @xmath217 for large values of @xmath177 and is close to @xmath218 or better , if @xmath211 . to state our general learning rates
, we need an assumption on a _ variance - expectation bound _ which is similar to definition [ noisecond ] in the special case of quantile regression .
[ assumption3 ] we assume that there exist an exponent @xmath219 $ ] and a positive constant @xmath220 such that @xmath221 assumption [ assumption3 ] always holds true for @xmath222 . if the triple @xmath223 satisfies some conditions , the exponent @xmath224 can be larger .
for example , when @xmath12 is the pinball loss ( [ pinloss ] ) and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath225 for some @xmath196 $ ] and @xmath226 as defined in @xcite , then @xmath227 .
[ mainratesthm ] suppose that @xmath228 is bounded by a constant @xmath229 almost surely . under assumptions [ assumption1 ] to [ assumption3 ] ,
if we take @xmath198 and @xmath230 for some @xmath231 , then for any @xmath232 , with confidence at least @xmath200 we have @xmath233 where @xmath234 is given by @xmath235 and @xmath202 is constant independent of @xmath203 or @xmath204 ( to be given explicitly in the proof ) .
we now add some theoretical and numerical comparisons on the goodness of our learning rates with those from the literature . as already mentioned in the introduction
, some reasons for the popularity of additive models are flexibility , increased interpretability , and ( often ) a reduced proneness of the curse of high dimensions .
hence it is important to check , whether the learning rate given in theorem [ mainratesthm ] under the assumption of an additive model favourably compares to ( essentially ) optimal learning rates without this assumption . in other words ,
we need to demonstrate that the main goal of this paper is achieved by theorem [ quantilethm ] and theorem [ mainratesthm ] , i.e. that an svm based on an additive kernel can provide a substantially better learning rate in high dimensions than an svm with a general kernel , say a classical gaussian rbf kernel , provided the assumption of an additive model is satisfied .
our learning rate in theorem [ quantilethm ] is new and optimal in the literature of svm for quantile regression .
most learning rates in the literature of svm for quantile regression are given for projected output functions @xmath236 , while it is well known that projections improve learning rates @xcite . here the projection operator @xmath237 is defined for any measurable function @xmath10 by @xmath238 sometimes this is called clipping .
such results are given in @xcite .
for example , under the assumptions that @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 , the approximation error condition ( [ approxerrorb ] ) is satisfied for some @xmath239 , and that for some constants @xmath240 , the sequence of eigenvalues @xmath241 of the integral operator @xmath117 satisfies @xmath242 for every @xmath243 , it was shown in @xcite that with confidence at least @xmath200 , @xmath244 where @xmath245 here the parameter @xmath246 measures the capacity of the rkhs @xmath247 and it plays a similar role as half of the parameter @xmath147 in assumption 2 . for a @xmath193 kernel and @xmath112
, one can choose @xmath246 and @xmath147 to be arbitrarily small and the above power index @xmath248 can be taken as @xmath249 .
the learning rate in theorem [ quantilethm ] may be improved by relaxing assumption 1 to a sobolev smoothness condition for @xmath121 and a regularity condition for the marginal distribution @xmath250 .
for example , one may use a gaussian kernel @xmath251 depending on the sample size @xmath203 and @xcite achieve the approximation error condition ( [ approxerrorb ] ) for some @xmath252 .
this is done for quantile regression in @xcite .
since we are mainly interested in additive models , we shall not discuss such an extension .
[ gaussmore ] let @xmath48 , @xmath49 $ ] and @xmath50 ^ 2.$ ] let @xmath51 and the additive kernel @xmath72 be given by ( [ gaussaddform ] ) with @xmath253 in example [ gaussadd ] as @xmath52.\ ] ] if the function @xmath121 is given by ( [ gaussfcn ] ) , @xmath191 almost surely for some constant @xmath192 , and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 for some @xmath196 $ ] , then by taking @xmath197 , for any @xmath145 and @xmath199 , ( [ quantilerates ] ) holds with confidence at least @xmath200 . it is unknown whether the above learning rate can be derived by existing approaches in the literature ( e.g. @xcite ) even after projection .
note that the kernel in the above example is independent of the sample size .
it would be interesting to see whether there exists some @xmath99 such that the function @xmath57 defined by ( [ gaussfcn ] ) lies in the range of the operator @xmath254 .
the existence of such a positive index would lead to the approximation error condition ( [ approxerrorb ] ) , see @xcite . let us now add some numerical comparisons on the goodness of our learning rates given by theorem [ mainratesthm ] with those given by @xcite .
their corollary 4.12 gives ( essentially ) minmax optimal learning rates for ( clipped ) svms in the context of nonparametric quantile regression using one gaussian rbf kernel on the whole input space under appropriate smoothness assumptions of the target function .
let us consider the case that the distribution @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 , where @xmath255 , and assume that both corollary 4.12 in @xcite and our theorem [ mainratesthm ] are applicable .
i.e. , we assume in particular that @xmath6 is a probability measure on @xmath256 $ ] and that the marginal distribution @xmath257 has a lebesgue density @xmath258 for some @xmath259 . furthermore , suppose that the optimal decision function @xmath260 has ( to make theorem [ mainratesthm ] applicable with @xmath261 $ ] ) the additive structure @xmath207 with each @xmath104 as stated in assumption [ assumption1 ] , where @xmath262 and @xmath263 , with minimal risk @xmath86 and additionally fulfills ( to make corollary 4.12 in @xcite applicable ) @xmath264 where @xmath265 $ ] and @xmath266 denotes a besov space with smoothness parameter @xmath267 .
the intuitive meaning of @xmath248 is , that increasing values of @xmath248 correspond to increased smoothness .
we refer to ( * ? ? ? * and p. 44 ) for details on besov spaces .
it is well - known that the besov space @xmath268 contains the sobolev space @xmath269 for @xmath270 , @xmath271 , and @xmath272 , and that @xmath273 .
we mention that if all @xmath41 are suitably chosen wendland kernels , their reproducing kernel hilbert spaces @xmath43 are sobolev spaces , see ( * ? ? ?
* thm . 10.35 , p. 160 ) .
furthermore , we use the same sequence of regularizing parameters as in ( * ? ? ?
4.9 , cor . 4.12 ) , i.e. , @xmath274 where @xmath275 , @xmath276 , @xmath277 $ ] , and @xmath278 is some user - defined positive constant independent of @xmath279 . for
reasons of simplicity , let us fix @xmath280 .
then ( * ? ? ?
4.12 ) gives learning rates for the risk of svms for @xmath159-quantile regression , if a single gaussian rbf - kernel on @xmath281 is used for @xmath159-quantile functions of @xmath177-average type @xmath170 with @xmath255 , which are of order @xmath282 hence the learning rate in theorem [ quantilethm ] is better than the one in ( * ? ? ?
4.12 ) in this situation , if @xmath283 provided the assumption of the additive model is valid .
table [ table1 ] lists the values of @xmath284 from ( [ explicitratescz2 ] ) for some finite values of the dimension @xmath216 , where @xmath285 .
all of these values of @xmath284 are positive with the exceptions if @xmath286 or @xmath287 .
this is in contrast to the corresponding exponent in the learning rate by ( * ? ?
* cor . 4.12 ) , because @xmath288 table [ table2 ] and figures [ figure1 ] to [ figure2 ] give additional information on the limit @xmath289 .
of course , higher values of the exponent indicates faster rates of convergence .
it is obvious , that an svm based on an additive kernel has a significantly faster rate of convergence in higher dimensions @xmath216 compared to svm based on a single gaussian rbf kernel defined on the whole input space , of course under the assumption that the additive model is valid .
the figures seem to indicate that our learning rate from theorem [ mainratesthm ] is probably not optimal for small dimensions . however , the main focus of the present paper is on high dimensions .
.[table1 ] the table lists the limits of the exponents @xmath290 from ( * ? ? ?
* cor . 4.12 ) and @xmath291 from theorem [ mainratesthm ] , respectively , if the regularizing parameter @xmath292 is chosen in an optimal manner for the nonparametric setup , i.e. @xmath293 , with @xmath294 for @xmath295 and @xmath296 .
recall that @xmath297 $ ] .
[ cols= " > , > , > , > " , ] | additive models play an important role in semiparametric statistics .
this paper gives learning rates for regularized kernel based methods for additive models .
these learning rates compare favourably in particular in high dimensions to recent results on optimal learning rates for purely nonparametric regularized kernel based quantile regression using the gaussian radial basis function kernel , provided the assumption of an additive model is valid .
additionally , a concrete example is presented to show that a gaussian function depending only on one variable lies in a reproducing kernel hilbert space generated by an additive gaussian kernel , but does not belong to the reproducing kernel hilbert space generated by the multivariate gaussian kernel of the same variance . *
key words and phrases . * additive model , kernel , quantile regression , semiparametric , rate of convergence , support vector machine . | [
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] |
the leptonic decays of a charged pseudoscalar meson @xmath7 are processes of the type @xmath8 , where @xmath9 , @xmath10 , or @xmath11 . because no strong interactions are present in the leptonic final state @xmath12 , such decays provide a clean way to probe the complex , strong interactions that bind the quark and antiquark within the initial - state meson . in these decays , strong interaction effects can be parametrized by a single quantity , @xmath13 , the pseudoscalar meson decay constant .
the leptonic decay rate can be measured by experiment , and the decay constant can be determined by the equation ( ignoring radiative corrections ) @xmath14 where @xmath15 is the fermi coupling constant , @xmath16 is the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite element , @xmath17 is the mass of the meson , and @xmath18 is the mass of the charged lepton .
the quantity @xmath13 describes the amplitude for the @xmath19 and @xmath20-quarks within the @xmath21 to have zero separation , a condition necessary for them to annihilate into the virtual @xmath22 boson that produces the @xmath12 pair .
the experimental determination of decay constants is one of the most important tests of calculations involving nonperturbative qcd .
such calculations have been performed using various models @xcite or using lattice qcd ( lqcd ) .
the latter is now generally considered to be the most reliable way to calculate the quantity .
knowledge of decay constants is important for describing several key processes , such as @xmath23 mixing , which depends on @xmath24 , a quantity that is also predicted by lqcd calculations .
experimental determination @xcite of @xmath24 with the leptonic decay of a @xmath25 meson is , however , very limited as the rate is highly suppressed due to the smallness of the magnitude of the relevant ckm matrix element @xmath26 .
the charm mesons , @xmath27 and @xmath28 , are better instruments to study the leptonic decays of heavy mesons since these decays are either less ckm suppressed or favored , _
i.e. _ , @xmath29 and @xmath30 are much larger than @xmath31 .
thus , the decay constants @xmath32 and @xmath33 determined from charm meson decays can be used to test and validate the necessary lqcd calculations applicable to the @xmath34-meson sector . among
the leptonic decays in the charm - quark sector , @xmath35 decays are more accessible since they are ckm favored .
furthermore , the large mass of the @xmath11 lepton removes the helicity suppression that is present in the decays to lighter leptons .
the existence of multiple neutrinos in the final state , however , makes measurement of this decay challenging .
physics beyond the standard model ( sm ) might also affect leptonic decays of charmed mesons .
depending on the non - sm features , the ratio of @xmath36 could be affected @xcite , as could the ratio @xcite @xmath37 .
any of the individual widths might be increased or decreased .
there is an indication of a discrepancy between the experimental determinations @xcite of @xmath33 and the most recent precision lqcd calculation @xcite .
this disagreement is particularly puzzling since the cleo - c determination @xcite of @xmath32 agrees well with the lqcd calculation @xcite of that quantity . some @xcite conjecture that this discrepancy may be explained by a charged higgs boson or a leptoquark . in this article
, we report an improved measurement of the absolute branching fraction of the leptonic decay @xmath0 ( charge - conjugate modes are implied ) , with @xmath1 , from which we determine the decay constant @xmath33 .
we use a data sample of @xmath38 events provided by the cornell electron storage ring ( cesr ) and collected by the cleo - c detector at the center - of - mass ( cm ) energy @xmath39 mev , near @xmath3 peak production @xcite .
the data sample consists of an integrated luminosity of @xmath40 @xmath41 containing @xmath42 @xmath3 pairs .
we have previously reported @xcite measurements of @xmath43 and @xmath0 with a subsample of these data . a companion article @xcite reports measurements of @xmath33 from @xmath43 and @xmath0 , with @xmath44 , using essentially the same data sample as the one used in this measurement .
the cleo - c detector @xcite is a general - purpose solenoidal detector with four concentric components utilized in this measurement : a small - radius six - layer stereo wire drift chamber , a 47-layer main drift chamber , a ring - imaging cherenkov ( rich ) detector , and an electromagnetic calorimeter consisting of 7800 csi(tl ) crystals .
the two drift chambers operate in a @xmath45 t magnetic field and provide charged particle tracking in a solid angle of @xmath46% of @xmath47 .
the chambers achieve a momentum resolution of @xmath48% at @xmath49 gev/@xmath50 .
the main drift chamber also provides specific - ionization ( @xmath51 ) measurements that discriminate between charged pions and kaons .
the rich detector covers approximately @xmath52% of @xmath47 and provides additional separation of pions and kaons at high momentum .
the photon energy resolution of the calorimeter is @xmath53% at @xmath54 gev and @xmath55% at @xmath56 mev .
electron identification is based on a likelihood variable that combines the information from the rich detector , @xmath51 , and the ratio of electromagnetic shower energy to track momentum ( @xmath57 ) .
we use a geant - based @xcite monte carlo ( mc ) simulation program to study efficiency of signal - event selection and background processes .
physics events are generated by evtgen @xcite , tuned with much improved knowledge of charm decays @xcite , and final - state radiation ( fsr ) is modeled by the photos @xcite program .
the modeling of initial - state radiation ( isr ) is based on cross sections for @xmath3 production at lower energies obtained from the cleo - c energy scan @xcite near the cm energy where we collect the sample .
the presence of two @xmath58 mesons in a @xmath3 event allows us to define a single - tag ( st ) sample in which a @xmath58 is reconstructed in a hadronic decay mode and a further double - tagged ( dt ) subsample in which an additional @xmath59 is required as a signature of @xmath60 decay , the @xmath59 being the daughter of the @xmath60 .
the @xmath61 reconstructed in the st sample can be either primary or secondary from @xmath62 ( or @xmath63 ) .
the st yield can be expressed as @xmath64 where @xmath65 is the produced number of @xmath3 pairs , @xmath66 is the branching fraction of hadronic modes used in the st sample , and @xmath67 is the st efficiency .
the @xmath68 counts the candidates , not events , and the factor of 2 comes from the sum of @xmath28 and @xmath61 tags .
our double - tag ( dt ) sample is formed from events with only a single charged track , identified as an @xmath69 , in addition to a st .
the yield can be expressed as @xmath70 where @xmath71 is the leptonic decay branching fraction , including the subbranching fraction of @xmath1 decay , @xmath72 is the efficiency of finding the st and the leptonic decay in the same event . from the st and dt yields we can obtain an absolute branching fraction of the leptonic decay @xmath71 , without needing to know the integrated luminosity or the produced number of @xmath3 pairs , @xmath73 where @xmath74 ( @xmath75 ) is the effective signal efficiency . because of the large solid angle acceptance with high segmentation of the cleo - c detector and the low multiplicity of the events with which we are concerned , @xmath76 , where @xmath77 is the leptonic decay efficiency .
hence , the ratio @xmath78 is insensitive to most systematic effects associated with the st , and the signal branching fraction @xmath71 obtained using this procedure is nearly independent of the efficiency of the tagging mode . to minimize systematic uncertainties , we tag using three two - body hadronic decay modes with only charged particles in the final state .
the three st modes and @xmath79 are shorthand labels for @xmath80 events within mass windows ( described below ) of the @xmath81 peak in @xmath82 and the @xmath83 peak in @xmath84 , respectively .
no attempt is made to separate these resonance components in the @xmath85 dalitz plot . ]
are @xmath86 , @xmath79 , and @xmath87 .
using these tag modes also helps to reduce the tag bias which would be caused by the correlation between the tag side and the signal side reconstruction if tag modes with high multiplicity and large background were used .
the effect of the tag bias @xmath88 can be expressed in terms of the signal efficiency @xmath74 defined by @xmath89 where @xmath90 is the st efficiency when the recoiling system is the signal leptonic decay with single @xmath59 in the other side of the tag .
as the general st efficiency @xmath67 , when the recoiling system is any possible @xmath91 decays , will be lower than the @xmath90 , sizable tag bias could be introduced if the multiplicity of the tag mode were high , or the tag mode were to include neutral particles in the final state . as shown in sec .
[ sec : results ] , this effect is negligible in our chosen clean tag modes .
the @xmath92 decay is reconstructed by combining oppositely charged tracks that originate from a common vertex and that have an invariant mass within @xmath93 mev of the nominal mass @xcite .
we require the resonance decay to satisfy the following mass windows around the nominal masses @xcite : @xmath94 ( @xmath95 mev ) and @xmath96 ( @xmath97 mev ) .
we require the momenta of charged particles to be @xmath56 mev or greater to suppress the slow pion background from @xmath98 decays ( through @xmath99 ) .
we identify a st by using the invariant mass of the tag @xmath100 and recoil mass against the tag @xmath101 .
the recoil mass is defined as @xmath102 where @xmath103 is the net four - momentum of the @xmath4 beam , taking the finite beam crossing angle into account ; @xmath104 is the four - momentum of the tag , with @xmath105 computed from @xmath106 and the nominal mass @xcite of the @xmath91 meson .
we require the recoil mass to be within @xmath107 mev of the @xmath108 mass @xcite .
this loose window allows both primary and secondary @xmath91 tags to be selected . to estimate the backgrounds in our st and dt yields from the wrong tag combinations ( incorrect combinations that , by chance , lie within the @xmath109 signal region ) , we use the tag invariant mass sidebands .
we define the signal region as @xmath110 mev @xmath111 mev , and the sideband regions as @xmath112 mev @xmath113 mev or @xmath114 mev @xmath115 mev , where @xmath116 is the difference between the tag mass and the nominal mass .
we fit the st @xmath109 distributions to the sum of double - gaussian signal function plus second - degree chebyshev polynomial background function to get the tag mass sideband scaling factor .
the invariant mass distributions of tag candidates for each tag mode are shown in fig .
[ fig : dm ] and the st yield and @xmath109 sideband scaling factor are summarized in table [ table : data - single ] .
we find @xmath117 summed over the three tag modes .
.[table : data - single ] summary of single - tag ( st ) yields , where @xmath118 is the yield in the st mass signal region , @xmath119 is the yield in the sideband region , @xmath120 is the sideband scaling factor , and @xmath68 is the scaled sideband - subtracted yield . [ cols="<,>,>,>,>",options="header " , ] we considered six semileptonic decays , @xmath121 @xmath122 , @xmath123 , @xmath124 , @xmath125 , @xmath126 , and @xmath127 , as the major sources of background in the @xmath128 signal region .
the second dominates the nonpeaking background , and the fourth ( with @xmath129 ) dominates the peaking background .
uncertainty in the signal yield due to nonpeaking background ( @xmath130 ) is assessed by varying the semileptonic decay branching fractions by the precision with which they are known @xcite .
imperfect knowledge of @xmath131 gives rise to a systematic uncertainty in our estimate of the amount of peaking background in the signal region , which has an effect on our branching fraction measurement of @xmath132 .
we study differences in efficiency , data vs mc events , due to the extra energy requirement , extra track veto , and @xmath133 requirement , by using samples from data and mc events , in which _ both _ the @xmath134 and @xmath2 satisfy our tag requirements , i.e. , `` double - tag '' events .
we then apply each of the above - mentioned requirements and compare loss in efficiency of data vs mc events . in this way
we obtain a correction of @xmath135 for the extra energy requirement and systematic uncertainties on each of the three requirements of @xmath136 ( all equal , by chance ) .
the non-@xmath69 background in the signal @xmath69 candidate sample is negligible ( @xmath137 ) due to the low probability ( @xmath138 per track ) that hadrons ( @xmath139 or @xmath140 ) are misidentified as @xmath69 @xcite .
uncertainty in these backgrounds produces a @xmath141 uncertainty in the measurement of @xmath142 .
the secondary @xmath69 backgrounds from charge symmetric processes , such as @xmath143 dalitz decay ( @xmath144 ) and @xmath145 conversion ( @xmath146 ) , are assessed by measuring the wrong - sign signal electron in events with @xmath147 . the uncertainty in the measurement from this source
is estimated to be @xmath148 .
other possible sources of systematic uncertainty include @xmath68 ( @xmath137 ) , tag bias ( @xmath149 ) , tracking efficiency ( @xmath148 ) , @xmath59 identification efficiency ( @xmath150 ) , and fsr ( @xmath150 ) . combining all contributions in quadrature ,
the total systematic uncertainty in the branching fraction measurement is estimated to be @xmath151 .
in summary , using the sample of @xmath152 tagged @xmath28 decays with the cleo - c detector we obtain the absolute branching fraction of the leptonic decay @xmath153 through @xmath154 @xmath155 where the first uncertainty is statistical and the second is systematic .
this result supersedes our previous measurement @xcite of the same branching fraction , which used a subsample of data used in this work .
the decay constant @xmath33 can be computed using eq .
( [ eq : f ] ) with known values @xcite @xmath156 gev@xmath157 , @xmath158 mev , @xmath159 mev , and @xmath160 s. we assume @xmath161 and use the value @xmath162 given in ref .
we obtain @xmath163 combining with our other determination @xcite of @xmath164 mev with @xmath43 and @xmath0 ( @xmath165 ) decays , we obtain @xmath166 this result is derived from absolute branching fractions only and is the most precise determination of the @xmath91 leptonic decay constant to date . our combined result is larger than the recent lqcd calculation @xmath167 mev @xcite by @xmath168 standard deviations .
the difference between data and lqcd for @xmath33 could be due to physics beyond the sm @xcite , unlikely statistical fluctuations in the experimental measurements or the lqcd calculation , or systematic uncertainties that are not understood in the lqcd calculation or the experimental measurements . combining with our other determination @xcite of @xmath169 , via @xmath44
, we obtain @xmath170 using this with our measurement @xcite of @xmath171 , we obtain the branching fraction ratio @xmath172 this is consistent with @xmath173 , the value predicted by the sm with lepton universality , as given in eq .
( [ eq : f ] ) with known masses @xcite .
we gratefully acknowledge the effort of the cesr staff in providing us with excellent luminosity and running conditions .
d. cronin - hennessy and a. ryd thank the a.p . sloan foundation .
this work was supported by the national science foundation , the u.s .
department of energy , the natural sciences and engineering research council of canada , and the u.k .
science and technology facilities council . c. amsler
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the transport properties of nonlinear non - equilibrium dynamical systems are far from well - understood@xcite .
consider in particular so - called ratchet systems which are asymmetric periodic potentials where an ensemble of particles experience directed transport@xcite .
the origins of the interest in this lie in considerations about extracting useful work from unbiased noisy fluctuations as seems to happen in biological systems@xcite .
recently attention has been focused on the behavior of deterministic chaotic ratchets@xcite as well as hamiltonian ratchets@xcite .
chaotic systems are defined as those which are sensitively dependent on initial conditions . whether chaotic or not , the behavior of nonlinear systems including the transition from regular to chaotic behavior is in general sensitively dependent on the parameters of the system .
that is , the phase - space structure is usually relatively complicated , consisting of stability islands embedded in chaotic seas , for examples , or of simultaneously co - existing attractors .
this can change significantly as parameters change .
for example , stability islands can merge into each other , or break apart , and the chaotic sea itself may get pinched off or otherwise changed , or attractors can change symmetry or bifurcate .
this means that the transport properties can change dramatically as well .
a few years ago , mateos@xcite considered a specific ratchet model with a periodically forced underdamped particle .
he looked at an ensemble of particles , specifically the velocity for the particles , averaged over time and the entire ensemble .
he showed that this quantity , which is an intuitively reasonable definition of ` the current ' , could be either positive or negative depending on the amplitude @xmath0 of the periodic forcing for the system . at the same time , there exist ranges in @xmath0 where the trajectory of an individual particle displays chaotic dynamics .
mateos conjectured a connection between these two phenomena , specifically that the reversal of current direction was correlated with a bifurcation from chaotic to periodic behavior in the trajectory dynamics .
even though it is unlikely that such a result would be universally valid across all chaotic deterministic ratchets , it would still be extremely useful to have general heuristic rules such as this .
these organizing principles would allow some handle on characterizing the many different kinds of behavior that are possible in such systems .
a later investigation@xcite of the mateos conjecture by barbi and salerno , however , argued that it was not a valid rule even in the specific system considered by mateos .
they presented results showing that it was possible to have current reversals in the absence of bifurcations from periodic to chaotic behavior .
they proposed an alternative origin for the current reversal , suggesting it was related to the different stability properties of the rotating periodic orbits of the system .
these latter results seem fundamentally sensible . however , this paper based its arguments about currents on the behavior of a _ single _ particle as opposed to an ensemble .
this implicitly assumes that the dynamics of the system are ergodic .
this is not true in general for chaotic systems of the type being considered . in particular , there can be extreme dependence of the result on the statistics of the ensemble being considered .
this has been pointed out in earlier studies @xcite which laid out a detailed methodology for understanding transport properties in such a mixed regular and chaotic system . depending on specific parameter value , the particular system under consideration has multiple coexisting periodic or chaotic attractors or a mixture of both .
it is hence appropriate to understand how a probability ensemble might behave in such a system .
the details of the dependence on the ensemble are particularly relevant to the issue of the possible experimental validation of these results , since experiments are always conducted , by virtue of finite - precision , over finite time and finite ensembles .
it is therefore interesting to probe the results of barbi and salerno with regard to the details of the ensemble used , and more formally , to see how ergodicity alters our considerations about the current , as we do in this paper .
we report here on studies on the properties of the current in a chaotic deterministic ratchet , specifically the same system as considered by mateos@xcite and barbi and salerno@xcite .
we consider the impact of different kinds of ensembles of particles on the current and show that the current depends significantly on the details of the initial ensemble .
we also show that it is important to discard transients in quantifying the current .
this is one of the central messages of this paper : broad heuristics are rare in chaotic systems , and hence it is critical to understand the ensemble - dependence in any study of the transport properties of chaotic ratchets .
having established this , we then proceed to discuss the connection between the bifurcation diagram for individual particles and the behavior of the current .
we find that while we disagree with many of the details of barbi and salerno s results , the broader conclusion still holds .
that is , it is indeed possible to have current reversals in the absence of bifurcations from chaos to periodic behavior as well as bifurcations without any accompanying current reversals .
the result of our investigation is therefore that the transport properties of a chaotic ratchet are not as simple as the initial conjecture .
however , we do find evidence for a generalized version of mateos s conjecture .
that is , in general , bifurcations for trajectory dynamics as a function of system parameter seem to be associated with abrupt changes in the current .
depending on the specific value of the current , these abrupt changes may lead the net current to reverse direction , but not necessarily so .
we start below with a preparatory discussion necessary to understand the details of the connection between bifurcations and current reversal , where we discuss the potential and phase - space for single trajectories for this system , where we also define a bifurcation diagram for this system . in the next section ,
we discuss the subtleties of establishing a connection between the behavior of individual trajectories and of ensembles .
after this , we are able to compare details of specific trajectory bifurcation curves with current curves , and thus justify our broader statements above , after which we conclude .
the goal of these studies is to understand the behavior of general chaotic ratchets .
the approach taken here is that to discover heuristic rules we must consider specific systems in great detail before generalizing .
we choose the same @xmath1-dimensional ratchet considered previously by mateos@xcite , as well as barbi and salerno@xcite .
we consider an ensemble of particles moving in an asymmetric periodic potential , driven by a periodic time - dependent external force , where the force has a zero time - average .
there is no noise in the system , so it is completely deterministic , although there is damping .
the equations of motion for an individual trajectory for such a system are given in dimensionless variables by @xmath2 where the periodic asymmetric potential can be written in the form @xmath3 + \frac{1}{4 } \sin [ 4\pi ( x -x_0 ) ] \bigg ] .\ ] ] in this equation @xmath4 have been introduced for convenience such that one potential minimum exists at the origin with @xmath5 and the term @xmath6 .
( a ) classical phase space for the unperturbed system . for @xmath7 , @xmath8 ,
two chaotic attractors emerge with @xmath9 ( b ) @xmath10 ( c ) and a period four attractor consisting of the four centers of the circles with @xmath11.,title="fig:",width=302 ] the phase - space of the undamped undriven ratchet the system corresponding to the unperturbed potential @xmath12 looks like a series of asymmetric pendula .
that is , individual trajectories have one of following possible time - asymptotic behaviors : ( i ) inside the potential wells , trajectories and all their properties oscillate , leading to zero net transport . outside the wells , the trajectories either ( ii ) librate to the right or ( iii ) to the left , with corresponding net transport depending upon initial conditions .
there are also ( iv ) trajectories on the separatrices between the oscillating and librating orbits , moving between unstable fixed points in infinite time , as well as the unstable and stable fixed points themselves , all of which constitute a set of negligible measure . when damping is introduced via the @xmath13-dependent term in eq .
[ eq : dyn ] , it makes the stable fixed points the only attractors for the system .
when the driving is turned on , the phase - space becomes chaotic with the usual phenomena of intertwining separatrices and resulting homoclinic tangles .
the dynamics of individual trajectories in such a system are now very complicated in general and depend sensitively on the choice of parameters and initial conditions .
we show snapshots of the development of this kind of chaos in the set of poincar sections fig .
( [ figure1]b , c ) together with a period - four orbit represented by the center of the circles . a broad characterization of the dynamics of the problem as a function of a parameter ( @xmath14 or @xmath15 ) emerges in a bifurcation diagram
. this can be constructed in several different and essentially equivalent ways .
the relatively standard form that we use proceeds as follows : first choose the bifurcation parameter ( let us say @xmath0 ) and correspondingly choose fixed values of @xmath16 , and start with a given value for @xmath17 .
now iterate an initial condition , recording the value of the particle s position @xmath18 at times @xmath19 from its integrated trajectory ( sometimes we record @xmath20 .
this is done stroboscopically at discrete times @xmath21 where @xmath22 and @xmath23 is an integer @xmath24 with @xmath25 the maximum number of observations made . of these , discard observations at times less than some cut - off time @xmath26 and plot the remaining points against @xmath27 .
it must be noted that discarding transient behavior is critical to get results which are independent of initial condition , and we shall emphasize this further below in the context of the net transport or current .
if the system has a fixed - point attractor then all of the data lie at one particular location @xmath28 . a periodic orbit with period @xmath29 ( that is , with period commensurate with the driving ) shows up with @xmath30 points occupying only @xmath31 different locations of @xmath32 for @xmath27 .
all other orbits , including periodic orbits of incommensurate period result in a simply - connected or multiply - connected dense set of points . for the next value @xmath33 , the last computed value of @xmath34 at @xmath35 are used as initial conditions , and previously , results are stored after cutoff and so on until @xmath36 .
that is , the bifurcation diagram is generated by sweeping the relevant parameter , in this case @xmath0 , from @xmath27 through some maximum value @xmath37 .
this procedure is intended to catch all coexisting attractors of the system with the specified parameter range .
note that several initial conditions are effectively used troughout the process , and a bifurcation diagram is not the behavior of a single trajectory .
we have made several plots , as a test , with different initial conditions and the diagrams obtained are identical .
we show several examples of this kind of bifurcation diagram below , where they are being compared with the corresponding behavior of the current .
having broadly understood the wide range of behavior for individual trajectories in this system , we now turn in the next section to a discussion of the non - equilibrium properties of a statistical ensemble of these trajectories , specifically the current for an ensemble .
the current @xmath38 for an ensemble in the system is defined in an intuitive manner by mateos@xcite as the time - average of the average velocity over an ensemble of initial conditions .
that is , an average over several initial conditions is performed at a given observation time @xmath39 to yield the average velocity over the particles @xmath40 this average velocity is then further time - averaged ; given the discrete time @xmath39 for observation this leads to a second sum @xmath41 where @xmath25 is the number of time - observations made . for this to be a relevant quantity to compare with bifurcation diagrams , @xmath38 should be independent of the quantities @xmath42 but still strongly dependent on @xmath43 .
a further parameter dependence that is being suppressed in the definition above is the shape and location of the ensemble being used .
that is , the transport properties of an ensemble in a chaotic system depend in general on the part of the phase - space being sampled .
it is therefore important to consider many different initial conditions to generate a current . the first straightforward result we show in fig .
( [ figure2 ] ) is that in the case of chaotic trajectories , a single trajectory easily displays behavior very different from that of many trajectories .
however , it turns out that in the regular regime , it is possible to use a single trajectory to get essentially the same result as obtained from many trajectories . further consider the bifurcation diagram in fig .
( [ figure3 ] ) where we superimpose the different curves resulting from varying the number of points in the initial ensemble .
first , the curve is significantly smoother as a function of @xmath0 for larger @xmath44 . even more relevant is the fact that the single trajectory data ( @xmath45 ) may show current reversals that do not exist in the large @xmath44 data .
current @xmath38 versus the number of trajectories @xmath44 for @xmath7 ; dashed lines correspond to a regular motion with @xmath46 while solid lines correspond to a chaotic motion with @xmath47 .
note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the @xmath44 exceeds a certain threshold , @xmath48.,title="fig:",width=302 ] current @xmath38 versus @xmath0 for different set of trajectories @xmath44 ; @xmath45 ( circles ) , @xmath49 ( square ) and @xmath50 ( dashed lines ) . note that a single trajectory suffices in the regular regime where all the curves match . in the chaotic regime
, as @xmath44 increases , the curves converge towards the dashed one.,title="fig:",width=302 ] also , note that single - trajectory current values are typically significantly greater than ensemble averages .
this arises from the fact that an arbitrarily chosen ensemble has particles with idiosyncratic behaviors which often average out . as our result , with these ensembles we see typical @xmath51 for example , while barbi and salerno report currents about @xmath52 times greater .
however , it is not true that only a few trajectories dominate the dynamics completely , else there would not be a saturation of the current as a function of @xmath44 .
all this is clear in fig .
( [ figure3 ] ) .
we note that the * net * drift of an ensemble can be a lot closer to @xmath53 than the behavior of an individual trajectory
. it should also be clear that there is a dependence of the current on the location of the initial ensemble , this being particularly true for small @xmath44 , of course .
the location is defined by its centroid @xmath54 . for @xmath45 , it is trivially true that the initial location matters to the asymptotic value of the time - averaged velocity , given that this is a non - ergodic and chaotic system .
further , considering a gaussian ensemble , say , the width of the ensemble also affects the details of the current , and can show , for instance , illusory current reversal , as seen in figs .
( [ current - bifur1],[current - bifur2 ] ) for example .
notice also that in fig .
( [ current - bifur1 ] ) , at @xmath55 and @xmath56 , the deviations between the different ensembles is particularly pronounced .
these points are close to bifurcation points where some sort of symmetry breaking is clearly occuring , which underlines our emphasis on the relevance of specifying ensemble characteristics in the neighborhood of unstable behavior .
however , why these specific bifurcations should stand out among all the bifurcations in the parameter range shown is not entirely clear . to understand how to incorporate this knowledge into calculations of the current ,
therefore , consider the fact that if we look at the classical phase space for the hamiltonian or underdamped @xmath57 motion , we see the typical structure of stable islands embedded in a chaotic sea which have quite complicated behavior@xcite . in such a situation , the dynamics always depends on the location of the initial conditions .
however , we are not in the hamiltonian situation when the damping is turned on in this case , the phase - space consists in general of attractors .
that is , if transient behavior is discarded , the current is less likely to depend significantly on the location of the initial conditions or on the spread of the initial conditions . in particular , in the chaotic regime of a non - hamiltonian system , the initial ensemble needs to be chosen larger than a certain threshold to ensure convergence .
however , in the regular regime , it is not important to take a large ensemble and a single trajectory can suffice , as long as we take care to discard the transients .
that is to say , in the computation of currents , the definition of the current needs to be modified to : @xmath58 where @xmath59 is some empirically obtained cut - off such that we get a converged current ( for instance , in our calculations , we obtained converged results with @xmath60 ) . when this modified form is used , the convergence ( ensemble - independence ) is more rapid as a function of @xmath61 and the width of the intial conditions .
armed with this background , we are now finally in a position to compare bifurcation diagrams with the current , as we do in the next section .
our results are presented in the set of figures fig .
( [ figure5 ] ) fig .
( [ rev - nobifur ] ) , in each of which we plot both the ensemble current and the bifurcation diagram as a function of the parameter @xmath0 .
the main point of these numerical results can be distilled into a series of heuristic statements which we state below ; these are labelled with roman numerals . for @xmath7 and @xmath8 , we plot current ( upper ) with @xmath62 and bifurcation diagram ( lower ) versus @xmath0 .
note that there is a * single * current reversal while there are many bifurcations visible in the same parameter range.,title="fig:",width=302 ] consider fig .
( [ figure5 ] ) , which shows the parameter range @xmath63 chosen relatively arbitrarily . in this figure , we see several period - doubling bifurcations leading to order - chaos transitions , such as for example in the approximate ranges @xmath64 . however , there is only one instance of current - reversal , at @xmath65 .
note , however , that the current is not without structure it changes fairly dramatically as a function of parameter .
this point is made even more clearly in fig .
( [ figure6 ] ) where the current remains consistently below @xmath53 , and hence there are in fact , no current reversals at all .
note again , however , that the current has considerable structure , even while remaining negative
. for @xmath66 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 .
notice the current stays consistently below @xmath53.,title="fig:",width=302 ] current and bifurcations versus @xmath0 . in ( a ) and ( b )
we show ensemble dependence , specifically in ( a ) the black curve is for an ensemble of trajectories starting centered at the stable fixed point @xmath67 with a root - mean - square gaussian width of @xmath68 , and the brown curve for trajectories starting from the unstable fixed point @xmath69 and of width @xmath68 . in ( b ) , all ensembles are centered at the stable fixed point , the black line for an ensemble of width @xmath68 , brown a width of @xmath70 and maroon with width @xmath71 .
( c ) is the comparison of the current @xmath38 without transients ( black ) and with transients ( brown ) along with the single - trajectory results in blue ( after barbi and salerno ) .
the initial conditions for the ensembles are centered at @xmath67 with a mean root square gaussian of width @xmath68 .
( d ) is the corresponding bifurcation diagram.,title="fig:",width=302 ] it is possible to find several examples of this at different parameters , leading to the negative conclusion , therefore , that * ( i ) not all bifurcations lead to current reversal*. however , we are searching for positive correlations , and at this point we have not precluded the more restricted statement that all current reversals are associated with bifurcations , which is in fact mateos conjecture .
we therefore now move onto comparing our results against the specific details of barbi and salerno s treatment of this conjecture . in particular , we look at their figs .
( 2,3a,3b ) , where they scan the parameter region @xmath72 .
the distinction between their results and ours is that we are using _
ensembles _ of particles , and are investigating the convergence of these results as a function of number of particles @xmath44 , the width of the ensemble in phase - space , as well as transience parameters @xmath73 . our data with larger @xmath44 yields different results in general , as we show in the recomputed versions of these figures , presented here in figs .
( [ current - bifur1],[current - bifur2 ] ) .
specifically , ( a ) the single - trajectory results are , not surprisingly , cleaner and can be more easily interpreted as part of transitions in the behavior of the stability properties of the periodic orbits .
the ensemble results on the other hand , even when converged , show statistical roughness .
( b ) the ensemble results are consistent with barbi and salerno in general , although disagreeing in several details .
for instance , ( c ) the bifurcation at @xmath74 has a much gentler impact on the ensemble current , which has been growing for a while , while the single - trajectory result changes abruptly . note , ( d ) the very interesting fact that the single - trajectory current completely misses the bifurcation - associated spike at @xmath75 .
further , ( e ) the barbi and salerno discussion of the behavior of the current in the range @xmath76 is seen to be flawed
our results are consistent with theirs , however , the current changes are seen to be consistent with bifurcations despite their statements to the contrary . on the other hand ( f ) , the ensemble current shows a case [ in fig .
( [ current - bifur2 ] ) , at @xmath77 of current reversal that does not seem to be associated with bifurcations .
in this spike , the current abruptly drops below @xmath53 and then rises above it again .
the single trajectory current completely ignores this particular effect , as can be seen .
the bifurcation diagram indicates that in this case the important transitions happen either before or after the spike .
all of this adds up to two statements : the first is a reiteration of the fact that there is significant information in the ensemble current that can not be obtained from the single - trajectory current .
the second is that the heuristic that arises from this is again a negative conclusion , that * ( ii ) not all current reversals are associated with bifurcations .
* where does this leave us in the search for ` positive ' results , that is , useful heuristics ?
one possible way of retaining the mateos conjecture is to weaken it , i.e. make it into the statement that * ( iii ) _ most _ current reversals are associated with bifurcations . * same as fig .
( [ current - bifur1 ] ) except for the range of @xmath0 considered.,title="fig:",width=302 ] for @xmath78 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 .
note in particular in this figure that eyeball tests can be misleading .
we see reversals without bifurcations in ( a ) whereas the zoomed version ( c ) shows that there are windows of periodic and chaotic regimes .
this is further evidence that jumps in the current correspond in general to bifurcation.,title="fig:",width=302 ] for @xmath7 and @xmath79 , current ( upper ) and bifurcation diagram ( lower ) versus @xmath0.,title="fig:",width=302 ] however , a * different * rule of thumb , previously not proposed , emerges from our studies .
this generalizes mateos conjecture to say that * ( iv ) bifurcations correspond to sudden current changes ( spikes or jumps)*. note that this means these changes in current are not necessarily reversals of direction .
if this current jump or spike goes through zero , this coincides with a current reversal , making the mateos conjecture a special case .
the physical basis of this argument is the fact that ensembles of particles in chaotic systems _ can _ have net directed transport but the details of this behavior depends relatively sensitively on the system parameters .
this parameter dependence is greatly exaggerated at the bifurcation point , when the dynamics of the underlying single - particle system undergoes a transition a period - doubling transition , for example , or one from chaos to regular behavior .
scanning the relevant figures , we see that this is a very useful rule of thumb . for example
, it completely captures the behaviour of fig .
( [ figure6 ] ) which can not be understood as either an example of the mateos conjecture , or even a failure thereof . as such
, this rule significantly enhances our ability to characterize changes in the behavior of the current as a function of parameter .
a further example of where this modified conjecture helps us is in looking at a seeming negation of the mateos conjecture , that is , an example where we seem to see current - reversal without bifurcation , visible in fig .
( [ hidden - bifur ] ) .
the current - reversals in that scan of parameter space seem to happen inside the chaotic regime and seemingly independent of bifurcation . however , this turns out to be a ` hidden ' bifurcation when we zoom in on the chaotic regime , we see hidden periodic windows .
this is therefore consistent with our statement that sudden current changes are associated with bifurcations .
each of the transitions from periodic behavior to chaos and back provides opportunities for the current to spike .
however , in not all such cases can these hidden bifurcations be found .
we can see an example of this in fig .
( [ rev - nobifur ] ) .
the current is seen to move smoothly across @xmath80 with seemingly no corresponding bifurcations , even when we do a careful zoom on the data , as in fig .
( [ hidden - bifur ] ) .
however , arguably , although subjective , this change is close to the bifurcation point .
this result , that there are situations where the heuristics simply do not seem to apply , are part of the open questions associated with this problem , of course .
we note , however , that we have seen that these broad arguments hold when we vary other parameters as well ( figures not shown here ) . in conclusion ,
in this paper we have taken the approach that it is useful to find general rules of thumb ( even if not universally valid ) to understand the complicated behavior of non - equilibrium nonlinear statistical mechanical systems . in the case of chaotic deterministic ratchets
, we have shown that it is important to factor out issues of size , location , spread , and transience in computing the ` current ' due to an ensemble before we search for such rules , and that the dependence on ensemble characteristics is most critical near certain bifurcation points .
we have then argued that the following heuristic characteristics hold : bifurcations in single - trajectory behavior often corresponds to sudden spikes or jumps in the current for an ensemble in the same system .
current reversals are a special case of this
. however , not all spikes or jumps correspond to a bifurcation , nor vice versa .
the open question is clearly to figure out if the reason for when these rules are violated or are valid can be made more concrete .
a.k . gratefully acknowledges t. barsch and kamal p. singh for stimulating discussions , the reimar lst grant and financial support from the alexander von humboldt foundation in bonn . a.k.p .
is grateful to carleton college for the ` sit , wallin , and class of 1949 ' sabbatical fellowships , and to the mpipks for hosting him for a sabbatical visit , which led to this collaboration .
useful discussions with j .-
m . rost on preliminary results are also acknowledged .
p. hnggi and bartussek , in nonlinear physics of complex systems , lecture notes in physics vol .
476 , edited by j. parisi , s.c .
mueller , and w. zimmermann ( springer verlag , berlin , 1996 ) , pp.294 - 308 ; r.d .
asturmian , science * 276 * , 917 ( 1997 ) ; f. jlicher , a. ajdari , and j. prost , rev . mod .
phys . * 69 * , 1269 ( 1997 ) ; c. dring , nuovo cimento d*17 * , 685 ( 1995 ) s. flach , o. yevtushenko , and y. zolotaryuk , phys
. rev .
lett . * 84 * , 2358 ( 2000 ) ; o. yevtushenko , s. flach , y. zolotaryuk , and a. a. ovchinnikov , europhys .
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e * 66 * , 041104 ( 2002 ) | in 84 , 258 ( 2000 ) , mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes .
this is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential .
barbi and salerno , in 62 , 1988 ( 2000 ) , have further investigated this claim and argue that , contrary to mateos claim , current reversals can occur also in the absence of bifurcations .
barbi and salerno s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system .
the behavior of ensembles can be quite different , depending upon their characteristics , which leaves their results open to question . in this paper we present results from studies showing how the current depends on the details of the ensemble
used to generate it , as well as conditions for convergent behavior ( that is , independent of the details of the ensemble ) .
we are then able to present the converged current as a function of parameters , in the same system as mateos as well as barbi and salerno .
we show evidence for current reversal without bifurcation , as well as bifurcation without current reversal .
we conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation , rather than current reversals , and show numerical evidence for our claims . | [
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] |
studies of laser beams propagating through turbulent atmospheres are important for many applications such as remote sensing , tracking , and long - distance optical communications .
howerver , fully coherent laser beams are very sensitive to fluctuations of the atmospheric refractive index .
the initially coherent laser beam acquires some properties of gaussian statistics in course of its propagation through the turbulence . as a result , the noise / signal ratio approaches unity for long - distance propagation .
( see , for example , refs.@xcite-@xcite ) .
this unfavourable effect limits the performance of communication channels . to mitigate this negative effect
the use of partially ( spatially ) coherent beams was proposed .
the coherent laser beam can be transformed into a partially coherent beam by means of a phase diffuser placed near the exit aperture .
this diffuser introduces an additional phase ( randomly varying in space and time ) to the wave front of the outgoing radiation .
statistical characteristics of the random phase determine the initial transverse coherence length of the beam .
it is shown in refs .
@xcite,@xcite that a considerable decrease in the noise / signal ratio can occur under following conditions : ( i ) the ratio of the initial transverse coherence length , @xmath0 , to the beam radius , @xmath1 , should be essentially smaller than unity ; and ( ii ) the characteristic time of phase variations , @xmath2 , should be much smaller than the integration time , @xmath3 , of the detector .
however , only limiting cases @xmath4 and @xmath5 have been considered in the literature .
( see , for example , refs .
@xcite,@xcite and ref .
@xcite , respectively ) .
it is evident that the inequality @xmath6 can be easily satisfied by choosing a detector with very long integration time . at the same time
, this kind of the detector can not distinguish different signals within the interval @xmath3 .
this means that the resolution of the receiving system might become too low for the case of large @xmath3 . on the other hand
, there is a technical restriction on phase diffusers : up to now their characteristic times , @xmath2 , are not smaller than @xmath7 . besides that , in some specific cases ( see , for example , ref .
@xcite ) , the spectral broadening of laser radiation due to the phase diffuser ( @xmath8 ) may become unacceptably high .
the factors mentioned above impose serious restrictions on the physical characteristics of phase diffusers which could be potentially useful for suppressing the intensity fluctuations .
an adequate choice of diffusers may be facilitated if we know in detail the effect of finite - time phase variation , introduced by them , on the photon statistics . in this case
, it is possible to control the performance of communication systems . in what follows , we will obtain theoretically the dependence of scintillation index on @xmath9 without any restrictions on the value of this ratio
this is the main purpose of our paper .
further analysis is based on the formalism developed in ref .
@xcite and modified here to understand the case of finite - time dynamics of the phase diffuser .
the detectors of the absorbed type do not sense the instantaneous intensity of electromagnetic waves @xmath10 .
they sense the intensity averaged over some finite interval @xmath3 i.e. @xmath11 usually , the averaging time @xmath3 ( the integration time of the detector ) is much smaller than the characteristic time of the turbulence variation , @xmath12 , ( @xmath13 ) .
therefore , the average value of the intensity can be obtained by further averaging of eq .
[ one ] over many measurements corresponding various realizations of the refractive - index configurations .
the scintillation index determining the mean - square fluctuations of the intensity is defined by @xmath14\bigg /\big
< \bar{i}\big > ^2=
\frac{\big < : \bar i(t ) ^2:\big>}{\big<\bar i \big>^2}-1,\ ] ] where the symbol @xmath15 indicates the normal ordering of the creation and annihilation operators which determine the intensity , @xmath10 .
( see more details in refs .
@xcite,@xcite ) . the brackets @xmath16 indicate quantum - mechanical and atmospheric averagings .
the intensity @xmath17 depends not only on @xmath18 , but also on the spatial variable @xmath19 . therefore , the detected intensity is the intensity @xmath20 averaged not only over @xmath18 as in eq .
[ one ] , but also over the detector aperture . for simplicity , we will restrict ourselves to calculations of the intensity correlations for coinciding spatial points that correspond to `` small '' detector aperture .
this simplification is quite reasonable for a long - distance propagation path of the beam . in the case of quasimonochromatic light
, we can choose @xmath20 in the form @xmath21 where @xmath22 and @xmath23 are the creation and annihilation operators of photons with momentum @xmath24 .
they are given in the heisenberg representation .
@xmath25 is the volume of the system .
it follows from eqs .
[ two],[three ] that @xmath26 can be obtained if one knows the average @xmath27 it is a complex problem to obtain this value for arbitrary turbulence strengths and propagation distances .
nevertheless , the following qualitative reasoning can help to do this in the case of strong turbulence .
we have mentioned that the laser light acquires the properties of gaussian statistics in the course of its propagation through the turbulent atmosphere . as a result , in the limit of infinitely long propagation path , @xmath28 , only
diagonal " terms , i.e. terms with ( i ) @xmath29 or ( ii ) @xmath30 , @xmath31 contribute to the right part of eq . [ four ] . for large but still finite @xmath28
, there exist small ranges of @xmath32 in case ( i ) and @xmath33 , @xmath34 in case ( ii ) contributing into the sum in eq .
the presence of the mentioned regions is due to the two possible ways of correlating of four different waves ( see ref .
@xcite ) which enter the right hand side of eq .
[ four ] . as explained in ref .
@xcite , the characteristic sizes of regions ( i ) and ( ii ) depend on the atmospheric broadening of beam radii as @xmath35 , thus decreasing with increasing @xmath28 . in the case of long - distance propagation , @xmath36 is much smaller than the component of photon wave - vectors perpendicular to the @xmath28 axis .
the last quantity grows with @xmath28 as @xmath37 .
( see ref .
@xcite ) . for this reason ,
the overlapping of regions ( i ) and ( ii ) can be neglected . in this case
eq . [ four ] can be rewritten in the convenient form : @xmath38 @xmath39 where the value @xmath40 , confining summation over @xmath41 , is chosen to be greater than @xmath42 but much smaller than the characteristic transverse wave vector of the photons ; this is consistent with the above explanations .
the two terms in the right - hand side correspond to the two regions of four - wave correlations .
the quantity @xmath43 entering the right side of eq . [ five ] is the operator of photon density in phase space ( the photon distribution function in @xmath44 space ) .
it was used in refs . @xcite,@xcite and
@xcite for the description of photon propagation in turbulent atmospheres . by analogy
, we can define the two - time distribution function @xmath45 then eq . [ five ] can be rewritten in terms of the distribution functions as @xmath46 let us represent @xmath47 in the form @xmath48 .
we assume that @xmath49 , as explained in the text after eq.[one ] . in this case
the hamiltonian of photons in a turbulent atmosphere can be considered to be independent of time . as a result , both functions defined by eqs .
[ six ] and [ seven ] satisfy the same kinetic equation , i.e. @xmath50 @xmath51 where @xmath52 is the photon velocity , @xmath53 is a random force , caused by the turbulence .
this force is equal to @xmath54 , where @xmath55 is the frequency of laser radiation .
@xmath56 is the refractive index of the atmosphere .
the general solution of the equation for @xmath48 can be written in the form @xmath57 where @xmath58 @xmath59 the functions @xmath60 and @xmath61 obey the equations of motion @xmath62 with the boundary conditions @xmath63 .
the instant @xmath64 is equal to @xmath65 , where @xmath66 is the speed of light .
@xmath64 is the time of the exit of photons from the source .
this choice of @xmath64 makes it possible to neglect the influence of the turbulence on the initial values of operators @xmath67 ( their dependence on time is as in vacuum ) .
the term for @xmath68 can be obtained from eq . [ twelve ] by putting @xmath69 . substituting both distribution functions into eq .
[ eight ] , we obtain @xmath70 @xmath71 @xmath72:\big>,\ ] ] where @xmath73 and @xmath74 are solutions of eqs .
[ twelve ] with the initial conditions @xmath63 and @xmath75 , respectively .
the operators on the right side of eq . [ thirteen ] are related through matching conditions with the amplitudes of the exiting laser radiation ( see ref .
@xcite ) by the relation @xmath76 where @xmath77 is the operator of the laser field which is assumed to be a single - mode field and the subscript ( @xmath78 ) means perpendicular to the @xmath28-axis component .
the function @xmath79 describes the profile of the laser mode , which is assumed to be gaussian - type function [ @xmath80 .
@xmath1 desribes the initial radius of the beam . to account for the effect of the phase diffuser
, a factor @xmath81 or @xmath82 should be inserted into the integrand of eq . [ fourteen ] .
the quantity @xmath83 is the random phase introduced by the phase diffuser .
a similar consideration is applicable to each of four photon operators entering both terms in square brackets of eq .
[ thirteen ] .
it can be easily seen that the factor @xmath84},\ ] ] describing the effect of phase screen on the beam , enters implicitly the integrand of eq .
[ thirteen ] ( the indices @xmath78 are omitted here for the sake of brevity ) .
there are integrations over variables @xmath85 as shown in eq .
[ fourteen ] .
furthermore , the brackets @xmath16 , which indicate averaging over different realizations of the atmosperic inhomogeneities , also indicate averaging over different states of the phase diffuser .
as long as both types of averaging do not correlate , the factor ( [ fifteen ] ) entering eq .
[ thirteen ] must be averaged over different instants , @xmath64 . to begin with ,
let us consider the simplest case of two phase correlations @xmath86}\big > .\ ] ] it is evident that in the case @xmath87 , as shown schematically in fig .
1 , the factor ( [ sixteen ] ) is sizable if only points @xmath19 and @xmath88 are close to one another .
two curves correspond to different instants @xmath18 and @xmath89 . ] therefore , the term given by eq .
[ sixteen ] can be replaced by @xmath90 where @xmath91 is considered to be a gaussian random variable with the mean - square values given by @xmath92 ^ 2\rangle = \langle [ \frac { \partial \varphi ( { \bf r},t_0)}{\partial y}]^2\rangle = 2\lambda _ c^{-2}$ ] , where @xmath93 is the correlation length of phase fluctuations . ( see fig.1 ) . as we see , in this case the effect of phase fluctuations can be described by the schell model @xcite-@xcite,@xcite-@xcite . a somewhat more complex situation is for the average value of @xmath94 given by eq . [ fifteen ] .
there is an effective phase correlation not only in the case of coincident times , but also for differing times . for @xmath95 , two different sets of coordinates contribute considerably to phase correlations .
this can be described mathematically as @xmath96}\big > \approx \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime},t_0)]}\big > \times\ ] ] @xmath97}\big > + \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime _
1},t_0+\tau ) ] } \big > \big < e^{i[\varphi ( { \bf r_1},t_0+\tau ) -\varphi ( { \bf r^\prime } , t_0)]}\big > .\ ] ] repeating the arguments leading to eq .
[ seventeen ] , we represent the difference in the last term @xmath98 as @xmath99 then , considering the random functions @xmath100 and @xmath101 as independent gaussian variables , we obtain a simple expression for @xmath102 . it is given by @xmath103}+ e^{-\lambda _ c^{-2}[({\bf r - r^\prime}_1)^2+({\bf r^\prime -r_1})^2]-2\nu^2\tau ^2},\ ] ] where @xmath104 ^ 2\rangle = 2\nu^2 $ ] . as we see , the effect of the phase screen can be described by two parameters , @xmath93 and @xmath105 , which characterize the spatial and temporal coherence of the laser beam . in the limiting case , @xmath106 , the second term in eq .
[ twenty ] vanishes and the problem is reduced to the case considered in refs .
@xcite,@xcite . in the opposite case , @xmath107 , both terms in eq .
[ twenty ] are important .
this is shown in ref .
@xcite . in
what follows , we will see that these two limiting cases have physical interpretations where where @xmath108 ( slow detector ) and @xmath109 ( fast detector ) , respectively .
there is a specific realization of the diffuser in which a random phase distribution moves across the beam .
( this situation can be modeled by a rotating transparent disk with large diameter and varying thickness . )
the phase depends here on the only variable @xmath110 , i.e. @xmath111 where @xmath112 is the velocity of the drift .
then we have @xmath113}+e^{-\lambda _ c^{-2}[({\bf r - r^\prime_1+v}\tau)^2+({\bf r^\prime -r_1+v}\tau)^2]}.\ ] ] comparing eqs .
[ twenty ] and [ twtw ] , we see that the quantity , @xmath114 , stands for the characteristic parameter describing the efficiency of the phase diffuser .
the criterion of slow " detector requires @xmath115 .
qualitatively , the two scenarios of phase variations , given by eqs . [ twenty ] and [ twtw ] , affect in a similar way the intensity fluctuations . in what follows
, we consider the first of them as the simplest one .
( this is because the spatial and temporal variables in @xmath102 , given by eq . [
twenty ] , are separable . )
_ vs _ propagation distance @xmath28 in the case of `` slow '' detector : @xmath116 .
the parameter @xmath117 indicates different initial coherence length . in the absence of phase diffuser @xmath118 ( solid line ) .
@xmath119 is the conventional parameter describing a strength of the atmospheric turbulence . ]
substituting the expressions for operators given by eq .
[ fourteen ] with account for the phase factors @xmath120 and averaging over time as shown in eq .
[ one ] , we obtain @xmath121 @xmath122\bigg > , \ ] ] where the notation @xmath16 after sums indicates averaging over different realizations of the atmospheric refractive index .
the parameter @xmath123 describes the initial coherence length modified by the phase diffuser .
other notations are defined by following relations @xmath124
@xmath125 @xmath126 further calculations follow the scheme described in ref @xcite .
2 illustrates the effect of the phase diffuser on scintillations in the limit of a slow " detector ( @xmath127 ) .
we can see a considerable decrease in @xmath128 caused by the phase diffuser . at the same time , the effect of the phase screen on @xmath128 becomes weaker for finite values of @xmath129 .
moreover , comparing the two upper curves in fig .
3 , we see the opposite effect : slow phase variations ( @xmath130 ) result in increased scintillations .
there is a simple explanation for this phenomenon : the noise generated by the turbulence is complemented by the noise arising from the random phase screen .
the integration time of the detector , @xmath3 , is not sufficiently large for averaging phase variations generated by the diffuser .
the function , @xmath131 , has a very simple form in the two limits : ( i)@xmath132 , when @xmath133 ; and ( ii ) @xmath134 , when @xmath109 .
then , in case ( i ) and for small values of the initial coherence [ @xmath135 , the asymptotic term for the scintillation index ( @xmath136 ) is given by @xmath137 the right - hand side of eq .
[ twfo ] differs from analogous one in ref .
@xcite by the value @xmath138 that is much less than unity but , nevertheless , can be comparable or even greater than @xmath139 . in case ( ii )
, the asymptotic value of @xmath26 is close to unity , coinciding with the results of refs . @xcite and @xcite .
this agrees with well known behavior of the scintillation index to approach unity for any source distribution , provided the response time of the recording instrument is short compared with the source coherence time .
( see , for example , survey @xcite ) .
a similar tendency can be seen in both figs . 3 and 4 : the curves with the smallest @xmath129 , used for numerical calculations ( @xmath130 ) , are close to the curves without diffuser " in spite of the small initial coherence length [ @xmath140 .
it can also be seen that all curves approach their asymptotic values very slowly .
describing diffuser dynamics .
the solid curve is calculated for @xmath118 ( without diffuser ) .
other curves are for @xmath141 . ] . ]
it follows from our analysis that the scintillation index is very sensitive to the diffuser parameters , @xmath0 and @xmath142 , for long propagation paths .
on the other hand , the characteristics of the irradience such as beam radius , @xmath143 , and angle - of - arrival spread , @xmath144 , do not depend on the presence of the phase diffuser for large values of @xmath28 . to see this , the following analysis is useful .
the beam radius expressed in terms of the distribution function is given by @xmath145 straightforward calculations using eq .
[ ten ] with @xmath69 ( see ref .
@xcite ) result in the following explicit form : @xmath146 where @xmath147 and @xmath148 is the inner radius of turbulent eddies , which in our previous calculations was assumed to be equal @xmath149 m . as we see , the third term does not depend on the diffuser parameters and it dominates when @xmath150 .
a similar situation holds for the angle - of - arrival spread , @xmath144 .
( this physical quantity is of great importance for the performance of communication systems based on frequency encoded information @xcite . )
it is defined by the distribution function as @xmath151 simple calculations , which are very similar to those while obtaining @xmath152 , result in @xmath153 ^ 2=\frac 2{r_1 ^ 2q_0 ^ 2}+12tz-\frac { 4z^2}{q_0 ^ 4r^2}(r_1^{-2}+3tq_0 ^ 2z)^2.\ ] ] for long propagation paths ,
. [ twei ] reduces to @xmath154 , which like @xmath152 does not depend on the diffuser parameters .
as we see , for large distances @xmath28 , the quantities @xmath152 and @xmath144 do not depend on @xmath93 and @xmath105 .
this contrasts with the case of the scintillation index .
so pronounced differences can be explained by differences in the physical nature of these characteristics .
it follows from eq .
[ two ] that the functional , @xmath26 , is quadratic in the distribution function , @xmath155 .
hence , four - wave correlations determine the value of scintillation index .
the main effect of a phase diffuser on @xmath26 is to destroy correlations between waves exited at different times .
( see more explanations in ref .
this is achieved at sufficiently small parameters @xmath93 and @xmath156 .
in contrast , @xmath152 and @xmath144 depend on two wave - correlations , both waves being given at the same instant .
therefore , the values of @xmath152 and @xmath144 do not depend on the rate of phase variations [ @xmath105 does not enter the factor ( [ seventeen ] ) describing the effect of phase diffuser ] .
moreover , these quantities become independent of @xmath93 at long propagation paths because light scattering on atmospheric inhomogeneities prevails in this case .
the plots in figs .
3 anf 4 show that the finite - time effect is quite sizable even for very slow " detectors ( @xmath157 ) .
our paper makes it possible to estimate the actual utility of phase diffusers in several physical regimes .
we have analyzed the effects of a diffuser on scintillations for the case of large - amplitude phase fluctuations .
this specific case is very convenient for theoretial analysis because only two parameters are required to describe the effects of the diffuser .
phase fluctuations may occur independently in space as well as in time .
also , our formalism can be applied for the physical situation in which a spatially random phase distribution drifts across the beam .
[ twtw ] . )
our results show the importance of both parameters , @xmath93 and @xmath142 , on the ability of a phase diffuser to suppress scintillations .
this work was carried out under the auspices of the national nuclear security administration of the u.s .
department of energy at los alamos national laboratory under contract no .
de - ac52 - 06na25396 .
we thank onr for supporting this research . | the effect of a random phase diffuser on fluctuations of laser light ( scintillations ) is studied .
not only spatial but also temporal phase variations introduced by the phase diffuser are analyzed .
the explicit dependence of the scintillation index on finite - time phase variations is obtained for long propagation paths .
it is shown that for large amplitudes of phase fluctuations , a finite - time effect decreases the ability of phase diffuser to suppress the scintillations . | [
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"the so - called `` nucleon spin crisis '' raised by the european muon collaboration ( emc ) measure(...TRUNCATED) | "with a special intention of clarifying the underlying spin contents of the nucleon , we investigate(...TRUNCATED) | [21603,10,8,78,3,18,718,3,2,206,2482,106,5404,5362,3,31,31,3279,57,8,14864,4035,106,3561,41,3,15,51,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [28,3,9,534,8762,13,17741,53,8,3,10067,5404,10223,13,8,206,2482,106,3,6,62,9127,8,879,1601,607,2580,(...TRUNCATED) |
"let @xmath1 . let @xmath2\\longrightarrow [ \n 0,\\pi_{p , q}/2]$ ] be the integral @xmath3 where @(...TRUNCATED) | "we improve the currently known thresholds for basisness of the family of periodically dilated @xmat(...TRUNCATED) | [21603,10,752,3320,226,3357,107,536,3,5,752,3320,226,3357,107,357,2,2961,3535,6770,784,8014,2,102,23(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [62,1172,8,1083,801,12709,7,21,1873,655,13,8,384,13,22654,3,26,173,920,3320,226,3357,107,9498,7,630,(...TRUNCATED) |
"the simulations we discuss here allowed us to obtain spectra of the shg response . \n we employed c(...TRUNCATED) | "we report on strong enhancement of mid - infrared second harmonic generation ( shg ) from sic nanop(...TRUNCATED) | [21603,10,8,11108,7,62,2497,270,2225,178,12,3442,3,7576,1313,13,8,6660,122,1773,3,5,62,8152,3,287,40(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [62,934,30,1101,15220,13,2076,3,18,16,89,11096,26,511,29610,3381,41,6660,122,3,61,45,108,75,13944,12(...TRUNCATED) |
"with significant research efforts being directed to the development of neurocomputers based on the (...TRUNCATED) | "synaptic memory is considered to be the main element responsible for learning and cognition in huma(...TRUNCATED) | [21603,10,28,1516,585,2231,271,6640,12,8,606,13,6567,25750,7,3,390,30,8,5014,2197,13,8,2241,3,6,3,9,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [8953,6789,447,2594,19,1702,12,36,8,711,3282,1966,21,1036,11,23179,4749,16,6917,3,5,2199,16910,529,3(...TRUNCATED) |
"the segmentation process as a whole can be thought of as consisting of two tasks : recognition and (...TRUNCATED) | "this paper investigates , using prior shape models and the concept of ball scale ( b - scale ) , wa(...TRUNCATED) | [21603,10,8,5508,257,433,38,3,9,829,54,36,816,13,38,5608,53,13,192,4145,3,10,5786,11,20,747,257,3,5,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [48,1040,9127,7,3,6,338,1884,2346,2250,11,8,2077,13,1996,2643,41,3,115,3,18,2643,3,61,3,6,1155,13,32(...TRUNCATED) |
"one surprising result that has come out of the more than 200 extrasolar planet discoveries to date (...TRUNCATED) | "long time coverage and high radial velocity precision have allowed for the discovery of additional (...TRUNCATED) | [21603,10,80,10306,741,24,65,369,91,13,8,72,145,2382,996,7,17401,4345,25175,12,833,19,8,1148,620,13,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [307,97,2591,11,306,3,5883,138,22924,11723,43,2225,21,8,9087,13,1151,4820,16,801,3,30351,1002,3,5,18(...TRUNCATED) |
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