POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 1000 the density uR, and then use integration by parts to eliminate 0100|B spatial derivatives on the flux EB. After some simplification (EB PO)00 1 0 (x,1) 000 dx e(x, nB(x, t) dx o(x, t) =(E2+B2+P2+Q2)(x,) dx uR(x, t) dx uR(x, t) We will denote this form(divided by 2)by uR(x, t) and, in order to distinguish it from the total energy density that dx x uR(x, t) dx uR(x, t) obeys a conservative law, will call it the free-energy density (or distribution). Though we do not engage in a statistical mechanics treatment, we justify the use of this term as fol- dx ur(x, t) lows. We can interpret Eq(96)as a phenomenological de- scription of a system in which uR(x, t) has the interpretation of being the sum of the densities associated with energy dxxo-(x, t) stored in the macroscopic fields and that stored in the coher ent motions of the molecular dipoles of the dispersive me- dx uR(x, t) dium(mechanical energy density). In this phenomenological treatment, clearly uR(x, t)cannot be interpreted as containing where energy deposited irreversibly in the medium via incoherent motions and their associated degrees of freedom The law of dissipation associated with this particular qua- ug=g-iQ2=;E2+;B2+;P2 (103) dratic form, or energy projection, is In Eq (102) the velocity is expressed as having two com- E2+B2+P2+Q2|+(cEB)=-2yQ ponents, the first not dependent on system parameters explic- itly and the second explicitly dependent upon the damping (99) rate y. For each time t the two terms are functionals of fund tions of position x. The first functional can be shown to pos- Here we have suppressed the coordinates x and t. sess extrema +c. This is done by showing that the absolute Note that if we define the free-energy density's velocity to value of the integrand of its numerator never exceeds(but be the ratio of the(suitably averaged) flux cEB to the(simi- can be equal to) the integrand of its denominator(which is E2+B2+P2+1Q2) non-negative)multiplied by c. Below we will show that the are guaranteed to get a luminal result since second term, which is multiplied by the damping rate y, is an unbounded functional in a relevant function space. Thus, in IcEB this function space, the center-of-mass velocity functional is (100) unbounded when damping is present but is bounded lum nally when damping is absent. Note that we do not presently Unfortunately this definition of the velocity generically has address the issue of the duration of superluminal behavior in the free-energy's center-of-mass motion, but only the issue almost nothing to do with the gross motion of the free- of superluminal system preparations, i.e., of whether the sys- energy distribution uR because the evolution of the free en ergy is dissipative. If one simply views the results of a nu tem can, in principle, be initially prepared so as to demon merical simulation of Eq.(96)by watching a movie of strate superluminal behavior in the motion of the free-energy uR(x, t) passing by, the perceived speed of the pulse can be arbitrarily large, depending on the system preparation. In the To that end we consider the velocity at t=0, and consider following, we make this observation concrete by showing a two-parameter family of system preparations, all members analytically that a pulses free energy"center-of-mass of which correspond to the same initial free energy. We ther xu(t), defined as how that, when damping is present, the initial center-of- mass velocity increases without bound as the difference in the two parameters increase. In order to motivate how this is dx x uR(x, t) accomplished we pause to comment on which details of the x(t) (101) structure of the second functional in Eq. (102)suggest that dx uR(x, t) dent by parallel uses of parenthesis and square brackets. The two terms in large square brackets have units of position (with the integration over all x)can move with any speed. The left term in square brackets mea the center of mass The velocity corresponding to this definition of position is (as normalized by the free energy)of the energy stored in the btained by time differentiating the center of mass. We use fields and in the displacements of the dipoles from equilib- the law of transport(99)to eliminate the time derivative of rium(dipole potential energy ). The right term in square 046610-11~EB PQ! S 1000 0100 0010 0001 D S E B P Q D ~x,t! 5~E21B21P21Q2!~x,t!. ~98! We will denote this form ~divided by 2! by uR(x,t) and, in order to distinguish it from the total energy density that obeys a conservative law, will call it the free-energy density ~or distribution!. Though we do not engage in a statistical mechanics treatment, we justify the use of this term as fol￾lows. We can interpret Eq. ~96! as a phenomenological de￾scription of a system in which uR(x,t) has the interpretation of being the sum of the densities associated with energy stored in the macroscopic fields and that stored in the coher￾ent motions of the molecular dipoles of the dispersive me￾dium ~mechanical energy density!. In this phenomenological treatment, clearly uR(x,t) cannot be interpreted as containing energy deposited irreversibly in the medium via incoherent motions and their associated degrees of freedom. The law of dissipation associated with this particular qua￾dratic form, or energy projection, is ] ]t S 1 2 E21 1 2 B21 1 2 P21 1 2 Q2 D 1 ] ]x ~cEB!522gS 1 2 Q2 D . ~99! Here we have suppressed the coordinates x and t. Note that if we define the free-energy density’s velocity to be the ratio of the ~suitably averaged! flux cEB to the ~simi￾larly averaged! density uR5( 1 2 E21 1 2 B21 1 2 P21 1 2 Q2), we are guaranteed to get a luminal result since ucEBu uR <c. ~100! Unfortunately this definition of the velocity generically has almost nothing to do with the gross motion of the free￾energy distribution uR because the evolution of the free en￾ergy is dissipative. If one simply views the results of a nu￾merical simulation of Eq. ~96! by watching a movie of uR(x,t) passing by, the perceived speed of the pulse can be arbitrarily large, depending on the system preparation. In the following, we make this observation concrete by showing analytically that a pulse’s free energy ‘‘center-of-mass’’ xuR (t), defined as xuR ~t!ª E dx x uR~x,t! E dx uR~x,t! , ~101! ~with the integration over all x) can move with any speed. The velocity corresponding to this definition of position is obtained by time differentiating the center of mass. We use the law of transport ~99! to eliminate the time derivative of the density uR , and then use integration by parts to eliminate spatial derivatives on the flux EB. After some simplification we obtain vuR ~t!5c E dx E~x,t!B~x,t! E dx uR~x,t! 1g H S E dx Q2 ~x,t! E dx uR~x,t! D 3F E dx x u¯ R~x,t! E dx uR~x,t! G 2S E dx u¯ R~x,t! E dx uR~x,t! D 3F E dx x Q2 ~x,t! E dx uR~x,t! GJ , ~102! where ¯uRªuR2 1 2 Q251 2 E21 1 2 B21 1 2 P2. ~103! In Eq. ~102! the velocity is expressed as having two com￾ponents, the first not dependent on system parameters explic￾itly and the second explicitly dependent upon the damping rate g. For each time t the two terms are functionals of func￾tions of position x. The first functional can be shown to pos￾sess extrema 6c. This is done by showing that the absolute value of the integrand of its numerator never exceeds ~but can be equal to! the integrand of its denominator ~which is non-negative! multiplied by c. Below we will show that the second term, which is multiplied by the damping rate g, is an unbounded functional in a relevant function space. Thus, in this function space, the center-of-mass velocity functional is unbounded when damping is present but is bounded lumi￾nally when damping is absent. Note that we do not presently address the issue of the duration of superluminal behavior in the free-energy’s center-of-mass motion, but only the issue of superluminal system preparations, i.e., of whether the sys￾tem can, in principle, be initially prepared so as to demon￾strate superluminal behavior in the motion of the free-energy center of mass. To that end we consider the velocity at t50, and consider a two-parameter family of system preparations, all members of which correspond to the same initial free energy. We then show that, when damping is present, the initial center-of￾mass velocity increases without bound as the difference in the two parameters increase. In order to motivate how this is accomplished we pause to comment on which details of the structure of the second functional in Eq. ~102! suggest that this can be done. We have tried to make this structure evi￾dent by parallel uses of parenthesis and square brackets. The two terms in large square brackets have units of position. The left term in square brackets measures the center of mass ~as normalized by the free energy! of the energy stored in the fields and in the displacements of the dipoles from equilib￾rium ~dipole potential energy!. The right term in square POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-11