Spohn H. Dynamics of Charged Particles and their Radiation Field
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Notes and references 53
There have been various attempts to improve on the oversimplistic version
(4.56) of the Lorentz model. Fermi (1922) argues that in a relativistic theory energy
and momentum have to be redefined. His argument has been rediscovered several
times and is explained in Rohrlich (1990). Poincar
´
e (1906) takes the elastic stresses
into account. We refer to Rohrlich (1960) and Yaghjian (1992), and the instructive
example by Schwinger (1983).
Section 4.2
Since the Lorentz model is defined through a Lagrangian, the total energy and
momentum are determined from Noether’s theorem for space-time translations.
The transformation as a four-vector is then automatically guaranteed, a property
which we used in the computation of the soliton mass.
Section 4.3
The limit of zero bare mass is discussed in Appel and Kiessling (2001).
5
Long-time asymptotics
Forany dynamical system one of the first qualitative issues is to understand
whether there are general patterns governing the long-time behavior. In this spirit
we plan to study the long-time asymptotics of the Abraham model with prescribed
external potentials. The basic mechanism at work is the loss of energy radiated to
infinity, which is proportional to ˙v(t)
2
according to Larmor’s formula. Since the
energy is bounded from below, we expect
lim
t→∞
˙v(t) = 0 (5.1)
under rather general conditions. In fact, one would also expect that the velocity
tends to a definite limit,
lim
t→∞
v(t) = v
∞
∈ V , (5.2)
which leaves us with two qualitatively rather different cases.
(i) v
∞
= 0. The charged particle comes to rest confined by the external potentials.
(ii) v
∞
= 0. The charge escapes into a region with zero external potentials and
travels there with constant velocity.
If we take also the asymptotics for t →−∞into account, then four familiar cases
arise: excitation by incident radiation and subsequent relaxation, (i) → (i); ioniza-
tion, (i) → (ii); capture through radiation losses, (ii) → (i); and scattering of light
from a freely moving charged particle, (ii) → (ii).
There must be a corresponding long-time asymptotic for the radiation field. It
consists of a part attached to the motion of the particle and a part scattered to
infinity. Thus a more complete description of the long-time solution is
Y (t)
∼
=
S
q(t),v(t)
+ (E
out
(t), B
out
(t), 0, 0) (5.3)
54
5.1 Radiation damping and the relaxation of the acceleration 55
for large t. Here S
q(t),v(t)
is the charge soliton at the current position and momen-
tum and E
out
(t), B
out
(t) are the solution of the homogeneous Maxwell equations
with appropriately adjusted initial conditions, the scattering data which depend on
Y (0).
At present two techniques are at hand for establishing a limit like (5.3). The first
one exploits the fact that energy cannot be radiated to infinity forever. This route
requires that all field modes are coupled to the particle as expressed by the
Wiener condition (W):
ϕ(k)>0 . (5.4)
The second route is based on a contraction method. It needs no extra condition and
gives explicit convergence rates. However, it requires |e| to be sufficiently small,
i.e. |e| < ¯e with a suitable ¯e depending only on the initial energy. Presumably (W )
and ¯e are artifacts of our mathematical technique.
5.1 Radiation damping and the relaxation of the acceleration
We will establish the limit (5.1) under the Wiener condition, but otherwise in com-
plete generality. The proof follows rather closely physical intuition and leads to an
equation of convolution type which has a definite long-time limit.
Let us consider a ball of radius R centered at the origin. At time t the sum of
the field energy in this ball and of the mechanical energy of the particle is given by
E
R
(t) = E(t) −
1
2
{|x|≥R}
d
3
x
E(x, t)
2
+ B(x, t)
2
(5.5)
provided R is sufficiently large. Using the conservation of total energy,
E(t) =
E
(0), E
R
changes in time as
d
dt
E
R
(t) =−R
2
d
2
ω
ω · [E(R
ω, t) × B(R
ω, t)] , (5.6)
where
ω is a vector on the unit sphere, d
2
ω the surface measure normalized to 4π,
and E× B the Poynting vector for the flux in energy at the surface of the ball under
consideration. Since the total energy is bounded from below, we conclude that
E
R
(R) − E
R
(R + t) =−
R+t
R
ds
d
ds
E
R
(s) ≤ C (5.7)
with the constant C =
E(0) −
¯
φ independent of R and t.
56 Long-time asymptotics
In (5.7) we first take the limit R →∞, which yields the energy radiated to
infinity during the time interval [0, t] through a large sphere centered at the origin.
Subsequently we take the limit t →∞to obtain the total radiated energy. To state
the result let us define
E
∞
(
ω, t) =−
e
4π
d
3
y ϕ(y − q(t +
ω · y)) (5.8)
×
(1 −
ω · v)
−1
˙v + (1 −
ω · v)
−2
(
ω ·˙v)(v −
ω)
t+
ω·y
which is a functional of the actual trajectory of the particle. Whatever its motion
we must have
∞
0
dt
d
2
ω |E
∞
(
ω, t)|
2
≤ C < ∞ . (5.9)
Note that the integrand in (5.9) is proportional to ˙v(t)
2
, which therefore is expected
to decay to zero for large t.
To establish (5.9) is somewhat tedious with pieces of the argument explained
in the section below and in section 8.5. One imagines that the trajectory t → q(t)
is given and solves the inhomogeneous Maxwell–Lorentz equations according to
(2.16), (2.17). If the time-zero fields are in
M
σ
, 0 <σ ≤ 1, see the definition
(2.49), then E
ini
(t) and B
ini
(t) decay as stated in (5.28). Therefore |
d
ds
E
R
(s)| <
CR
2
(1 + s)
−2−2σ
and the contribution to (5.7) from the initial fields vanishes in
the limit R →∞.Next one has to study the asymptotics of the retarded fields,
which is carried out in section 8.5. There ε is fixed, and for our purpose we may
set ε = 1. In addition in (8.48) the sphere of radius R is centered at q
ε
(t), rather
than at the origin. This means, in the present context one can use the asymptotics
(8.51), (8.52) as R →∞with q
ε
(t) replaced by 0. Combining both arguments
proves that (5.9) follows from (5.7) in the limit R →∞.
The real task is to extract from (5.9) that the acceleration vanishes for long
times.
Theorem 5.1 (Long-time limit of the acceleration).For the Abraham model sat-
isfying (C), (P), and the Wiener condition (W ) let the initial data be Y (0) =
(E
0
, B
0
, q
0
, v
0
) ∈ M
σ
with 0 <σ ≤ 1. Then
lim
t→∞
˙v(t) = 0 . (5.10)
Proof:Byenergy conservation |v(t)|≤ ¯v<1. Inserting in (2.41) and using (P)
we conclude that |˙v(t)|≤C. Differentiating (2.41) and using again (P) also
|¨v(t)|≤C uniformly in t. Therefore E
∞
(
ω, t) is Lipschitz continuous jointly in
5.1 Radiation damping and the relaxation of the acceleration 57
ω, t. Since the energy dissipation (5.9) is bounded, this implies
lim
t→∞
E
∞
(
ω, t) = 0 (5.11)
uniformly in
ω.
We analyze the structure of the integrand in (5.8). The retarded argument de-
pends only on y
=
ω · y. Therefore the integration over y
⊥
= y − y
ω can be
carried out and we are left with a one-dimensional integral of convolution type.
We set ϕ
a
(x
3
) =
dx
1
dx
2
ϕ(x). Then
E
∞
(
ω, t) =
e
4π
dy
ϕ
a
(y
− q
(t + y
))
×
(1 −
ω · v)
−2
ω × ((
ω − v) ×˙v)
t+y
=
e
4π
dsϕ
a
(t − (s − q
(s)))
×
(1 −
ω · v)
−2
ω × ((
ω − v) ×˙v)
s
. (5.12)
Since |˙q
(s)| < 1, we can substitute θ = s − q
(s) and obtain the convolution rep-
resentation
E
∞
(
ω, t) =
dθϕ
a
(t − θ)g
ω
(θ) = ϕ
a
∗ g
ω
(t), (5.13)
where
g
ω
(θ) =
e
4π
(1 −
ω · v)
−2
ω × ((
ω − v) ×˙v)
s(θ )
. (5.14)
From (5.11) we know that lim
t→∞
ϕ
a
∗ g
ω
(t) = 0. If ϕ(k
0
) = 0 for some k
0
, hence
ϕ violating the Wiener condition, then we could choose g
ω
(θ) periodic with fre-
quency |k
0
| and still have ϕ
a
∗ g
ω
(t) = 0. At this point no further progress seems
to be possible. However under the Wiener condition (W ) and with the smoothness
of g
ω
(θ) already established, Pitt’s extension to the Tauberian theorem of Wiener
assures us that
lim
θ→∞
g
ω
(θ) = 0 , (5.15)
which, since θ(t) →∞as t →∞, implies
lim
t→∞
ω × ((
ω − v(t)) ×˙v(t)) = 0 (5.16)
for every
ω in the unit sphere. Replacing
ω by −
ω and summing both expressions
yields
ω × (
ω ×˙v(t)) → 0ast →∞. Since this is true for every
ω,the claim
follows.
2
58 Long-time asymptotics
Note that by fiat Theorem 5.1 avoids any claims as regards the convergence of
(q(t), v(t)) as t →∞.
Since the acceleration vanishes for large times, the comoving electromagnetic
fields will adjust locally to the appropriate charge soliton. We established already
that E
ini
(t) and B
ini
(t) decay. Thus one only has to consider the retarded fields
E
ret
(x + q(t), t), B
ret
(x + q(t), t) relative to the position of the particle and com-
pare them with the soliton fields E
v(t)
(x), B
v(t)
(x) at the current velocity. For
this purpose one uses the representations (4.31), (4.32) for the charge soliton and
(2.16), (2.17) for the retarded fields. We insert the explicit form (2.15) of the prop-
agator. This yields
E
v
(x) = e
d
3
y (4π|x − y|)
−1
|x − y|
−1
ϕ(y − v|x − y|)
n
+ v ·∇ϕ(y − v|x − y|)(v −
n)
, (5.17)
B
v
(x) = e
d
3
y(4π|x − y|)
−1
n ×
−|x − y|
−1
ϕ(y −|x − y|v)v
+ v ·∇ϕ(y −|x − y|v)v
, (5.18)
where
n = (x − y)/|x − y|. Similarly for the retarded fields
E
ret
(x + q(t), t) =
d
3
y (4π|x − y|)
−1
|x − y|
−1
ϕ(y + q(t) − q(τ ))
n
+ v(τ ) ·∇ϕ(y + q(t) − q(τ ))(v(τ ) −
n)
− ϕ(y + q(t) − q(τ ))˙v(τ )
, (5.19)
B
ret
(x + q(t), t) =
d
3
y(4π|x − y|)
−1
n ×
−|x − y|
−1
ϕ(y + q(t)
− q(τ ))v(τ ) + v(τ) ·∇ϕ(y + q(t) − q(τ ))v(τ )
− ϕ(y + q(t) − q(τ ))˙v(τ )
, (5.20)
where τ = t −|x − y| and t ≥ t
ϕ
= 2R
ϕ
/(1 −¯v).
We compare the fields locally and use the result that lim
t→∞
˙v(t) = 0. Then, for
any fixed R > 0,
lim
t→∞
{|x|≤R}
d
3
x
E(x + q(t), t) − E
v(t)
(x)
2
+
B(x + q(t), t) − B
v(t)
(x)
2
= 0 . (5.21)
The scattered fields are not covered by (5.21) and will be studied in section 5.3.
5.2 Convergence to the soliton manifold 59
5.2 Convergence to the soliton manifold
In the case of zero external potentials, in essence any solution Y (t) rapidly con-
verges to the soliton manifold S as t →∞,inparticular v(t) → v
∞
. Such behav-
ior will be of importance in the discussion of the adiabatic limit, see chapter 6,
where it will be explained that in the matching to a comparison dynamics one
cannot use the naive v(0) but instead must take v
∞
.For hydrodynamic boundary
value problems such a property is known as the slip condition, since the extrap-
olation from the bulk does not coincide with the boundary conditions imposed
externally.
To prove the envisaged behavior we need a little preparation. Firstly we must
have some decay and smoothness of the initial fields at infinity. We already intro-
duced such a set of “good” initial data,
M
σ
,compare with (2.49), and therefore
require here Y (0) ∈
M
σ
, 0 <σ ≤ 1. Secondly, we need a notion for two field
configurations being close to each other. At a given time and far away from the
particle the fields are determined by their initial data. Only close to the particle are
they Coulombic. Therefore it is natural to measure closeness in the local energy
norm defined by
(E, B)
2
R
=
1
2
{|x|≤R}
d
3
x
E(x)
2
+ B(x)
2
(5.22)
for given radius R.
The true solution is Y (t) = (E(x, t), B(x, t), q(t), v(t)) which is to be com-
pared with the charge soliton approximation
E
v(t)
(x − q(t)), B
v(t)
(x − q(t)),
q(t), v(t)
.Weset Z
1
(x, t) = E(x, t) − E
v(t)
(x − q(t)), Z
2
(x, t) = B(x, t) −
B
v(t)
(x − q(t)), Z = (Z
1
, Z
2
) and want to establish that Z (·+q(t), t)
R
→ 0
for large times at fixed R.
Proposition 5.2 (Long-time limit for the velocity).For the Abraham model with
zero external potentials and satisfying (C) let |e|≤¯e with sufficiently small ¯e and
let the initial data be Y (0) ∈
M
σ
for some σ ∈ (0, 1]. Then for every R > 0 we
have
Z (·+q(t), t)
R
≤ C
R
(1 +|t|)
−1−σ
. (5.23)
In addition, the acceleration is bounded as
|˙v(t)|≤C(1 +|t|)
−1−σ
(5.24)
and there exists a v
∞
∈ V such that
lim
t→∞
v(t) = v
∞
. (5.25)
60 Long-time asymptotics
Proof: Using the Maxwell equations together with the identities (v ·∇) E
v
=
−∇ × B
v
+ eϕv,(v ·∇)B
v
=∇×E
v
one obtains
d
dt
Z(t) = AZ(t) − g(t ), (5.26)
where A is defined in (2.18) and g (t) has the components (˙v(t) ·∇
v
)E
v
(x − q(t)),
(˙v(t) ·∇
v
)B
v
(x − q(t)), and therefore
Z(t) = U(t)Z(0) −
t
0
ds U(t − s)g(s) (5.27)
with U(t) = e
At
.
For the first term we note that Z
1
(x, 0) = E
0
(x) − E
v
0
(x − q
0
), Z
2
(x, 0) =
B
0
(x) − B
v
0
(x − q
0
) ∈ M
σ
by assumption. Using the solution of the inhomo-
geneous Maxwell–Lorentz equations in position space and the bound (2.49) one
has
|Z
1
(x, t)|+|Z
2
(x, t)|≤Ct
−2
d
3
y δ(|x − y|−t)
|Z
1
(y, 0)|+|Z
2
(y, 0)|
+ Ct
−1
d
3
y δ(|x − y|−t)
|∇ Z
1
(y, 0)|
+|∇Z
2
(y, 0)|
≤ Ct
−2
d
3
y δ(|x − y|−t)(1 +|y|)
−1−σ
+Ct
−1
d
3
y δ(|x − y|−t)(1 +|y|)
−2−σ
≤ C (1 + t)
−1−σ
. (5.28)
The integrand in the second term of (5.27) will be estimated in section 7.3 with the
bound
U(t − s)g(s)
R
ϕ
≤ C( ¯v)e
2
(1 + (t − s)
2
)
−1
Z(·+q(s), s)
R
ϕ
; (5.29)
compare with (7.36).
We choose R ≥ R
ϕ
. From (5.29) and (5.28)
Z(·+q(t), t )
R
≤ C(1 + t)
−1−σ
+ C( ¯v)e
2
t
0
ds (1 + (t − s)
2
)
−1
Z(·+q(s), s)
R
. (5.30)
5.3 Scattering theory 61
Let κ = sup
t≥0
(1 + t)
1+σ
Z(·+q(t), t )
R
. Then
κ ≤ C + C( ¯v)e
2
t
0
ds (1 + (t − s)
2
)
−1
(1 + s)
−1−σ
κ, (5.31)
which implies κ<∞ provided C( ¯v) e
2
is sufficiently small.
To estimate the decay rate for the acceleration we start from Newton’s equations
of motion in the form
d
dt
m
b
γ v(t)
= e
E
ϕ
(q(t)) − E
v(t)ϕ
(0) + v × (B
ϕ
(q(t)) − B
v(t)ϕ
(0))
,
(5.32)
which uses the fact that the force from the soliton field vanishes. By energy con-
servation |v(t)|≤ ¯v<1. Therefore (5.32) implies
|˙v(t)|≤CeZ(·+q(t), t)
R
ϕ
(5.33)
and (5.24) follows from (5.23). Since v(t) = v(0) +
t
0
ds ˙v(s), one has |v(t) −
v
∞
|≤C (1 +|t|)
−σ
. 2
5.3 Scattering theory
We still have to provide an analysis of the scattered wave. Our results are somewhat
fragmentary and we start with an easy and sufficient integrability condition.
Theorem 5.3 (Existence of scattering solutions).For the Abraham model satis-
fying (C) and (P) let Y (t) ∈
M be a solution. If
∞
0
dt|˙v(t)| < ∞ , (5.34)
then there exist scattering data (E
sc
, B
sc
) such that
lim
t→∞
E(t) − E
v(t)
(·−q(t)) − E
sc
(t)
+B(t) − B
v(t)
(·−q(t)) − B
sc
(t)
= 0 , (5.35)
where (E
sc
(t), B
sc
(t)) = U(t)(E
sc
, B
sc
) propagate according to the homoge-
neous Maxwell–Lorentz equations.
Note that in (5.35) the difference is in the global energy norm and therefore
carries the information on the scattered wave.
62 Long-time asymptotics
Proof: The difference in (5.35) is Z(t) by definition. (5.26) remains valid in the
presence of external forces, which means that
Z(t) = U(t)
Z(0) −
t
0
dsU(−s)g(s)
. (5.36)
We set
E
sc
(x) = E
0
(x) − E
v
0
(x − q
0
) −
∞
0
dt ( ˙v(t) ·∇
v
)E
v
(x − q(t)) ,
B
sc
(x) = B
0
(x) − B
v
0
(x − q
0
) −
∞
0
dt ( ˙v(t) ·∇
v
)B
v
(x − q(t)) . (5.37)
Since |v(t)|≤ ¯v<1, the integrands have uniformly bounded energy norm. Thus
by assumption (5.34) the integrals converge in
M and define (E
sc
, B
sc
) ∈ M.
Hence (5.35) follows.
2
There are two cases of interest for which the integrability condition (5.34) can
be checked.
(i) Compton scattering (zero external potential). If |e|≤ ¯e, then by (5.24)
|˙v(t)|≤C(1 +|t|)
−1−σ
which implies (5.34). For a freely moving charge the
asymptotic motion is rectilinear and the scattered waves propagate according to
the free Maxwell equations. Such a result also applies to a charge reaching an
essentially potential-free region. The standard example is a charge scattered by
an infinitely heavy nucleus. For sufficiently long times the incident fields have de-
cayed already and we assume that the charge has reached, with its velocity pointing
outwards, a large sphere centered at the nucleus. Then the external force decays as
1/t
2
which combined with Theorem 5.3 yields the desired asymptotics.
(ii) Rayleigh scattering (bounded motion). Under the Wiener condition (W )we
already know that lim
t→∞
˙v(t) = 0. If in addition the motion is bounded,
|q(t)|≤ ¯q (5.38)
for all t,then necessarily
lim
t→∞
v(t) = 0 , (5.39)
i.e. the particle comes to rest. Inserting in Newton’s equations of motion (2.34) and
using the fact that the fields become locally soliton-like, we infer that
lim
t→∞
∇φ
ex
(q(t)) = 0 . (5.40)
Let us define the set
A of critical points for the potential φ
ex
, A ={q |∇φ
ex
(q) =
0}.By(5.40), q(t) approaches
A as a set. If A happens to be a discrete set, then,
by the continuity of solutions in t, q(t) has to converge to some definite q
∗
∈ A.