investigation of diffusive transport is of highest importance in many areas of physics and related sciences .the most fundamental characteristics of diffusion is provided by two quantities the diffusion coefficient and the drift velocity .the knowledge of the latter is especially important in studies of non - equilibrium phenomena , where usually .this includes , among others , diffusion in disordered media subject to an external field or the so called molecular `` pumps '' and `` motors '' , responsible for transport of various chemicals in biological cells .however , no universal _ and _ practical theoretical method of determining and for arbitrary systems has been developed , and although many solutions for some particular physical models have been proposed , in more complicated cases one often has to resort to approximations or numerical methods .the first , to our knowledge , attempt towards determining and in an arbitrary system was made by derrida , who considered a one dimensional , periodic lattice of a period and arbitrary hopping rates between nearest - neighbor sites .he managed to give exact expressions for the velocity and the diffusion constant as functions of the hopping rates .he then extended his method to a periodic -dimensional system , but his solution turns out to be rather complicated .an alternative method of calculating was also developed by kehr __ . of the two quantities , and ,the latter is of course much harder to find .a surprisingly simple formula for one - dimensional systems with transition rates satisfying the detailed balance condition was derived by lyo and richards , and recently independently reinvented by kutner and wichmann , where is the period of the lattice , denote the equilibrium site occupancy probabilities , and are transition rates from site to . however , this equation is valid only when the random walker hops between nearest - neighbor sites . a method of determining in arbitrary periodic systems satisfying the detailed balance conditionwas then proposed by braun and sholl .description of several other techniques , developed for some special types of random environments , can be found in review articles .all the above - mentioned methods are based on the following scheme :  1 .reduce the infinite system to a single elementary cell with periodic boundary conditions .2 .  write down the master equation .3 .  using it calculate the steady - state properties of this system ; in particular , find the steady - state site occupation probabilities or propagators .4 .  given these quantities ,find and .the third step is here critical and can be carried out explicitly only for relatively small systems or for models possessing some special properties , e.g , one - dimensional lattices with jumps restricted to the nearest - neighbor sites .the main goal of this paper is to develop a simple and , at the same time , general method of calculating and explicitly without performing detailed analysis of the steady state . in our approachthe third step reads  1 .find the matrix representing the fourier transform of the master equation .  which is a rather straightforward operation .the technique we propose is valid for both equilibrium and non - equilibrium systems .it can also be quite easily implemented in computer - algebra or numerical programming languages .following the first step , our considerations will be restricted to periodic systems , such as crystals or molecular motors .a general problem of diffusion in an arbitrary disordered system remains an open question .one might be tempted to treated it by taking the limit .this method can sometimes provide a hint about anomalous type of diffusion in a given aperiodic system .however , except for some simple models , it has not been established whether the limits and commute and thus , whether this approach is generally correct.  we will pay a special attention to a one - dimensional system of finite period with arbitrary jump rates between any of its sites whose distance is less than .this is perhaps the simplest model of a periodic system where all elements of the transition rate matrix can take on nonzero values .its investigation can be therefore carried out with relatively simple mathematical formalism . a solution to a general, multi - particle and multi - dimensional problem requires the same mathematical methods , but the notation would have to be complex .what is even more important , any periodic system with finite number of sites in its elementary cell can be mapped on such a one - dimensional system ; geometry of the original problem is relevant mainly in constructing the matrix and can be taken into account quite easily .once the explicit form of has been found , one can compute and using the methods we derive here for this simple one - dimensional system .in our calculations we will use both continuous - time and discrete - time formalisms .one could expect that since we calculate and in a stationary state , where all quantities are independent of time , both approaches should yield the same result .however , as was found by derrida , the form of the diffusion coefficient as a function of the transition rates can depend on whether time flows continuously , or not .this is an important issue , since using discrete - time models belongs to favorite techniques of computer physics ( in particular the cellular - automata , and the exact - enumeration methods ) , and one needs to know whether results obtained in this way are correct .derrida showed that in his ( grossly simplified ) model the relation between the diffusion constants obtained in the continuous - time ( ) and discrete - time ( ) formalisms reads .we will show that this relation is general and will prove it for any periodic system with arbitrary transition rates .the paper is organized as follows . using the fourier transform of the master equation ,see refs . , in section [ secgen1 ] we analyze a one - dimensional system with arbitrary period and transition rates .we find a simple technique of calculating and in terms of the 3 lowest coefficients of the characteristic polynomial of the fourier - transformed transfer - matrix , .the most important properties of its spectrum at has been already described in refs . , but only for systems obeying the detailed balance condition .therefore , in sections [ sectioncont ] and [ sectiondiscrete ] we analyze in detail the spectrum of in a general case , assuming that the time is a continuous or a discrete quantity , respectively .then , in section [ comparison ] we compare the two approaches and explain the different forms of diffusion coefficients derived in each of them . in section [ sectotgen ]we generalize our approach to systems in arbitrary space dimension , and in section [ secnn ] we present a particularly simple method of calculating and for one - dimensional systems with transitions between the nearest - neighbor sites . in section [ secappl ]we derive explicit forms of and in some commonly used models .in particular , one of the examples studied there explains how to apply our technique to many - body problems .finally , section [ secconcl ] is devoted to conclusions .consider a one - dimensional lattice with its sites located at , . at time we put a particle at site .the particle can then jump between the lattice sites .transitions are assumed to represent a continuous ( e.g. , a poisson ) markov process in time .the ( constant in time ) transition rate from a site to a site will be denoted by .we assume that the system is periodic in space and denote its period by , of course transition rates between two different sites can not be negative , if . for simplicitywe also restrict our considerations to the case where direct transitions between sites and are allowed only if .let denote the probability density of finding the particle at site at time .the evolution of this quantity is governed by the master equation ,\ ] ] and the initial condition reads we can now treat the whole lattice as if it consisted of sub - lattices and define the probability densities of finding a particle at a given sublattice at site at time as where is a generalized kronecker s delta , if and otherwise. we can now rewrite ( [ cmel ] ) as a system of linear differential equations for , , .\ ] ] the major advantage of ( [ cmell ] ) as compared with ( [ cmel ] ) is that its coefficients do not depend on .therefore we can calculate its fourier transform , ,\ ] ] where .  the system of equations ( [ fcmell ] )can be written in a compact form using an matrix , where      \gamma_{lj }   { \rm e}^{ik(x_j - x_l ) }     + \gamma_{l , j - l } { \rm e}^{ik(x_{j - l}-x_l ) } ,     & j > l     \\[1ex ]     -\sum_{m\neq l } \gamma_{ml } ,     & j = l .\end{array }   \right.\end{aligned}\ ] ]  we will impose only one restriction on the form of transition rates .we will demand that be irreducible ( by a permutation of indices ) , i.e. , in the stationary state the particle can be found at any of the sublattices defined in ( [ theta ] ) with a probability . physically this condition means that in the long - time limit the system does not split up into several non - interacting subsystems .mathematically irreducibility means that has exactly one eigenvector whose all components are strictly positive .our approach is thus quite general we do not require that the transition rates should satisfy the detailed balance condition , the fourier - transformed transition rate matrix , , need not be symmetric or even diagonalizable at , and some of its eigenvalues can be complex .a general solution to ( [ fcmellambda ] ) reads  , \ ; l= 0,\ldots , l-1,\ ] ] where the coefficients are polynomials in and can be determined using the initial condition .the degree of is smaller than the multiplicity of .we assume that the eigenvalues are ordered in accordance with the descending magnitude of their real parts at , i.e. , . since ,there is , and so ( [ fpl_kt ] ) yields ,\ ] ] where are polynomials in ( actually can depend on time only if is degenerated ) .  in a one - dimensional system the stationary drift velocity and the diffusion coefficient given by    & = &     \lim_{t\to\infty }      i \frac {          \left .\frac{\partial \tilde{p}(k , t)}{\partial k }           \right|_{k=0 }         } { t } ,      \\[1ex ]  \label{generaldefd }  d & = &  \lim_{t\to\infty }       \frac{\langle x^2 \rangle - \langle x \rangle^2}{2 t }        \nonumber  \\[1ex ]   & = &      \lim_{t\to\infty }         \frac {         -\left .\frac{\partial^2\tilde{p}(k , t ) } { \partial k^2 }            \right|_{k=0 }           + \left (             \left .\frac{\partial \tilde{p}(k , t ) } { \partial k }             \right|_{k=0 }            \right)^2                      } { 2t},\end{aligned}\ ] ] where .thus , if we could calculate the eigenvalues of , we would be able , in principle , to calculate and .the diagonalization of , however , is a formidable task feasible only in some special cases ( e.g. for small or when special relations have been imposed on its elements ) .fortunately , as we will see below , if our aim is restricted only to determining and as functions of the transition rates , we need not calculate explicitly even a single eigenvalue of the transition rate matrix !let be a constant such that .since is irreducible , so is .moreover , because all elements of are nonnegative , we can apply to it the frobenius theorem and conclude that has a positive eigenvalue , which is a simple root of the characteristic equation , and the moduli of all other eigenvalues are at most . because the dominant eigenvalue of an irreducible , nonnegative matrix lies between the largest and smallest column sums , and in our case these sumsare all equal to , we find that the dominant eigenvalue . since the spectrum of is shifted , with respect to the spectrum of , by , we conclude that for the matrix has exactly one dominating eigenvalue and the real parts of all other eigenvalues are negative , thus , in the limit this single eigenvalue dominates the sum in r.h.s of ( [ fpl_kt ] ) at . in determining the forms of and we can therefore employ a simple approximation .\ ] ] moreover , since , there is which , owing to ( [ l0 ] ) , implies upon inserting ( [ fpkt0 ] ) into ( [ generaldefj ] ) , ( [ generaldefd ] ) and using ( [ l0 ] ) , ( [ h0 ] ) we conclude that   \label{defd1l }  d & = &  \left .-\frac{1}{2}\frac{\partial^2\lambda_0 } { \partial k^2 }       \right|_{k=0 } .\end{aligned}\ ] ]  let denote the characteristic polynomial of the matrix .let denote its coefficients at , .we thus have ^j = 0.\ ] ] on differentiating it with respect to and using the first part of ( [ l0 ] ) we arrive at where we used a short - hand notation and .note that , which follows from ( [ l0 ] ) . differentiating ( [ w ] ) twice yields where and .we thus finally arrive at our major result   \label{sold1c }  d & = &     \frac{c_0 '' - 2c_2 v^2 - 2 i c_1'v}{2 c_1}.   \end{aligned}\ ] ]  functions can depend on and the transition rates in a very complicated way .we were able to find only two general properties , both following immediately from ( [ l0 ] ) : and .explicit forms of , , , and for some particular models will be given below , in sections [ secnn ] and [ secappl ] .the discrete - time formulation of the problem is basically similar to the continuous one , but there are some major differences , too . for simplicity we will use the same notation as in the previous section , but one should remember that almost all functions employed in the discrete - time formulation of the problem differ from those we dealt with in section [ sectioncont ] . where it will be necessary to compare quantities computed within each approach, we will attach a superscript `` d '' to the quantity derived in the discrete - time formalism .  in the discrete - time version of the problemthe master equation ( expressed in terms of the probabilities of finding a particle at time at a site belonging to a sublattice ) reads ,\ ] ] where , and are dimensionless probabilities satisfying the usual condition .note that in the continuous formulation of the problem were unbounded from above , dimensional quantities ( of dimension [ t ) and we called them `` transition rates '' .last , but not least , in the present approach time assumes only integer values .  upon taking the fourier transform of ( [ dmel ] )we arrive at ,\ ] ] which can be rewritten using a stochastic matrix  where the only difference between and its continuous - time counterpart lies in their diagonal elements therefore the eigenvalues of are related to the eigenvalues of through a simple formula   applying the theorem of frobenius to , which is a real , nonnegative matrix , we conclude that and the moduli of all other eigenvalues do not exceed 1 .however , in contrast to the continuous - time formalism , now there can be eigenvalues , say , , such that ; for convenience we assume that .although this can happen only if all diagonal elements of vanish , we do not exclude this exceptional case from our considerations ( see ref .the frobenius theorem ensures us that all these dominating eigenvalues are distinct , and so in the normal jordan representation of the size of the corresponding jordan blocks is 1 .this suffices to assert that for and ^t,\ ] ] where are some functions of .  because equation ( [ one ] ) is valid both in continuous - time and discrete - time formalisms , upon comparing it with ( [ dfpkt ] )we conclude that except for all other coefficients in ( [ dfpkt ] ) must vanish at , actually , since is an integer , this is not a trivial statement ; a proof is based on the fact that all dominant eigenvalues are distinct roots of the equation , see refs. . upon inserting ( [ dfpkt ] ) into ( [ generaldefj ] ) and ( [ generaldefd ] ) and then using ( [ l0=1 ] ) and ( [ dh1 ] ) we arrive at   \label{ddefd1l }  d^{\rm d } & = &  \left .-\frac{1}{2 }        \left [          \frac{\partial^2\lambda^{\rm d}_0(k ) } { \partial k^2 } -          \left (           \frac{\partial\lambda^{\rm d}_0(k ) } { \partial k }          \right)^2        \right ]       \right|_{k=0 } - \psi(0,t),\end{aligned}\ ] ] where note that , by definition , if , i.e. , if is the only dominating eigenvalue of the transition matrix . moreover , because the existence of the diffusion constant is guaranteed by the central limit theorem , must be independent of time , at least in the limit , for any .this , in turn , requires that even if  ( see the comment under eq .( [ dh1 ] ) ) . therefore , to determine and one needs only to investigate the properties of a single eigenvalue at , i.e. , eq.([dfpkt ] ) can be replaced with ^t.\ ] ]  finally , using ( [ defj1l ] ) , ( [ defd1l ] ) , ( [ lambdy ] ) , and ( [ ddefj1l ] ) ( [ psi0 ] ) we conclude that the drift velocities and diffusion coefficients obtained in continuous and discrete formulations of the problem are related to each other by simple , general formulae  \label{ddtau=1 }   d^{\rm d } & = & d -\case{1}{2 } v^2 .\end{aligned}\ ] ] equation ( [ ddtau=1 ] ) holds , however , only if the interval between successive jumps in the discrete formulation of the problem equals 1 , which has been assumed in our calculations for simplicity . for a general value of relation between and reads which has a correct dimensional form ( all its terms have a dimension [ ) .   comparing equations ( [ fpkt0 ] ) and ( [ dfpkt1 ] )we find that the main mathematical differences between the continuous- and discrete - time formalisms are related to different functional forms of at .while in the continuous - time approach this function is given by an exponential , in the discrete - time model we deal with a power function .in particular , notice that yields a term proportional to .this gives rise to an additional term linear in , which is responsible for the difference between ( [ ddefd1l ] ) and ( [ defd1l ] ) , and hence for the term in ( [ dd ] ) .the presence of the time unit in ( [ dd ] ) suggests that the key to understanding the physical reasons for the difference between and lies in the dimensional analysis . in the continuous - time approach the transition rates are _ dimensional _ quantities that scale with time as [ t .therefore , multiplying them all by some positive constant corresponds to changing the time unit , and so the resulting diffusion coefficient will be equal to the product of and the original value of the diffusion coefficient , .similarly , .this explains why in the continuous - time approach we could safely assume .however , in the discrete formalism there is no such a simple relation between the time interval and the jump probabilities , which are _dimensionless_. suppose , for example , that at times we toss a coin .changing the frequency of tossing , or , will have no impact on the probability of the coin falling heads up .the proper way of taking the continuous - time limit in the discrete - time model is to assume that the jump probabilities are related to the continuous - time transition rates by a simple formula and taking the limit . in other words , to get the continuous - time limit , we need to apply the `` alpha - transformation '' to the transition probabilities with an infinitesimally small value of .equation ( [ dd ] ) shows that is actually a sum of two terms one that scales linearly under the `` -transformation '' ( ) , and the other one which scales quadratically ( ) . in the limit the continuous - time diffusion coefficient is thus of order , while is of order ) and can be neglected .consequently , as could be expected , the relative difference between diffusion coefficients calculated within each approach vanishes , as .a decrease of the diffusion coefficient in the discrete - time formalism can be also interpreted as a consequence of the fact that a discrete process tends to be `` less random '' that a continuous one .this is clearly seen in a limiting case of a one - dimensional lattice ( with lattice constant and time unit ) where all probabilities of jumping to the left vanish ( ) and all probabilities of jumping to the right are equal ( ) .a discrete process with this choice of probabilities is completely deterministic . at each time interval the diffusing particle with probability 1 hops to the right , hence the diffusion constant vanishes . on the other hand ,if we consider a continuous - time process with and , we can see that the motion of the particle is now by no means deterministic .although we know that on average there will be one jump to the right per unit time , we do not know when it will actually occur .this uncertainty introduces randomness to the process , and the corresponding diffusion coefficient equals .note finally that because the left - hand side of ( [ dd ] ) represents a diffusion coefficient , it can not be negative , and so this has several interesting consequences .first , if in a periodic system there is a stationary drift ( , then the continuous - time diffusion coefficient is bounded from below . that such a bound exists can indirectly imply bounds on other physical quantities , e.g. , the maximal force exerted by a molecular motor .second , _ suppose _ that ( [ dd ] ) , and hence ( [ d > v2 ] ) , is also valid in _ infinite _ ( aperiodic ) random systems .whenever the diffusion is sublinear , i.e , whenever , there is .equation ( [ dd ] ) would then imply that , or .this would mean that whenever diffusion is sublinear ( subdiffusion ) , the drift is also sublinear ( `` subdrift '' ) or vanishes altogether .the asymmetric hopping model with bond disorder can serve as an example of an infinite system where such relation is actually observed .suppose we want to calculate the drift velocity and the diffusion tensor * d * in a system with euclidean coordinates .if the system can be divided into a _ finite _ number of subsystems with constant transition rates between each two of them , our major results ( [ l0 ] ) ( [ h0 ] ) and ( [ dfpkt1 ] ) remain valid irrespective of the geometry of the system .such division is always possible for periodic systems with finite number of sites in the elementary cell .the matrix is the most `` sensitive '' to the geometry of the system under consideration .in calculating its explicit form one can use an equation similar to ( [ deflambdak ] ) , remembering , however , that and are now vectors in a -dimensional space .one should also take into account all possible transitions between sublattices , and this can be done by considering all possible jumps starting at any site belonging to some elementary cell and ending in the same cell or in one of its nearest - neighbor cells .the components of the velocity vector and the diffusion tensor are given by   { \bf d}_{\mu\sigma }      & = &       \lim_{t\to\infty }        \frac { \langle x_\mu x_\sigma \rangle            -\langle x_\mu \rangle \langle x_\sigma \rangle}{2 t }      =      \lim_{t\to\infty }         \frac{1}{2 t }      \left .\left (           -  \frac{\partial^2\tilde{p}(\vec{k},t ) } {                \partial k_\mu \partial k_\sigma }            +             \frac{\partial \tilde{p}(\vec{k},t)}{\partial k_\mu }             \frac{\partial \tilde{p}(\vec{k},t)}{\partial k_\sigma }          \right )        \right|_{\vec{k}=0}.\end{aligned}\ ] ] using ( [ l0 ] ) ( [ h0 ] ) and ( [ dfpkt1 ] ) we conclude that   \label{d - defd1l }  { \bf d}_{\mu\sigma } & = &      \left.       -\frac{1}{2 }       \frac{\partial^2\lambda_0 } { \partial k_\mu \partial k_\sigma }      \right|_{\vec{k}=0 } ,  \\[1ex ]  \label{d - ddefd1l }  { \bf d}^{\rm d}_{\mu\sigma } & = &       \left .-\frac{1}{2 }        \left [          \frac{\partial^2\lambda^{\rm d}_0}{\partial k_\mu\partial k_\sigma }          -           \frac{\partial\lambda^{\rm d}_0}{\partial k_\mu }           \frac{\partial\lambda^{\rm d}_0}{\partial k_\sigma }        \right ]       \right|_{\vec{k}=0},\end{aligned}\ ] ] where . actually , since is a generating function for cumulants of ( see ) , the first two of the above formulae are a natural consequence of ( [ fpkt0 ] ) and ( [ h0 ] ) .  just as in the one - dimensional caseany component of or can be expressed in terms of derivatives of the three lowest coefficients of the characteristic polynomial of the matrix at .in particular ,   \label{d - d }    { \bf d}_{\mu\sigma } & = &    \left .\frac {        c_0^{(\mu)(\sigma ) }      -2c_2v_\mu v_\sigma      - i\left (         c_1^{(\mu ) }   v_\sigma        + c_1^{(\sigma ) } v_\mu       \right )       } { 2c_1 }    \right|_{\vec{k}=0 } ,    \\[1ex ]  { \bf d}_{\mu\sigma}^{\rm d } & = &        { \bf d}_{\mu\sigma }       -\frac{1}{2}v_\mu v_\sigma .\end{aligned}\ ] ] where we used a short - hand notation .if in a one - dimensional system only transitions to the nearest - neighbors are allowed , and if the distance between nearest - neighbor sites is equal 1 , our formulae ( [ solj1c ] ) and ( [ sold1c ] ) can be simplified by noting that in this particular case  \label{c_0pp } c_0 '' & = & l^2\left (          \prod_{j=0}^{l-1}\gamma^{\rightarrow}_j         + \prod_{j=0}^{l-1}\gamma^{\leftarrow}_j        \right),\\[1ex ]  \label{c_1p } c_1 '  & = & 0,\end{aligned}\ ] ] where and are the transition rates to the right and left from site , respectively ; and .thus , to determine and we only need to compute the two terms of the characteristic polynomial of : and .although and could be , at least in principle , found by calculating , being the identity matrix , this could lead to serious computational overhead even for of order 10 .a more efficient method exploits the fact that are polynomials in and , and can be expressed as where , , , and if at least one of the following conditions is satisfied :    & \exists_{m }  \;\ ;  \gamma_m = \delta_m = 1 , &    \nonumber\\[1ex ]    & \exists_{m }  \;\ ;  \gamma_m = \delta_{m+1 } = 1 ; &  \end{aligned}\ ] ] otherwise .for example , for there is [ l=3 ]    & & + { \gamma^{\leftarrow}}_1{\gamma^{\leftarrow}}_2 + { \gamma^{\leftarrow}}_1{\gamma^{\rightarrow}}_2 + { \gamma^{\rightarrow}}_1{\gamma^{\rightarrow}}_2  \nonumber \\[1ex ]    & & + { \gamma^{\leftarrow}}_0{\gamma^{\leftarrow}}_1 + { \gamma^{\leftarrow}}_0{\gamma^{\rightarrow}}_1 + { \gamma^{\rightarrow}}_0{\gamma^{\rightarrow}}_1 , \nonumber \\[1ex ]   c_2 & = & { \gamma^{\leftarrow}}_0 + { \gamma^{\leftarrow}}_1 + { \gamma^{\leftarrow}}_2+{\gamma^{\rightarrow}}_0 + { \gamma^{\rightarrow}}_1 + { \gamma^{\rightarrow}}_2 . \end{aligned}\ ] ] although formally the sum in ( [ clpoly ] ) consists of terms , in practice only for a few of them . in particular , for a given value of the sum in ( [ clpoly ] ) consist of non - vanishing terms for and terms for .thus it should be possible to calculate algebraically for of order 20 , and numerically for of order at least 100 .it is worth noting that since can be computed as sums of positive values , by using ( [ clpoly ] ) rather than calculating the determinant of we can avoid large numerical errors which often appear when computing determinants of large matrices .it is not difficult to show that our general formulae ( [ solj1c ] ) and ( [ sold1c ] ) are consistent with eq .( [ kutner ] ) derived for a general case of 1d systems at equilibrium with nearest - neighbor transitions . to this endit suffices to notice that the equilibrium site occupation probabilities can be expressed in terms of the jump probabilities as on inserting it into ( [ kutner ] ) and then taking into account ( [ clpoly ] ) and the trivial condition one arrives at ( [ sold1c ] ) .another interesting consequence of ( [ c_0p ] ) - ( [ c_1p ] ) is that diffusion in systems with transitions to nearest - neighbours is always normal . to see it note that since , is always finite and thus no superdiffusion is possible .the other type of anomaly , subdiffusion , would require that .owing to ( [ d > v2 ] ) this would imply which , upon taking into account ( [ sold1c ] ) , would require .however , the explicit form of , see ( [ c_0pp ] ) , guarantees that except for a degenerated case where the diffusing particle is confined to a finite region of the lattice ( for some and ) .  one final remark .some authors , when considering a one - dimensional system with nearest - neighbor transitions , prefer to reduce the problem to diffusion on a finite ring and investigate the probability current rather than the drift velocity .the former is defined as and in the steady state does not depend on the site it is measured at . for single - particle systems and are related to each other through a simple formula  . in this caseour approach can thus be applied for calculating both and .applying our approach to a particle diffusing in a one - dimensional system with a lattice constant , time unit , and the period we immediately arrive at a well known result /2 . \end{aligned}\ ] ] for we find where .note also that the solution for can be easily derived using ( [ c1c2l3 ] ) .   consider a one - dimensional system of period with site energies , where is an integer , , and is a constant .such a pattern is know as a discrete sawtooth potential .we assume that the transition rate from a site to is given by and the rate of jumping from to reads where is the boltzmann factor and represents a bias due to an external force .note that with this choice of the transition rates , for the system satisfies the detailed balance condition .  using ( [ clpoly ] ) and some combinatoricsone can prove that for   [ n1sawtooth ]   where controls the anisotropy of the potential , , /(x^2 - 1)^2 |i_2-i_1| +   to corresponds to a single jump of one of two diffusing particles .for example , the transition rate from to is , and from to is 0 .consequently , if the lattice constant , the matrix takes on the following form , \ ] ]  the drift velocity and the diffusion coefficient can be found for arbitrary transition rates , but the results are _ very _ lengthy .however , in a particular case of the sawtooth potential the transition rates and are given by relatively simple formulae ( [ sright ] ) and ( [ sleft ] ) , respectively .then \nonumber\\   c_1 '  & = &  8i g^{-5 } ( g^8-r^8 ) ( r^8 + 4g^2 + 3 )  \nonumber\\  c_2  & = &  g^{-4 } [        37g^8 + 69g^6 + 58g^4 + 31g^2 + 9 \nonumber\\     & & + r^8(23g^6 + 49g^4 + 53g^2 + 19 )      \nonumber\\     & & + r^{16}(3g^4 + 8g^2 + 9 )                ] , \end{aligned}\ ] ] where and were defined below eq .( [ n1sawtooth ] ) .  just like in the previous example, , , and can be now determined using ( [ solj1c ] ) , ( [ sold1c ] ) , and ( [ dd ] ) .the properties of the velocity will be studied in detail elsewhere ( see ref . ) . here , in fig .[ fig2 ] we present the ratio of the diffusion coefficient calculated for a 2-particle system ( ) to that calculated for a single - particle system ( ) , for , as a function of the bias .this ratio measures the change of diffusivity due to interactions with the second particle . as could be expected , , i.e. the presence of the second particle decreases the diffusivity of the first one .this effect is not strong , however , for we find that . as in the previous example , for large bias , the ratio becomes independent of the potential anisotropy .we have developed an effective technique of calculating the drift velocity and the diffusion tensor in arbitrary periodic systems . a novel feature of our approach , as compared to previously proposed methods , consists in the fact that we need not examine in detail the steady - state properties of the system . in particular , we do not need to solve linear equations of relatively high order to find the stationary probability distribution of the random walker over different sublattices .instead we calculate and directly .the only quantity we really need in order to carry out our calculations is the matrix .its explicit form , however , can be found almost trivially ; all one needs to know are transition rates and distances from any site of an elementary cell to any other site in the same or adjacent cell .once the form of has been determined , one should calculate the three lowest terms of its characteristic polynomial .then and can be determined using our formulae ( [ d - j ] ) and ( [ d - d ] ) , respectively .our approach is very general and can be applied practically to any periodic system in an arbitrary space dimension .it is particularly well suited for calculations employing computer - algebra systems , e.g. , maple or reduce , or for numerical analysis . in this contextone should also note that the coefficients of the characteristic polynomial of can be _ always _ expressed as polynomials in , , with _ positive _ coefficients , and thus can be calculated numerically with extremely high accuracy . while this is not necessarily true in the case of their derivatives, we believe that our approach enables one to calculate and even for systems where the corresponding steady - state problem is numerically ill - conditioned .we were able to prove a general relation between the diffusion constants calculated using continuous - time and discrete - time approaches .our analysis shows that in _ any _ periodic system ( including , for example , lattice gases studied in refs . ) they differ by a term , where denotes the steady - state drift velocity along a given direction .since the limits and were shown to commute , at least in some simple models , we expect that this relation holds also for infinite random systems. it would be interesting to know whether such a simple and apparently universal relation can be applied in contexts other than those studied here .we also showed how our technique can be applied to a simple many - body problem . while , owing to mathematical complexity , we do not expect that in this way one will be able to calculate explicitly transport coefficients in non - equilibrium systems containing more than a few particles , our conclusion about the universal difference between diffusion coefficients calculated in continuous- and discrete - time models should still hold .this conjecture is based on the fact that diffusion of several particles can be interpreted as diffusion of a single particle in the corresponding multidimensional phase space , and for the latter problem the relation between and has been established rigorously in section [ sectotgen ] .thus , for the future work , two problems are of primary importance : the role of periodic boundary conditions in the limit of asymptotically infinite period , , and detailed analysis of possible applications of our method to many - body systems .

Abstract: we develop a practical method of computing the stationary drift velocity and the diffusion coefficient of a particle ( or a few particles ) in a periodic system with arbitrary transition rates . we solve this problem both in a physically relevant continuous - time approach as well as for models with discrete - time kinetics , which are often used in computer simulations . we show that both approaches yield the same value of the drift , but the difference between the diffusion coefficients obtained in each of them equals . generalization to spaces of arbitrary dimension and several applications of the method are also presented .


electromagnetic acoustic transduction is a technique used to generate and detect mechanical waves in magnetoelastic conductors for the purpose of nondestructive testing . the devices , called electromagnetic acoustic transducers ( emats ) or electromagnetic acoustic receivers ( emars ) , function by exploiting the coupling between electromagnetic and mechanical effects in a deformable continuum .the transducer mechanism ( as shown in the schematic in figure 1 ) consists of a current - carrying coil suspended on the specimen under test which is magnetized by a large static magnetic field .an ac current is passed through the coil which generates a time - varying magnetic field in the specimen .this , on interaction with the static magnetic field , leads to an electromagnetic body force that causes the generation of mechanical waves in the specimen . in the case of emars ,the existing mechanical waves in the specimen , on interaction with the static magnetic field , produce time - varying magnetic field in the vicinity .this generates an ac current in the current - carrying coil which can be measured to analyze the mechanical waves in the specimen .thus , it is very important to understand the exact coupling between mechanical and electromagnetic fields for an accurate mathematical modelling of the process .several orientations of the specimen and the coil for emats have been studied in the recent literature .much of the work ( including ) is based , among other literature , on the papers by ludwig et al . , ogi , and thompson on the theory and numerical simulation of emats .recently , shapoorabadi et al . have presented the governing equations of emats using a formulation involving the magnetic vector potential .the list of references is by no means exhaustive and there are many more papers and theses on this subject .however , the existing theory of emats is , in our view , incomplete in some respects .many theoretical models have been developed and experimental results been obtained for only a sinusoidal steady state of current and displacement ( such as ) .a large volume of literature assumes linear elastic deformations in the specimen and a linear coupling between the electromagnetic and mechanical phenomena ( such as , ) .this is , however , not the case in reality with the coupling being nonlinear , as has been presented in the classical texts by pao , eringen and maugin , and in the recent paper by maugin .moreover , with the development of soft polymeric electro- and magnetoelastic smart materials ( , ) in the recent years , one needs to take into account the possibility of finite deformations . in some materials ,heat generation caused by electric currents may lead to significant changes in magneto - elastic properties , hence these effects also need to be taken into account .  in this paper , we present the equations that govern wave propagation in a finitely deformed solid in the presence of electromagnetic fields . on the existing finite deformation , magnetization , and electric polarization , time dependent small increments in the electric ,magnetic , temperature , and deformation fields are allowed .we provide a general constitutive formulation based on a free energy function that depends on the initial deformation , underlying electromagnetic fields , and temperature .this is based on a generalization of the constitutive formulations of electroelasticity and magnetoelasticity provided by dorfmann and ogden . the same formulation of magnetoelasticity has been used by destrade and ogden , and author and co - workers to study some wave propagation problems in magnetoelasticity .  in section 2 ,we recapitulate the basic governing equations of electromagnetism in continua and the laws of thermodynamics as given by pao and eringen and maugin .these are used to derive constitutive relations from a free energy function .we then allow for small time - dependent increments in section 4 to study wave propagation .several ` moduli ' tensors are also introduced in this section to quantify the couplings in elastic , electric , magnetic , and temperature fields .it is expected that the general equations provided here will be solved using numerical schemes such as fem for specific boundary problems related to emats .the undeformed stress - free _ reference configuration _ of a continuous elastic body is denoted by and its boundary by .let , the _ current configuration _ , be the region occupied by the body at time and its boundary .the material points of body are identified by the position vector in which becomes the position vector in .the time - dependent deformation ( or motion ) of the body is described by an invertible mapping that maps points from to points in such that .the function and its inverse are assumed to be sufficiently regular in space and time .the velocity and acceleration of a material particle at are defined by where the subscript following a comma denotes the material time derivative . in this paper; grad , div , and curl denote the standard differential operators in the reference configuration while grad , div , and curl denote the same in the current configuration .the _ deformation gradient tensor _ is defined as and its determinant is denoted by , with .associated with is the right cauchy - green tensor . for an incompressible material the constraint needs to be satisfied .the well - known maxwell s equations governing electromagnetic fields in a deformable continua ( the region in figure 1 ) are given by the dipole - current circuit model of pao as where is the magnetic induction vector , is the electric field vector , and are , respectively , the magnetization and electric polarization of the continuum ; is the electric current density , is the electric charge density , and are the electrical pemittivity and magnetic permeability of the vacuum respectively .we have used the following field relations in the above equations   and are the regions occupied by the continuum material and a wire of the current - carrying coil , respectively .the corresponding boundaries are denoted by and . ]all the electromagnetic quantities above are defined in the current ( eulerian ) configuration .they can also be expressed in the reference configuration ( lagrangian form ) given by using which the maxwell s equations can be written in lagrangian form as   in the current - carrying coil of rigid material ( region in figure 1 ) , the governing equations for the electromagnetic fields are where the constants and are the relative electric permittivity and the relative magnetic permeability of the current - carrying coil , respectively .  outside the material , in a vacuum ,the governing equations for electromagnetic fields are where we denote a physical quantity in vacuum by a superscript * .  at the boundary ,the following conditions need to be satisfied ( see , for example , ) where is the unit normal to the boundary , is the particle velocity , is the value of at the boundary , is the surface electric charge density , and is lagrangian description of the surface current density .we note that is not defined outside the material and the values of and calculated on the boundary are used in the above equations .the boundary conditions are simpler at and given by    balance of linear momentum , in the absence of a mechanical body force , is given by where is the cauchy stress tensor , is the mass density , and the electromagnetic body force as given in is .\ ] ]  balance of angular momentum gives where is the third order permutation tensor with components and and is the electromagnetic body couple vector .the above balance equations can be written in lagrangian form using the nominal stress tensor as where and are lagrangian counterparts of the corresponding vectors and are given by ^{\mathrm t } ( j^{-1 } \mathbf{fp}_l ) \nonumber \\+ \mathbf{f}^{- \mathrm t } \left [ \mbox{grad}(j^{-1 } \mathbf{fb}_l ) \right]^{\mathrm t } ( \mathbf{f}^{-\mathrm t } \mathbf{m}_l ) + \frac{\partial}{\partial t } \left [ j^{-2 } \left(\mathbf{fp}_l\right ) \times \left ( \mathbf{fb}_l \right ) \right ] \nonumber\\  + j^{-1 } \mbox{div } \left [ j^{-1 } \mathbf{v } \otimes \left\ { ( \mathbf{fp}_l ) \times ( \mathbf{fb}_l ) \right\ } \right ] ,  \end{aligned}\ ] ]   on any part of the boundary where the traction is prescribed , the boundary condition may be given as where is the lagrangian representation of the traction force .first law of thermodynamics gives the balance of energy as where is the internal energy , is the heat flux at the surface , is the volumetric heat generation , the symbol denotes a scalar product between two second order tensors given in component form as , and is the electromagnetic power ( see , for example , pao ) given by where for a dynamic problem we have defined the effective field variables as   let be the absolute temperature , then we can rewrite as here is the specific heat capacity . on defining the pullback versions of the physical quantities the above equation can be written in lagrangian form as here , the lagrangian form of electromagnetic power is given as \cdot \left ( \mathbf{f}^{-\mathrm t } \mathbf{e}_{el } \right ) \nonumber \\ - j \mathbf{f}^{-\mathrm t } \mathbf{m}_{el } \cdot \left [ \frac{\partial}{\partial t } \left ( j^{-1 } \mathbf{fb}_l \right ) + \mbox{grad } \left ( j^{-1 } \mathbf{fb}_l\right ) \mathbf{v }  \right].\end{aligned}\ ] ]  if is the entropy density , then the second law of thermodynamics is given by the clausius duhem inequality as   substituting equation into and following the standard coleman - noll procedure ( see , for e.g. , ) , we arrive at the constitutive relations where is the free energy per unit volume related to ( see , for example , for details ) as   we note that in the case of an incompressible material , we have an additional constraint and the constitutive law for stress is modified to  being the lagrange multiplier associated with the constraint of incompressibility .we assume the constitutive laws governing heat flow and electric current flow to be given by the fourier s law and the ohm s law , respectively , as where and are positive - definite symmetric second order tensors quantifying the thermal conductivity and the electrical conductivity , respectively .we require tensorial values for the conductivities since in general for anisotropic materials , the conductivity depends on the direction of heat flow or electric current . for isotropic materials ,these assume the values and .the above equations can be written in lagrangian form as the initial motion and underlying electromagnetic fields , we consider an incremental mechanical motion , and increments in electromagnetic fields which are denoted by a superposed dot .we emphasize here the departure from convention in using a superposed dot to denote an increment rather than a time - derivative for the sake of brevity .the incremented forms of the lagrangian maxwell s equations and are given as  = { \accentset{\mbox{\large\bfseries .}}{\rho}}_{\mathrm e } + \mbox{div}\ , { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_l,\ ] ] - \varepsilon_0 \mbox{curl }  \left [ { \accentset{\mbox{\large\bfseries .}}{\mathbf{v } } } \times ( j \mathbf{c}^{-1 } \mathbf{e}_l ) \right . \nonumber \\ \left. + \mathbf{v } \times \left\ { j ( \mbox{div } \ , \mathbf{u } ) \mathbf{c}^{-1 } \mathbf{e}_l + j \mathbf{c}^{-1 } { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_l - j \mathbf{f}^{-1 } ( \mathbf{l+l}^{\mathrm t } ) \mathbf{f}^{-\mathrm t } \mathbf{e}_l  \right\ } \right ]  \nonumber \\ - \varepsilon_0 \left [ j \left\ {  j ( \mbox{div } \ , \mathbf{u } ) \mathbf{c}^{-1 } \mathbf{e}_l + j \mathbf{c}^{-1 } { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_l - j \mathbf{f}^{-1 } ( \mathbf{l+l}^{\mathrm t } ) \mathbf{f}^{-\mathrm t } \mathbf{e}_l \right\ } \right]_{,t }  \nonumber \\ = { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_{l ,t } + \mbox{curl } \ , { \accentset{\mbox{\large\bfseries .}}{\mathbf{m}}}_{el } & + { \accentset{\mbox{\large\bfseries .}}{\mathbf{j}}}_{\mathrm e}.\end{aligned}\ ] ]  the incremented lagrangian quantities are ` pushed forward ' to the eulerian configuration and denoted by a subscript ` 0 ' after ` ' or ` ' .the push - forward relations for the incremented fields are given ( similar to ) as using which we can write the eulerian form of the above incremental equations as { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } - { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0,t } , \ ] ]  - \varepsilon_0 \mbox{curl } \left ( \mathbf{u}_{,t } \times \mathbf{e } + \mathbf{v } \times \hat{\mathbf{e } } \right ) -  \varepsilon_0 \hat{\mathbf{e}}_{,t } & \nonumber \\  = \mbox{curl } \ , { \accentset{\mbox{\large\bfseries .}}{\mathbf{m}}}_{el0 }  +  { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_{l0,t } + \left [ ( \mbox{div}\ , \mathbf{v } ) \mathbf{i } - \boldsymbol{\gamma } \right ] { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_{l0 } +  & { \accentset{\mbox{\large\bfseries .}}{\mathbf{j}}}_{\mathrm e0 } , \label{ch2 up bal 4}\end{aligned}\ ] ] where   the incremented momentum and angular momentum balance equations are given as  where the increments in electromagnetic body force and moment are given by  & _ e = j^-1 \ { - ( ) _ e ^- t _l + _ e ^- t _ l - _ e ^t ^- t _l + _ e ^-t _ l } &  ^{\mathrm t } ( \mathbf{fp}_l )  \nonumber\\ + j^{-1 } \mathbf{f}^{- \mathrm t } \left [ \mbox{grad } ( - \mathbf{l}^{\mathrm t } \mathbf{f}^{- \mathrm t } \mathbf{e}_l + \mathbf{f}^{- \mathrm t } { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_l ) \right]^{\mathrm t } ( \mathbf{fp}_l ) \nonumber\ ] ]  ^{\mathrm t } \left [ - ( \mbox{div}\ , \mathbf{u } ) \mathbf{fp}_l + \mathbf{lfp}_l + \mathbf{f } { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_l \right]\nonumber\ ] ]    ^{\mathrm t } ( \mathbf{f}^{-\mathrm t } \mathbf{m}_l ) \nonumber\ ] ]  ^{\mathrm t } ( \mathbf{f}^{-\mathrm t } \mathbf{m}_l ) \nonumber\ ] ]  ^{\mathrm t } \left ( - \mathbf{l}^{\mathrm t } \mathbf{f}^{- \mathrm t } \mathbf{m}_l + \mathbf{f}^{- \mathrm t } { \accentset{\mbox{\large\bfseries .}}{\mathbf{m}}}_l \right ) \nonumber\ ] ]      - j^{-1 } ( \mbox{div}\ , \mathbf{u } ) \ ,\mbox{div } \left [ j^{-1 }  \mathbf{v } \otimes \left\ { ( \mathbf{fp}_l \times ( \mathbf{fb}_l ) \right\ } \right ] \nonumber\ ] ]    ,\ ] ]  & _ e = - j^-1 ( ) \ { ( _ l ) ( ^-t _ l ) + ( ^- t _ el ) ( _ l ) } &      when updated to eulerian form , the balance equations and become  where and are the push - forward forms of the incremental body force and moment , respectively , and are given by  & _ e0 = - ( ) _ e + _ e0 - _ e ^t + _ e _ l0 - ^t ( ) ^t &  ^{\mathrm t } \mathbf{p } + \left ( \mbox{grad}\ , \mathbf{e } \right)^{\mathrm t } \left [ - ( \mbox{div}\ , \mathbf{u } ) \mathbf{p } + \mathbf{lp } + { \accentset{\mbox{\large\bfseries .}}{\mathbf{p}}}_{l0 }  \right ] \nonumber\ ] ]    ^{\mathrm t } \mathbf{m } + \left ( \mbox{grad}\ , \mathbf{b } \right)^{\mathrm t } \left (  - \mathbf{l}^{\mathrm t } \mathbf{m } + { \accentset{\mbox{\large\bfseries .}}{\mathbf{m}}}_{l0 } \right ) \nonumber\ ] ]   \nonumber\ ] ]   + \mbox{div } \left [  \left ( { \accentset{\mbox{\large\bfseries .}}{\mathbf{v } } } - \mathbf{lv } \right ) \otimes ( \mathbf{p } \times \mathbf{b } ) - ( \mbox{div}\ , \mathbf{u } )  \mathbf{v } \otimes ( \mathbf{p } \times \mathbf{b } )  \right .\nonumber\ ] ]  ,\ ] ]  & _ e0 = - ( ) ( + _ e ) + ( + _ l0 ) + ( - ^t + _ l0 ) &    the heat equation can be incremented to give =  { \accentset{\mbox{\large\bfseries .}}{q}}_l + { \accentset{\mbox{\large\bfseries .}}{w}}_{e } + { \accentset{\mbox{\large\bfseries .}}{\mathbf{t } } } \colon \mbox{grad } ( \mathbf{fv } )  \nonumber \\ + \mathbf{t } \colon \mbox{grad } ( \mathbf{lfv } + \mathbf{f } { \accentset{\mbox{\large\bfseries .}}{\mathbf{v}}}),\end{aligned}\ ] ] which when updated to the eulerian configuration becomes = { \accentset{\mbox{\large\bfseries .}}{q}}_{l0 } + { \accentset{\mbox{\large\bfseries .}}{w}}_{e0 } + { \accentset{\mbox{\large\bfseries.}}{\mathbf{t}}}_0 \colon \mbox{grad}\ , \mathbf{v } + \boldsymbol{\tau } \colon \mbox{grad}\ , { \accentset{\mbox{\large\bfseries .}}{\mathbf{v}}}.\ ] ] we have used the push - forward relations , and to effect the above transformation .the increment in the electromagnetic power is given by  & _ e = ( _ el + _ el ) ( ^-t _ el ) + ( _ el ) ( - ^t ^-t _ el + ^-t _ el ) &   \nonumber\ ] ]    \cdot \left ( - \mathbf{l}^{\mathrm t } \mathbf{f}^{-\mathrm t } \mathbf{e}_{el } + \mathbf{f}^{-\mathrm t } \cdot { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el } \right ) \nonumber\ ] ]     - j \mathbf{f}^{-\mathrm t } \mathbf{m}_{el } \cdot \left [ \frac{\partial}{\partial t } j^{-1 } \left ( -(\mbox{div}\ , \mathbf{u } ) \mathbf{fb}_l + \mathbf{lfb}_l + \mathbf{f } { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_l  \right ) \right . \nonumber\ ] ]  .\ ] ]  this is given in eulerian form as  & _ e0 = ( _ e + _el0 ) _ e + _ e ( - ^t _ e + _ el0 ) &   \cdot \mathbf{e}_e + \rho \frac{d}{dt } \left ( \frac{\mathbf{p}}{\rho } \right ) \cdot \left ( -\mathbf{l}^{\mathrm t } \mathbf{e}_e + { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } \right ) \nonumber\ ] ]    & & . +( ) ( - ) ] . &   on incrementing the constitutive equations , we obtain and where we have defined the moduli tensors as   here is a fourth - order tensor , are third - order tensors , are second - order tensors , and are first - order tensors ( or vectors ) . to put them in perspective , for a problem involving simpler quasimagnetostatic case as presented in and , only , and are required .products in and are defined in component form as and the following relations hold   on updating ( push - forward to eulerian configuration ) the incremented constitutive equations and , we obtain  and with and the updated moduli tensors defined in component form as   incrementing and updating the constitutive equations , we get ,\ ] ] and .\ ] ]  on substituting the incremented updated constitutive equations , , , , and into the incremented updated balance equations , , , , and , we obtain &  = { \accentset{\mbox{\large\bfseries .}}{\rho}}_{e0 } \nonumber \\  - \mbox{div } & \left ( \boldsymbol{\mathcal b}_0^{\mathrm t } \mathbf{l } + \boldsymbol{\mathcal g}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } + \boldsymbol{\mathcal h}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } + \boldsymbol{\mathcal i}_0 { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 } \right),\end{aligned}\ ] ] - \varepsilon_0 \mbox{curl } \left ( \mathbf{u}_{,t } \times \mathbf{e } + \mathbf{v } \times \hat{\mathbf{e } } \right ) - \varepsilon_0 \hat{\mathbf{e}}_{,t } \nonumber \\ = - \mbox{curl } \left ( \boldsymbol{\mathcal c}_0^{\mathrm t } \mathbf{l } + \boldsymbol{\mathcal h}_0^{\mathrm t } { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } + \boldsymbol{\mathcal m}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } + \boldsymbol{\mathcal n}_0 { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 }\right )  + { \accentset{\mbox{\large\bfseries .}}{\mathbf{j}}}_{\mathrm e 0 }  \nonumber \\ - \left [ ( \mbox{div}\ , \mathbf{u } ) \mathbf{i } - \boldsymbol{\gamma } \right ] \left ( \boldsymbol{\mathcal b}_0^{\mathrm t } \mathbf{l } + \boldsymbol{\mathcal g}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } + \boldsymbol{\mathcal h}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } + \boldsymbol{\mathcal i}_0 { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 }  \right ) \nonumber \\  - \left (  \boldsymbol{\mathcal b}_0^{\mathrm t } \mathbf{l } + \boldsymbol{\mathcal g}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } + \boldsymbol{\mathcal h}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } + \boldsymbol{\mathcal i}_0 { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 }  \right)_{,t},\end{aligned}\ ] ]   \nonumber\\ = -\rho c_p \frac{\partial}{\partial t } \left [ { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 } - ( \mbox{div}\ , \mathbf{u } ) \vartheta \right ]  + { \accentset{\mbox{\large\bfseries .}}{q}}_{l0 } + { \accentset{\mbox{\large\bfseries .}}{w}}_{e0 }  + \boldsymbol{\tau } \colon \mbox{grad}\ , \mathbf{u}_{,t } \nonumber\\ + \left ( \boldsymbol{\mathcal a}_0 \mathbf{l } + \boldsymbol{\mathcal b}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{e}}}_{el0 } + \boldsymbol{\mathcal c}_0 { \accentset{\mbox{\large\bfseries .}}{\mathbf{b}}}_{l0 } + \boldsymbol{\mathcal d}_0 { \accentset{\mbox{\large\bfseries .}}{\vartheta}}_{l0 }  \right ) \colon \mbox{grad}\ , \mathbf{v},\end{aligned}\ ] ] along with . the governing equations of the incremental fields in the region are while in vacuum , we have    at the boundary of the continuum and vacuum , the incremental form of the boundary conditions is given as    we recollect the nanson s formula connecting reference and current area elements and , where is the unit outward normal to .the surface current and electric charge densities in the two configurations are related by and . using these relations and the relations , we rewrite the above incremental boundary conditions in the current configuration on as   where we have defined and .  at the boundary , incremental form of the boundary conditionsis given as   thus , a generic emat problem requires the solution of equations and in , equations in , and equations in vacuum using the boundary conditions at and at .this paper reviews the existing theory of electromagnetic interactions with a solid continuum .in particular , we analyze the problem of wave propagation in a finitely deformed elastic solid with an underlying electric , magnetic , and temperature field . unlike the existing literature , we generalize a nonlinear theory of elasticity to include electromagnetic effects . on a finite deformation , magnetic , and electric field ,the equations are linearized to consider wave propagation and several moduli tensors are introduced .in addition to filling a gap in literature , it is expected that this work will lead to an accurate mathematical modelling of the ` electromagnetic acoustic transducers ' in particular .this work was undertaken when the author was at the university of glasgow supported by a university postgraduate scholarship and a uk ors scholarship .thanks are also extended to prof .ray w. ogden , frs of the university of glasgow , uk for his helpful guidance during the initial phase of this work .

Abstract: in this paper , we present the general governing equations of electrodynamics and continuum mechanics that need to be considered while mathematically modelling the behaviour of electromagnetic acoustic transducers ( emats ) . we consider the existence of finite deformations for soft materials and the possibility of electric currents , temperature gradients , and internal heat generation due to dissipation . starting with maxwell s equations of electromagnetism and balance laws of nonlinear elasticity , we present the governing equations and boundary conditions in incremental form in order to solve wave propagation problems of boundary value type .   * keywords * : emat , nondestructive testing , nonlinear magnetoelasticity , nonlinear electroelasticity , electromechanical coupling , wave propagation


a frequent object of interest is the ceteris paribus effect of on when observed is an individual choice variable partly determined by preferences or technology .panel data holds out the hope of controlling for individual preferences or technology by using multiple observations for a single economic agent .this hope is particularly difficult to realize with discrete or other nonseparable models and/or multidimensional individual effects .these models are , by nature , not additively separable in unobserved individual effects , making them challenging to identify and estimate .a fundamental idea for using panel data to identify the ceteris paribus effect of on is to use changes in over time . in order for changes over time in to correspond to ceteris paribus effects , the distribution of variables other than must not vary over time .this restriction is like time being randomly assigned or `` time is an instrument . '' in this paper we consider identification via such time homogeneity conditions .they are also the basis of many previous panel results , including chamberlain ( 1982 ) , manski ( 1987 ) , and honore ( 1992 ) .recently time homogeneity has been used as the basis for identification and estimation of nonseparable models by chernozhukov , fernandez - val , hahn , newey ( 2013 ) , evdokimov ( 2010 ) , graham and powell ( 2012 ) , and hoderlein and white ( 2012 ) . because economic data often exhibits drift over time , we also allow for some time effects , while maintaining underlying time homogeneity conditions .  in this paperwe give identification and estimation results for quantile effects with time homogeneity and continuous regressors .the effects of interest are derivatives of quantile structural functions of the model .we find that these derivatives are identified with two time periods for stayers " , i.e. conditional on being equal in two time periods .time homogeneity is too strong for many econometric applications where time trends are evident in the data .we weaken homogeneity by allowing for location and scale time effects .allowing for such time effects makes identification and estimation more complicated but more widely applicable .we also give analogous results for conditional mean effects under weaker identification conditions than previously .quantile identification under time homogeneity is based on differences of quantiles .it is also interesting to consider whether quantiles of differences can help identify effects of interest .we do not find that time homogeneity alone can lead to identification from quantiles of differences .we do give quantile difference identification results that restrict the distribution of individual effects conditional on , similarly to chamberlain ( 1980 ) , altonji and matzkin ( 2005 ) , and bester and hansen ( 2009 ) .in our opinion these added restrictions make quantiles of differences less appealing .we therefore focus for the rest of the paper , including the application , on differences of quantiles .to illustrate we provide an application to engel curve estimation .the engel curve describes how demand changes with expenditure .we use data from the 2007 and 2009 waves of the panel study of income dynamics ( psid ) .endogeneity in the estimation of engel curves arises because the decision to consume a commodity may occur simultaneously with the allocation of income between consumption and savings .in contrast with the previous cross sectional literature , we do not rely on a two - stage budgeting argument that justifies the use of labor income as an instrument for expenditure . instead , we assume that the engel curve relationships are time homogeneous up to location and scale time effects , which leads to identification of structural effects from panel data .an alternative approach to identification in panel data is to impose restrictions on the conditional distribution of the individual effect given .this approach leads to nonparametric generalizations of chamberlain s ( 1980 ) correlated random effects model .as shown by chamberlain ( 1984 ) , altonji and matzkin ( 2005 ) , bester and hansen ( 2009 ) , and others , this kind of condition leads to identification of various effects .in particular , altonji and matzkin ( 2005 ) show identification of an average derivative conditional on the regressor equal to a specific value , an effect they call the local average response ( lar ) . in this paperwe take a different approach , preferring to impose time homogeneity rather than restrict the relationship between observed regressors and unobserved individual effects .we refer to hsiao ( 2003 ) for a broader perspective of panel data models .section 2 describes the model and gives an average derivative result .section 3 gives the quantile identification result that follows from time homogeneity .section 4 considers how quantiles of differences can be used to identify the effect of on .section 5 explains how we allow for time effects .estimation and inference are briefly discussed in section 6 , and the empirical example is given in section 7 .the appendix contains the proofs of the main results .the data consist of observations on and ^{\prime } ] over the whole population using the movers with .we identify an average slope over a smaller population for a fully nonlinear , nonparametric specification .the following result makes the previous derivation precise , including conditions for differentiating under integrals .[ theorem : mean ] suppose that assumptions [ assum : model ] and ass : distrinv are satisfied , , and that ( where ) resp .the conditional density of given are continuously differentiable in resp . for fixed . given ,suppose that for some , then ( [ eq : resultmean1 ] ) and ( [ eq : resultmean2 ] ) hold true .this result has slightly weaker conditions than that of hoderlein and white ( 2012 ) .here we drop their assumption that is independent of conditional on .the result given here allows for to be correlated with as long as the marginal distribution of conditional on does not vary with we maintain these weaker conditions as we consider identification of quantile effects in the next section .turning now to the identification of the quantile effects given above , let denote the conditional quantile of conditional on .it will be the solution to pair  ] , and .  in this modelthe effects of interest vary with time .we consider the time - averaged conditional mean effect : + \bar \sigma(x ) e[\partial_x \phi(x , u_t ) \mid x_1 = x_2 = x],\ ] ] and the time - averaged conditional quantile effect : ,\ ] ] where /2, ] and is the conditional quantile of given   the conditional mean effect is related to the time - averaged casf : ,\ ] ] through + \bar \sigma(x ) e[\partial_x \phi(x , u_t ) \mid x_1 = x_2 = x],\ ] ] under the conditions that permit interchanging the derivative and expectation .similarly , the conditional quantile effect is related to the time - averaged cqsf , , that gives the -quantile of conditional on , through .\ ] ]  let , ]   and  are continuously differentiable in and the conditional density of given  is continuously differentiable in . given , suppose that for some , then , , ] .the identification of the conditional mean effect does not require any normalization . note that we now have just one equation for identifying the conditional mean effect .we find a similar result for quantiles .[ th : cqsf_te ] suppose that assumptions [ ass : distrinv ] , ass : regquantile , and [ ass : model2 ] are satisfied , and are continuously differentiable in and  is continuously differentiable in , and the conditional density of given is positive on the interior of its support .then for all  exists and is continuously differentiable at such that \\ = [ \partial_{x_1 } q_1(\tau \mid x , x ) - \partial_{x_1 } q_2(\tau \mid x , x)/\sigma(x)]/2 + [ \partial_{x_2 } q_2(\tau \mid x , x ) - \sigma(x ) \partial_{x_2 } q_1(\tau \mid x , x)]/2,\end{gathered}\ ] ] /[q_1(\tau_1 \mid x , x ) - q_1(\tau_2 \mid x , x)], ] .  as in theorem[ th : casf_te ] , the time effects are identified up to location and scale normalizations , whereas the conditional quantile effects are identified without any normalization . here , however , instead of conditional mean and variance restrictions , we use quantile restrictions to identify the time effects up to the normalizations .these effects are over identified by many possible quantiles  and .for example , for  and the scale is identified by a ratio of conditional interdecile ranges across time and the location is identified by a difference of conditional medians across time .we note that graham and powell ( 2012 ) allowed for random time effects in location and slope rather than location and scale effects that could depend on .the conditional mean and quantile effects of interest are identified by special cases of the functionals : and respectively , where and are known smooth functions , is a region of regressor values of interest , and is a region of regressor values and quantiles of interest .we consider the estimators of and based on the plug - in rule : and where , , and are nonparametric series estimators of , , and .  to describe the series estimators , let denote a vector of approximating functions , such as tensor products of univariate polynomial or spline series terms of the components of , and let .then , where denotes any generalized inverse inverse of the matrix ; ^ 2\ ] ] is a series version of the ( kernel ) conditional variance estimator of fan and yao ( 1998 ) ; and where is the koenker and bassett ( 1978 ) quantile regression estimator [y_{it } -\boldsymbol{p}_i^{\prime } b].\ ] ]  following praestgaard and wellner ( 1993 ) , hahn ( 1995 ) , and chamberlain and imbens ( 2003 ) , we use weighted bootstrap for inference . to describe this method ,let be an i.i.d .sequence of nonnegative random variables from a distribution with mean and variance equal to one ( e.g. , the standard exponential distribution ) , independent of the data .the weighted bootstrap uses the components of as random sampling weights in the construction of the bootstrap version of the series estimators .thus , the bootstrap versions of and are and where is the bootstrap version of  ^ 2\ ] ] is the bootstrap version of and is the bootstrap version of , with [y_{it } -\boldsymbol{p}_i^{\prime } b].\ ] ]  belloni , chernozhukov , chetverikov , and kato ( 2013 ) and chernozhukov , lee , and rosen ( 2013 ) developed functional distributional theory and bootstrap consistency for series estimators of functionals of the conditional mean function , and belloni , chernozhukov , and fernandez - val ( 2011 ) developed similar theory for series estimators of functionals of the conditional quantile function .we can use these results to construct analytical or bootstrap confidence bands for the effects that have uniform asymptotic coverage over regressor values and quantiles .for example , the end - point functions of a confidence band for have the form where and are consistent estimators of the asymptotic variance function of  ] for conditional on the data .( ii ) compute a bootstrap estimate of such as the bootstrap standard deviation : ^ 2/(b-1)\}^{1/2} ] for , where is the -sample quantile of .( 3 ) compute realizations of the bootstrap version of the maximal t - statistic for ( iii ) form a -confidence band for using ( [ eq : bands ] ) setting to the -sample quantile of .the validity of algorithm [ alg : bands ] follows from the results in belloni , chernozhukov , and fernandez - val ( 2011 ) and the delta method .we can construct uniform bands for the conditional mean effects with a similar algorithm replacing by , adjusting all the steps accordingly , and relying on the results of belloni , chernozhukov , chetverikov , and kato ( 2013 ) and chernozhukov , lee , and rosen ( 2013 ) .in this section , we illustrate the results with an empirical application on estimation of engel curves with panel data .the engel curve relationship describes how a household s demand for a commodity changes as the household s expenditure increases .lewbel ( 2006 ) provides a recent survey of the extensive literature on engel curve estimation .we use data from the 2007 and 2009 waves of the panel study of income dynamics ( psid ) .since 2005 , the psid gathers information on household expenditure for different categories of commodities .the psid does not collect information on total expenditure .we construct the total expenditure on nondurable goods and services by adding all the expenses in housing , utilities , phone , child care , food at home , food out from home , car , transportation , schooling , clothing , leisure , and health .we exclude expenses in mortgage , home insurance , car insurance , and health insurance because these categories have many missing values .our sample contains households formed by couples without children , where the head of the household was 20 to 65 year - old in 2009 , and that provided information about all the relevant categories of expenditure in 2007 and 2009 .we focus on the commodities food at home and leisure for comparability with recent studies ( e.g. , blundell , chen , and kristensen ( 2007 ) , chen and pouzo ( 2009 , 2013 ) , and imbens and newey ( 2009 ) ) .the expenditure share on a commodity is constructed by dividing the expenditure in this commodity by the total expenditure in nondurable goods and services .endogeneity in the estimation of engel curves arises because the decision to consume a commodity may occur simultaneously with the allocation of income between consumption and savings .in contrast with the previous cross sectional literature , we do not rely on a two - stage budgeting argument that justifies the use of labor income as an instrument for expenditure . instead , we assume that the engel curve relationships are time homogeneous up to location and scale time effects , and rely on the availability of panel data . specifically , we estimate where is the observed share of total expenditure on food at home or leisure , is the logarithm of total expenditure in dollars of 2005 , and are location and scale time effects , is a vector of unobserved household heterogeneity that satisfies time homogeneity and captures both differences in preferences and idiosyncratic household shocks , corresponds to 2007 , and corresponds to 2009. the inclusion of time effects might be important to account for temporal changes in preferences and relative prices across commodities .for example , the price index of nondurable goods increased by between 2007 and 2009 , whereas the price indexes for food and leisure increased by and during the same period .we allow these time effects to vary with total expenditure , what gives flexibility to the model .this model does put some restrictions on interactions between prices and heterogeneity , implying that price changes only shift the location and scale of the distribution of demand .table [ table : desc_stat ] reports descriptive statistics for the variables used in the analysis .both total expenditure and expenditure shares display within and between household variation , with means and standard deviations that remain stable between 2007 and 2009 .the low percentage of within variation in expenditure indicates that there might be a substantial number of households with zero or little change in expenditure across years .figure [ fig : density ] plots histogram and kernel estimates of the density of the change in expenditure between 2007 and 2009 .the kernel estimates are obtained using a gaussian kernel with silverman s rule of thumb for the bandwidth .the estimates confirm that there is a high density of households with zero change in expenditure .our methods will identify mean and quantile effects for these households with .lccccccc & & & + & mean & std .& within ( % ) & mean & std . dev .& mean & std .+ log expenditure & 10.13 & 0.56 & 21 & 10.19 & 0.57 & 10.07 & 0.55 + food share & 0.19 & 0.10 & 28 & 0.19 & 0.10 & 0.20 & 0.10 + leisure share & 0.10 & 0.10 & 26 & 0.10 & 0.09 & 0.10 & 0.10 +  we estimate the location time effects , scale time effects , conditional mean effects , and conditional quantile effects using sample analogs of the expressions in theorems [ th : casf_te ] and [ th : cqsf_te ] .in particular , we estimate the conditional expectation , variance , and quantile functions by the nonparametric series methods described in section sec : estimation .we consider two different specifications for the series basis in all the estimators : a quadratic orthogonal polynomial and a cubic b - spline with three knots at the minimum , median and maximum of total log - expenditure in the data set . both specifications are additively separable in the total log - expenditures of 2007 and 2009 .we also compute cross sectional estimates that do not account for endogeneity .they are obtained by averaging the nonparametric series estimates in 2007 and 2009 that use the same specification of series basis as the panel estimates but only condition on contemporaneous expenditure . for inference, we construct 90% confidence bands around the estimates by weighted bootstrap with exponential weights and 499 repetitions .these bands are uniform in that they cover the entire function of interest with 90% probability asymptotically .figures [ fig : te2_asf_food_in ] and [ fig : te2_asf_leisure ] show the estimates and confidence bands for the time effects functions : based on theorem [ th : cqsf_te ] with  and , where is the interval of values between the 0.10 and 0.90 sample quantiles of log - total expenditure .we find that we can not reject the hypothesis that there are no location and scale time effects for food at home , whereas we find significant evidence of time effects for leisure with both series specifications . in resultsnot reported , we find similar estimates and confidence bands for the time effects functions based on conditional means and variances using theorem [ th : casf_te ] .  figure [ fig : dqsf ] plots the estimates and confidence bands for the time - averaged conditional quantile effects or cqsf derivates integrated over the values of : \right\ } \mu(dx),\ ] ] for where is the empirical measure of log - expenditure , and . ] does not depend on by assumption [ ass : distrinv ] .the proof of the third result is similar to the proof of theorem theorem : mean replacing by .in particular , is continuously differentiable , , in a neighborhood of and for  where and .subtracting and using assumption [ ass : distrinv ] , evaluating at gives a similar argument yields the result follows by averaging the previous expressions and using that does not depend on by assumption [ ass : distrinv ] .let and let where .the first result follows by a similar argument to the proof of theorem theorem : quant .in particular , by lemma [ lemm : quantilemain ] and ( eq : dommain_te ) , is continuously differentiable with derivatives thus , is continuously differentiable with positive derivative with respect to . by the implicit function theorem , there is a unique solution to  which is differentiable with partial derivatives where is the partial derivative w.r.t .the components of in , .evaluating at , and plugging in the expressions for the derivatives yields ^{-1 } \int 1(\phi ( x , u)\leq q(\tau , x ) ) \partial_{x_s } f(u|{\boldsymbol{x } } ) du , \end{gathered}\ ] ] where we use that by invariance of quantiles to monotone transformations , and /\sigma_t(x ) |{\boldsymbol{x } } ) / \sigma_t(x)$ ] by a change of variables . subtracting and using assumption [ ass : distrinv ] and and that does not depend on by assumption [ ass : distrinv ] .browning , m. and j. carro ( 2007 ) , `` heterogeneity and microeconometrics modeling , '' in blundell , r. , w.k .newey , t. persson ( eds . ) , _ _ advances in theory and econometrics , vol .3__ cambridge : cambridge university press .let then by the fundamental theorem of calculus , is differentiable in with derivative that is continuous in and . consider by the inverse and implicit function theorems, is continuously differentiable in and with ^{-1 } , \\ \partial_x\phi^{-1}(x , h , y ) & = & -\frac{\partial_x \phi ( x , h , \phi^{-1}(x , h , y))}{\partial_e \phi ( x , h , \phi^{-1}(x , h , y ) ) } \\ & = & - \partial_x \phi ( x , h , \phi^{-1}(x , h , y))\ , \partial_y \phi ^{-1}(x , h , y).\end{aligned}\ ] ] then by assumption [ ass : regquantile ] both and are continuous in and and bounded .therefore , are both bounded and continuous in and where the last equality in each equation follows by a standard change of variables argument . from the boundedness assumptions on and on in assumption [ ass : regquantile ] ,it follows that is partially differentiable in and with partial derivatives continuous in , which can be computed by differentiating under the integral in ( [ eq : hilflemone ] ) . in order to establish the expressions in the lemma , insert ( eq : hilflemtwo ) into the partial derivatives of ( [ eq : hilflemone ] ) w.r.t . and , and note that .the first expression is then immediate . for the second ,note that given ( for a fixed ) , , so that  , and set by the form of the model and the conditional independence assumption . below we show that from assumption [ ass : specialass ] , is continuously partially differentiable with derivatives from the positivity of the conditional density of given and the implicit function theorem , the conditional quantile given by exists and is differentiable in with derivative where we used ( [ hilf : generalagain ] ) , thus proving the theorem .it remains to prove ( [ hilf : generalagain ] ) , which is analogous to lemma [ lemm : quantilemain ] .let , so that is differentiable in with derivative that is continuous in .consider by the inverse and implicit function theorems , is continuously differentiable in and with then by assumption [ ass : specialass ] both and are continuous in and and bounded .therefore , are both bounded and continuous in and where the last equality in each equation follows by a change of variables argument together with conditional independence of and given . from the boundedness assumptions on and on in assumption [ ass : specialass ] , it follows that is partially differentiable with continuous partial derivatives which can be computed by differentiating under the integral in ( [ eq : hilflemone2 ] ) .now insert ( [ eq : hilflemtwo2 ] ) into ( [ eq : hilflemone2 ] ) and note that by conditional independence .the first expression is then immediate . for the second ,note that given ( for a fixed ) , , so that

Abstract: this paper considers identification and estimation of ceteris paribus effects of continuous regressors in nonseparable panel models with time homogeneity . the effects of interest are derivatives of the average and quantile structural functions of the model . we find that these derivatives are identified with two time periods for stayers " , i.e. for individuals with the same regressor values in two time periods . we show that the identification results carry over to models that allow location and scale time effects . we propose nonparametric series methods and a weighted bootstrap scheme to estimate and make inference on the identified effects . the bootstrap proposed allows uniform inference for function - valued parameters such as quantile effects uniformly over a region of quantile indices and/or regressor values . an empirical application to engel curve estimation with panel data illustrates the results .   * keywords : * panel data , nonseparable model , average effect , quantile effect , engel curve