KBlueLeaf/taica-nlp-hw1 · Datasets at Hugging Face

text string
( [ 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 d17 , 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 .
lett . * 54 * , 141 ( 2001 ) ; s. denisov et al .
e * 66 * , 041104 ( 2002 )
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 .

( [ 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 .

lett . * 54 * , 141 ( 2001 ) ; s. denisov et al .

e * 66 * , 041104 ( 2002 )

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 .