_ superseparability _ may be defined as the polar opposite of entanglement : two identical charged bosons , with state vectors that have considerable spatio - temporal overlap , are unable to feel each other s presence because the state vectors lie in disjoint hilbert spaces and can not be superposed .mathematically , this possibility appears to be contained in von neumann s hilbert space formulation of quantum mechanics , but whether or not it is realized in nature can only be ascertained by experiment .this note describes the principle of a possible experiment based on the magnetic aharonov - bohm effect .  the mathematical phenomena that suggest the experiment are subtleties hidden in the notion of self - adjointness for unbounded operators .experience shows that these subtleties can generally be disregarded in practical applications of quantum mechanics ; among physicists , only the mathematically minded are likely to be familiar with them .for this reason , a brief review of the basic definitions and results , due mostly to von neumann , is provided in section [ sec - self - adj ] .it follows an essential review of the historical background in section [ sec - hist ] .the paper begins with a description of the experimental scheme in section [ sec - scheme ] ; the theory of the experiment is given , after the historical and mathematical excursions , in section [ sec - reeh ] .if the experiment turns out to be feasible , its results be they positive or negative will be consequential , and implications of the possible results are discussed briefly , in a non - speculative manner , in section [ sec - discuss ] .the purpose of the experiment is to determine whether or not two beams of identical bosons , prepared in inequivalent representations of the ccr , can interfere with each other . a possible scheme for such an experiment is shown in figure 1 .a coherent beam of charged bosons ( e.g. , -particles or deuterons ) from a source is split into two by a beam - splitter .one beam goes through the chamber , the other through the chamber .neither chamber contains a magnetic field , and the two are electromagnetically isolated from each other .they contain the magnetic flux lines respectively ( perpendicular to the plane of the paper ) ; at least one of these fluxes is continuously variable over a certain range .the experiment consists of observing changes in the interference pattern at the detector as is varied .units < 1mm,1 mm >x from -75 to 45 , y from -20 to 15  from -30 -10 to 30 -10 from -30 10 to 30 10 from -30 0 to -10 0 from 10 0 to 30.4 0 from -10 0.3 to -10 -2.3 from -10.4 -2.3 to 0.4 -2.3 from 0 -2.3 to 0 2.3 from -0.4 2.3 to 10.4 2.3 from 10 2.3 to 10 -0.3 from -30 -10.3 to -30 -8 from -30 -6 to -30 6 from -30 8 to -30 10.3 from 30 -10.3 to 30 -2 from 30 2 to 30 10.3 [ b1 ] at -4.5 0 [ b1 ] at 5.5 0 [ b1 ] at -60 0  from -45 -10 to -45 10 from 43 -5 to 43 5  = 0.5pt  -45 8.5 43 0 -45 -8.5 / -45 8.5 -60 0 -45 -8.5 / at -61.5 -4 at -45 -13 at 43 -8.3 at -6.5 2.5 at 4 -1.5 at 0 -7 at 0 7  as will be shown in section [ sec - reeh ] , von neumann s hilbert space formulation of quantum mechanics suggests that the two beams should interfere _ only _ when is an integer , where is the charge of the boson , which will be for deuterons and and for -particles .( we use units in which for non - integral values of , the two beams will belong to disjoint hilbert spaces and should not if the phrase _ disjoint hilbert spaces _ has physical meaning be able to interfere with each other . for this reason , the scheme of the figureis called a _noninterferometer_. it is also possible that the above interpretation of quantum mechanics is invalid , and that both beams belong to the same hilbert space . in that casethe interference pattern should merely shift , as is varied , returning to the original state when has changed exactly by unity ; the fringe shift should be periodic , with period .if the experiment is realizable , the case will correspond to the standard aharonov - bohm effect , so that this effect may be used to test the electromagnetic isolation of the chambers and .the reader is referred to the monograph by peshkin and tonomura for a historical account of the aharonov - bohm effect , and to the review article and monograph for the decisive experiments .it should be recalled that if a flux is quantized , then , where is an integer , so that for deuterons and for -particles .therefore superseparability will never be observed with -particles if both fluxes are quantized .we shall end this section with a few reservations and a remark .the configuration described above is an ideal which may be hard to realize in the laboratory ; that is why we have called it the ` scheme ' of an experiment .it is unlikely that the chambers and can be perfectly isolated from each other .it may not be possible to prepare a state with a sharp value of if the flux is not quantized .finally , the theory of section [ sec - reeh ] would be applicable only if the interiors of chambers and are , for purposes of the experiment , reasonable approximations to the punctured plane .on the other hand , we know if only by hindsight that the aharonov - bohm effect can be observed under conditions that are less than ideal .therefore the possibility that superseparability may also be testable under less than ideal conditions should not be ruled out of hand .we begin by recalling two basic facts .( i ) the born - jordan commutation relation = -{\mathrm{i}}\,\!i ] , if and are unbounded .the basic structures of quantum mechanics , namely matrix mechanics , wave mechanics and transformation theory were laid down in 192527 , but unbounded operators began to be explored only in 19291930 . the ` first quantum revolution ' ( this term is due to aspect )was completed while unbounded operators were still terra incognita even to mathematicians .in retrospect , one is struck by the fact that transformation theory could be developed with scant understanding of the operators that were to be transformed . by what magic was this achieved ?the answer lies in an ansatz due to hermann weyl and a theorem proven by von neumann .  in 1928, weyl published his book _ gruppentheorie und quantenmechanik _ . in this bookhe replaced the canonical commutation relations ( ccr ) for degrees of freedom by a -parameter lie group , which had the ccr as its lie algebra .this group has become known as the _ weyl group _ , and we shall denote it by . we shall give the argument for ; the general case merely requires a cumbersome modification of the notation ( see , pp .272276 ) .let and define , formally , from the properties of the exponential function , it follows that write , and .formal computation yields the result by definition , the weyl group consists of the set of elements ( [ weyl - group-1 ] ) , with multiplication defined by ( [ weyl - group-2 ] ) and ( [ weyl - group-3 ] ) .the element is the identity of the group .the group is nonabelian and noncompact , with as the group manifold , and is a lie group .the same is true of the weyl group for degrees of freedom , except that its group manifold is .being noncompact , the weyl groups have no finite dimensional unitary representations . in a unitary representation , the elements and of are represented by unitary operators and on the hilbert space , and similar statements hold for . a result known as stone s theorem asserts that a one - parameter group of unitaries on a hilbert space has an infinitesimal generator : , where is self - adjoint .it is bounded if is compact ( , the circle ) and unbounded if is not compact ( ) .a representation of defines , uniquely , a representation of its lie algebra the ccr by self - adjoint operators . in the representationso defined , at least one member of any canonical pair is represented by an unbounded operator .. the exponential of the unbounded self - adjoint operator needs definition , but we shall content ourselves with the statement that it turns out to have the expected properties . ]  in 1930 von neumann proved that , for finite , the weyl group has only one irreducible unitary representation .he gave the name _ schrdinger operators _ to the representatives of the canonical variables , , and titled his paper ` die eindeutigkeit schrdingersche operatoren ' . his result has become known as ` von neumann s uniqueness theorem ' .if the ccr were equivalent to the weyl group , it would explain why quantum mechanics could be developed ahead of the theory of unbounded operators without falling into gross error .a lie group defines a unique lie algebra , but the converse is not true .the simplest examples are the covering groups of compact non - simply - connected lie groups. examples of this phenomenon that are relevant to elementary particle physics were unearthed by michel as early as 1962 .the canonical commutation relations are _ not _ abstractly equivalent to the weyl group ; as we shall see below , the will not even generate a lie group unless they are represented by self - adjoint operators .however , the requirement of self - adjointness can not be met in some simple and realizable physical situations .let be a hilbert space and an operator on it .if there exists a positive number such that for all , then is said to be _bounded_. if no such exists , then is said to be _unbounded_. an unbounded operator is not defined everywhere on ; the subset on which it is defined is called the _ domain _ of .if is not dense in then is not ( yet ) mathematically manageable , and one generally assumes that is _ densely defined _, i.e. , is dense in .  in the rest of this section we shall deal only with unbounded operators , andtherefore the adjective ` unbounded ' will be omitted .an operator is called _ closed _ if the set of ordered pairs is a closed subset of .an operator is an _ extension _ of if and for ; one writes .an operator is called _ closable _ if it has a closed extension . every closable operator has a smallest closed extension , which is denoted by .  in matrix theory ,the adjoint is defined by . in operator theory, one has to take domains into consideration .let such that for all , and define by .then is precisely the set of these .one can show that if is densely defined , then is closed .furthermore , is densely defined if and only if is closable , and if it is , then .  if and for all , then is called _symmetric_. if and for all , then is called _ self - adjoint_. self - adjoint operators form a subclass of symmetric operators .a symmetric operator may have no self - adjoint extension , it may have many self - adjoint extensions , or it may have only one . in the last case ,it is called _ essentially self - adjoint_. one can show that if is essentially self - adjoint , then its closure is self - adjoint , i.e. , is the unique self - adjoint extension of .the fundamental differences between symmetric and self - adjoint operators are :  1 .the spectrum of a self - adjoint operator is a subset of the real line , whereas the spectrum of a symmetric operator is a subset of the complex plane ; a symmetric operator is self - adjoint if and only if its spectrum is a subset of the real line . 2 .a self - adjoint operator can be exponentiated , i.e. , if is self - adjoint then is defined for all ; a symmetric operator which is not self - adjoint _ can not _ be exponentiated .the representation problem for the ccr ( one degree of freedom ) may now be formulated as follows : _ find all pairs of essentially self - adjoint operators , densely defined on a common domain , such that \varphi = -{\mathrm{i}}\varphi ] .consider now the equation it is a linear homogeneous differential equation of the first order which can be solved explicitly for any , and the same holds for the equation .the solutions do not have compact support . by exploiting these solutions , reeh established the following results :  1 .  the operators and are not self - adjoint ; they are essentially self - adjoint .2 .  let and be their self - adjoint extensions , and define then [\epsilon(y ) - \epsilon(y - b)]}\,i ,  \ ] ] where is the identity operator , and         -1&\;\;t<1 .\end{array }        \right.\ ] ]  note that the product [\ldots]$ ] in the exponent on the right - hand side of ( [ reeh - comm ] ) can only assume the values , so that the entire right - hand side can only assume the values .it follows that if is an integer , then the right - hand side of ( [ reeh - comm ] ) equals the identity operator _ for all admissible _ , but _ not _ if is not an integer ; in this case the group generated by the operators is no longer isomorphic with the weyl group . clearly , the groups generated by these operators for are not isomorphic with each other if is not an integer , and therefore the representations of the ccr ( for two degrees of freedom ) they define are not unitarily equivalent .  the experiment suggested in section [ sec - scheme ]is designed to determine whether this mathematical inequivalence has observable physical consequences .if superseparability is observed with charged boson beams , then irrespective of the psychological effect of the observation it will confirm that the notion of inequivalent irreducible representations of the ccr ( for a finite number of degrees of freedom ) is physically meaningful ; vectors from two inequivalent representations can not be superposed upon each other .it should then prompt the investigation of other possible effects that arise from the existence of inequivalent irreducible representations of the ccr .note that the discussion is at the level of the first quantization .2 .  if , however , the phenomenon is _ not observed _ , we shall have to conclude that something basic is lacking in our understanding of the linear space that underlies quantum mechanics : the question that david hilbert asked rolf nevanlinna in the late 1920 s _ tell me , rolf , what is this hilbert space that the young people are talking about ? _ may not yet have been answered to the physicist s satisfaction .3 .  the theoretical considerations of section [ sec - reeh ] do not apply to fermions .the creation - annihilation operators for a fermion are bounded , and the canonical anticommutation relations for a finite number of degrees of freedom have only one irreducible unitary reprsentation .this was proved by jordan and wigner in their very first paper on anticommutation relations in 1928 .therefore one should not expect to find the phenomenon of superseparability among fermions . if the experiment is performed with electrons and superseparability _is _ observed , it will pose new and unsettling problems for theoretical physics .i would like to thank t eisenberg , n panchapakesan , h reeh , h roos , g l sewell and a shimony for their comments on earlier versions of this paper .aspect a ( 2004 ) introduction : john bell and the second quantum revolution , in _ speakable and unspeakable in quantum mechanics _ , by j s bell , second edition ( cambridge : cambridge university press ) pp xvii xxxix   heisenberg w ( 1930 ) _ the physical principles of quantum theory _ ( chicago : university of chicago press ) english translation of _ die physikalischen prinzipien der quantenmechanik _ 1930 ( leipzig : hirzel - verlag )  michel l ( 1964 ) invariance in quantum mechanics and group extension , in _ group - theoretical concepts and methods in elementary particle physics _ lectures of the istanbul summer school of theoretical physics 1962 ed f grsey ( new york : gordon and breach ) pp 135 - 200 .von neumann j ( 1932 ) _ mathematische grundlagen der quantenmechanik _( berlin : julius springer ) english translation ( revised by the author ) _ mathematical foundations of quantum mechanics _ 1955 ( princeton : princeton university press )  weyl h ( 1950 ? ) _ the theory of groups and quantum mechanics _ ( new york : dover press ) translated from the 2nd german edition [ the first german edition of _ gruppentheorie und quantenmechanik _ was published in 1928 ]

Abstract: in 1988 , reeh showed that the representation of the canonical commutation relations that corresponds to the aharonov - bohm effect depends on the magnetic flux . it can be integrated to a representation of the weyl group only if the flux is quantized . it follows from reeh s analysis that representations for are generally inequivalent . as a result , two identical charged bosons may be found in inequivalent representations of the ccr . if unitary inequivalence is a restriction on superposability , then these two particles should not feel each other s presence even when they are in close physical proximity ! if they _ do _ feel each other s presence , then unitary inequivalence is _ not _ a restriction on superposability , and the question arises : what does unitary inequivalence mean ? this paper suggests an experiment that can distinguish between these two possibilities and provides a brief account of the theory behind it , which depends upon the subtle notion of self - adjointness of unbounded operators .   _ you can not occupy two places in space simultaneously . that is axiomatic . _


the investigation of the optical response of disordered matter is an intricate problem . the most common tools to extract information from the scattering data are the trial - and - error methods .however , due to a large number of relevant parameters , the results are tedious to obtain and may prove to be ambiguous .for this reason , direct methods allowing computation of system parameters from the scattering data are highly desirable .  in the single - scattering case ,a few direct methods are known .for example , the fractal dimension of a finite aggregate of particles is given by a special scaling law of the _ structure factor _, , in the fractal regime : here , is the magnitude of the momentum transfer , is the typical microscopic particle size , and is the fractal dimension of the aggregate .this fractal scaling law is valid in the range , where is the mean radius of gyration .the above relation in -space reflects the corresponding scaling law of the pair - correlation function , , in real space : valid in the range .  from the introduction of x - ray generators a century ago to the present synchrotron facilities ,the structure factor has turned out to be a major tool to characterize correlations between the particle positions over many length scales .the relation ( [ 0 ] ) is widely used in small - angle x - ray scattering ( saxs ) to study the morphology of fractal aggregates or fractal surfaces .we can also obtain the specific surface of the system from the _ porod regime _ , and the mean gyration radius from the _ guinier regime _ .on the other hand , most light - scattering experiments involve multiple - scattering processes , which are intractable using simple mathematical tools .however , previous studies have shown that eq .( [ 0 ] ) may remain valid for scattering measurements even though multiple - scattering events are suspected to be present .experimentalists widely use the above single - scattering scaling law for the characterization of various fractal aggregates , though they also claim that minor differences exist between the experimental data and the transmission electron microscopy ( tem ) analysis of the fractal dimension .other results suggest that eq .( [ 0 ] ) might not be valid when multiple - scattering is present . a theoretical argument due to berry and percival claims that the scaling relation in eq .( [ 0 ] ) should fail for aggregates with mass - fractal dimension .however , it is not clear what relation should replace eq .( [ 0 ] ) in that case .results of numerical computations suggest that eq .( [ 0 ] ) is valid for the case of , but with an effective exponent different from the fractal dimension .  in this paper , we present detailed theoretical and numerical results to clarify this important issue .the paper is organized as follows . in section [ s2 ], we introduce the static structure factor with multiple - scattering condition .section [ s3 ] describes the mean - field multiple - scattering theory for the dipolar and multipolar regimes . in section [ s4 ] ,we compare theoretical and numerical results for three different types of cluster aggregates . in section [ sec5 ] ,we discuss the scaling behavior of the structure factor .finally , we conclude with a summary in section [ s6 ] .consider a monochromatic electromagnetic wave of amplitude and wavelength , illuminating an aggregate of spherical particles of radius .we assume elastic light - scattering , with the wave - vector ( of modulus ) defining the propagation direction of the incident beam .the scattered wave - vector , ( with the same modulus ) , defines the observation direction . for the general case of a randomly - oriented aggregate ,the intensity of the scattered wave depends on , where is the magnitude of the scattering vector .assuming separation of the optical properties and the spatial distribution of the particles , is related to the structure factor as the form factor being the intensity scattered by a single particle .we make the following observations : + 1 ) the structure factor is a positive function which can be conveniently normalized such that .this function contains information about the spatial distribution of the particles . for the case of a single particle ( _ i.e. _ , ) ,+ 2 ) the scaling factor is related to the forward - scattering scaling as in most cases , this function behaves as a power - law : , with an exponent  .the simplest way to define the structure factor is then via the normalized ratio of the scattered intensity from the particles ( ) and the scattered intensity from a single particle ( ) : now , being written as the product of a function of the optical parameters and a function involving the spatial distribution of the particles , the definition ( [ sf2 ] ) results in a quantity which depends essentially on the mass distribution of the aggregate .+ the rayleigh - debye - gans ( rdg ) theory provides a framework in which the separation between the optical and geometrical properties is realized .indeed , the rdg theory tells us that the single - scattering of the incoming wave by a collection of electromagnetic dipoles is the dominant process due to the weak electric polarizability of the particles inside the aggregate .the structure factor in eq .( [ sf2 ] ) is then written as the square modulus of the fourier transform of the density distribution of the scattering system : in eq .( [ 2 ] ) , denotes the position of the particle in the aggregate . in eq .( [ sg ] ) , is the pair - correlation function , and is the particle number density in a given volume .+ for a fractal scatterer of radius of gyration , which is an aggregate of spherical particles of radius , the main features of the rdg structure factor are as follows + 1 ) is the _ guinier regime _ , which depends only on the parameter .+ 2 ) is the _ fractal regime _ , which is characterized by eq .( [ 0 ] ) .we focus on this regime in the next section .+ 3 ) is the _ porod regime _ , which is not relevant for the present paper .for experimental studies , it is important to bridge the gap between the simple single - scattering approximation and the complicated multiple - scattering computations . in this context , the _ mean - field _ multiple - scattering approximation provides a mathematically tractable method , which includes the main features of the scattering problem . in this approximation ,all the particles of the aggregate are expected to radiate the same electromagnetic field , differing only in the phase .however , the mean - field approach is not consistent with localization phenomena such as coherent scattering .let us consider an aggregate made up of particles of radius and complex refractive index .the optical size parameter of the aggregate is .then , the dipolar regime corresponds to  in this regime , the multiple - scattering field is similar to the single - scattering field with the renormalized amplitude  here , is the material constant given by \right\ } \label{chi}.\ ] ] further , the complex - valued coefficient is a known function of ( not of ) , given by eq .( [ a ] ) . \label{a } .\end{aligned}\ ] ]  from eq .( [ renorm ] ) , the precise condition for the single - scattering approximation to be valid is written as otherwise , the renormalized coefficient has to be accounted for in the scattering formulae . as the coefficient in eq .( [ renorm ] ) does not depend upon , we deduce an important consequence , i.e. , eq .( [ sg ] ) remains valid in the mean - field _dipolar _ multiple - scattering approximation .the mean - field scattering theory in the multipolar case is comparatively more complicated than the dipolar case .it gives two different factors to renormalize the coefficients in the expansion of the electromagnetic field in terms of the spherical vector wave functions .consequently , it is natural to question the validity of the simple eq .( [ sg ] ) in this case .actually , the structure factor can be written in this case as ( see eq . ( 25 ) of ref . ) where is the intensity of the wave scattered by an isolated sphere , and is the corresponding intensity with the renormalized multipolar coefficients. then , consists of the rdg structure factor ( [ 2 ] ) corrected by a ( possibly complicated ) function of .thus , the fractal scaling relation ( [ 0 ] ) is not guaranteed to work in this framework . + it is shown in ref . that the left - hand term of the condition in eq .( [ cond1 ] ) tends to a constant ( independent of the aggregate size ) when , i.e. , fractal dimensions corresponding to very fluffy aggregates .if this constant is small enough , single - scattering is the dominant process regardless of the cluster size . on the other hand ,when corresponding to dense aggregates , the left - hand term .thus , there is a typical cluster size beyond which multiple - scattering can not be neglected for .in this section , we study the structure factors of two classes of fractal aggregates of spheres , viz . , _ diffusion - limited cluster aggregates _ or dlca with fractal dimension ; and _ diffusion - limited aggregates _ or dla with . for numerical computation, we use the multiple scattering code tmatrix , which takes into account the multipolar terms of the scattered wave .[ fig1](a)-(c ) shows tmatrix scattering data [ vs. on a log - log plot ] obtained for the dlca model .the data is angularly averaged over 20 independent random off - lattice dlca aggregates of spherical particles .we considered three different values of the optical size parameter : ( a ) ; ( b ) ; ( c ) , corresponding to wavelengths respectively larger than , of the same order , and smaller than the aggregate gyration radius .moreover , three different values of the refractive index ( ) were used .one can check the validity of the mean - field approximation by direct comparison with the exact tmatrix results . for this, we analyze the probability distributions of the particle scattering cross - sections , for all the particles in an aggregate ( the index representing the sphere label ) .these distributions are shown in fig .[ fig2 ] for both dlcas and dlas- the latter will be discussed in sec .the main conclusions are as follows . in all the cases , except for the largest optical size parameter and largest refractive index , the mean values agree well with the dominant modes .thus , the mean - field approach is valid in those cases .this is also consistent with the small values of the quantity ( see table [ table1 ] ) , which arises in eq .( [ cond1 ] ) . for the case , the mean - field assumption is not valid , as is clear from the wide distribution of the individual , and the relatively high value ( ) of ..values of [ see eq .( [ cond1 ] ) ] for dlca clusters of particles .we consider various combinations of and .the dipolar single - scattering theory applies for the values given in boldface . [ cols="^,^,^,^",options="header " , ]   the structure factors for the dla cluster are shown in fig .one can draw the following conclusions , which will be discussed in greater detail in sec .+  \1 ) for [ fig .[ fig3](a ) ] , dipolar single - scattering is the relevant process , in agreement with table [ table2 ] .the tmatrix structure factor is numerically indistinguishable from the single - scattering calculated using eq .( [ 2 ] ) and plotted on the figure as the solid line .note that the fractal regime is not easily accessible in these numerical experiments , due to the natural limitation in the range of the -values ( the maximum value of is , realized in the backward scattering direction ) .+ 2 ) for [ fig .[ fig3](b ) ] , dipolar single - scattering is valid for the weak scattering cases with ( comparison can be done with the solid line or with the corresponding single - scattering dla structure factor as reproduced in ) .hence , an excellent agreement between the tmatrix results and the single - scattering data is observed .however , the fractal scaling law in eq .( [ 0 ] ) , which should appear in the range is hardly seen .the case is also in agreement with the single - scattering result , but through the explanation given in section [ s31 ] as the _ dipolar _ mean - field multiple scattering case holds ( see table [ table2 ] and fig .[ fig1](a ) ) .+ 3 ) for , when , one can note that a power - law behavior : appears with an anomalous exponent , substantially different from the fractal dimension . this behavior was recently discussed in detail by oh and sorensen and katyal et al .we conclude that it is a _ robust _ feature , which is not limited to the dipolar single - scattering case . since the scaling behavior is seen in both single - scattering and multiple - scattering conditions , this is a signature of the local arrangement of the particles in the dla clusters .+ 4 ) for the parameters , we expect relevant multipolar mean - field multiple scattering ( see table [ table2 ] and fig . [ fig1](b ) ) .however , the major features of are similar to the case , though appearing at systematically smaller values of , i.e. , for larger characteristic lengths . in other words , extracting the value of the gyration radius from the guinier regime [ would result in an over - estimation by some of its correct value .the anomalous power - law : appears again in this case with the same exponent as for the above case .+ 5 ) for [ fig .[ fig3](c ) ] , the change with increasing refractive index , is similar to the case , but occurs at smaller values of .the case is comparable to the single - scattering result , though it exhibits a shorter guinier regime .the case emphasizes the appearance of an anomalous power - law for .+ 6 ) the behavior arising for is completely different . here, the mean - field approximation is no longer valid ( see table [ table2 ] and fig . [ fig1](d ) ) . in that case, one can see strong scattering in both the forward and the backward directions ( small and large values of ) , which destroys the possible scaling law for .the tmatrix data analyzed in section [ s4 ] shows different scenarios for scattering aggregates with fractal dimension and .this is in agreement with a qualitative argument by berry and percival who stated that the electromagnetic scattering features _ must _ differ essentially in these cases .this statement holds true whether the geometric projection of the aggregate on the plane perpendicular to the incident wave direction is compact ( ) , or still a fractal ( ) .the case exhibits rather simple behavior . the scaling relation of the generalized structure factor , , is valid in the fractal regime ( ) for moderate multiple - scattering [ and moderate multipolar feature ( ) .this confirms the common belief regarding the robustness of the fractal scaling law for .in the most extreme case that we studied , i.e. , , the backscattering signal is strongly enhanced .this is related to coherent backscattering of a random system made of small particles , resulting in weak localization of the wave in the backward direction .this phenomenon leads to a backscattering cone of enhanced intensity , in particular for fractal aggregates , as seen in fig .[ fig1](c ) , and to a smaller extent in fig .[ fig1](b ) ( ) . as it corresponds to a localization event ,coherent backscattering is not part of the mean - field theory developed in , thus it is not taken into account in the condition ( [ cond1 ] ) .for , we conclude that the structure factor can be fully explained with the help of the guinier regime , the fractal regime , and the coherent backscattering feature .in particular , the fractal dimension can be safely extracted from the relation ( [ 0 ] ) , even if multiple - scattering components are not small .we have emphazised the robustness of the power law exponent in the vs. plot for . the next question that arisesis related to the sensitivity of the fractal domain to a possible enlargement of the backscattering regime .in fact , the backscattering cone angle increases with the mean free length of the wave scattering in the system .thus , the largest backscattering cones are expected to occur for the smallest fractal dimensions , for which the particles forming the aggregate are far apart .therefore , we performed tmatrix calculations for a linear chain ( ) of spherical particles with an angular averaging , for the same parameters as used previously .the results are shown in fig .[ fig4 ] , and should be compared with the exact result of the single - scattering case , which is for the large aggregates of size and .indeed , multiple scattering is expected to be irrelevant in this case because of the low density of the aggregate , or equivalently the small values of , which are about 3 times smaller than the values for the dlca aggregates . however , the results plotted in fig . [ fig4 ] show a discrepancy between the single - scattering result ( which is a straight line of slope on the figures ) and the multiple - scattering results - coherent backscattering leads to an enhanced scattering signal upto , hiding the fractal regime .interestingly , the behavior for can be well represented as an anomalous power - law scaling : with an exponent .we should emphasize here that this feature is due to the coherent scattering , because it is neither due to special morphology of the aggregate ( the anomalous scaling law is not seen in the single - scattering data ) nor due to the ordinary multiple - scattering process ( because of the small values of ) .we now have the tools to analyze the seemingly more complicated case ( shown in fig . [ fig3 ] ) . for small values of the optical size parameter ( ) ,the multiple - scattering process is irrelevant due to the small values of ( see table [ table2 ] ) .therefore , the behavior is the same as for the single - scattering case .the guinier regime is dominant because of the small size of the aggregate . for , one should also recover the single - scattering structure factor because of the small values of .this is indeed the case while comparing fig .[ fig3](b ) and the corresponding single - scattering figures in .the unexpected power - law scaling with the exponent , as seen for , is due to the local morphology of the aggregates , since this is a feature of the single - scattering pattern ( thus of the fourier transform of the correlation function ) .because of the slope being similar to the dlca fractal slope , it is conjectured that a dlca - like local arrangement of particles is present inside the dla clusters .this can be checked directly for the small- behavior of the correlation function  .the present scattering data also strengthens the above conclusion , though the precise value of the slope may be difficult to estimate .for example , the same exponent has been estimated to be ( instead of ) for composite clusters made of dlca ( ) at short length - scales , and dla ( ) at long length - scales ( with parameters , , and using the dda code ) .the latter case is comparable to our case . as explained in ref . , the hump seen for is due to a misfit between the guinier regime for the small values of [ which uses the fractal dimension to obtain the overall gyration radius . also , the dlca - like fractal regime [ uses the fractal dimension for the large .this hump in the structure factor results in a kind of _ corona _ as appearing for a porous sphere of fractal dimension with a high refractive index .the general appearance of the hump for the dla clusters provides another proof that the guiner and fractal regimes are robust against moderate multiple - scattering processes .to summarize , one can note two special behaviors for larger values of , as described below .+ 1 ) for , our data is consistent with the fractal dimension , followed by an anomalous scaling law ( slope ) .the same behavior occurs for the case ( which yields the same value of ) .the cause of this effect is probably due to the multiple scattering process as evident from the relatively large value of .however , further investigation is required to understand its origin and behavior . in this particular example, we can also see the onset of coherent backscattering .+ 2 ) for the case , the most striking feature is the coherent backscattering enhancement developing for the largest values of the refractive index , for both the dla [ fig . [ fig1](c ) ] and dlca [ fig . [ fig3](c ) ] cases .we have analyzed tmatrix scattering results for three different classes of fractal aggregates ( the linear chain , the dlca and dla clusters ) for various optical size parameters and refractive index in the domain of common applications .we conclude that the fractal regime , allowing for direct estimation of the fractal dimension , is robust against moderate multiple scattering .however , its range ( in the -values ) can be significantly reduced either by a specific local arrangement of the particles inside the scatterer ( dla case ) or by a coherent backscattering process ( for small fractal dimensions ) .we provide an explicit criterion to check whether a scattering experiment is in the single or multiple - scattering regime .we emphasize that the criterion is valid only in the _ fractal regime _ , and not in the forward or backward scattering directions .thus , we now have a tool to extract the fractal dimension in a controlled way from the scattering data in the fractal regime .

Abstract: in the single - scattering theory of electromagnetic radiation , the _ fractal regime _ is a definite range in the photon momentum - transfer , which is characterized by the scaling - law behavior of the structure factor : . this allows a straightforward estimation of the fractal dimension of aggregates in _ small - angle x - ray scattering _ ( saxs ) experiments . however , this behavior is not commonly studied in optical scattering experiments because of the lack of information on its domain of validity . in the present work , we propose a definition of the multiple - scattering structure factor , which naturally generalizes the single - scattering function . we show that the mean - field theory of electromagnetic scattering provides an explicit condition to interpret the significance of multiple scattering . in this paper , we investigate and discuss electromagnetic scattering by three classes of fractal aggregates . the results obtained from the tmatrix method show that the fractal scaling range is divided into two domains : 1 ) a genuine fractal regime , which is robust ; 2 ) a possible anomalous scaling regime , , with exponent independent of , and related to the way the scattering mechanism uses the local morphology of the scatterer . the recognition , and an analysis , of the latter domain is of importance because it may result in significant reduction of the fractal regime , and brings into question the proper mechanism in the build - up of multiple - scattering .  fractal , structure factor , muliple scattering , single scattering , scaling law


in this study , we present numerical calculations that suggest that , by exploiting key differences in mechanical properties between cancerous and normal cells , _ oncolysis _ , or bursting of cancerous cells , can be induced selectively by means of carefully tuned ultrasound harmonic excitation while simultaneously leaving normal cells intact .we refer to this procedure as _oncotripsy_. specifically , by studying the vibrational response of cancerous and healthy cells , we find that , by carefully choosing the frequency of the harmonic excitation , lysis of the nucleolus membrane of cancerous cells can be induced selectively and at no risk to the healthy cells .numerous studies suggest that aberrations in both cellular morphology and material properties of different cell constituents are indications of various forms of cancerous tissues .for instance , a criterion for malignancy is the size difference between normal nuclei , with an average diameter of to microns , and malignant nuclei , which can reach a diameter of over microns .early studies have shown that the nuclear - nucleolar volume ratios in normal tissues and benign as well as malignant tumors do not differ quantitatively .nucleoli volumes of normal tissues , however , are found to be significantly smaller than the volume of nucleoli in cancerous tissues .similarly , the mechanical stiffness of various cell components has been found to vary significantly in healthy and diseased tissues . in ,the stiffness of live metastatic cancer cells was investigated using atomic force microscopy , showing that cancer cells are more than softer than healthy cells .other cancer types , including lung , breast and pancreas cancer , display similar stiffness characteristics . furthermore , using a magnetic tweezer , found that cancer cells with the lowest invasion and migratory potential are five times stiffer than cancer cells with the highest potential .likewise , increasing stiffness of the extracellular matrix ( ecm ) was reported to promote hepatocellular carcinoma ( hcc ) cell proliferation , thus being a strong predictor for hcc development . moreover, enhanced cell contractility due to increased matrix stiffness results in an enhanced transformation of mammary epithelial cells as shown in .conversely , a decrease in tissue stiffness has been found to impede malignant growth in a murine model of breast cancer .various experimental techniques have been utilized in order to quantitatively assess the material properties of individual cell constituents in both healthy and diseased tissues .the inhomogeneity in stiffness of the living cell nucleus in normal human osteoblasts has been investigated by using a non - invasive sensing system . as shown in that study ,the stiffness of the nucleolus is relatively higher compared to that of other nuclear domains .similarly , a difference in mass density between nucleolus and nucleoplasm in the xenopus oocyte nucleus was determined by by recourse to refractive indices .the elastic modulus of both isolated chromosomes and entire nuclei in epithelial cells are given by and , respectively .specifically , showed that mitotic chromosomes behave linear elastically up to 200% extension .experiments of additionally measured the network elastic modulus of the nuclear envelope , independently of the nucleoplasm , by means of micropipette aspiration , suggesting that the nuclear envelope is much stiffer and stronger than the plasma membranes of cells .in addition , wrinkling phenomena near the entrance of the micropipette were indicative of the solid - like behavior of the envelope . estimated the elastic moduli of both cytoplasm and nucleus of hepatocellular carcinoma cells based on force - displacement curves obtained from atomic force microscopy .in addition , used micropipette aspiration techniques in order to further elucidate the viscoelastic behavior of human hepatocytes and hepatocellular carcinoma cells . based on their study , concluded that a change in the viscoelastic properties of cancer cells could affect metastasis and tumor cell invasion .the increased compliance of cancerous and pre - cancerous cells was also investigated by , who used atomic force microscopy to determine the mechanical stiffness of normal , metaplastic and dysplastic cells , showing a decrease in young s modulus from normal to cancerous cells .the scope of the present work , and the structure of the present paper , are as follows .we begin by defining the geometric model and summarizing the material model and material parameters used in finite - element analyses .subsequently , the accuracy of the finite - element model is assessed by means of a comparison between numerical and analytical solutions for the eigenmodes of a spherical free - standing cell .we then present eigenfrequencies and eigenmodes of a free - standing ellipsoidal cell , followed by a bloch wave analysis to model tissue consisting of a periodic arrangement of cells embedded in an extracellular matrix . finally , resonant growth rates are calculated that reveal that cancerous cells can selectively be targeted by ultrasound harmonic excitation .the transient response at resonance of healthy and cancerous cells is presented in the fully nonlinear range by way of verification and extension of the findings of the harmonic modal analysis .we close with a discussion of results .in this section , we investigate the dynamical response of healthy and cancerous cells under harmonic excitation .we begin by briefly outlining the underlying geometric and material parameters used in our analysis , followed by a verification of the finite element model used for modal analysis .we then calculate the eigenfrequencies and eigenmodes of both free - standing and periodic distributions of cells . in this latter case , we determine the full dispersion relation by means of a standard bloch wave analysis . finally , we present resonant growth rates and simulate the transient response of both cancerous and healthy cells excited at resonance in a fully - nonlinear setting by means of implicit dynamics calculations .the nucleus , the largest cellular organelle , occupies about of the total cell volume in mammalian cells .it contains the nucleolus , which is embedded in the nucleoplasm , a viscous solid similar in composition to the cytosol surrounding the nucleus . in this study ,the cytosol is modeled in combination with other organelles contained within the plasma membrane , such as mitochondria and plastids , which together form the cytoplasm . for simplicity , we idealize the plasma mebrane , nuclear envelope , cytoplasm , nucleoplasm , and nucleolus as being of spheroidal shape .we model the plasma membrane , a lipid bilayer composed of two regular layers of lipid molecules , in combination with the actin cytoskeleton providing mechanical strength as a membrane with a thickness of nm .similarly , we model the nuclear envelope , a double lipid bilayer membrane , in combination with the nuclear lamin meshwork lending it structural support as a nm thick membrane .we define the cytoplasm , nucleoplasm , and nucleolus as spheres with radii of m , m , and m and subsequently scale them by a factor of in two dimensions in order to obtain the desired spheroidal shape .we assume an average nuclear diameter of about m , as reported in .diameters for both cytoplasm and nucleolus follow from and , who report nucleus - to - cell and nucleus - to - nucleolus volume ratios of and , respectively .the geometry with all cell constituents as used in subsequent finite element analyses is illustrated in figure [ fig : geometry ] . in order to further elucidate the effect of an increasing nucleus - to - cell volume ratio , as observed experimentally , we consider a range of geometries with increasing nuclear and nucleolar volumes .for all of these geometries , we hold fixed the volume of the cytoplasm .we assume a constant nuclear - to - nucleolar volume ratio for both healthy and cancerous cells , as observed by .    cell - to - cell differences and experimental uncertainties notwithstanding , the preponderance of the observational evidence suggests that the cytoplasm , nucleus and nucleolus are ordered in the sense of increasing stiffness . neglecting viscous effects, we model the elasticity of the different cell constituents by means of the mooney - rivlin - type strain energy density of the form , \ ] ] where denotes the deformation gradient , is the jacobian of the deformation , and , and are material parameters . for both cytoplasm and nucleus in cancerous cells , material parameters corresponding to the data reported by chosen and summarized in table [ tab : parameters ] .we additionally infer the elastic moduli of the nucleolus from based on a comparison of the relative stiffnesses of the nucleoli and other nuclear domains . for membrane elements of the plasma membrane and nuclear envelope ,we choose material parameters corresponding to the cytoplasm and nucleoplasm , respectively .furthermore , we infer matrix parameters from the shear moduli reported by for normal and fibrotic livers . for all parameters ,we resort to small - strain elastic moduli conversions , with a poisson s ratio of to simulate a nearly incompressible material , in order to match experimental values with constitutive parameters .we vary the stiffness of both cellular components and extra - cellular matrix ( ecm ) within a pathophysiologically - relevant range in order to investigate the effect of cell softening and ecm stiffening on eigenfrequencies . finally , we assume both cytoplasm and nucleoplasm to have a mass density of / , a value reported by as an average cell density , and we set the density of the nucleolus to / ..set of constitutive parameters ( bulk modulus and shear moduli and ) used in the eigenfrequency analyses . [ cols="^,^,^,^",options="header " , ]   the spectral gap , or gap in the lowest eigenfrequencies , between healthy and cancerous cells and tissues provides a first hint of sharp differences in the response of healthy and cancerous tissue to harmonic excitation .in particular , the preceding analysis shows that the fundamental frequencies of the cancerous tissue may be in close proximity to eigenfrequencies of the healty tissue , which appears to undermine the objective of selective excitation of the cancerous tissue .however , a complete picture requires consideration of the relative energy absorption characteristics and growth rates of resonant modes . to this end , we consider the modal decomposition of the displacement field where are eigenvectors obeying the orthogonality and normalization condition ( [ e45dls ] ) and are time - dependent modal amplitudes obeying the modal equations of motion in this equation , is the corresponding eigenfrequency , is the external force vector and is the corresponding modal force . for a harmonic excitation of frequency , eq . ( [ kc3v7h ] )further specializes to where now is a constant modal force amplitude . at resonance , , the amplitude of the transient solution starting from quiescent conditions grows linearly in time and the transient solution follows as we thus conclude that the growth rate of resonant modes is   figure [ fig : fn ] shows the growth properties of for two different cases . in the first case , a cancerous cell is excited at its resonant frequency of / s , whereas the healthy cell is excited at its closest resonance frequency of / s . in the second case ,eigenfrequencies of / s and / s are investigated .the simulations reveal that the growth rate of the resonant response of the cancerous cells is much faster than that of the healthy cells , which opens a window for selectively targeting the former .the preceding analysis has been carried out with a view to understanding the resonant response of cells and tissues under harmonic excitation in the harmonic range . in this section, we seek to confirm and extend the conclusions of the harmonic analysis by carrying out fully nonlinear implicit dynamics simulations of the transient response of healthy and cancerous cells under resonant harmonic excitation . in this analysis , a geometry of ratio is considered , figure [ fig : mesh ] , together with material parameters of table [ tab : parameters ] .we restrict attention to the long wavelength limit , i. e. to ultrasound radiation of wavelengths larger than the cell size . in keeping with this limit, we enforce harmonic displacement boundary conditions directly as shown in figure [ fig : geometry ] in order to mechanically excite the cell .the strength of the harmonic excitation used in the calculations is m .  in the simulations , we track the transient amplification of the cell response up to failure .we assume that failure occurs when the stress in the cytoskeletal polymer network , which constitutes the structural support for cell membranes , reaches a threshold strength value . found that the macroscopic network strength can be traced to the microscopic interaction potential of cross - linking molecules and other cytoskeletal components such as actin filaments . here, we assume a rupture strength of the order of based on strength values of a single actin / cross - linking protein bond reported in .figure [ fig : transient ] shows the fully - nonlinear transient response of healthy and cancerous cells at the resonant frequency of the latter .it can be seen from the figure that stresses in both the plasma membrane and nuclear envelope of the cancerous cell grow at a much faster rate than in healthy cells . for the harmonic excitation under consideration , the strength of the nuclear envelope of the cancerous cell reaches the rupture strength at time , while , at the same time , the level of stress in the healthy cells is much lower .figure [ fig : kineticpotential ] furthermore illustrates the kinetic and potential energy of the nuclear envelope during excitation at resonance of both healthy and cancerous cells .  from transient response simulations , the energy that needs to be supplied until the point of rupture is reached is where is the applied traction on the boundary , is the displacement vector , is the force acting on surface node at time , and is the corresponding displacement vector .for a cell geometry with a ratio of and a cancerous potential of , calculations give a value of for the energy per cell required for lysis .assuming an average cell size of 20 m , a time to lysis of 70 and a tumor of 1 cm in size , this energy requirement translates into a power density requirement in the range of 0.8 w/ .   and a cancerous potential of .shown is the frobenius norm of first piola - kirchhoff stress tensor.,scaledwidth=100.0% ]in this study , we have presented numerical calculations that suggest that spectral gaps between hepatocellular carcinoma and healthy cells can be exploited to selectively bring the cancerous cells to lysis through the application of carefully tuned ultrasound harmonic excitation , while keeping healthy cells intact .we refer to this procedure as _ oncotripsy_. a normal mode analysis in the harmonic range reveals the existence of a healthy - to - cancerous spectral gap in ground frequency of the order of rad / s , or khz .further analysis of the growth rates of the transient response of the cells to harmonic excitation reveals that lysis of cancerous cells can be achieved without damage to healthy cells .these findings point to oncotripsy as a novel opportunity for cancer treatment via the application of carefully tuned ultrasound pulses in the frequency range of 80 khz , duration in the range of 70 and power density in the range of 0.8 w/ .this type of ultrasound actuation can be readily delivered , e. g. , by means of commercial low - frequency and low - intensity ultrasonic transducers .evidently , the present numerical calculations serve only as preliminary evidence of the viability of oncotripsy , and further extensive laboratory studies would be required in order to confirm and refine the present findings and definitively establish the viability of the procedure .

Abstract: we investigate a method of selectively targeting cancer cells by means of ultrasound harmonic excitation at their resonance frequency , which we refer to as _ oncotripsy_. the geometric model of the cells takes into account the cytoplasm , nucleus and nucleolus , as well as the plasma membrane and nuclear envelope . material properties are varied within a pathophysiologically - relevant range . a first modal analysis reveals the existence of a spectral gap between the natural frequencies and , most importantly , resonant growth rates of healthy and cancerous cells . the results of the modal analysis are verified by simulating the fully - nonlinear transient response of healthy and cancerous cells at resonance . the fully nonlinear analysis confirms that cancerous cells can be selectively taken to lysis by the application of carefully tuned ultrasound harmonic excitation while simultaneously leaving healthy cells intact .   oncotripsy , modal analysis , resonance , cell necrosis


networks composed of nodes and edges have been attracting a lot of research interest recently  . unlike conventional networks such as a periodic regular network and a random network , many real - world networks have complex structure .many of them belong to the class of so - called scale - free ( sf ) networks . denoting the degree of a node as the number of edges attached to it , a sf is characterized with the power - law degree distribution here called the degree distribution exponent .the power - law degree distribution implies that the sf network has an inhomogeneous structure . on the one hand , it is a challenging problem to characterize and understand the organization principle of the sf network . on the other hand ,it is also interesting to study thermodynamic or dynamic systems on such an inhomogeneous structure .various physical problems have been studied .examples include the ferromagnetic phase transitions in the ising model  , the non - equilibrium phase transition in the epidemic spreading model  , the random walk process  , the dynamic scaling in pair - annihilation process  , and so on . in those studies ,the -dependent scaling properties have been studied .  in the present work ,we address the question how the global structure of the underlying sf network influences the dynamic scaling behavior of stochastic particle systems . for that purposewe study the random walk process and the pair - annihilation process , and investigate their dynamic scaling property .both systems are studied on the _ tree _ structure sf ( tsf ) networks and the _ looped _ structure sf ( lsf ) networks , respectively .  to be specific, we study the stochastic systems on the dorogovtsev - mendes - samukhin ( dms ) network  , which is a generalization of the barabsi - albert ( ba ) network  .it is a _ non - deterministic _ model for a growing network : each time step , a new node is added , and linked with nodes which are selected among existing nodes with the probability given by . here the parameter is called an initial attractiveness .the ba model corresponds to the case of the dms network .the resulting network is scale - free and the degree distribution exponent is given by with the parameters and , one can vary the value of the degree exponent . at the same time , one can also generate a tsf network with or a lsf network with .hence , we can study the effect of the degree distribution exponent and the global network structure systematically .this paper is organized as follows : in sec .[ sec:2 ] , we present the results on the random walk process .section [ sec:3 ] is devoted to the scaling property of the pair - annihilation process .summary will be given in sec .[ sec:4 ] .as a basic and fundamental stochastic process , the random walk process  on complex networks has been attracting a lot of research interest  . in this workwe concentrate on relaxation dynamics on sf networks .  on a network with nodes ,a random walker is assigned to a starting node denoted by at time .then , at each unit time step , it hops to one of the neighboring nodes selected randomly with the equal probability . defining as the probability to find the walker at node at time , one finds that it evolves in time as with the initial condition . here is the adjacency matrix whose elements are if two nodes and are connected ( disconnected ) .  the relaxation dynamics is studied with the so - called return probability . for a given network , one can solve numerically the master equation eq .( [ master_eq ] ) iteratively to obtain .we then average it over different realizations of the networks .  on a network with loops, the probability distribution converges to the stationary one in the limit  .hence , the return probability converges to . on the other hand , on a tree network ,the probability distribution is oscillating in time , so is the return probability .the random walker can not return to a starting node in odd time steps , which means that at odd . in this case , we only measure the return probability at even time steps , which converges to stationary value .  in a sf network ,the node degree is distributed so broadly that one can define an exponent for each node describing the degree scaling with the network size : for instance , in the dms network with the degree distribution exponent , a _ peripheral node _ with the minimum degree has , while the _ hub _ with the maximum degree has . then , the return probability in the stationary state scales with the network size as   ( open symbols ) and  ( filled symbols ) in the dms networks with and with in ( a ) and in ( b).,width=264 ]  due to the broad degree distribution , the return probability may have a different scaling property at different starting node with different values of .so , we measured the return probability for the hub and for a peripheral node on the dms networks .we present the numerical data for the return probability and in fig .we compare the data obtained from the tsf networks and the lsf networks .the data show that the return probability behaves distinctly depending on the global structure of the network , which will be discussed in detail in the following .we perform the quantitative analysis of the scaling property in the tsf . from fig .[ fig1 ] , one finds that the return probability relaxes much slower in the tsf network with .we estimated the relaxation time using the condition with a constant .the relaxation time is plotted as a function of the network size in fig . [ fig2 ] .it shows that the relaxation time estimated from and follows a power - law scaling with the same dynamic exponent   for  ( open symbols ) and  ( filled symbols ) on the tree structure sf networks .the solid line has the slope 1.,width=264 ]  we provide a theoretical argument for the numerical result from the analysis of the mean first passage time ( mfpt ) .the mfpt problem has been studied on complex networks recently  , and some rigorous results are known  . consider the mfpt , denoted by , from an arbitrary node  ( degree ) to one of its neighboring node .let denote the neighbors of the node .then , following ref . , the mfpt on _ tree _ networks satisfies the recursion relation applying the recursion relation repeatedly until one arrives at dangling ends , one can find the explicit solution for . without the link between and , the tree network would be decomposed into two parts . denoting the number of nodes in the side by , the mfpt is simply given by hence , for a typical adjacent nodes and , one has that . for non - adjacent nodes and ,the mfpt is given by the sum of the mfpt s given by eq .( [ t_ji ] ) along the path between them .hence , the mfpt between a typical node pair is given by , where is the mean diameter of the networks .for sf networks , the mean diameter scales at most logarithmically with the network size  .therefore , we conclude that the relaxation time follows the power - law scaling with the dynamic exponent .the power - law scaling of the relaxation time suggests that the return probability decays algebraically as with the decay exponent . for a node with the degree scaling ,the decay exponent can be deduced from the finite - size - scaling ansatz with the dynamic exponent . for large ,the return probability should converge to the stationary value  ( see eq .( [ r_s_scaling ] ) ) .it requires that the scaling function should behave as .the power - law scaling for requires that the scaling function should behave as , which yields that .therefore , the finite - size - scaling ansatz predicts that the decay exponent is given by   ( open symbols ) and  ( filled symbols ) in the tsf networks with  ( a ) and  ( b ) ., width=264 ]  it is interesting to note that the decay exponent varies with the degree scaling exponent of the starting node .we confirm the scaling behavior with the scaling plot of versus for the hub and the peripheral node in fig .[ fig3 ] . for , for the hub and for the peripheral node , which yields that and .similarly , for , one expects that and .one finds that all data from different network sizes collapse very well in the scaling plot , which supports the result in eq .( [ delta_s ] ) .it is also interesting that the return probability decays as with the exponent at all nodes .it indicates that the random walks are recurrent in the limit  .   from fig .[ fig1 ] , one can see that the return probability decays much faster in the lsf networks .the downward curvature in the log - log plot implies that the decay is faster than a power - law decay .the relaxation time measurement also indicates a faster decay .we found that the relaxation time , measured using the condition that , grows at most logarithmically with the network size .this is in contrast to the power - law growth in the tsf networks . in this subsection, we address a question how the return probability decays in time in looped sf networks .  in the random network, it is known that the return probability follows a stretched exponential decay as with a constant and the exponent  .it was reported that the return probability in the small - world network follows the stretched exponential decay , too  .it might suggest that the looped sf network follow the stretched exponential decay , too .obtained from on the dms networks with and with  ( open symbols ) and  ( filled symbols ) . is plotted against in ( a ) and in ( b).,width=264 ]  on the dms networks with , we measured numerically and fitted the data to the form in eq .( [ stretched_exp ] ) to estimate .if the return probability follows an exponential decay , one would have . on the other hand, one would have , if it follows the stretched exponential decay . in fig .[ fig4 ] , we present the data for for obtained on the dms networks with various values of and up to sizes .similar behaviors are observed in for . at small values of , the exponent seems to depend on and to be smaller than .one may interpret that as an evidence of the stretched exponential decay with a non - universal exponent .however , we observe that there is a very strong finite size effect .the plot of versus in fig .[ fig4 ] ( b ) shows that the finite size effect is significant even for . with this strong finite size effect, one can not exclude the possibility of the exponential decay with .we suggest that an analytic approach be necessary to conclude whether the return probability follows the exponential or the stretched exponential decay .  before closing this section , we remark on the previous studies on the random walks on _ deterministic _ sf networks  . unlike the non - deterministic lsf networks studied in this work , the deterministic lsf networks  behave similarly as the tsf networks . for example , in the hierarchical network  which has a looped structure , the relaxation time scales algebraically as  .the reason why the hierarchical network behaves as tsf networks is clear .the network has a symmetry , due to which the random walks on it can be mapped to the walks on tsf networks  .therefore , our general conclusion should not be applied to the deterministic lsf networks with high symmetry .the pair - annihilation process is a diffusion - limited reaction - diffusion process . in this process , each node in a given network may be occupied by a particle ( denoted by ) or empty ( denoted by ) .the particles perform random walks on the network , and annihilate pairwise whenever they meet at a same node ( ) .the pair - annihilation process on sf networks was studied numerically by gallos and argyrakis  .they found that the particle density decays algebraically with the -dependent decay exponent .this is contrasted with the -dimensional periodic lattice case where .namely , the particle density decays faster in sf networks .  later on , catanzaro _ et al . developed a mean field theory for the pair - annihilation process , which will be reviewed briefly hereafter .let us define as the average particle density at nodes with degree .it is related to the total density through the relation . in a mean field level, one can write down the rate equation for as \rho(t ) \ , \ ] ] where is the mean degree .multiplying and summing over , one finds that \ .\ ] ] then , catanzaro _ et al ._ made a quasistatic approximation neglecting the time derivative in eq .( [ rho_k ] ) .the approximation assumes that the particles adjust themselves so efficiently that their distribution remains close to a stationary one at a given value of .it leads to the relation substituting it in eq .( [ rho ] ) and solving the resulting equation , catanzaro _ et al . _obtained that the density decay follows a power law in the limit as the decay exponent is given by and the exponent for the logarithmic correction is given by for and for .this result is qualitatively consistent with the simulation results of refs . .  in the previous works , only the -dependent scaling behaviors have been studied .however , the study on the random walks in sec .[ sec:2 ] suggests that the scaling behavior may also depend on the global structure of the underlying sf network . in this section, we present the results of our numerical works on the pair - annihilation process on the dms networks with  ( tree structure ) and  ( looped structure ) .comparing the two different cases , we will show that the global structure does also matter for the scaling behavior of the pair - annihilation process .   and .the degree distribution exponent is .( b ) density decay exponent  ( circles ) and  ( squares ) defined in eq .( [ rho_decay ] ) for the dms networks with  ( open symbols ) and  ( filled symbols).,width=264 ]  we have performed the monte carlo simulations on the dms networks of size with various values of and . the fully - occupied state is taken as the initial configuration . in fig .[ fig5 ] ( a ) , the numerical data from the tsf networks ( ) and the lsf networks ( ) are compared .the data show that the particle density decay follows the power law in both cases but with a different exponent .we estimate the decay exponent by fitting the data to the form in eq .( [ rho_decay ] ) , and plot and as a function of in fig .[ fig5 ] ( b ) . at ,our result is qualitatively consistent with the previous works ; for , and varies with for . and the logarithmic correction is prominent at .however , at , we obtain a completely different result ; at all values of and the logarithmic correction is present at all values of .these numerical results show that the global structure of the underling sf networks affects the scaling behavior of the pair - annihilation process .we speculate the origin for the different scaling behaviors . on the lsf networks, the scaling behavior seems to be consistent with the analytic mean field result in eq .( [ alpha ] ) . in the analytic approachone adopts the quasistatic approximation  leading to the particle distribution given by eq .( [ rho_k_assume ] ) .it can be rewritten as note that for small .vs. at different time steps  ( top),, ( bottom ) .the data are taken from the dms network with  ( a ) and  ( b ) .the network size is and the degree distribution exponent is in both plots .the dashed lines have the slope 1.,width=264 ]  we investigate the particle distribution numerically . in fig .[ fig6 ] ( a ) , we plot against for the dms network with the looped structure ( ) .we find that the particle distribution is fully consistent with eq .( [ crossover ] ) .this supports that the analytic approach is appropriate on the lsf networks .however , the particle distribution deviates from eq .( [ crossover ] ) on the tsf network .figure [ fig6 ] ( b ) shows that there are three different regimes : for , for , and for .the scaling exponent in the intermediate regime is found to be less than 1 and to vary with .this feature is inconsistent with the quasistatic approximation .the quasistatic approximation assumes that particles rearrange themselves quickly upon the change in the total particle density via diffusion .it requires that the diffusion should be a fast process . in the previous section, we have shown that the diffusion is a slow process in the lsf networks .this explains why the quasistatic approximation is invalid in the tsf network whereas it is valid in the lsf network .in summary , we have investigated the scaling properties of the random walk and the pair - annihilation processes on non - deterministic sf networks with the tree structure and the looped structure , respectively . in the random walk process on tsf networks, we find that the relaxation time scales as with the network size and that the return probability decay follows the power law with the node - dependent exponent .the lsf network does not display the power - law scalings . in the pair - annihilation process , we find that the exponent describing the particle density decay is different in tsf and lsf networks .our results show that the global structure of the sf network , as well as the degree distribution exponent , is the important ingredient in understanding the dynamic scaling behaviors .r. albert and a .-barabsi , rev .phys . * 74 * , 47 ( 2002 ) ; s.n .dorogovtsev and j.f.f .mendes , adv .phys . * 51 * , 1079 ( 2002 ) ; m.e.j .newman , siam rev .* 45 * , 167 ( 2003 ) .a. v. goltsev , s. n. dorogovtsev , and j. f. f. mendes , phys .e * 67 * , 026123 ( 2003 ) .r. pastor - 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Abstract: we investigate the dynamic scaling properties of stochastic particle systems on a non - deterministic scale - free network . it has been known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying scale - free network . our study shows that it also depends on the global structure of the underlying network . in random walks on the tree structure scale - free network , we find that the relaxation time follows a power - law scaling with the network size . and the random walker return probability decays algebraically with the decay exponent which varies from node to node . on the other hand , in random walks on the looped scale - free network , they do not show the power - law scaling . we also study a pair - annihilation process on the scale - free network with the tree and the looped structure , respectively . we find that the particle density decays algebraically in time both cases , but with the different exponent .