Notas de Materngtica - Saber ULA

Universidad de 10s Andes Facultad de Ciencias Departamento de Matemdtica

Tools for Stochastic Electrodynamics

Gianfranco Spavieri and Luis Nieves

Notas de Materngtica Serie: Pre-Print No. 195

MCrida

- Venezuela 1999

Notas de MatemLtica, No. 195 MBrida. 1999.

Tools for Stochastic Electrodynamics Guia de Matemitica para la Electrodinimica Clbica y EstocLtica Gianfranco Spavieri and Luis Nieves

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Fourier series Fourier integral Three dimensional Fourier transform and antitransform A convenient expression of the Dirac delta function 6(r) Parseval's theorem Another form for the Fourier transform A stochastic vector field function of r and t expressed by its Fourier transform 9. Correlation function and power spectral density 10. Application to the electric noise 11. Diffusion coefficients 12. Mean square dispersion 13. The stochastic harmonic oscillator 14. The diffusion of a free particle in SED 15. Diffusion in SED plus spin 16. Markovian diffusion and corresponding power spectral noise 17. Spin motion 18. The harmonic oscillator in stochastic electrodynamics

1.

Introduction

The application of stochastic processes to physics becomes more and more important. One of the branches of electrodynamics where stochastic processess are relevant is the so-called Stochastic Electrodynamics (SED). T h e original aim of SED was to derive quantum mechanics from classical physics, the probabilistic nature of quantum mechanics being related t o stochastic behavior of charges particles and fields. Later, developments of SED led it t a become a branch of electrodynamics in itself. In particular, there is an alternative approach, called "stochastic electrodynamics plus spin" [I], t o atomic and particles physics, and even to the foundation of the relativity theory, which heavily uses the stochastic processes. However, the mathematical tools used in SED have general validity and may be useful in other areas of physics and also in engineering. On the other hand, there are no books and papers which treat simply, yet rigorously, the mathematical tools necessary for SED in a way dedicated t o it. Moreover, the limits L of integration are often included in the Fourier transform f so that ?diverges for stochastic problems. Even Boyer [2], one of the founders of SED, and his followers [3] use this inclusion so that he is forced t o state that the average of the product of two cosines gives a Dirac delta function S which is not a pure number (it is dimensional). Moreover, Boyer's average has an infinite value where the argument of the S vanishes, instead of having the correct value 0.5. It is therefore useful t o summarize the mathematical tools in a way dedicated t o SED. Only the first two pages of next section are taken from a standard textbook [4].

2.

Fourier series

Consider a function f (x) of the real variable x, defined on the interval - L 5 x < L. It is an important mathematical result that f ( x ) , when suitably restricted, can be expanded in a series of trigonometric functions. Thus [4]

where


3

and a, and b, are properly chosen numerical coefficients. We can determine an by multiplying f (x) by cos(m I( x)/L and integrating from -L t o +L. In fact. since

where 6,,

= I for n = m and 6,,

= 0 for n

# m and

only one term in the sum is nonzero, and we find that

In a similar way we find bn by multiplying f (x) by sin(m I( x)/L and integrating from -L t o +L. Using Eq. (4) and dx sin ( m I( x) sin ( n K x) = L 6,,

,

(6)

we again find that only one term in the sum is nonzero and

The series of Eq. ( I ) ,with coefficients an and bn given by Eqs. (5) and (7), is known as the Fourier series. Had we considered a function f (t) of time defined in the interval -T 5 t < T we would have obtained the same expression substituting K x for o t , where o = 2nlperiod = 2n/(2T) = TIT . (8) It is useful t o write the Fourier series as a series of complex exponentials instead of trigonometric functions. To accomplish this, we write Eq. (1) as

Letting n

+ -n

in the last sum of terms, we obtain


4

where

for n > 0 , forn=O, for n < 0

1

5 (an - ibn) 1

+ ib-,)

(11)

with a n = a_, and bn = b-,. Multiplying through both sides of Eq. (10) by exp(-i mKx) and integrating gives dx eiKx (n-m) +W

eiKx (n-m)

n=-cu

E

=

Cn2L

sin [n (n - m)] n(n-m)

n=-cu

whence

This equation could also be obtained by Eqs. (5) and (7). Another approach consists in writing Eq. (1) in the form

x

+cu f (x) =

en cos(nI 0 for n = 0 for n < 0


5

+

Since Re {exp [i(nKx .9,)]) = Re {exp [i(-nKx - .9,)]), the last term of (13) becomes, letting n + -n,

Consequently, we have

Finally, we may write the equality

with Cn and cn related to each other by Eq. (15). All what obtained in this section remains valid for a periodic function f (x) with period 2 L.

3.

Fourier integral

The form (10) of the Fourier series [with the coefficients given by Eq. (12)] is particularly useful when we wish to extend the interval -L 5 x < L to the entire x axis by considering the limit L + oo. Define k=nK=n-

n L

'

Ann n Arc=--L L

sinceAn=l,

C(k)=Cn (19)

and write Eqs. (10) and (12) as

with the understanding that k = nn/L and n takes on all the integral values from n = -oo to n = +oo. In the limit for L + oo so that Ak = n/L + 0, the sum in Eq. (20) becomes an integral. Moreover, setting


6

Eqs. (20) and (21) become f (x) =

Ifdk f ( k ) e"'

j(k) = 2n

1 -m

,

dx f (3) e - l k X .

These integrals are known as Fourier integrals. The functions f (3) and f (k) are said t o be the Fourier transform of one another. The Fourier transform is one of the most powerful mathematical tools in the repertoire of theoretical physics. Had we used t and w as in Eq. (8) (instead of x and k), we would have obtained

T h e introduction of the frequency v = w/(2n) and the position dwf (w) = dv j ( v ) [so that j ( v ) = 2 n f (w)] further simplifies Eqs. (25) and (26)

where

An alternative position, more convenient than (22), is

By it, and instead of (23) and (24), we have )

= L-tm

()

112 + m

j ( k ) eik'ak = lim

k=-m

and j ( k ) = lim

L-tm

L-tm

(E)

112

lm

L 1 dx f (x) e-ikx 2 ( a ~ ) 1 / 2J - L

+m

dk j ( k ) e"x

.

,

7


4.

Three dimensional Fourier transform and antitransform

It is straightforward to extend Fourier transforms to functions of several variables. For instance, consider f ( r ) = f (x, y, z ) . We may transform with respect to each of the variables, generalizing Eq. (23)

= Similarly

m-z

d3k f (k) eik"

-

. d3r f (r) e-ik'r

1

.

If we use the three-dimensional position corresponding to (29), we generalize Eqs. (30) and (31)

and y(k) = lim L+OO

( nL ) - 3 / 2 8

d3r f (r) e-ik'r

.

To include L in f (k), as done in Eq. (24), is convenient if f (k) is finite for L + oo. But if f(x) does not vanish at infinity, then f ( k ) diverges. For example, in a steady-state random process with zero average, Eq. (24) leads to an T(k) diverging as L1I2 (for L + oo). In the same case, but three-dimensional, Eq. (33) diverges as L3I2. On the contrary, Eqs. (29) and (35) give finite results. In any case what matters is the ensemble average square value which, for ergodic processes, is equal to the square value averaged over a large volume. Expressing f (r) by its Fourier transform (34) we obtain 1

d3r f (r)

d3kt

j% (kt) ei(k-k').r 7

where j'(k) is the complex conjugate of ?(k) since, to perform the square of a complex number, we must multiply it by its complex conjugate.


8

5.

A convenient expression of the Dirac delta function 6(r)

If we take f ( x ) = 6 ( x ) in Eq. (24) we obtain the Fourier transform a ( k ) of 6(x) 1 6 ( k )= - . (37) 2n Then, its antitransform (23) gives the wanted expression

Similarly, if we take f ( r ) = 6 ( r ) in Eq. (35) we obtain

X(k) = lim

L-+m

1 8 ( nL ) 3 / 2 '

(39)

Substituting Eq. (39) into Eq. (34) gives

Obviously, L disappears and (40) is the exact three-dimensional form of (38).

6.

Parseval's theorem

If we substitute k for r into Eq. (40) we obtain

Using (41) into (36) gives

( f 2 c r ) ) = lim

L+00

I+m -m

3 k 1 f .( k l )6 ( k - k l ) =

mf

d 3 k 7 ( k ),

-00

(42) which is Parseval's theorem, allowed, in its simplest form ( 4 2 ) ,by the position (33). Had we included L completely in T ( k ) [as done in (22)],then L would not appear in (35) and a factor L-3 would remain in (42). In general, if we include a factor D in (33), the same D appears a t the denominator of (34) and a t the nominator of (35), so t h a t the product of the coefficients of the integrals (34) and (35) always gives ( 2 ~ ) - ~ .


9

The average squared value of a function (usually proportional to an energy) may be written as the integral over propagator space of a function p(k) called the "power spectral density"

(f ( r ) )=

I+m

d3k p(k) .

-00

Comparing (42) with (43), we obtain

i.e:, the power spectral density is equal to the square of the Fourier transform. Some authors [5, 61 introduce the Fourier transform by Eq. (22), instead of Eq. (29). Consequently, their r ( k ) diverges for a stationary stochastic process and Eq. (42) should include L-3 in its r.h.s. However Eq. (43) is universally written in the same way so that they [5, 61 arrive at the relationship p(k) = lim L- 3 -2f * (k) L+00

which, implying some infinite, is questionable, as just questioned by those authors [5, 61. It is possible to show that Eq. (45) is valid but, in any case, the position (29) eliminates any problem at its very bases.

7.

Another form for the Fourier transform

Let us find the expression of the Fourier antitrasform corresponding to (18). By Eq. (19), with cn/2 for C,, and

Eq. (18) becomes f (z) = L+00 lim

=

L+W lim

)(: )(:

112

+00

g(k) ~e {e'[kx+J(k)l Ak k=-00 112

/__

+00

}

}

dk g(k) Re {e'[kx+J(k)I

.

(47)


10

8.

A stochastic vector field function of r and t expressed by its Fourier transform

Consider the random electric field E ( r , t) radiated by all spinning particles of our expanding universe. It is stochastic since the orientations of the spin axes and the positions of the particles on their spin orbit are random. If we fix the time, for instance t = 0, we may use Eq. (47) put in vector form

E(r,O) = L+OO lim

(~)312~fmm

d3kg(k) Re{e i[k.r+J(k)])

(48)

This Fourier antitrasform, if E is real, has a real transform g(k), differently from (34) which reads,in present notations, E ( r , 0) = lim L+OO

(4)m fIm 312

d3k T(k) eik.'

,

with T(k) a complex function. The relation between T(k) and g ( k ) can be derived from (15) and (18) f(k) =

+

g ( k ) [cos 6(k) i sin 6(k)] for k g ( k ) [cos 6(k) - isin 6(k)] for k

>0 < 0.

(50)

When 6 ( k ) = 0 and g(k) = ( T / L ) ~ we / ~ obtain again (40) from (48), (49), and (50). We may also interpret (48) as the superposition of e.m. waves. For t # 0 the whole phase depends systematically on t and we must add -wk t = -k c t to the exponent. Since the particles are very far compared to the radius of the spin orbit, the velocity field (which is radial) is negligible with respect to the radiation field so that each wave component is transversal and its direction can be expressed by two unit vectors gA(k,A) depending on the two possible polarizations A. We can therefore write

having assumed h(k) independent of X for statistical reasons. Consequently, a random, or transverse, field can be expressed as

(4) Cm fIm 312

E ( r , t ) = lim L+W

2

~ = l

I

d3ki(k, A) h(k) Re [ei(k'r-kctfd(kJ)


11

which generalizes Eq.(4) of Boyer [2] to the case that h is anisotropic. Actually, in our case h depends on k and not only on k. Boyer [2] writes the notation of real part Re outside the integral (the Re of a sum, or of an integral, is equal to the sum of the real parts). The additional factor ( L / T ) ~ / ~ does not appear in Eq.(4) of Boyer who has implicitly included it into his h(k) thus implying an extremely high value of h(k), diverging for L + co. It is therefore much convenient to keep this factor as done in (52). Let us now calculate the average energy density U taking into account that, for e.m.wave, E = B, so that

To calculate E2we may either consider the square of real parts (implying cosines as done by Boyer), or eliminate Re in E l denote it by Ec and then use 1 1 (54) (E2) = (E:) = 2 (E, . EZ) where EE is the complex conjugate of E,. Boyer [2] has used the first choice while we use here the second choice. We therefore obtain by Eq. (52), (53), and (54)

x

d3k' i ( k , A) . i ( k l , A') h(k) h(kl)

x exp {i [(k - k') . r - (k - k') ct + 6 ( k , A)

- 6(k1,A')]

Since L no longer appears in (55), we may take the limit for L (55) becomes, using (41),

)

. (55)

+ co so that

The integral gives zero unless A = A', in which case the exponential gives unity so that


12

having summed over polarizations in the last step since the integral no longer depends on A. The second step of Eq.(8) of Boyer [2] is equal to Eq. (57) with the exception of h2(k)for h2(k). The k dependence in our case prevents the last two steps of Eq. (8) of Boyer. Instead of using (54) we can use Boyer's method thus deriving from (52)

u

= -I I 4r8L3

I). (k)3 2 -

d3k

X=lX'=l

x h(k) h(kl) cos [k . r - k ct -

d3k

d3k' i ( k , A) . i ( k l ,A')

+ 6 ( k , A)] cos [k' . r - k' ct + 6(k1,A')]

KT

d3k' i ( k , A) . O(kl,A') h(k) h(kl)

d3r 1 {COS[(k - k') r ( 2 4 32 a

-L

I+m -00

-O0

+ (k' - k) ct + 6 ( k , A) - .9(k1,A')]

+ cos [(k + k') . r - (k' + k) ct + 8 ( k , A) + .9(k1,A')]

) .

(58)

Equation (41) gives 3S(k- k') Sxx1for the first cosine and would give i S ( k + k') for the second cosine if .9 = 0. But for k = -k' it is 6 ( k , A)+.9(-k, A) # 0 and cos [.9(k, A) .9(-k, A)] remains whose average value is zero. Consequently, we obtain (57) again. Notice that, in general,

+

1 +O0 1 d3r - {COS[(k - k') . r] ( 2 ~ ) ~ 2 1 = - [S(k - k') + 6 ( k + k 1 ) ] . 2 -

9.

K,

+ cos [(k + k')

. r] )

Correlation function and power spectral density

Consider a vector field E ( r ) , either independent o f t or a t a fixed time t = 0, which is a stochastic variable with r. Its (auto)correlation function is defined as C ( R ) = (E(r) .E*(r

+ R ) ) = L+m lim

1

KL

(2L)3

L

d3r E*(r) - E ( r

+ R ) . (60)


13

Substituting Eq. (48) into (60), using Eq. (54), a n d passing t o t h e limit for L + oo gives

which, by Eqs. (41) a n d (44), reduces t o

Regarding Eq.(61) as a Fourier antitrasform we c a n obtain p(k) as t h e Fourie r transform. Notice t h a t L does not appear in (61) so t h a t it is similar t o (32) whose Fourier transform is given by (33). In o u r case we have

which gives t h e power spectral density p(k) once known t h e correlation function C ( R ) . Since h2(k) = p(k) is a n even function of k,, k,, k,, we may reduce t h e integrations in Eqs.(61) and (62) from 0 t o oo. In fact

x

Similarly

[L + dk,

dk,] eikz " p(k)

L+OO

14


Let us find the reduction of Eqs. (61)-(64) in the one-dimensional case, for instance when the correlation C is a function of time t

The so called Wiener-Khintchine relationship (66) is the one used in the electronic noise, when C ( t ) is the correlation function either of the current ' I ( t ) flowing inside of, or of the voltage V(t) across a resistor. Then Eq. (66) gibes the power spectral density of the electronic noise.

10.

Application to the electric noise

Consider a conductor with plane geometry, having length L and two conducting planes as its ends. Inside the conductor there are N free electrons, the s-th one having velocity v, and charge e. Take the x axis superimposed to the symmetry axis of the conductor. Then Ramo's theorem gives [7], for the current I ( t ) flowing inside the conductor [I(t) is the same flowing in the wire feeding the conductor], if v, > wo, we put y = w the other terms so that ( 1 1 1 ) becomes

- wo

and w = wo in

(for SED), we have

The corresponding average energy is 1

-(m(vi)

2

h 1 + ~ ( x f )=) A ( x i ) = m wo2 = -hwo 2mwo 2

,

(115)

which is the zero-point energy. For the dispersion of a harmonic oscillator in SED see the Appendix. 14.

The diffusion of a free particle in SED

If A = 0 in Eq. (102) and - B v is substituted by (110),we have the equation of motion of a free particle in SED. On the x axis

To put A = 0 means t o put wo = 0 in (111) which diverges for w + 0 , thus showing that the diffusion increases with T . In this case it is important to examine the dispersion by Eq. (93). By the same procedure of Eq. (101) we derive -4 P x = w-2 PdxIdt = Pd2~ldt2 . (117) We obtain by (104) and (116)


23

whence

where p ( w ) is given by ( 1 1 3 ) . Then Eqs. ( 9 3 ) , ( 1 17), and (118) give

The solution has been given by Eq. ( 8 ) of Ref. [12] which, in our notations, reads

where R L = e 2 ( m c 2 ) - ' = 2.818 x 10-l3 cm is the Lorentz radius and Re = h ( m e ) - ' = 3.861 x lo-'' cm the electron spin radius, r0 = 2 R L / 3 c , and, for r / r O>> 1 ,

E; being the exponential integral

For r / r O>> 1 we derive from (120) and (121)

where (Y is the fine-structure constant. This diffusion is so slow that even after one second it is

i.e., ( ~ ~ ( 1 ) ) ' 2' ~1.492 x lo-" radius.

cm, viz. one thousandth of an atomic

24

Tools lor Stochastic Electrodynamics

15. Diffusion in SED plus spin If all the charged particles have a gyration motion which explains all the spin properties (as shown in Ref. [I]), the center of the gyration motion responds only t o forces parallel t o the spin axis ii so that the equation of motion, neglecting the radiation damping, is

When an electromagnetic wave impinges on a particle with spin and with ' E parallel t o n , then a is maximum, the velocity acquired very high, but th'e total displacement very low since the half wave with E antiparallel t o n compensates the displacement due t o the preceding half wave with E parallel t o n. However, if n rotates with w, the wave with w acts strongly on the particle since it can happen that only half wave is effective (if, during the first half wave, n is roughly parallel t o E ) and the subsequent half wave is ineffective (if, during this second half wave, n is roughly perpendicularly t o E ) . There is no longer the compensation and the particle flies away with the velocity acquired in a half wave. This is the basis for the explanation of diffraction by one or twoslits [I, 131. Once undergone the transversal impulse in the confined region near the slits, the subsequent motion is still almost rectilinear and uniform with the so called "balistic diffusion", with (Ax2) a t2. The superposition of SED diffusion as given by (123) is negligible.

16.

Markovian diffusion and corresponding power spectral noise

A typical Markovian process is that of the drift and diffusion of gas molecules or of free electrons in slightly ionized gases. The steady-state process is t h a t of the velocities and not of the positions. Another example is the Brownian motion in which a small body receives random impulses from the much smaller molecules of a gas or of a liquid. Let us schematize the motion by free flights I and instantaneous collisions and denote by X the mean free flight (I). If I = A and v is the same for all the molecules, the maximum memory of a flight is t,, = X/v. Actually, the free flights have a distribution

I- -1

1 dn s ( ~ )= no dl = X-'

exp

(-k)

,

so t h a t CdXldt(t)= (vX(0) VX(~)) = (':(o))

ex~(-tv/X)'

(I26]

25


Then by (66) we obtain

iii 1

P(W) =

+0°

- (v;(o)) 27r

-l

+0°

dt (v:(~)) exp

(-):

{[ -v cos(wt) exp (-I); dt--

cos(wt) v2 w2 ~2 exp

cos(wt)

+0°

0

(-:)I .

If we factorize v2/(w2X2) in the last integral, it is equal t o the second step of Eq.(127) so that

which is the famous Lorentzian. For w 117, (A.30) and (A.31) tend to AX' E - y h r n - ' ~ , (A.32) and

np2N y mtwoT/2 .


35

T h e linear dependence of t h e fluctuations on T is typical of a Brownian motion. Consequently, t h e probability distributions are determined by t h e mean square values only, i.e.,

and

where (x2) and (p2) are given by (A.28) and (A.29), respectively. If we multiply (A.34) with (A.35) and consider the joint probability as factorizable in t h e two variables x and p, we obtain

-

1

(--

exp 2~ ((2" > P ~ ) ) ~ / ~

-)

x 2 - p2 2 (x2) 2 (P2)

.

(A.36)

For t h e harmonic oscillator in t h e ground state, such joint probability is given, neglecting t h e Lamb shifts appearing in (A.28) and (A.29)

In Ref. [16], some factors 2 in excess appear in Eqs. (5.24) and (6.2). They are t o be present in t h e wave function 11, but not in pO(x) = 11,11,*. Also, a factor 2 is in excess in Eq. (4.11) of Ref. [16], another 2 is a t t h e numerator instead of at t h e denominator of Eq. (4.12), and still another 2 is missing in Eq. (4.17). Summarizing, for T > 2r/ywo, i.e., for w