Casimir Friction Force Between Polarizable Media

0.13 MB·English

by  Johan S. Høye

CASIMIR FRICTION FORCE BETWEEN POLARIZABLE
MEDIA
Johan S. Høye1
2
1
0 Department of Physics, Norwegian University of Science and Technology,
2
N-7491 Trondheim, Norway

n
a
J Iver Brevik2

8
1 Department of Energy and Process Engineering, Norwegian University of

Science and Technology, N-7491 Trondheim, Norway
]
h
p
-
t Abstract
n
a
u ThisworkisacontinuationofourrecentseriesofpapersonCasimir
q friction, for a pair of particles of low relative particle velocity. Each
[
particleismodeledasasimpleharmonicoscillator. Ourbasicmethod,

1 as before, is the use of quantum mechanical statistical mechanics, in-
v
volvingtheKuboformula,atfinitetemperature. Inthisworkwebegin
0
3 by analyzing the Casimir friction between two particles polarizable in
8 all spatial directions, this being a generalization of our study in EPL
3 91, 60003 (2010), which was restricted to a pair of particles with lon-
.
1
gitudinal polarization only. For simplicity the particles are taken to
0
2 interact via the electrostatic dipole-dipole interaction. Thereafter, we
1 considertheCasimirfrictionbetweenoneparticleandadielectrichalf-
:
v space, andalsothefriction between twodielectric half-spaces. Finally,
Xi we consider general polarizabilities (beyond the simple one-oscillator
form), and show how friction occurs at finite temperature when finite
r
a
frequency regions of the imaginary parts of polarizabilities overlap.
PACS numbers: 05.40.-a, 05.20.-y, 34.20.Gj, 42.50.Lc
1
[email protected]
2
[email protected]
1
1 Introduction
Casimir friction - a subclass of the Casimir field of research - has emerged to
be a topic of considerable current interest. It is basically a non-contact kind
of friction that may be related to electromagnetic fluctuations. The effect is
smallundernormalcircumstances, andmaymoreoverbedifficult todealwith
theoretically since it involves energy dissipation, necessitating in turn the use
of complex valued permittivities in macroscopic electrodynamics. Once one
leaves the state of thermal equilibrium one will have to face fundamental
problems. Thus, in connection with macroscopic electrodynamics a very
instructivewayofapproachistomakeuseofthespectralsummationmethod,
but this method is based upon real eigenfrequencies in the spectral problem
and how to deal with such a system in the case of dissipation is at present
unclear. Cf., for instance, the discussion on this point in Refs.[1] and [2].
Most of the previous works in this area are based on the macroscopic
dielectric model with permittivity properties. See, for instance, Refs. [3, 4,
5, 6, 7, 8, 9]. There exists, however, a different strategy in order to deal with
Casimir friction, namely to consider the statistical mechanics for harmonic
oscillators at finite temperature T moving with constant velocity v relative
to each other. We argue that such a microscopic model, in spite of its sim-
plicity compared with a full-fledged macroscopic model, has nevertheless the
capacity of providing physical insight in the problem. And this is the kind
of model that we shall consider in the following. We shall make an exten-
sion of our considerations in recent papers [10, 11, 12]; cf. also the arXiv
report [13]. And this is again based upon a study of ours back in 1992 [14],
dealing with the same kind of system. It contains a generalization to the
time-dependent case of the statistical mechanical Kubo formalism spelled
out for the time-independent case in Ref. [15].
Whereas our previous works were limited to the case of two microscopic
oscillators, we shall here generalize the formalism so as to deal with one
oscillator outside a slab (a collection of oscillators). Also, the generalization
to two slabs in relative constant motion is easily achievable.
We emphasize that we are considering dilute media only. The general-
ization to media of arbitrary density ought to be tractable with the actual
method, but has to our knowledge not been treated yet. Our results are
that the energy change ∆E because of Casimir friction is finite in general.
This corresponds to a finite friction force. At zero temperature the formal-
ism yields ∆E → 0, however; this being due to our assumption of a slowly
2
varying coupling. For rapidly varying couplings, there will be a finite friction
force also at T = 0 [11, 16].
We mention finally that the microscopic approach has been analyzed by
other investigators also, especially by Barton in recent papers [16, 17, 18].
The equivalence between Barton’s results and our own results is not so easy
to see by mere inspection since the methods are different, but were shown
explicitly to be equivalent in one of our recent papers [12].
2 Friction force between two oscillators
Consider the quantum mechanical system of two polarizable particles whose
reference state is that of uncoupled motion corresponding to a Hamiltonian
H . The equilibrium situation is then perturbed by a time independent
0
term −AF(t) where A is a time independent operator and F(t) is a classical
time dependent function. For a pair of polarizable particles perturbed with
dipole-dipole interaction we have
−AF(t) = ψ s s , (1)
ij 1i 2j
where thesummation convention for repeatedindices iandj is implied. Here
s and s are the components of the fluctuation dipole moments of the two
1i 2j
particles (i,j = 1,2,3). With the electrostatic dipole-dipole interaction,
3x x δ
i j ij
ψ = − − , (2)
ij r5 r3
(cid:18) (cid:19)
where r = r(t) with components x = x (t) is the separation between the par-
i i
ticles. In an earlier work we also studied the situation with time-dependent
interaction (with retardation effects) [19], but we will avoid this added com-
plexity here.
The situation with which we shall mainly be concerned in the follow-
ing is when the relative velocity v of the oscillators is constant. Then the
interaction will vary as
∂
−AF(t) = ψ (r )+ ψ (r ) v t+... s s , (3)
ij 0 ij 0 l 1i 2j
∂x
(cid:20) (cid:18) l (cid:19) (cid:21)
and the components of the force B between the oscillators are
∂
B = − (ψ s s ). (4)
l ij 1i 2j
∂x
l
3
The equilibrium situation with both particles at rest is represented by the
first term in (2). It gives rise to the (reversible) equilibrium force. Thus the
friction is connected with the second term. To simplify, we shall here neglect
thefirst termbywhichthetwo oscillatorswill befullyuncorrelatedintheirin
their relative position r = r . Thus we can write −AF(t) → −A F (t) where
0 l l
A = B and F (t) = v t. The friction force will be a small perturbation upon
l l l l
the equilibrium situation, and it leads to a response ∆hB (t)i in the thermal
l
average of B . According to Kubo [20, 15, 14]
l
∞
∆hB (t)i = φ (t−t′)F (t′)dt′, (5)
l BAlq q
−∞
Z
where the response function is
1
φ (t) = Trρ[A ,B (t)]. (6)
BAlq i~ q l
Here ρ is the density matrix and B (t) is the Heisenberg operator B (t) =
l l
eitH/~B e−itH/~ where β like A is time independent. With Eqs. (3) and (4)
l
and with F (t) = v t expression (6) can be rewritten as
l l
φ (t) = G φ (t), (7)
BAlq lqijnm ijnm
where
∂ψ ∂ψ
ij nm
G = , (8)
lqijnm
∂x ∂x
l q
φ (t) = Tr{ρC (t)}, (9)
ijnm ijnm
1
C (t) = [s s ,s (t)s (t)] (10)
ijnm i~ 1i 2j 1n 2m
(the i in the denominator is the imaginary unit).
Here as in Refs. [14] and [10] the perturbing interaction (1) will be con-
sidered weak. This will also hold for dilute dielectric media by which thermal
averages of products, containing s and s as factors, to leading order fac-
1i 2j
torize. Further, with isotropy or scalar polarizability the components of each
of the fluctuating dipole moments are uncorrelated.
From now on we find it convenient to utilize imaginary time, which was
used in Sec. 4 of Ref. [14]. With this we get
g (λ) = Tr[ρs (t)s (t)s s ] = g (λ)g (λ), (11)
ijnm 1n 2m 1i 2j 1in 2jm
4
g (λ) = hs (t)s i = g (λ)δ , (a = 1,2), (12)
apq aq ap a qp
where g(λ) is the correlation function and angular brackets denote thermal
averages (h..i = Tr[ρ...]). The λ is imaginary time given by
t
λ = i , (13)
~
so that for an operator B
B(t) = eλHBe−λH. (14)
With this,
1
φ (t) = [g (β +λ)−g (λ)]
ijnm i~ ijnm ijnm
1
= [g (β +λ)g (β +λ)−g (λ)g (λ)]δ δ . (15)
i~ 1 2 1 2 in jm
The response function φ (t) corresponds to the retarded Green function,
ijnm
in the usual language of quantum field theory.
Earlier we unfortunately made a mistake by defining λ in Eq. (13) with
opposite sign [14]. However, this did not influence the results of previous
applications since the operators A and B were equal apart from prefactors.
In Appendix B of Ref. [14] it was found that
˜
φ(ω) = g˜(K), (16)
so that the Fourier transforms of the response function φ and the correlation
function g are equal. Explicitly,
∞
φ˜(ω) = φ(t)e−iωtdt (φ(t) = 0 for t < 0),
0
Z
β
g˜(K) = g(λ)eiKλdλ, (17)
0
Z
and
K = i~ω. (18)
[ In Ref. [14] K = −i~ω was used due to the mistake mentioned.] Equality
(16) holds in the common region |Im(K)| < C with C > 0 and Im(ω) < 0,
i.e., Re(K) > 0 where both functions are analytic.
5
It may be remarked that Eqs. (12)- (18) above correspond in a quantum
field theoretical language to the statement that the spectral correlation func-
tion (frequency ω) is equal to the imaginary part of the spectral retarded
Green function multiplied with coth(1β~ω). See, for instance, Eq. (76.6)
2
in Ref. [21]. In the present case the function g˜ (ω) which is the real time
+
Fourier transform of g (t) = 1 (g(it/~)+g(−it/~)), will be the spectral cor-
+ 2~
˜
relation function while the φ(−ω) (with Fourier transform (17)) will be the
spectral retarded Green function (for ω > 0 as g˜ (ω) is symmetric in ω while
+
˜
Im[φ(−ω)] is antisymmetric).
With Eqs. (11) and (12) we have
g (λ) = g(λ)δ δ ,
ijnm in jm
g(λ) = g (λ)g (λ). (19)
1 2
Thus in K- space the g˜(K) can be written as the convolution
1
g˜(K) = g˜ (K )g˜ (K −K ), (20)
1 0 2 0
β
XK0
which is Eq. (4.15) of Ref. [14].
An advantage of using imaginary time is that g˜ (K) can be identified
a
with the frequency dependent polarizability α of oscillator a (= 1,2). For
aK
a simple harmonic oscillator with eigenfrequency ω one has [15]
a
α (~ω )2
a a
g˜ (K) = α = , (21)
a aK K2 +(~ω )2
a
where α is the zero-frequency polarizability. [ The g˜(K) given by Eq. (4.10)
a i
of Ref. [14] will differ from the one of Eq. (21) by a factor e2 where e is the
electron charge since here the s is identified with a dipole moment.]
a
With the above one finds for the Fourier transform of (9)
˜ ˜
φ (ω) = φ(ω)δ δ , (22)
ijnm in jm
˜
with φ(ω) = g˜(K). This follows by use of Eqs. (15) and (19) from which
g˜ (K) = g˜(K)δ δ , and when account is taken of Eq. (16). The g˜(K)
ijnm in jm
is moreover given by the convolution (20) where for two simple harmonic
oscillators the g˜ (K) (a = 1,2) are given by Eq. (21).
a
6
To obtain the perturbing force (5) the expression (7) should be evaluated.
With (22) one can write
φ (t) = φ(t)δ δ , (23)
ijnm in jm
by which
φ (t) = G φ(t), (24)
BAlq lq
G = G = T T , (25)
lq lqijij lij qij
∂ψ
ij
T = . (26)
lij
∂x
l
With ψ given by Eq. (2) one finds
ij
15x x x 3(x δ +x δ +x δ )
i j l i lj j il l ij
T = − , (27)
lij r7 r5
by which
2 2
15 270 9
4 2
G = T T = r − r + x x
lq lij qij r7 r12 r5 l q
" #
(cid:18) (cid:19) (cid:18) (cid:19)
2
3 18 36x x
2 l q
+ 2r δ = δ + . (28)
r5 lq r8 lq r10
(cid:18) (cid:19)
With the above expressions the result for the perturbing force (5) will be
precisely the same as obtained in Ref. [10], except from subscripts l and q.
One may simply insert Eq. (24) into Eq. (5) with G given by Eq. (28), φ(t)
lq
following from the relations (16)-(23).
Thus, like in Eqs. (8) and (9) of Ref. [10] we here get the perturbing force
F = ∆hB (t)i = F +F , (29)
l l rl fl
where
∞
F = G v t φ(u)du (30)
rl lq q
0
Z
is part of the reversible force. The part of the force representing friction is
∞
F = −G v φ(u)udu. (31)
fl lq q
0
Z
7
The Fourier transformedversion of thisequation is, like Eq. (11)of Ref. [10]),
˜
∂φ(ω)
F = −iG v . (32)
fl lq q
∂ω ω=0
(cid:12)
The expression for φ˜(ω) is evaluated below Eq(cid:12). (38) in the next section, by
(cid:12)
which the explicit expression (43) is obtained for F .
fl
Now the dissipated energy may be obtained; this requiring a perturbing
interaction to last a finite amount of time to make it unique. Thus the force
expression may be modified to
F (t) → q (t), (33)
l l
where q (t) is interpretable as a position. The dissipated energy ∆E is then
l d
givenbyexpression (27)ofRef.[10]. Inthepresent casethiswillbe(A = B )
l l
∞ t
∆E = q˙(t) φ (t−t′)q (t′)dt′ dt. (34)
d l AAlq q
Z−∞ (cid:20)Z−∞ (cid:21)
3 Friction between particle and a half-space,
and between two half-spaces
Armed with the result (28) for G it is now straightforward to extend the
lq
results in the previous section to the situation where a polarizable particle
moves parallel to a resting dielectric half-space. We assume then that the
particle density ρ in the half-space is low so that the forces are additive. Let
the half-space be located at z ≥ z such that its surface is parallel to the
0
xy plane at the vertical position z = z . The dielectric particle is located at
0
the origin and moves with constant velocity v = v along the x axis. The
x 1
resulting friction then follows by integrating G with l = q = 1 over the
lq
dielectric half-space. With expression (28) we find
G = ρ G dxdydz. (35)
h 11
Zz≥z0
Symmetry with respect to the x and y coordinates means that the x2 in
Eq. (28) can be replaced with 1(x2 +y2), and we can use cylindrical coordi-
2
nates with ρ2 = x2 +y2 and dxdy = 2πρdρ. Thus with r2 = ρ2 +z2,
∞ 1 ρ2 15π
G dxdy = 36π + ρdρ = , (36)
11
r8 r10 2z6
Z Z0 (cid:18) (cid:19)
8
by which
∞ 15π 3πρ
G = ρ dz = . (37)
h 2z6 2z5
Zz0 0
Finally, for two dielectric half-spaces moving parallel relative to each other
one can obtain the friction force per unit area from
∞ 3π
G = ρ G dz = ρ ρ , (38)
2 h 0 8d4 1 2
Zd
where d is the gap width and ρ and ρ are the (low) particle densities in the
1 2
two half spaces.
˜
Thefunctionφ(ω) = g˜(K)isneededtoobtainthefriction. Performingthe
summation in Eq. (20), inserting the expression (21) we obtain, in agreement
with Eq. (4.16) in [14],
g˜(K) = Hf(K), (39)
with
E E α α
1 2 1 2
H = ,
4sinh(1βE )sinh(1βE )
2 1 2 2
Σ sinh(1βΣ ) Σ sinh(1βΣ )
f(K) = 1 2 1 + 2 2 2 . (40)
K2 +Σ2 K2 +Σ2
1 2
Here Σ and Σ are defined as
1 2
Σ = E +E , Σ = E −E , E = ~ω (i = 1,2).
1 1 2 2 1 2 i i
When the velocity is small and constant (or very slowly varying), only the
limit K → 0 is needed. One further sees that the contribution requires that
Σ → 0. Because of this, the f(K) becomes a δ-function (plus a constant)
2
π
f(K) = − βKδ(Σ ), (41)
2
2
like Eq. (4.18) of Ref. [14]. [Here Re(K) > 0 in view of the correction
mentioned below Eq. (15).] To obtain the friction force like (32) for a pair
of particles and thus the more general situation, the derivative of f(K) with
respect to ω is needed. With δ(Σ ) = δ(~(ω − ω )) = δ(ω − ω )/~ and
2 1 2 1 2
K = i~ω one finds
∂ ∂f(K) π
i f(K) = −~ = βδ(ω −ω ). (42)
1 2
∂ω ∂K 2
9
For a pair of polarizable particles the friction force (32) thus becomes
πβ
F = −G v H δ(ω −ω ), (43)
fl lq q 1 2
2
with H given by Eq. (40) and G by Eq. (28).
lq
For the more simple model studied in Ref. [10] the G v is replaced by
lq q
G = (∇ψ)(v · ∇ψ). The result (19) of that reference is recovered if the
(~ω )2α (a = 1,2) is replaced with ~2/m in the expression (40) for H.
a a a
This replacement follows from a corresponding change in g˜ (K) as given by
a
Eq. (21) to the one given by Eq. (4.10) in Ref. [14].
It is now straightforward to obtain the friction F between a polarizable
h
particle and a half-space. One can simply replace the G v in Eq. (43) with
lq q
G v, where v is the velocity of the particle parallel to the plane and G is
h h
given by expression (37). Thus
3πρ πβ
F = − H δ(ω −ω ). (44)
h 2z5 2 1 2
0
Likewise, with two half-spaces moving relative to each other the G is re-
h
placed with G given by Eq. (38) to obtain the friction force per unit area
3π πβ
F = − ρ ρ H δ(ω −ω ). (45)
8d4 1 2 2 1 2
4 General polarizability
For simple harmonic oscillators the polarizability is given by Eq. (21). How-
ever, itcanbeamoregeneral functionofK thatmayberegardedasresulting
from a sum of harmonic oscillators. Thus we may write
2
h(K ) = g˜ (K) = α , (46)
a aK
where it can be shown [22] that the function h(K2) satisfies the relation
α (m2)m2
2 a 2
h(K ) = d(m ), (47)
K2 +m2
Z
with
1
α (m2)m2 = − Im[h(−m2 +iγ)], (m = ~ω = −iK, γ → 0+). (48)
a
π
10