this paper studies the quantized control problem for switched systems . for linear time - invariant systems , various approaches to quantized controlhave been developed : lyapunov - based methods  , optimization with norm  , etc .in contrast , few results of quantized control are generalized to switched systems in spite of a wide range of their applications .recently , based on the results in for a single mode , liberzon  has developed an encoding and control strategy achieving the global asymptotic stability of sampled - data switched systems with quantized state feedback .also , quantized state feedback stabilization is discussed for discrete - time markov jump linear systems in . however , stabilization of switched systems by quantized _ output _feedback has not yet explored .here we consider a continuous - time switching linear system , whose quantized outputs and switching signal are transmitted to the controller at all times .the objective of this paper is to extend the encoding method of for non - switched systems .the key point of the earlier studies is that certain level sets of a lyapunov function are invariant regions .the difficulty of switched systems is that such level sets change at every switching time .therefore , at the `` zooming - in '' stage , non - switched systems require only periodic reduction of the `` zoom '' parameter of quantizers , whereas in switched systems , we need to adjust the parameter after each switch .we assume that the average dwell time  of the switching signal is large enough , and develop an output encoding for global asymptotic stabilization by using multiple lyapunov functions .this paper is organized as follows . in sectionii , we explain the components of the closed - loop system one by one and then give the main result , theorem [ thm : stability_theorem ] .section iii is devoted to its proof .we present a numerical example in section iv and finally conclude this paper in section v.  _ notation : _ let and denote the smallest and the largest eigenvalue of .let denote the transpose of .the euclidean norm of is denoted by .the euclidean induced norm of is defined by , which equals the largest singular value of .  for a piecewise continuous function , its left - sided limit at denoted by .in this section , we first define switched systems and construct quantizers and controllers based on the non - switched case in . next we present the main result , theorem [ thm : stability_theorem ] .this theorem guarantees the existence of a quantizer leading to the globally asymptotic stability of the closed - loop system under an average dwell time assumption .consider the switched linear system where is the state , is the control input , and is the output . for a finite index set , is right - continuous and piecewise constant .we call _ switching signal _ and the discontinuities of _ switching times_. let denote by the number of discontinuities of on the interval  ] such that holds for .   choose so that holds , and define .we consider with calculated by and .the discussion above ensures .fix so that is satisfied .let us first investigate the case without switching on the interval  ] for , then we update in the same way .we now study the switched case .let be switching times on the interval  ] .the argument on the existence of at the `` zooming - out '' stage implies that the time , at which the stage changes from `` zooming - out '' to `` zooming - in '' , satisfies for every switching signal .fix . by , , and hence we see from that achieving can be chosen so that where is defined by note that is independent on switching signals .  by ,if satisfies then we have let be the smallest integer satisfying .define and by , we have for  ] , so and on the same interval .combining this with , we obtain for .thus for every , and hence by .in addition , since for all and since satisfies , it follows that that also lies in .since is an invariant set for , we have latexmath:[\[\label{eq : z_eps2 }  lyapunov stability can be achieved .  through lemma [ lem : adt_upperbound ] , we implicitly use the average dwell time property to obtain the upper bound in .consider the continuous - time switched system with the following two modes : as the feedback gain and the observar gain of each mode , we take let the quantizer be uniform - type , and define the parameters and of the quantizer by , also , define and in and in by ,  , where means a diagonal matrix whose diagonal elements starting in the upper left corner are . then we obtain in , in , in , and in .[ fig : cont_simulation ] ( a ) and ( b ) show that the output and the -norm of the state of the switched system  with ^{\top}$ ] and . in this example , the `` zooming - out '' stage finished at .we see the non - smooth behaviors of and at the switching times .in particular , we observe from the behaviors of at that , not surprisingly , adjustments of at some swithcing times are conservative .the stabilization of continuous - time switched linear systems by quantized output feedback has been studied .we have proposed an output encoding method for globally asymptotic stability .the encoding method is rooted in the non - switched case , and an additional adjustment of the zoom parameter is needed at every switching time in the zooming - in stage .we have discussed the effect of switching by using multiple lyapunov functions and an average dwell time assumption .

Abstract: this paper studies the problem of stabilizing a continuous - time switched linear system by quantized output feedback . we assume that the quantized outputs and the switching signal are available to the controller at all time . we develop an encoding strategy by using multiple lyapunov functions and an average dwell time property . the encoding strategy is based on the results in the case of a single mode , and it requires an additional adjustment of the `` zoom '' parameter at every switching time .


the retrieval of the state of a physical system is an important problem in classical as well as in quantum physics , and yet it is a subject which is seldom discussed in textbooks .the state of a system in _ classical _ physics is described by a density in phase space , , which could be determined ( i.e. , reconstructed ) by the directly measurable conditional probability density of the position for a given momentum , , and the measurable probability density of , , through the relation thus we may envision the state as being specified by the set of measurable quantities and . an alternative approach for determining a classical state involves measuring a linear combination of position and momentum ( the constants and are introduced for the purpose of fixing dimensions ) sometimes termed , for electromagnetic - field state measurements , rotated quadratures " ( ref . , p. 136 ) .the probability for the new variable for all values of can then be used to reconstruct the phase - space density  .this procedure is similar to the familiar one employed in medical imaging for the reconstruction of a two - dimensional ( 2d ) configuration density , known as the computer aided tomography ( cat ) scan : one simply replaces the two - dimensional configuration - space variables of the cat method by the two - dimensional phase - space variables .  in _ quantum _ physics , a system may be prepared in a pure state described by a vector in hilbert space , or , more generally , in a mixed state described by a density operator ( ref . , p. 204 and , p. 72 ) .the problem of state retrieval involves the inverse inquiry , i.e. , what are the measurable quantities whose values will suffice to determine the quantum state .historically , this question may be traced back to the pauli query whether one can reconstruct the wave function , amplitude and phase , for a one - particle system , from the probability of its position , i.e. , , and that for its momentum , ; here is the wave function in the momentum representation , the tilde indicating the fourier transform .we now know that , in general , this is not possible : we need more information than these two distributions .the literature on this subject , which is still of current interest , has grown enormously ever since .here we have made a selection out of these approaches , with the idea of providing a link with the classical reconstruction scheme .the classical approach based on , is , of course , untenable in quantum physics , where a fixed momentum precludes a well - defined position probability .a similar observation is applicable to the direct approach of measuring the joint probability of and .however , it is remarkable that the alternative method based on measuring defined in phase space can be taken over to _ quantum _ mechanics ( ref . , p. 143 , p. 101 ) .but then the question arises : how can that be , if there is no such thing as a joint probability density in quantum mechanics ?it turns out that the answer one obtains by following this procedure is a function defined in phase space which , although is not a bona - fide probability density ( it is real , but not - necessarily non - negative , and has sometimes been named a quasi - probability " ) , contains all the information needed to compute any quantum mechanical expectation value we please , just as if we were given the complex wave function , or the density operator .this concept of quasi - probability was invented by wigner in the early days of quantum mechanics , with the purpose of finding the quantum - mechanical corrections to thermodynamic functions , and is known as the _ wigner function_. thus retrieving the wigner function using this tomographic method is a true quantum - state reconstruction , and to explain how this is achieved , and its relation with the classical tomographic approach , constitutes the main goal of the present paper .the main results for this approach are to be found in eq .( [ inverse radon q , p ] ) below for the classical case , and in eq .( [ inverse radon q , p qm ] ) for the quantum - mechanical one .there is another concept which has been very useful in the task of reconstructing a quantum state . to give a trivial example ,consider the eigenvectors of position and momentum : if the state vector of a system is an eigenstate of momentum , the system is _ equally likely _ to be found in any of the eigenstates of position .pairs of bases with a similar property have been extensively studied and are known as _ mutually unbiased bases _ ( mub ) .it turns out that mub constitute a powerful tool for state reconstruction , since it is possible to express the density operator that defines the state of the system in terms of a complete orthonormal set of operators .we will explain the mub approach to the problem of state reconstruction and show that the result [ see eq .( [ wfctn of rho(mub ) ] ) below ] is consistent with that found with the method explained above , based on tomography in phase space and the wigner function .even more important , we shall find that the two approaches correspond , essentially , to employing two ways of handling the same complete set of operators , thus providing a unified description of both methods .the paper is organized as follows . in the next sectionwe review the cat scan method , as employed for the reconstruction of a classical 2d density . in section [ classical reconstruction ]we present a scheme for classical state reconstruction in phase space similar to the cat method employed in configuration space . in section [ quantum reconstruction ]we explain how the classical scheme can be taken over to quantum mechanics , and explain the role played by wigner function .we then present in sec .[ mub ] the alternative method for quantum state reconstruction based on the notion of mub .so far , the discussion has been restricted to quantum systems described in a continuous , infinite - dimensional hilbert space , because of our desire to make an analogy with classical physics .however , there have been many contributions to the problem of state reconstruction for quantum systems described in a finite - dimensional hilbert space employing the notion of mub .although these systems do not have a classical counterpart , still they allow us to draw an illuminating parallel with the various concepts that have been introduced for a continuous hilbert space .this fact motivates the brief discussion on the role of mub for a finite - dimensional hilbert space presented in sec .[ finite mub ] . finally , we give our conclusions in sec .[ concl ] . to avoid cluttering of the main text, we include some details of the mathematical derivations in a number of appendices .we wish to emphasize that the main goal for writing this paper is to give a pedagogical presentation of a subject which has been studied for many years and is still of current interest . with this motivation ,we use a language that can be followed by a physics graduate student .we do hope that the analysis is in a form that allows its incorporation in a graduate quantum mechanics course .first we review briefly the method mentioned in the introduction , the cat scan , that is used for the reconstruction of a two - dimensional ( 2d ) configuration density .the mathematical procedure can be translated directly to retrieve a classical 2d phase - space density and , even more interesting for us , it can be taken over to quantum mechanics .  in a 2d cat - scan fine pencil beam of x - rays passes through a sample , shown as the shaded area in fig .[ cat ] , along the line of sight " defined by  is the position vector of a point on the line of sight " and a unit vector perpendicular to the line of sight , forming an angle with the -axis , so that and can be written as , , * i * and * j * being unit vectors along the and axes , respectively .then the equation for the line of sight becomes the line of sight is offset by the amount from the rotated axis .= 0.63=0.4  the beam is attenuated by scattering and absorption produced by the various parts of the sample encountered along the path .assuming that the attenuation at is proportional to the sample density , the total attenuation will be proportional to where we have used an arbitrary offset value designated by , and and have been defined in eq .( [ xtheta ] ) .now it is important to remark that knowing the response of the sample given by for all and directions , we can reconstruct the density of the sample .the mathematics of this problem was actually developed by j. radon at the beginning of the twentieth century for the study of astronomical data .in fact , the function of eq .( [ radon ] ) is known in the literature as the _ radon transform _ of the density .thus the task is to invert the radon transform to find the sample density .it is shown in appendix [ invert_radon ] that the sample density can be expressed in terms of the response of the sample , , for all and directions defined above ( see ref . , pp .144 ) , as where stands for the cauchy principal value of the integral .indeed , eq . ( [ inverse radon ] ) is the inverse radon transform of .  to gain some insight into the structure of the sample response , it is illustrative to consider the particular case in which the sample density is isotropic , i.e. , dependent only on the distance from the origin and independent of the angle. if we write and in polar coordinates as , , eq .( [ radon ] ) for the response takes the form showing that is independent of for the isotropic case . by direct substitution , one may also observe that in this case is symmetric , i.e. , .= 0.4=0.4  it may also be pointed out that the sample density must be a _ non - negative _ quantity , although this fact is not explicitly manifest in eq . ( [ inverse radon ] ) .it is thus useful to verify this property in some particular example . for this purpose, we choose the isotropic case studied in the last paragraph .for example , at the origin of coordinates , , eq . ( [ inverse radon ] )gives since is a symmetric function of , is antisymmetric . since the quantity appearing in eq .( [ inverse radon 1 ] ) has precisely this same property , as illustrated in fig .[ rho(x ) ] , the resulting density at the origin of coordinates is positive .a state in _ classical _ statistical physics is determined by a probability density in phase space . in this paperwe shall always consider , for simplicity , one - particle systems with one degree of freedom .we write the probability density in phase space as which , for convenience in our comparison with quantum mechanics , will be normalized as after the discussion given in the previous section on cat in 2d configuration space , it is clear that a similar method can be applied in 2d phase space : if we consider the linear combination of position and momentum given in eq . ( [ xtheta ] ) , the probabilities for the new variable for all values of can then be used to reconstruct  .  before proceeding , we indicate our choice for the constants and which were introduced to fix dimensions .we choose , , where and represent any convenient scales for position and momentum . subsequently renaming the _ dimensionless _ quantities and again as and , respectively , the transformation of eq .( [ xtheta ] ) reads we go back to the probability density of the variable . if we designate it as , where represents an arbitrary value of , we have , just as in eq .( [ radon ] ) the goal is to find in terms of by inverting eq .( [ rho(x)_vs_rhoq , p ] ) . proceeding as in the previous section and appendix [ invert_radon ] , we find the equivalent of eq .( [ inverse radon ] ) as    as mentioned in the introduction , the above method based on measuring defined in phase space can be taken over to _ quantum _ mechanics ( ref . , p. 143 ) .this leads to a _ quasi - probability density _ defined in phase space known as the wigner function .  inwhat follows we shall take units in which .consider an arbitrary hermitean operator .we define its _ wigner transform _ as  for the case where the operator is the density operator defining the state of the system , we speak of the _ wigner function _ of the state , which has the normalization property similar to the normalization of eq .( [ int rho cl ] ) adopted for the classical distribution .it is well known that wigner function for a state may be negative in some parts of phase space .thus it does not qualify as a true probability density and is referred to as a quasi - probability density .an illustration of the fact that it plays in quantum mechanics a role analogous to that played by the classical probability density is the similarity of eqs .( [ rhoqm(x)_wt 2 ] ) and ( [ inverse radon q , p qm ] ) given below with eqs .( [ rho(x)_vs_rhoq , p ] ) and ( [ inverse radon q , p ] ) , respectively .an important property of wigner function , obtained from the definition ( [ wt ] ) , is  for any two operators and .this implies that the trace of the product of two operators in hilbert space can be evaluated as an integral in phase space of the corresponding wigner transforms . the normalization of eq .( [ normal - wfrho ] ) is consistent with the property ( [ tr(ab ) ] ) , taking and . the statistical expectation value of an observable , obtained by using eq .( [ tr(ab ) ] ) , can be expressed as i.e. , as an integral in phase space of wigner function for the state times wigner transform of the observable . with these results ,wigner function of the state and the wigner transform of observables can be employed to do qm in phase space " .it is also a simple exercise to show that the above definition of the wigner function of the state is equivalent to the inverse fourier transform of the characteristic function of the density operator ( ref . , eqs .( 3.12 ) , ( 3.16 ) ) , i.e. ,    \ ; . \label{char_fctn_rho}\end{aligned}\ ] ]  [ wt = invft and char_fctn ]  now consider the observable which is the qm counterpart of the classical quantity of eq . ( [ xtheta 1 cm ] ) .this observable satisfies the eigenvalue equation where denotes an eigenvalue and the corresponding eigenvector .our program is as follows .if the system is prepared in the state defined by the density operator , we first consider the probability density that a measurement of the observable will give the value : this probability density will be initially expressed in terms of , eq . ( [ rhoqm(x ) ] ) below , and then in terms of the wigner function in phase space , eq .( [ rhoqm(x)_wt 2 ] ) below .the final goal is to invert " this relation and show that we can retrieve the wigner function in terms of .  the probability density is given by the standard qm expression making use of eq .( [ < a > ] ) , we write in this expression , is the wigner transform of the projector , which is calculated in appendix [ wt of projector(theta ) ] with the result then eq . ( [ rhoqm(x)_wt 1 ] ) takes the form this last equation is the qm counterpart of eq .( [ rho(x)_vs_rhoq , p ] ) for the classical probability density .it shows explicitly that what plays the role of the classical probability density in phase space is now the quasi - probability density given by the wigner function .thus in order to invert eq .( [ rhoqm(x)_wt 2 ] ) we just copy the result in eq .( [ inverse radon q , p ] ) and write in terms of as this equation allows reconstructing the qm state , in the sense that from the observable probability density the wigner function of the density operator can be extracted ; its knowldedge , in turn , is equivalent to that of the state itself .this completes our analysis that shows a close analogy between the classical and quantum state reconstruction : both require the use of the inverse radon transform .we now turn to an alternative quantum - state reconstruction scheme which does not require the use of the radon transform .mutually unbiased bases ( mub ) in concept were introduced by schwinger in his studies of vectorial bases for hilbert spaces that exhibit `` maximal degree of incompatibility '' .the eigenvectors of and , and , respectively , are example of such bases . the information - theoretical oriented appellation `` mutual unbiased bases '' was introduced by wootters .consider two complete and orthonormal vectorial bases , , , whose vectors will be designated by and , respectively . the two bases are said to be mub if and only if , for , where is a constant _ independent _ of and ( see ref .this property means that the absolute value of the scalar product of vectors from _ different _ bases is independent of the vectorial label within either basis .this implies that if a system is measured to be in one of the states , say , of , it is equally likely to be found in any of the states of any other basis , when and are mub .the value of may depend on the bases , , which indeed is the case for a continuous hilbert space . for a hilbert space with a finite dimensionality , .the concept of mub is found to be of interest in several fields .for instance , the ideas are useful in a variety of cryptographic protocols and signal analysis . in what follows we outline a scheme for state reconstruction based on mub which is an alternative to the one presented in the previous section .we begin with a review of the properties of the operator and its eigenstates , eqs .( [ xtheta_qm ] ) , ( [ e - value eqn for xtheta_qm ] ) , and show that the bases , ( , fixed ) are mub .we repeat the definition ( [ xtheta_qm ] ) of the operator and introduce the new operator as  and are _ canonically conjugate _, i.e. , = i ] .when is a prime number , the operations of multiplication and division , modulo , can be defined consistently . as a simple example , we find , for , that =2 ] .  as a result, we may replace uniquely , up to a power of , all the operators of the form , with , by , with .we can readily prove that with  ] introducing inside the matrix element of eq .( [ ft - wpxc ] ) a complete set of eigenstates of position and of momentum right after the first and second exponentials , respectively , we find the inverse fourier transform of this last expression gives the result of eq .( [ wt_p_x_theta ] ) .to prove eq .( [ psi_x , theta(x ) 2 ] ) we impose the requirements of eq . ( [ norm , m.els.x , p , theta to 0,plane_wave ] ) on the solution ,( [ psi_x , theta(x ) 1 ] ) .\1 ) the ortho - normalization condition , eq . ( [ orthonormal ] ) , imposed on the wave function of eq . ( [ psi_x , theta(x ) 1 ] ) gives , for the function ,  is an arbitrary phase , dependent on and .the wave function becomes }. \label{psi_(x , theta)(x ' ) 3}\ ] ] 2 ) the first requirement in eq .( [ m.els .x and p ] ) for the matrix elements of with the wave function of eq .( [ psi_(x , theta)(x ) 3 ] ) is automatically fulfilled , since i ) our starting point has been the eigenvalue equation , eq .( [ e - value eqn for xtheta_qm ] ) , and ii ) the wave function of eq .( [ psi_(x , theta)(x ) 3 ] ) satisfies the orthonormalization condition , eq .( [ orthonormal ] ) .\3 ) we first compute the matrix element of which appears on the left - hand side of the second requirement in eq .( [ m.els .x and p ] ) . using the definition of given in eq .( [ xtheta , ptheta ] ) and eq .( [ psi_(x , theta)(x ) 3 ] ) we have , in the coordinate representation \psi_{x_2 , \theta}(x ) dx , \nonumber \\ & = & \int \psi_{x_1 , \theta}^{*}(x )  \left[-\frac{1}{\sin \theta } x + \frac{\cos \theta}{\sin \theta}x_2\right ] \psi_{x_2 , \theta}(x ) dx , \nonumber \\ & = &  -i \ ; e^{i\left [ \phi_{\theta}(x_2)-\phi_{\theta}(x_1)\right ] } \delta ' ( x_1-x_2 ) + \frac{\cos \theta}{\sin \theta } \ ; x_2 \;\delta ( x_1-x_2 ) .\label{p12 1}\end{aligned}\ ] ] we write , for the above exponential , the taylor expansion } & = & 1 + i\left[(x_2-x_1)\phi'_{\theta}(x_1 ) + \frac{(x_2-x_1)^2}{2!}\phi''_{\theta}(x_1 ) + \cdots\right ] \nonumber \\ & & -\frac{1}{2!}\left[(x_2-x_1)^2(\phi'_{\theta}(x_1))^2 +  2\frac{(x_2-x_1)^3}{2!}\phi'_{\theta}(x_1)\phi''_{\theta}(x_1)+ \cdots  \right ] \nonumber \\ & & + \cdots \ ; , \label{taylor exponential}\end{aligned}\ ] ] where the primes mean derivatives with respect to the argument .we use the -function identities , , to write the matrix element , eq .( [ p12 1 ] ) , as + \frac{\cos \theta}{\sin\theta } x_1 \delta ( x_1-x_2 ) .\label{p12 1a}\end{aligned}\ ] ] in order to satisfy the second requirement in eq .( [ m.els . x and p ] )we thus need with the solution where is an arbitrary function of .the wave function of eq .( [ psi_(x , theta)(x ) 3 ] ) then becomes } \ ; .\label{psi_(x , theta)(x ' ) 4}\ ] ] 4 ) choosing , for the phase , we satisfy the requirements of eq .( [ lim theta to 0 and plane_wave ] ) .we finally find the wave function of eq .( [ psi_x , theta(x ) 2 ] ) .in this appendix we compute the matrix elements of the operator in the original basis .we have    where are the one - dimensional harmonic oscillator wave functions ( see ref . , p. 61 , eq .( 5.24 ) , where the variable has been replaced by the dimensionless , as used in this paper )  being hermite polynomials .we compute the sum in eq .( [ x u+ x b ] ) using the identity ( ref . , p. 781, problem 6.12 ) , \label{morse - feshbach identity}\ ] ] with the result } } { \sqrt{2 \pi ( -i ) \sin \theta } } .\label{x ' u+ x 2}\ ] ] this result is identical to that of eq .( [ < xu+x > ] ) if we choose , for the square root , the branch will suffice to evaluate the quantity ; this is the probability to find in a unit interval around the value when the system has been prepared in the state .we find    where in the last line we have used eq .( [ tr(ab ) ] ) to express our probability in terms of wigner transforms .the wigner transform of the projector is found from eq .( [ wt_p_x_theta ] ) and that for the projector is simply .we thus write the last equation as    on the one hand , this integral can be evaluated directly , giving just as in eq .( [ |<xtheta|x theta>| ] ) with and . on the other hand ,the appearance of the factor in the denominator of the result ( [ |<x|x theta>|2 3 ] ) can be understood by using an intuitive geometrical argument starting from ( [ |<x|x theta>|2 2 b ] ) , as follows .we approximate the delta function occurring in eq .( [ |<x|x theta>|2 2 b ] ) by the step the delta function being attained in the limit .the non - zero region is indicated as the shaded area in fig .[ geom_inter_s ] .the segment along the -axis , over which we are integrating , contained inside that area , is .the integral in eq .( [ |<x|x theta>|2 2 b ] ) is thus given by thus reproducing the formal result of eq .( [ |<x|x theta>|2 3 ] ) .m. moshinsky and c. quesne , oscillator systems " , in _ symmetry properties of nuclei _, proc . of the xv solvay conference on physics , 233 - 300 ( gordon and breach , london , 1974 ) ; linear canonical transformations and their unitary representations " , journ .* 12 * , 1772 - 1780 ( 1971 ) .a. vourdas , phase space methods for finite quantum systems " , rep .phys . * 40 * , 367 - 371 ( 1997 ) ; a. b. klimov , l. l. sanchez - soto and h. de guise , multicomplementary operators via finite fourier transform " , j. phys .a : math . gen . * 38 * , 2747 - 2761 ( 2005 ) ; m. planat and h. c. rosu , mutually unbiased phase states , phase uncertainties and gauss sums " , europ .j. * d 36 * , 133 - 139 ( 2005 ) ; a. b. klimov , c. munos and j. l. romero , geometric approach to the discrete wigner function " , quant - ph/0605113 , 25 p. ( 2005 ) , lectures on _ quantum optics and quantum information _ , las trancas , chile , nov .21 - 25 , 2005 .

Abstract: we review the problem of state reconstruction in classical and in quantum physics , which is rarely considered at the textbook level . we review a method for retrieving a classical state in phase space , similar to that used in medical imaging known as computer aided tomography . we explain how this method can be taken over to quantum mechanics , where it leads to a description of the quantum state in terms of the wigner function which , although may take on negative values , plays the role of the probability density in phase space in classical physics . we explain another approach to quantum state reconstruction based on the notion of mutually unbiased bases , and indicate the relation between these two approaches . both are for a continuous , infinite - dimensional hilbert space . we then study the finite - dimensional case and show how the second method , based on mutually unbiased bases , can be used for state reconstruction .


during the last years , many systems have been analyzed from the perspective of graph theory .it turns out that seemingly diverse systems such as the internet , the world wide web ( www ) , metabolic and protein interaction networks and food webs , to mention a few examples , share many topological properties . among these properties ,the fact that one can go from one node ( or element ) of the network to another node passing by just a few others is perhaps the most popular property , known as `` six degrees of separation '' or small - world ( sw ) property .the sw feature has been shown to improve the performance of many dynamical processes as compared to regular lattices ; a direct consequence of the existence of key shortcuts that speed up the communication between otherwise distant nodes and of the shorter path length among any two nodes on the net .however , it has also been recognized that there are at least two types of networks fulfilling the sw property but radically different as soon as dynamical processes are ran on top of them .the first type can be called `` exponential networks '' since the probability of finding a node with connectivity ( or degree ) different from the average connectivity decays exponentially fast for large  .the second kind of networks comprises those referred to as `` scale - free '' ( sf ) networks .for these networks , the probability that a given node is connected to other nodes follows a power - law of the form , with the remarkable feature that for most real - world networks .the heterogeneity of the connectivity distribution in scale - free networks greatly impacts the dynamics of processes that they support .one of the most remarkable examples is that an epidemic disease will pervade in an infinite - size sf network regardless of its spreading rate .the change in the behavior of the processes is so radical in this case that it has been claimed that the standard epidemiological framework should be carefully revisited .this might be bad news for epidemiologists , and those fighting natural and computer viruses .on the other hand , in a number of important technological and commercial applications , it is desirable to spread the `` epidemic '' as fast and as efficient as possible , not to prevent it from spreading .important examples of such applications are epidemic ( or rumor - based ) protocols for data dissemination and resource discovery on the internet , and marketing campaigns using rumor - like strategies ( viral marketing ) .the above applications , and their dynamics , have passed almost unnoticed to the physics community working on complex networks despite the fact that they have been extensively studied by computer scientists and sociologists .the problem here consists of designing an epidemic ( or rumor - mongering ) algorithm in such a way that the dissemination of data or information from any node of a network reaches the largest possible number of remaining nodes .note that in this case , in contrast to epidemic modeling , one is free to design the rules of epidemic infection in order to reach the desired result , instead of having to model an existing process .furthermore , in a number of applications , such as peer - to - peer file sharing systems built on top of the internet and grid computing , the connectivity distribution of the nodes can also be changed in order to maximize the performance of such protocols .  in this paperwe study in detail the dynamics of a generic rumor model on complex scale - free topologies through analytic and numerical studies , and investigate the impact of the interaction rules on the efficiency and reliability of the rumor process .we first solve the model analytically for the case of exponential networks in the infinite time limit and then introduce a stochastic approach to deal with the numerical solution of the mean - field rate equations characterizing the system s dynamics .the method is used to obtain accurate results for several quantities when the topology of random sf networks is taken into account , without using large and expensive monte carlo ( mc ) simulations .the rest of the paper is organized as follows .section ii is devoted to introducing the rumor model and to derive the mean - field rate equations used throughout the paper . in sectioniii we deal with the stochastic approach , and compare its performance with analytical and mc calculations in homogeneous systems . we extend the method to the case of power - law distributed networks and present the results obtained for this kind of networks in section iv and v. finally , the paper is rounded off in the last section , where conclusions are given .the rumor model is defined as follows .each of the elements of the network can be in three different states .following the original terminology and the epidemiological literature , these three classes correspond to ignorant , spreader and stifler nodes .ignorants are those individuals who have not heard the rumor and hence they are susceptible to be informed .the second class comprises active individuals that are spreading the rumor .finally , stiflers are those who know the rumor but that are no longer spreading it .the spreading process evolves by directed contacts of the spreaders with others in the population .when a spreader meets an ignorant the last one turns into a new spreader with probability .the decay of the spreading process may be due to a mechanism of `` forgetting '' or because spreaders learn that the rumor has lost its `` news value '' .we assume this latter hypothesis as the most plausible so that the contacting spreaders become stiflers with probability if they encounter another spreader or a stifler .note that as we are designing our rumor strategy in such a way that the fraction of the population which ultimately learns the rumor be the maximum possible , we have assumed that contacts of the type spreader - spreader are directed , that is , only the contacting individual loses the interest in propagating the rumor further .therefore , there is no double transition to the stifler class .  in a homogeneous system ,the original rumor model due to daley and kendall can be described in terms of the densities of ignorants , spreaders , and stiflers , , , and , respectively , as a function of time . besides , we have the normalization condition , in order to obtain an analytical insight and a way to later test our numerical approach , we first study the rumor model on top of exponentially distributed networks .these include models of random graphs as well as the watts and strogatz ( ws ) small - world model .this model produces a network made up of nodes with at least links to other nodes .the resulting connectivity distribution in the random graph limit of the model takes the form which gives an average connectivity .hence , the probability that a node has a degree decays exponentially fast and the network can be regarded as homogeneous .the mean - field rate equations for the evolution of the three densities satisfy the following set of coupled differential equations : , \label{eq3}\\  \frac{d r(t)}{d t } & = & \alpha { \langle k \rangle}s(t ) [ s(t)+r(t ) ] , \label{eq4}\end{aligned}\ ] ] with the initial conditions , and . the above equations state that the density of spreaders increases at a rate proportional to the spreading rate , the average number of contacts of each individual and to the densities of ignorant and spreader individuals , and , respectively. on the other hand , the annihilation mechanism considers that spreaders decay into the stifler class at a rate times the density of spreaders and of non - ignorant individuals .the system of differential equations ( [ eq2]-[eq4 ] ) can be analytically solved in the infinite time limit when . using equation ( [ eq1 ] ) , we have that . introducing the new variable we obtain the transcendental equation,   equation ( [ eq5 ] ) always admits the trivial solution , but at the same time it also has another physically relevant solution _ for all _ values of the parameters and .this can be easily appreciated since the condition , reduces to .that is , there is no `` rumor threshold '' contrary to the case of epidemic spreading .this strikingly different behavior does not come from any difference in the growth mechanism of  two are actually the same , but from the disparate rules for the decay of the spreading process .on the other hand , this result also points out that a mathematical model for the spreading of rumors can be constructed in many different ways .the results of this paper , however , indicate that the presence of spreader annihilation terms due to spreader - spreader and spreader - stifler interactions is very relevant for practical implementations .we shall come back to this point later on .recently , we have introduced a numerical technique to deal with the mean - field rate equations appearing in epidemic - like models .it solves the differential equations by calculating the passage probabilities for the different transitions .the main advantage of this method , as compared to mc simulations , is its modest memory and cpu time requirements for large system sizes . besides, we do not have to generate any network .instead , we produce a sequence of integers distributed according to the desired connectivity distribution .the numerical procedure here proceeds as follows . at each time step until the end of the rumor spreading process , the following steps are performed :  .density of stiflers at the end of the rumor spreading process .results are shown for 5 different values of for each method considered .monte carlo ( mc ) simulations were performed in a ws network with and nodes .the same system size was used in the stochastic numerical approach ( sna ) . [ cols="^,^,^,^",options="header " , ]   in figs .[ figure4 ] and [ figure5 ] we have represented the time ( in units of ) profiles of and for each decay term and several values of . from the figures ,it is clear that while the final density of stiflers increases when modifying the original decay rules , the time needed to reach such high levels of reliability also increases .this is due to the fact that the tails of the densities of spreaders decay more slowly than before . in particular , it is noticeable that when only spreader - spreader interactions are taken into account in the decay mechanism , the lifetime of the propagation process is more than two times longer than for the other two settings .this means that this implementation is not very suitable for practical applications as the costs associated to the process rise as well . on the other hand, the performance of the spreader - stifler setting seems to depend on the value of in such a way that it is more efficient at both the reliability level and time consumption for a large , but not in the middle region of the parameter space . in summary, the present results support that the original model works quite well under any condition , while other variants can be considered depending on the value of used and the type of applications they are designed for .in this paper , we have analyzed the spreading dynamics of rumor models in complex heterogeneous networks .we have first introduced a useful stochastic method that allows us to obtain meaningful time profiles for the quantities characterizing the propagation process .the method is based on the numerical solution of the mean - field rate equations describing the model , and contrary to monte carlo simulations , there is no need of generating explicitly the network .this allows to save memory and a fast exploration of the whole evolution diagram of the process .the kind of processes studied here are of great practical importance since epidemic data dissemination might become the standard practice in multiple technological applications .the results show that there is a fragile balance between different levels of reliability and the costs ( in terms of time ) associated to them . in this sense ,our study may open new paths in the use of rumor - mongering process for replicated database maintenance , reliable group communication and peer to peer networks .besides , as shown here , the behavior and features of the different algorithms one may implement are not trivial and depend on the type of mechanisms used for both the creation and the annihilation terms .it is worth noting here that we have studied the simplest possible set of rumor algorithms , but other ingredients such as memory must be incorporated in more elaborated models .  of further interestwould be a more careful exploration of the possibility of using dynamical shortcuts for a more efficient spreading of the updates .our results suggest that it would be more economic to start from hubs and then kill off the updating process when a given level of reliability is reached than starting at random and letting the process dies out by itself .preliminary studies of more elaborated models aimed at implementing a practical protocol confirm our results .this feature is specially relevant for the understanding and modeling of social phenomena such as the spreading of new ideas or the design of efficient marketing campaigns .we would like to thank a. vespignani for many useful discussions and comments .y. m. thanks m. vzquez - prada for helpful comments and the hospitality of bt exact , u.k . where parts of this work were carried out .y. m. acknowledges financial support from the secretara de estado de educacin y universidades ( spain , sb2000 - 0357 ) . this work has been partially supported by the spanish dgicyt project bfm2002 - 01798 .a. j. demers , d. h. greene , c. hauser , w. irish , and j. larson , _ epidemic algorithms for replicated database maintenance_. in proc . of the sixth annual acm symposium on principles of distributed computing , vancouver , canada , 1987.              note that for the variant , the initial conditions should include the existence of at least one stifler .this has been taken into account by randomly selecting a node among all the remaining ignorants .hence the initial conditions are now , , and .

Abstract: we derive the mean - field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks . these equations are solved numerically by means of a stochastic approach . first , we present analytical and monte carlo calculations for homogeneous networks and compare the results with those obtained by the numerical method . then , we study the spreading process in detail for random scale - free networks . the time profiles for several quantities are numerically computed , which allow us to distinguish among different variants of rumor spreading algorithms . our conclusions are directed to possible applications in replicated database maintenance , peer to peer communication networks and social spreading phenomena .


coarse - graining is at the heart of equilibrium statistical mechanics .it is used to describe systems with a macroscopic number of degrees of freedom with the help of a much smaller number of variables .a familiar example is the description of a gas in terms of its slowly varying local density rather than the position of all its molecules  .coarse - graining is particularly important in the study of phase transitions , and more generally , interacting systems .it is a crucial building block in renormalization group theory  . even far from a phase transition the combination of coarse - graining with the identification of relevant and irrelevant variablescan sometimes be used to replace seemingly complicated interactions by ( approximate ) simpler ones .the theory of systems out of equilibrium is less developed and , in particular , the possible uses of coarse - graining in such a theory are not as clear .one approach , recently reviewed by derrida  , uses large deviation theory to address the fluctuations of a local density or some other coarse - grained field .this leads to a functional of the local density that characterizes the steady state and fluctuations around it .  in recent years, a different approach has been employed to study systems far from equilibrium . in this approach oneinvestigates symmetries , related to time - reversal , that are associated with the probabilities of observing sequences of events .such investigations have led to the discovery of a number of interesting results , collectively known as _ fluctuation relations _ , including some that provide information related to the likelihood to observe so - called `` second law violations ''  .loosely speaking , fluctuation relations have the form = q t + o(t),\ ] ] where is the probability to observe a change in entropy , is the observation time , and is the observed entropy creation rate . [ the notation indicates a correction that grows more slowly than , that is . ]these results are quite general , they remain valid far from thermal equilibrium , and they are part of an active field of research focused on the application of the laws of thermodynamics to microscopic systems  .fluctuation relations are derived by considering statistical distributions of microscopic trajectories of the system of interest . for any such trajectory ,we assign a value to a quantity , which we interpret physically as a measure of entropy production . by comparing the probability of observing a given trajectory ( ) with that of its time - reversed counterpart ( ) ,the symmetry represented by equation ( [ eq : ft_intro ] ) emerges ( see section [ fluct ] ) .this approach , however , runs into difficulties when coarse - graining is involved .consider an experiment in which some degrees of freedom can not be observed , perhaps because they evolve on time scales much faster than the response time of the detector . in this caseour observation of the system gives us a _ coarse - grained _ i.e. a `` smeared '' , or locally averaged trajectory , rather than a full microscopic record of the evolving state of the system .it is not at all evident that such coarse - grained trajectories satisfy the same symmetry relations as their fully microscopic counterparts .moreover , important information might be lost due to coarse graining ; specifically , the expression for entropy production , , might contain contributions from those degrees of freedom that are not resolved by the detector . under these circumstances, can we define some sort of `` coarse - grained entropy production '' , , which satisfies a fluctuation relation ?our goal in this paper is to give partial answers to such questions .we will examine a simple system for which an assumed separation of time scales leads to a simple coarse - graining transformation .the system we study is a stochastic jump process whose probability distribution satisfies a master equation .fluctuation relations for such jump processes were first derived by lebowitz and spohn  , following kurchan s analysis of langevin processes  .these two papers laid the groundwork for much subsequent work on stochastic fluctuation relations , which has recently been reviewed by harris and schtz  . in the present paper , we impose a separation of time scales and argue that , when the slow component of the dynamics is accurately measured , a non - trivial fluctuation relation , which differs from the one valid for the microscopic system , is ( approximately ) satisfied . this relation involves a coarse - grained entropy production , which depends only on the slow component of the evolution .we note that in the context of fluctuation theorems , an explicit separation of time scales has appeared as well in the work of zamponi _ et.al ._  .these authors have studied a langevin particle interacting with a nonequilibrium thermal environment consisting of slow and fast components at different temperatures ( see also ) .for such a system a fluctuation - dissipation relation between spontaneous and externally induced fluctuations can be used to define an _effective temperature_. the system was shown to satisfy a generalized fluctuation theorem formulated in terms of this effective temperature .  in section [ coarse ]we will present the model used and apply a coarse - graining transformation , replacing it by a simpler system . the (approximate ) coarse - grained system is also markovian and therefore satisfies a fluctuation relation of its own . in section [ fluct ]we use the existence of a coarse - grained counterpart to define coarse - grained trajectories , and a coarse - grained entropy production .these trajectories can be heuristically viewed as measurements obtained with a limited - resolution detector .we show that the coarse - grained entropy production approximately satisfies a fluctuation relation ( for the original system ) .a simple example is numerically studied in section [ example ] , in order to illuminate the results obtained in previous sections .the results are summarized in section [ disc ] .while there are many physically relevant and interesting examples of systems with natural coarse - grained counterparts , we will focus on stochastic jump processes governed by master equations .after briefly reviewing general properties of such processes , we present models with special structure , which makes them amenable to coarse - graining .we then use simple perturbation theory to motivate a coarse - graining transformation that replaces the system of interest by a smaller , reduced system .  consider a system composed of a finite number of distinct states .the system can make transitions between these states , and these transitions are assumed to be described by a markov process . specifically , given two states and , the rate is the probability per unit time to make a transition to , from a current state . while not all transitions are allowed ( some of the rates vanish ) , we do assume that if the transition from to is allowed [ then also the reverse transition is allowed [ .this assumption is analogous , but not equivalent , to microscopic reversibility for deterministic systems .it is useful to depict such a system with the help of a graph , in which each node represents a state and each link represents a pair of non - vanishing transitions .for instance , the graph shown in figure [ examplegraph ]  illustrates a five - state system with six non - vanishing links . notethat we allow multiple links between a given pair of states , as between states 2 and 3 .in other words , there may exist information that allows one to distinguish between different ways to make a transition .this implies that four rates namely , and  are needed to describe all transitions between states and .such a process satisfies a master equation where denotes the normalized probability to find the system in the state at time .it is useful to think of as a vector whose components are labeled by the index , and as a matrix .a non - diagonal element is given by the sum of the rates from to , e.g.  specifies the net probability rate of transitions out of the state . to ensure conservation of probability , the rates satisfy we assume that the graph is _ connected _, that is , it is possible to go from any state to any other state using the links . in that case, the solution of equation ( [ eqmaster ] ) decays with time to a unique stationary solution satisfying  the stationary probability current measures the net flow from to , via the transition , in the stationary state . if for every transition , then _ detailed balance _ is satisfied , and it is natural to think of the system as being in a state of thermal equilibrium . for a generic transition matrix , however , the currents are non - vanishing , and then we consider the system to be in an out - of - equilibrium steady state  .systems described by master equations are , generally , relatively easy to handle mathematically .however , our interest is in systems for which it is useful to coarse - grain the dynamics .this leads us to examine systems with special structure .specifically , we will assume that the states of our system are assorted into clusters , with `` strong '' links within a given cluster , , and `` weak '' links between states belonging to different clusters , , as illustrated in figure [ bforecoarse ] for , .we further assume that each separate cluster is ( internally ) connected , and also that the entire system is connected .finally , we assume that there is at most one microscopic transition connecting any pair of states in the system .( it will become clear that coarse - graining does not preserve this property . ) a system satisfying these conditions is characterized by two widely separated time scales , corresponding to the fast dynamics of relaxation within each cluster ( ) , and the slow redistribution of probability among the clusters ( ) .the microscopic evolution of our system is fully described by a sequence of transitions from state to state , , along with the times at which these transitions occur .we introduce coarse graining by imagining that we monitor this evolution using a low - resolution apparatus , capable of distinguishing different clusters , but not different states within a cluster .the dynamics we observe then consists of transitions from cluster to cluster , , which provides only partial information regarding the underlying microscopic trajectory . in general this low - resolution dynamics is not a markov process , and can not be described by a master equation of the form given by equation ( [ eqmaster ] ) .we aim to investigate whether this coarse - grained sequence satisfies a meaningful fluctuation theorem .let us imagine , for a moment , that such a coarse - graining transformation maps the original system onto a smaller system , whose states correspond to the clusters of the original system .for instance , the system depicted in figure [ bforecoarse ] is mapped onto a 3-state system .note that there may be several different transitions between a pair of clusters .the ability of the low - resolution apparatus to distinguish between these transitions determines the structure of the simpler system .for instance , the system depicted in figure [ bforecoarse ] will be mapped to the one depicted in figure [ afterc]a when the apparatus can distinguish between the transitions connecting the same clusters . alternatively , when the apparatus can only distinguish the clusters , butcan not be used to identify the transitions , the system will be mapped to the one depicted in figure [ afterc]b .  in order to give a more quantitative meaning to the notion of coarse - graining , we now develop a perturbation theory that maps our original -state stochastic jump process onto a _ reduced _-state stochastic jump process ; each state of the reduced system represents one of the clusters in the original system . such a mapping is useful only if it preserves relevant dynamical properties . in our casethis means the reduced dynamics ought to provide a good approximation of the low - resolution dynamics of the original system . in what followswe show that the structure that we have assumed above leads to a separation between fast and slow decay modes of equation ( [ eqmaster ] ) .the latter are then used to set the transition rates of the reduced , -state process , in such a way that this process does , indeed , faithfully reproduce the low - resolution dynamics of the original system , to leading order in .  for systems with the cluster structure depicted in figure[ bforecoarse ] , we decompose the transition matrix , : here is block - diagonal , with each block corresponding to one cluster [ see equation ( [ eq : matrixr0 ] ) for an example ] .the non - diagonal elements of are the weak links connecting states belonging to different clusters , while the diagonal elements of are adjusted so that equation ( [ condition ] ) holds separately for and .we assume that the dependence of on resides entirely in the factor that multiplies in the above decomposition .thus all non - vanishing elements of and are assumed to be order unity .we then treat as a small perturbation to .  to keep considerations simple , let us assume that both and have full sets of left and right eigenvectors , satisfying bi - orthogonality .thus for we have with .the same relations hold among the eigenvalues and the left and right eigenvectors of , denoted by , and .the transition matrix has one vanishing eigenvalue , .all other eigenvalues are negative .the corresponding right eigenvector is proportional to the steady state distribution , and the left eigenvector expresses conservation of probability . since bi - orthogonality does not fix the normalization of these eigenvectors , we choose , for convenience , and .  to carry out our perturbation analysis , we expand eigenvectors and eigenvalues in powers of , and we match powers of in equation ( [ fulleigen ] ) . at leading orderwe get and similarly for . making use of the block - diagonal structure of , and diagonalizing each block separately , we obtain eigenvectors and that inherit this structure : each eigenvector has non - vanishing elements only in one block , corresponding to one cluster .the block - diagonal structure of describes mutually isolated clusters , each of which supports its own steady state and corresponding vanishing eigenvalue .thus has vanishing eigenvalues , for .the corresponding right eigenvectors , , are composed of blocks of vanishing elements with the steady state of the cluster , inserted at the appropriate place .( here runs over states belonging to the cluster . )the corresponding left eigenvectors , , have a similar structure , with the elements replaced by s .the remaining eigenvectors of have negative eigenvalues of order unity . in first - order perturbation theory ,the term in equation ( [ eq : r.r0.er1 ] ) shifts each eigenvalue by an amount of order .while this represents a relatively small change in the non - vanishing eigenvalues , it crucially lifts the degeneracy among the first ( vanishing ) eigenvalues .these are replaced by a single vanishing eigenvalue ( corresponding to the steady state of the fully connected system ) and negative eigenvalues of order , which describe the long time decay of the system .assuming for simplicity that the degeneracy is fully lifted at first order , higher order corrections will lead only to small further changes in the eigenvalues and eigenvectors .therefore , in the following we will concentrate on the application of first - order perturbation theory to the degenerate sector , .degenerate perturbation theory begins with the matrix whose eigenvalues represent the first - order corrections , , to the vanishing eigenvalues of . because of the correspondence between the clusters of the system and the left / right eigenvectors in the above equation , the quantity represents the transition rate between clusters and . in the followingwe emphasize this identification by using indices to specify the clusters .the slow decay modes of the system are determined by the matrix whose dimensionality is the number of clusters , .this matrix satisfies where we have used . in other words satisfies the conservation - of - probability condition expressed ( for the full system ) by equation ( [ condition ] ) .it is thus meaningful to interpret the equation , as describing a markov jump process .we now discuss the physical meaning that can be assigned to and .the separation of time scales between the slow and the fast decay modes , implies that after the decay of the latter the probability distribution satisfies .thus the probability inside each cluster is a product of its local steady state distribution , , and the total probability to find the system in the cluster , .equation ( [ coarsemaster ] ) approximately describes the time evolution of the latter .more precisely , equation ( [ coarsemaster ] ) governs a _ reduced _ ( -state ) markov jump process , and _ this reduced process provides an approximate description of the transitions between clusters within the original system_. note that the individual elements of differ from those of : for a transition out of a cluster , with the rate appearing in , the probability current is .this contributes a rate in the appropriate place in .the procedure outlined above is not the only way to define a reduced system .an alternative approach would involve measuring the transition rates between clusters in the steady state of the full system , and then constructing a reduced , -state markov jump process governed by precisely these transition rates .we expect that this `` empirical construction '' based on the observed cluster - to - cluster transition rates and the `` perturbative construction '' leading to equation ( [ coarsemaster ] ) , will give reduced systems that become equivalent in the limit .  to summarize , we have used perturbation theory to give quantitative meaning to a coarse - graining transformation , in which a markov jump process among states , arranged into clusters , is replaced by a reduced markov jump process among states meant to represent those clusters . in the following section , where we consider fluctuations , it is important to bear in mind that the reduced dynamics only approximately reproduce the statistics of transitions between clusters of the original system .of course , one might improve the approximation by extending the perturbation expansion to higher orders of ; see , e.g. the appendix of reference  .however , higher - order corrections to the eigenvectors of a particular degenerate sector typically involve _ all _ unperturbed eigenvectors , not only those within that sector .thus it is not generally possible to reduce the dimensionality of the system when considering higher orders in .in this section we ask whether the structure described in section [ coarse ] leads to a non - trivial fluctuation relation .we start by briefly reviewing the fluctuation relation for general stochastic jump processes  .we will then turn to systems amenable to coarse - graining .fluctuation relations are obtained by comparing probability densities of microscopic paths and their time reversed counterparts . a path , or a history , , will denote a stochastic trajectory , represented by an ordered sequence of transitions at given times ( between say and ) , the s in equation ( [ defsmallg ] ) denote the transitions between states .they are redundant for the original system , but would be needed after coarse - graining , where several transitions between clusters are possible .we have chosen to explicitly denote the transitions in order to use similar notation for the original and coarse - grained system . the conditional probability density ( or weight )corresponding to a path is \prod_{j=1}^n r_{\alpha_j } ( \sigma_j | \sigma_{j-1}).\ ] ] in equation ( [ weight ] ) denotes the overall rate of probability flow out of the state , and we set and .the probability density of a path is obtained by multiplying by the probability to sample its initial conditions , .to each path we can define a time reversed counterpart , starting at at , and ending at at . since the path and its counterpart reside in the same states for the same lengths of time their exponential survival probability factors in equation ( [ weight ] ) are identical . as a result taking the logarithm of both sides yields the right side includes a boundary term , , arising from the initial and final states , and a sum of contributions from the transitions .if we interpret as the amount of entropy that is produced when the system makes a transition from to  , then is the net entropy production associated with the trajectory , and we have   identities of the form of equation ( [ eq : preft ] ) are at the heart of both _ steady - state _ and _ transient _ fluctuation relations  .the distinction between the two cases , roughly , amounts to whether one chooses to consider the limit ( in which case the relative contribution of can be neglected , at least for finite systems  ) , or rather to argue that the boundary term represents the net change in the entropy of the system ( in which case the sum on the right represents the entropy produced in the system and its surroundings  ) .in either situation the desired fluctuation relation follows in a more or less straightforward manner .  for our purposesthe distinction between the transient and steady state cases is not of central importance .rather , we aim to investigate whether a relationship similar to equation ( [ eq : preft ] ) can be obtained for systems of the type studied in section [ coarse ] . to clarify this issue ,let us return to the situation introduced briefly in section [ coarse ] , in which we monitor the evolution of our system using a low - resolution apparatus that distinguishes between clusters , but not between states within a cluster .under these circumstances we do not observe the full trajectory , , but rather a coarse - grained version ( defined below ) that describes only transitions between clusters .can we expect this trajectory to satisfy a fluctuation relation ? in what follows we use the perturbation analysis of section [ coarse ] to define a coarse - grained entropy production , , that differs from above , and is expressed in terms of the _ observed _ transition rates .we argue that this quantity approximately satisfies a fluctuation relation , and that the approximation improves as the separation of time scales becomes more pronounced ( i.e. as ) .the deviations from this ( approximate ) relation are more pronounced for extreme values of , which select rare trajectories that , in some sense , do not respect the separation of time scales .these deviations reflect the non - markovian nature of the observed , coarse - grained dynamics , and can be viewed as evidence of the existence of unobserved degrees of freedom .the reduced , -state system defined in section [ coarse ] is a stochastic , markov jump process that satisfies an exact fluctuation relation .the entropy produced during a transition from to is given in this case by and the net entropy production associated with a trajectory is just a sum of such contributions . to simplify notations ,we assume here , and in what follows , the scenario where different transitions between the same clusters are distinguishable .the following considerations can be easily modified to apply to coarse - grained dynamics with indistinguishable transitions .let us now return to the full , -state system , but assume the conditions of a low - resolution apparatus .suppose the evolution of the system from to is described by a trajectory .to this trajectory let us assign a _coarse - grained _ entropy production  ] .the only source of errors will then appear in sums over terms of the form .the number of terms in the sum is proportional to the number of transitions between clusters , which in turn scales as .let us consider first the corrections for a typical trajectory .the corrections are a combination of typical terms with small errors and rare terms with a large error , as discussed in the previous paragraph .the resulting errors in estimating the entropy production is small , and scales as .the steady state fluctuation relation does not deal only with typical trajectories .in fact , one examines trajectories whose entropy production rate is ( approximately ) fixed .this will pick out trajectories which may be very unlikely .we _ assume _ that for a finite range of entropy production rates the relevant trajectories are such that the time differences for which the approximation made in equation ( [ approxmu ] ) break down are still unlikely ( with probability which scales as ) . with this assumption ,the coarse grained - entropy satisfies for some finite range of values hint that the error term in equation ( [ steadystatefr ] ) depends on in a non - trivial manner . ] .note that one can not expect this relation to hold for any coarse - grained entropy production rate . taking this rate to be ``large enough '' will eventually pick trajectories whose typical time differences are of order .these trajectories will never spend enough time in clusters to equilibrate .in this section we use a simple example to illustrate fluctuation relations in a coarse - grained setting , and to gain some intuition for the admittedly abstract considerations of the previous section .we want to use the simplest system possible whose reduced counterpart can exhibit out - of - equilibrium steady states .this implies at least three clusters of states .consider a system with eight states , , organized into three clusters , .the transition matrix describing the full system is with and ( the non - vanishing rates are in bold - face . )numbering the states of the full system from 1 to 8 , the three clusters are : , , and .we see that has the structure described in section [ coarse ] : all off - diagonal elements of are between states within a given cluster , and all off - diagonal elements of are between states belonging to different clusters .we begin our analysis of this system by constructing the reduced , three - state markov system .  the steady state distributions for the ( unconnected ) clusters appearing in given by the null vectors of the three matrices along the block diagonal : , and . from these weeasily construct the eigenvectors spanning the null space of . using equation ( [ defr1 ] )we then obtain this transition matrix defines the dynamics of the three - state reduced system , and approximately describes the cluster - to - cluster evolution of the full , eight - state system .  to evaluate the probability distribution of entropy production rates in the limit of large observation time , , direct simulations are of limited use , as the asymptotic probability to observe a specific _ average _ entropy production rate decays exponentially with . thus obtaining good statisticsbecomes problematic .following lebowitz and spohn  ( see also ) , we consider the large deviation function and its legendre transform . here denotes the observed entropy production rate , while is the probability distribution of this entropy production . to differentiate between the large deviation function of the reduced and coarse - grained system we will denote the former ( latter ) by a superscript ( )lebowitz and spohn have shown that is the largest eigenvalue of an operator which governs the time evolution of averages of the type .the diagonal elements of are those of , while non - diagonal elements are multiplied by a factor related to where the entropy change is that of the transition corresponding to this non - diagonal element .we refer for references and for the derivation .  for the reduced system ,whose transition matrix is given by equation ( [ cgsystem ] ) , this operator is given by this operator has the property , which implies the steady state ( or gallavotti - cohen ) fluctuation relation  .note that the dependence of on is trivial . to cancel this dependencewe define to be the maximal eigenvalue of divided by .let us turn back to the original system .we have defined a coarse - grained entropy which changes only during stochastic transitions connecting different clusters , see equation ( [ coarseentropy ] ) .one can apply the formalism developed by lebowitz and spohn to study the distribution function of entropy production for the coarse - grained dynamics .the derivation follows closely that of .again , the legendre transform of the large deviation function is given by the largest eigenvalue of an operator ( divided by ) .the relevant operator is note that only elements which connect different clusters depend on .this results from the fact that the coarse - grained entropy changes only during these transitions .it is important to notice that .this means that the coarse - grained entropy does not satisfy an _exact _ fluctuation relation .it does , however satisfy an approximate one .consider for small values of .when we find that .we have already studied the eigenvalues and eigenvectors of in section [ coarse ] , see also the discussion leading to equation ( [ cgsystem ] ) .the largest eigenvalue is with three corresponding pairs of left and right eigenvectors .all other eigenvalues are negative and of order unity . for small one can study the eigenvalues of using degenerate perturbation theory .the largest eigenvalue must be part of the degenerate sector .moreover , performing the leading order perturbation theory leads to as the matrix whose eigenvalues are the leading order approximation for the eigenvalues of that belong to the degenerate sector .therefore , the function , obtained by computing the largest eigenvalue of ( [ calh ] ) approximately satisfies ( with corrections of order ) .this holds not only for the largest eigenvalue but to all the eigenvalues in the degenerate sector . this perturbation theory for the legendre transform of the large deviation function can be considered as another justification for the existence of an ( approximate ) fluctuation relation for the coarse - grained entropy .we have calculated numerically the largest eigenvalue of , as a function of and compared it to the largest eigenvalue obtained from .the results are depicted in figure [ ldvfig ] .it is clear that the eigenvalue corresponding to the coarse - grained system is a good approximation to the one obtained for the reduced system .moreover , the approximation improves for smaller .the function for the reduced system vanishes at and has the exact symmetry as expected .the corresponding functions obtained for the coarse - grained system start to deviate from that function for larger values of .this is related to the fact that the perturbation theory can not be uniformly valid .for instance , examining equation ( [ calh ] ) , it is clear that , for any , one can find so that , say , . as a result , terms which were assumed to be a small perturbation cease to be small .this can be loosely interpreted as picking contributions from highly unlikely trajectories , which may break our assumption of local relaxation .the legendre transform of the function is the large deviation function which describes the asymptotic ( with time ) behavior of the probability distribution of the entropy production .the legendre transform can be computed numerically . for both systems studied here the maximum of found to be at .this is the most likely value of the coarse - grained entropy production rate in a long enough experiment , in agreement with the entropy production rate calculated for the steady state of the reduced system , .the fact that this value does not vanish indicates that the system relaxes to non - equilibrium steady state , in which detailed balance is violated .  to verify that the coarse - grained entropy ( approximately ) follow the gallavotti - cohen fluctuation relation we have calculated and compared it to the expected linear behavior .the results are depicted in figure [ prodfig ] .it is clear that fluctuation relation is a good approximation for the coarse - grained entropy .the approximation improves for smaller values of , as expected .note that the approximation deteriorates for larger values of entropy production rates .one may speculate that this is a result of the increased weight of trajectories that do not spend enough time in the clusters to relax to the local stationary state .the numerical results , presented in this section , were for a coarse - grained entropy defined using a reduced system , which in turn was obtained from the perturbation theory of section [ coarse ] .we have mentioned that there is another way to define a reduced system , with the help of the steady state of the microscopic system .the rates of this empirically - constructed reduced system differ from the ones used so far by higher orders of .the different rates will result in a different definition of the coarse - grained entropy .one may wonder whether this coarse - grained entropy will satisfy the fluctuation relation with much smaller deviations .we have repeated the numerical calculation performed in this section for this definition of coarse - grained entropy ( with , results not shown ) .the deviation from the fluctuation relation were found to be similar to those presented in figure [ prodfig ] .in this paper , we have studied stochastic jump processes which can be coarse - grained .the coarse - grained dynamics can be viewed as the dynamics of the system measured using a low resolution apparatus .a perturbation theory , based on a separation of time scales , was used to define a reduced system .the dynamics of this reduced system , which is markovian ( and therefore satisfies an exact fluctuation relation ) , approximates the coarse - grained system .this reduced system then motivates the definition of a coarse - grained entropy for the coarse - grained dynamics .this entropy was found to approximately satisfy a fluctuation relation .deviations from this relation are more pronounced for large coarse - grained entropy production rates .this deviation can be interpreted as a result of the existence of internal degrees of freedom , which lead to deviations from markovian behavior of the coarse - grained entropy .let us consider an experiment whose goal is to measure fluctuation relations .the considerations in this paper may be useful in the interpretation of the results of such an experiment . if the results exhibit deviations from the fluctuation relation one may suspect the existence of unobserved degrees of freedom .( however , other mechanisms , leading to deviations from the expected linear behavior , exist  . ) note that , even if there are internal degrees of freedom , it is entirely possible that the entropy production rates , needed for observing deviations from the fluctuation relation , are so rare that such events will not be measured during the experiment .the results found in this paper lead to several new questions .for instance , a better understanding of the deviation of the coarse - grained entropy production from the fluctuation relation is needed .the definition of the coarse - grained entropy used in this paper was motivated by the definition applied to microscopic systems .one may wonder whether it is possible to define another coarse - grained entropy , using only quantities measured by the low resolution apparatus , which will _ exactly _ satisfy the fluctuation relation .it is also of interest to extend the considerations , obtained here for stochastic systems , to deterministic systems .finally , we have seen that the existence of a coarse - grained counterpart leads to a physically motivated coarse - grained entropy exhibiting a fluctuation relation .one may wonder whether the reverse is also true .does the existence of a quantity , which has some physical interpretation , and satisfies a non - trivial fluctuation relation , suggest that the system can be simplified in some way ?such questions are left for future work .we are grateful for start - up funds provided by the university of maryland .

Abstract: we consider the application of fluctuation relations to the dynamics of coarse - grained systems , as might arise in a hypothetical experiment in which a system is monitored with a low - resolution measuring apparatus . we analyze a stochastic , markovian jump process with a specific structure that lends itself naturally to coarse - graining . a perturbative analysis yields a reduced stochastic jump process that approximates the coarse - grained dynamics of the original system . this leads to a non - trivial fluctuation relation that is approximately satisfied by the coarse - grained dynamics . we illustrate our results by computing the large deviations of a particular stochastic jump process . our results highlight the possibility that observed deviations from fluctuation relations might be due to the presence of unobserved degrees of freedom .