S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 brackets measures the center of mass(as normalized by the this initial velocity we choose initial data for which the total free energy)of the kinetic energy associated with the motion initial kinetic energy is equal to its complement On the other of the dipoles. Note that the two energy projections just dis- hand, if we choose the associated distributions to be identical cussed sum to the free energy. Consequently we will speak then the differences in the associated centers of (reduced) of them as being compliments with respect to the free en- mass will be zero. The simplest way to get a nontrivial and ergy. In contrast to the terms measuring positions, the two interesting result is to choose the two distributions to be terms in large parenthesis are homogeneous and so are unit- translates of each other: For some even and square integrable less. They measure the relative amounts of energy associated function f(x)different from 0 and for some positions x and with the two centers of mass just described. As such they x2 define mi(x) and m2(x)by may be described as weights for the associated"masses Note that the weights are paired (via multiplication) with mI(x):=f(x-xi), and (107) their complement centers. Thus, since the notion of comple- ment is with respect to the free energy, and since the free m2(x)=f2(x-x2) energy is the distribution used to normalize the two centers of mass, the two products of these four objects give the cen- Then we get M=M=Sdxf(x)=M>, and then ters of reduced mass of the two complimentary energy pro- jections. Thus we see that the second functional in Eq. (102) dxx mix)=x M gives a measure of the difference in the centers of reduced mass of the kinetic energy and its free-energy complement Thus the"additional' velocity of the free-energy distribu- Then Eq (106)reduces at (=0 to tion(i.e, the component of its velocity that is attributable to damping) is just the damping rate multiplied by a difference dx E(x, O)B(x, 0) in(representative) positions of two complementary distribu U(O)=c +(x2-x1).(110 dx uR(x,o) To make these connections more obvious we introduce some notation. Relabel the kinetic energy density g2(x, t)/ In order to make this expression more explicit we can,for as mi(x)and relabel its complement R(x, t)=uR(x, t) example, further decompose the initial energy densities as 02(r, t)/2 as m2(x). Also relabel the corresponding ener- follows. Choose E(x,0)=B(x,0) gies(the integrals of the densities)by the same symbols but [and @(x,0)=v2f(x-xDl With this two-parameter family with capital letters and, of course, without reference to the of choices for the initial data, the initial velocity of the free position x, energy center of mass(110)reduces to dx mi(x), and (104) c y (0)= (111) M2:= dx m2(r) (105)(Here we have expressed the difference in the centers of reduced mass of the kinetic energy and its complement, x2 x1, as Ax. )It now becomes clear that if y is not zero, the Then the center-of-mass velocity of the free energy can be speed of the free-energy center of mass can be increased arbitrarily, as long as theposition'' of the kinetic energy distribution can be made to lead (or lag) that of its free en- Mime(x) ergy complement by arbitrarily large distances. Furthermore dx e(x, B(x, t) dx x this notion of position becomes more natural and precise as Uu.(1)=c the variance of f(x)reduces. In fact there is nothing to keep dx uR(x, t) us from considering the limit in which f(x)+VS(x): in this imit we still get the (not luminal) result(111)for the initial center-of-mass velocity of the free ene dr r mi(r)Ms 1+M2 V, SUMMARY M1+M2 In this paper the liminality of both local and global no- tions of total energy transport in very general(anisotropic, Here we see that the integrands used to define the centers of inhomogeneous, passive) media was established. In lieu of mass appear analogous to the classical expression for re- specific microscopic models, these results were established duced mass in the two-body problem using only the macroscopic limitations of causality and pas We now consider the velocity at t=0 and a corresponding sivity. Specifically these estimates were obtained by (1)de- class of system preparations or initial conditions that show veloping total dynamical energy densities for these media that the velocity at this instant can be arbitrarily large if and (i.e. conserved, positive definite, quadratic forms) and the only if y does not vanish. To find the simplest expression for by (2) considering the time evolution of their associated 046610-12brackets measures the center of mass ~as normalized by the free energy! of the kinetic energy associated with the motion of the dipoles. Note that the two energy projections just dis￾cussed sum to the free energy. Consequently we will speak of them as being compliments with respect to the free en￾ergy. In contrast to the terms measuring positions, the two terms in large parenthesis are homogeneous and so are unit￾less. They measure the relative amounts of energy associated with the two centers of mass just described. As such they may be described as weights for the associated ‘‘masses.’’ Note that the weights are paired ~via multiplication! with their complement centers. Thus, since the notion of comple￾ment is with respect to the free energy, and since the free energy is the distribution used to normalize the two centers of mass, the two products of these four objects give the cen￾ters of reduced mass of the two complimentary energy pro￾jections. Thus we see that the second functional in Eq. ~102! gives a measure of the difference in the centers of reduced mass of the kinetic energy and its free-energy complement. Thus the ‘‘additional’’ velocity of the free-energy distribu￾tion ~i.e., the component of its velocity that is attributable to damping! is just the damping rate multiplied by a difference in ~representative! positions of two complementary distribu￾tions. To make these connections more obvious we introduce some notation. Relabel the kinetic energy density Q2(x,t)/2 as m1 t (x) and relabel its complement ¯uR(x,t)5uR(x,t) 2Q2(x,t)/2 as m2 t (x). Also relabel the corresponding ener￾gies ~the integrals of the densities! by the same symbols but with capital letters and, of course, without reference to the position x, M1 t ª E dx m1 t ~x!, and ~104! M2 t ª E dx m2 t ~x!. ~105! Then the center-of-mass velocity of the free energy can be expressed as vuR ~t!5c E dx E~x,t!B~x,t! E dx uR~x,t! 12gH E dx x M1 t m2 t ~x! M1 t 1M2 t M1 t 1M2 t 2 E dx x m1 t ~x!M2 t M1 t 1M2 t M1 t 1M2 t J . ~106! Here we see that the integrands used to define the centers of mass appear analogous to the classical expression for re￾duced mass in the two-body problem. We now consider the velocity at t50 and a corresponding class of system preparations or initial conditions that show that the velocity at this instant can be arbitrarily large if and only if g does not vanish. To find the simplest expression for this initial velocity we choose initial data for which the total initial kinetic energy is equal to its complement. On the other hand, if we choose the associated distributions to be identical then the differences in the associated centers of ~reduced! mass will be zero. The simplest way to get a nontrivial and interesting result is to choose the two distributions to be translates of each other: For some even and square integrable function f(x) different from 0 and for some positions x1 and x2 define m1 0 (x) and m2 0 (x) by m1 0 ~x!ªf 2 ~x2x1!, and ~107! m2 0 ~x!ªf 2 ~x2x2!. ~108! Then we get M1 0 5M2 0 5*dx f 2(x)5M.0, and then E dx x mi~x!5xiM, i51,2. ~109! Then Eq. ~106! reduces at t50 to vuR ~0!5c E dx E~x,0!B~x,0! E dx uR~x,0! 1 g 2 ~x22x1!. ~110! In order to make this expression more explicit we can, for example, further decompose the initial energy densities as follows. Choose E(x,0)5B(x,0)5 f(x2x2) and P(x,0)50 @and Q(x,0)5A2 f(x2x1)#. With this two-parameter family of choices for the initial data, the initial velocity of the free energy center of mass ~110! reduces to vuR ~0!5 c 2 1 g 2 Dx. ~111! ~Here we have expressed the difference in the centers of reduced mass of the kinetic energy and its complement, x2 2x1, as Dx.! It now becomes clear that if g is not zero, the speed of the free-energy center of mass can be increased arbitrarily, as long as the ‘‘position’’ of the kinetic energy distribution can be made to lead ~or lag! that of its free en￾ergy complement by arbitrarily large distances. Furthermore this notion of position becomes more natural and precise as the variance of f(x) reduces. In fact there is nothing to keep us from considering the limit in which f(x)→Ad(x): in this limit we still get the ~not luminal! result ~111! for the initial center-of-mass velocity of the free energy. V. SUMMARY In this paper the luminality of both local and global no￾tions of total energy transport in very general ~anisotropic, inhomogeneous, passive! media was established. In lieu of specific microscopic models, these results were established using only the macroscopic limitations of causality and pas￾sivity. Specifically these estimates were obtained by ~1! de￾veloping total dynamical energy densities for these media ~i.e. conserved, positive definite, quadratic forms! and then by ~2! considering the time evolution of their associated fi- S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-12