fulltext

A free particle under the action of a zero-point field

and a small perturbation

M. BATTEZZATI(*)

Istituto di Cibernetica e Biofisica del CNR - Via de Marini 6, 16149 Genova, Italy (ricevuto il 4 Marzo 1996; approvato il 25 Marzo 1997)

Summary. — The main scope of this article is to give an exposition, which is both

simplified and satisfactory, of the results that were obtained in BATTEZZATI M., Nuovo Cimento B, 108 (1993) 559, which was concerned with the same problem, namely, the steady-state solutions for the probability density distribution of a unidimensional system under the action of a zero-point field and a small static perturbation. These solutions belong to a diffusion equation, whose parameters are to be evaluated by the procedures which were described in full details in previous articles by the same author. A semi-Eulerian representation is used for the velocity field, whose deterministic part, which has been proved to satisfy a Hamilton-Jacobi-Riccati equation, represents the drift velocity of the system. The full equations of motion in Lagrangian representation are here solved in the presence of a zero-point field without external potential, so as to obtain a representation of the response functions satsfying the prerequisites which are necessary in order to evaluate the diffusion coefficient by the present method. The further developments considering a nonvanishing potential are therefore obtained through a limiting procedure, exploiting the typical stability theorems of classical mechanics. Consequently, it is shown that, in the presence of a static perturbation, a resonance phenomenon is mainly responsible for diffusion. Nonresonant terms reproduce fairly well the Lamb shifts obtained from quantum electrodynamics.

PACS 11.10 – Field theory.

PACS 03.20 – Classical mechanics of discrete systems: general mathematical aspects. PACS 12.20.Ds – Specific calculations.

PACS 05.40 – Fluctuation phenomena, random processes, and Brownian motion.

1. – Position of the problem

The problem to be discussed here is analogous to that one which was treated in [1], therefore only the main points need to be recalled here. A free particle in an

(*) Present address: Istituto di Cosmo-Geofisica del CNR, Corso Fiume 4, 10133 Torino, Italy. 1479

1480

electromagnetic zero-point field (ZPF), or a zero-point field of different nature, is considered in a three-dimensional configuration space. Consequently, the energy spectrum of the stochastic force results, from the Wiener-Khinchin theorem, to be proportional to

fi j(v) 4mˇtcNvN3di j

(1.1) _{i , j 41, 2, 3 .}

fi j(v) is the Fourier transform of the autocorrelation function of the stochastic

electric force Ei, i 41, 2, 3, fi j(t 2s) 4 aEi(t) Ej(s)b 4



2Q 1Q fi j(v) e2iv(t 2 s) dv 2 p . (1.2)

m is the mass of the particle, ˇ is Planck’s constant divided by 2 p, tcis a time constant

for the interaction, v is the frequency, t and s are time coordinates, and di jdenotes the

Kronecker symbol. In terms of the constants specific of the particle’s e.m. nature, there results tc4 2 3 e2 mc3 , (1.3)

where e is the e.m. charge, and c is the velocity of light in free space.

The brackets in eq. (1.2) denote, as usual, the stochastic averages over the realizations of the stochastic force E(t) of the quantities enclosed therein.

According to Dirac [2], the motion of the particle is described by the following ordinary differential equation of the third-order with respect to time derivatives, for the variable x indicating the position of the particle:

2tcx ... 1xO1 1 m ¯n ¯x 4 1 mk(t) , (1.4)

where n(x) is the static perturbation, here supposed to be small, and k(t) the Lorentz force on the moving particle from the stochastic field, which is Gaussian with zero mean.

In the following the renormalized stochastic electric force b(t) will be used, whose autocorrelation function has the spectral density

fb(v)i j4 ˇtc m NvN3 1 1t2 cv2 di j4 fb(v) di j. (1.5) There results [3] fb(t) 42 ˇ mptc 1 t2 2 1 t2 c fa(t) , (1.6) fa(v) 4 ˇ_tc m NvN 1 1t2cv2 . (1.7)

two integrals over the real axis of the variable h , for DQ (t) E 0, respectively: fb(t) 4 ˇ mptc



2Q 0 dh he6ht 2 ˇ pmtc



– 2Q 0 dh he 6ht 1 2t2 ch2 4 2 ˇ_tc pm



– 2Q 0 dh h 3 e6ht 1 2t2 ch2 . (1.8)

2. – Position of the problem: the equations of motion of the particle

The system to be considered here consists of a particle in a ZPF as described in sect. 1, with the further assumption of separability in Cartesian coordinates. Then the one-dimensional canonical coordinate and momentum are denoted by q and p, respectively. The potential energy profile is assumed to be flat inside the coordinate interval ( 0 , l), and changing suddenly to a high value U at the extremities. Then the limiting cases U KQ and eventually lKQ can be considered. The ideally flat potential profile can be assumed corrugated by a small periodic static perturbation n(q) (which eventually includes also the effect of the boundaries). That will be proved to be essential in order to ensure stability for the system in steady states.

The equations of motion in the variables q , p can be deduced from the Hamilton-Jacobi equation 1 2 m

g

P 2 e cA(t)

h

2 1 n(q) 2 qk(t) 4 2 ¯ ¯tS, (2.1)

P _{being the generalized momentum, S the action, and defining (eOc) A(t)42mt}cq

O . As k(t) is reducible, to first-order in tc, to the sole electric component of the force,

eq. (1.4) can be solved by putting, along each dimension,

p(q , t) 4p(q)1pA(t)

(2.2)

which, upon substitution, yields the following equations:

2 tc m2

(

p(q) 2 p 9(q)1p(q) p 8(q)2

)

₁ 1 mp(q) p 8(q)1n8(q) 4g(q) , (2.3)

g

1 2 tcd dt

hg

d dtpA1 1 mp 8(q) pA(t)

h

4 k(t) 1 tc m

(

p 9(q) p(q)1p 8(q) 2

)

pA(t) 2g(q) , (2.38)

where g(q) is an arbitrary smooth function. Then the system of ordinary differential equations of first order in two variables follows, to O(tc2),

(2.4)

.

`

/

`

´

dq dt 4 1 m

(

p(q) 1pA(t)

)

, d dtpA42

g

1 mp 8(q)1b(q)1 t2 c m d dtn 9(q)

h

pA(t) 2g – (q) 1

g

1 2 t 2 c mn 9(q)

h

mb(t)

1482 with b(q) 4 tc mn 9(q)1O(t 2 c) , (2.5) g_{(q) 4} tc mg 8(q) p(q)1g(q)1O(t 2 c) , (2.58) b(t) 4 1 mtc



t Q k(a) exp

k

t 2a tc

l

da . (2.6)

Equation (2.4) can be solved formally by the following expression:

(2.7) pA(t)

k

_{1 1} t 2 c mn 9

(

q(t)

)

l

4 pA0(t) 2



2Q t ds pA0(s)

g

¯ ¯s 2 i v – (s) 2b–(s)

h

_{g(t , s) 2} 2



2Q t ds g(t , s)

k

_{1 1} t 2 c mn 9

(

q(s)

)

l

g –

₍

q(s)

)

,

i v–(s) and b–(s) being arbitrary smooth functions and

g

d dt 1 i v – (t) 1b–(t)

h

pA0(t) 4mb(t) , (2.8) g(t , s) 4exp

y

2 1 m



s t p 8

(

q(a)

)

da 2



s t b

(

q(a)

)

da

z

. (2.9)

The initial conditions must be accounted for by specifying the initial value of the coordinate at the time t0, and the initial value of momentum at the time t 80, which

subsequently K2Q. The problem of finding that trajectory which connects two given points q08 and q0 at the times t08 and t0 is solvable. The action function is Hamilton’s

principal function S(q0, t0; q08 , t08 )]k(s)( and its derivatives yield the momenta at the

starting point and at the point of arrival. The boundary condition ¯S

¯_{q 8}0

4 2 p(q 80) ,

(2.10)

corresponds manifestly, through (2.2), to requiring pA(t08 ) 4 0 . The starting point q08 of

that trajectory, if it exists, is obtained from the solutions to eq. (2.10), for given q0, t0.

Therefore, the solution of the present problem is given by that solution of the time-dependent HJ equation which assumes the boundary values W(q0) at the time t08.

Then it is possible to select that trajectory which ends in (q0, t0), if it exists for a given

]k(s), t08EsEt0(, among the characteristic curves with the given boundary conditions.

From eqs. (2.4), (2.7) the diffusion equation in configuration space can be obtained by stochastically averaging, for the two-time conditional probability density P2(q, tOq0, t0).

The stochastic averages are conveniently calculated by using Novikov’s theorem, though the original proof of this theorem requires the analyticity of the solutions to the

equation of the motion as functions of the values of the stochastic forces at different instants of time. This analyticity is not generally true because the system can be nonintegrable, even in one dimension, when time-dependent perturbations are

added [4]. However, more general proofs of the theorem which do not require

analyticity have been given [5], by functional integration techniques.

3. – The evaluation of the diffusion coefficient using Kubo’s expansion of the response functions

It is the purpose of this paragraph to yield a general proof of the results concerning the value of the diffusion coefficient of a one-dimensional particle in a ZPF, by means of a representation of the response functions by a cumulant expansion [6]. The diffusion coefficient, defined by means of the average value of the fluctuating component of the velocity multiplied by a d-function of the coordinate, has been shown to be given by the approximate formula (3.1) D×(t, t0)ad

(

q(t) 2q

)

b 4 4



t0 t dt

»

d

(

q(t) 2q

)

exp

y

₂



t t b

(

q(a)

)

da

z



2Q t ds



– 2Q t ds g(t , s) g(t , s)

«

fb(s 2s) .

The operator notation reminds that D×(t, t0) is actually an operator acting upon the

variable q. Equation (3.1) has been obtained in the frozen-trajectory approximation (FTA) to the response function dq(t) OdpA0(s) [7]. In the present context there results

(3.2) d 2 dt2 dq(t) dpA0(s) 1 b

(

q(t)

)

d dt dq(t) dpA0(s) 1 V

(

q(t)

)

2 dq(t) dpA0(s) 4 4

g

d dt 1 1 mp 8

(

q(t)

)

1 b

(

q(t)

)

h

1 m

u

dpA(t) dpA0(s)

v

q ,

where, for the exact and FTA response functions, one has, respectively, (3.3) V

(

q(t)

)

2 4 1 mn 9

(

q(t)

)

1 d dtb

(

q(t)

)

1 O(t 2 c) , (3.38) VFTA

₍

_q(t)

₎

2 4 2 1 m2

[

p 9

(

q(t)

)

pA(t) 1p 9

(

q(t)

)

p

(

q(t)

)

1 p 8

(

q(t)

)

2 1 1mb

(

q(t)

)

_{p 8}

(

q(t)

)]

; it follows from eqs. (2.3), (2.5), (2.58) that the r.h.s. of eqs. (3.3), (3.38) are nearly the same if the term 21Om2

p 9

(

q(t)

)

pA(t) is averaged over the realizations of the stochastic

forces.

The r.h.s. of eq. (3.1) can thus be evaluated by first expanding the exponential function, then taking the logarithm of the resulting expression and expanding again.

1484

Therefore the product of exponentials results in

(3.4)

»

d

(

q(t) 2q

)

Q Q exp

y

2 1 m



s t p 8

(

q(a)

)

da 2 1 m



s t p 8

(

q(a)

)

da 2



s t b

(

q(a)

)

da 2



s t b

(

q(a)

)

da

z

«

` `_ad

₍

_{q(t) 2q}

₎

_{b exp}

y

₂



t t b – (a) da 2



s t [i v–_{(a) 1b}–_{(a) ] da 2}



s t [i v–_{(a) 1b}–(a) ] da

z

Q Q exp

y

!

n42 Q (21)n n!

»y

1 m



s t p 8

(

q(a)

)

da 1 1 m



s t p 8

(

q(a)

)

da

z

n

«

c

z

,

where the stochastic averages a R bc denote the cumulant averages as defined, for

instance, in ref. [6], iv–(a) and b–(a) being defined accordingly. The cumulant average of order n is defined as the sum of all the averaged terms appearing in the expansion of the logarithm of the averaged exponential function, which are made of products of n factors ( 1 Om) p 8(q)1b(q). Although the logarithmic expansion has a limited convergence radius, the cumulant expansion of the averaged exponential can be considered valid whenever the argument assumes a finite value. In eq. (3.4) small terms involving higher powers of b

(

q(a)

)

have been neglected, as well as long-range correlations thereof, consequently, the term exp

[

s

ttb

–

(a) da

]

can be factorized as shown. The averaged cumulants can be conditioned,

(3.5)

o

d

(

q(t) 2q

)

1 mp 8

(

q(a)

)

p

4 i v –_(a)ad

₍

q(t) 2q

)

b , (3.58)

o

1 mp 8

(

q(a)

)

1 mp 8

(

q(a 8)

)

p

c 4 4

k

1

m2ad

(

q(t) 2q

)

p 8

(

q(a)

)

p 8

(

q(a 8)

)

b 2iv

–_{(a) i v}–

(a 8) ad

(

_{q(t) 2q}

)

b

l

ad

(

_{q(t) 2q}

)

b21

and so on.

For a stationary process in which all the correlations vanish in the limit t c t0, Kubo

has shown that the limiting form of the expression (3.4) is a simple exponential function. This is also a consequence of Birkhoff’s and Koopman’s theorems, which hold for autonomous systems (attempts to extend these ergodic properties to weakly interacting systems however exist [8]). The ZPF can be considered as equivalent to a collection of coupled harmonic oscillators, on a hyper-surface of constant energy for the particle-plus-bath system. This model has been displayed in ref. [9] in some detail.

The short-time behaviour of the expression (3.4) can be estimated as follows, provided p(q) satisfies the Hamilton-Jacobi-Yasue-Riccati (HJYR) equation [10]. Upon

differentiating twice, this yields (3.6) 1 mp 8(q) 2 1 1 mp(q) p 9(q)1n9(q)1D0pR(q) 4 4 2 tc

k

d dq 1 mp(q) n 9(q)1D0n S(q)

l

1 O(t 2 c) .

Defining D0as the constant part of D×(q), from the relation

o

1 mpA(t) p 9

(

q(t)

)

p

4 aD ×

₍

q(t)

)

t p R

(

q(t)

)

_{b 1O(t}c) BD0apR

(

q(t)

)

b (3.7)

there follows, because of stationarity, 1

map 8

(

q(t)

)

2

b B2 an9

(

q(t)

)

b (3.8)

and in the same way, proceeding as in ref. [7] (3.9)

o

d

(

q(t) 2q

)

1 mpA(t) p 9

(

q(t)

)

p

4 ad

(

q(t) 2q

)

D –

(

t , t , t0; ]k(s)() pR

(

q(t)

)

b 1 1ad 8

(

q(t) 2q

)

–D

(

t , t , t0; ]k(s)(

)

p 9

(

q(t)

)

b 4 4 D0ad

(

q(t) 2q

)

p R

(

q(t)

)

b 1 ad8

(

q(t) 2q

)

D –

(

t , t , t0; ]k(s)(

)

p 9

(

q(t)

)

b 1O(tc) and, consequently, (3.10) 1 mad

(

q(t) 2q

)

p 8

(

q(t)

)

2 b 4 4 2 ad

(

q(t) 2q

)

n 9

(

q(t)

)

b 2

o

d

(

q(t) 2q

)

d dtp 8

(

q(t)

)

p

1 1ad 8

(

q(t) 2q

)

–D

(

t , t , t0; ]k(s)(

)

p 9

(

q(t)

)

b 1O(tc) .

The notations have been employed as in ref. [7]. From eq. (3.8), and from eq. (3.10), by neglecting the r.h.s., it follows that for s AsAt, in the absence of external potential (n 40), (3.11)

»

exp

y

₂ 1 m



s t p 8

(

q(a)

)

da 2 1 m



s t p 8

(

q(a)

)

da

z«

_B B exp

y

2



s t i v–_{(a) da 2}



s t i v–_{(a) da 2} 1 2



s t da



s t da 8 v–(a) v–_{(a 8)2}

1486 21 2



s t da



s t da 8 v–(a) v–_{(a 8)2}



s t da



s t da 8 v–(a) v–_{(a 8)}

z

₄ 4 1 2



s t i v–_{(a) da 2}



s t i v–_{(a) da 1O(Nt2sN}3_{, Nt2sN}3)

which proves that the iv(a) 4 (1Om) p8

(

q(a)

)

is slow modulated for short time intervals [6]. In fact, the r.h.s. of (3.11) can be deduced by assuming a stationary probability distribution for iv(a) 4 (1Om) p8

(

q(a)

)

. Then, the diffusion coefficient in the limit t 2t0c1 Ob can be calculated from (3.1) by writing, according to notations

employed in [10, 7], (3.12) a D

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` D×(t, t0)ad

(

q(t) 2q

)

b 4 4



t0 t dt exp

y

₂



t t b–(a) da

z



2Q t ds



– 2Q 1Q ds ad

(

q(t) 2q

)

g(t , s) g(t , s)b Q Q fb(s 2s)2 a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b .

This representation results to be appropriate whenever the average of the product of

g-functions on the r.h.s. is analytic in the lower half-plane of the complex variable s. In

this case, the autocorrelation function of the variable b(t) is conveniently represented by the following expansion in powers of tc:

fb(s 2s) 4

!

n 42 Q _ˇ 2 pm

y

( 2 n 21)! tc2 n 23 (s 2s1ie)2 n 1 ( 2 n 21)! tc2 n 23 (s 2s2ie)2 n

z

. (3.13)

Therefore, the integral over ds can be evaluated by the method of residues applied to each separate term in (3.13). Although the expansion (3.13) is not convergent, nevertheless it yields the correct value for scalar products



2Q 1Q

f (t) fb(t) dt

between fb(t) and every analytical function f (t) defined on the real axis, whose

spectrum is bounded by the inequality

NvN E 1

tc

. (3.14)

The integral over ds in eq. (3.12) is thus evaluated by completing the path of integration through an arc of a semicircle centred in the point s and lying entirely in the lower half-plane, on the left of a straight line from t to infinity with angular coefficient Ntg uNE1, u being the angle between the real axis and the straight line. Considering that the dominant contribution to the integral is given by the region around the poles of (3.13), and keeping account of the first term of the expansion only,

there follows (3.15) a D

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 4 4 2iˇtc m



t0 t dt exp

y

₂



t t b – (a) da

z



2Q t ds

y

¯ 3 ¯s3 1 t 2 c ¯5 ¯s5 1 R

z

Q Q ad

(

_{q(t) 2q}

)

g(t , s) g(t , s)bs 4s2 a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` ` 2iˇtc m

{



t0 t dt exp

y

₂



t t b–(a) da

z

¯ 2 ¯s2ad

(

q(t) 2q

)

g(t , t) g(t , s)bs 4t1 1



t0 t dt exp

y

₂



t t b – (a) da

z



2Q t ds

g

₂ ¯ ¯s ¯2 ¯s2

h

ad

(

q(t) 2q

)

g(t , s) g(t , s)bs 4s

}

2 2a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 42 iˇtc 2 m



t0 t dt exp

y

₂



t t b – (a) da

z

Q Q

{

2 ¯ 2 ¯s2ad

(

q(t) 2q

)

g(t , s)bs 4t2 ¯ ¯s ¯ ¯s ad

(

q(t) 2q

)

g(t , s) g(t , s)bs 4s4t

}

2 2a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 42 iˇtc 2 m



t0 t dt exp

y

₂



t t b – (a) da

z

Q Q

o

d

(

q(t) 2q

)

k

g

1 mp 8

(

q(t)

)

1 b

(

q(t)

)

h

2 1 2 d dt

g

1 mp 8

(

q(t)

)

1 b

(

q(t)

)

h

l

p

2 2a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` ` iˇtc 2 m



t0 t dt exp

y

₂tc m



t t daad

(

q(t) 2q

)

n 9

(

q(a)

)

b ad

(

q(t) 2q

)

b

z

Q Q

k

ad

(

q(t) 2q

)

n 9

(

q(t)

)

b 1 ¯ ¯q ad

(

q(t) 2q

)

D –

(

t , t ; t0; ]k(s)(

)

p 9

(

q(t)

)

b

l

2 2a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` iˇ 2 m

y

1 2exp

y

2



t0 t b – (a) da

zz

ad

(

q(t) 2q

)

b 1 1iˇtc 2 m ¯ ¯q



t0 t dt exp

y

₂



t t b – (a) da

z

ad

(

q(t) 2q

)

Q Q D–

(

t , t , t0; ]k(s)(

)

p 9

(

q(t)

)

b 2 a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b .

1488

The expansion has been stopped to O(tc2) because second-order terms in b(q) have not

been retained. To leading order this is, however, identical to the result which was obtained in ref. [10]. The main advantage of the present procedure is that it allows to factorize out of the stochastic averages the damping factor exp

[

2

s

ttb

–

(a) da

]

in eq. (3.12), which yields the dominant contribution to the resonant integral over dt . The results show clearly that it is independent of the detailed structure of the dissipative forces, being only determined, in stationary conditions, by the average ab(q)b. Equations (3.8) and (3.10) show that, since in stationary conditions p(q) can be required to be purely imaginary on the real axis, the phase average of the second derivative of the potential energy is positive, which corresponds to physical intuition, and moreover guarantees the stability of the system through the sign of the exponent in the dissipative factor of (2.9). The second cumulant in the dissipative factor has been neglected because it is second order in b(q) and second order in n(q). Moreover, this term should be small in a steady state, if the ergodic property holds, because then only the first cumulant average, which includes the phase average, survives in the limit of large time.

The remaining terms in the result (3.15) have been evaluated in refs. [10] and [1], where the case of a small periodic perturbation superimposed to a flat potential profile is considered. To lowest order with respect to the ZPF, secular terms oscillating with zero mean value appear, multiplying the first derivative of the density. If however higher-order fluctuations of ZPF are taken into account, the oscillations result to be bounded. However, it is generally consistent to truncate the Kramers-Moyal expansion after the second term; consequently, this term is not considered here. The last term in (3.15) was calculated in detail in [10]. It results to be absolutely convergent only provided the response function is bounded. Specifically if, for some real number M,

(3.16)

N

»

d

(

q(t) 2q

)

exp

y

₂



t t

b

(

q(a)

)

da

z

g(t , s) g(t , s)

«

N

_{E MNad}

(

q(t) 2q

)

b N inside the lower half-plane of the variable s, along the path of integration defined above

(

see eq. (3.15)

)

, then there results

Na R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b NE5k3 M

ˇ

mNad

(

q(t) 2q

)

b N .

(3.17)

Expanding the averaged function which is inside the brackets in eq. (3.16) in a series of exponentials in the variables t 2s, t2s, with coefficients g(t, t; v–m, v–n), the

following result is obtained provided the poles of the Laplace transforms are invariant with respect to time for t K1Q:

(3.18) ad

(

q(t) 2q

)

R

(

t , t0; ]k(s)(

)

b C C 2 ˇ_tc pm m , n

!

v –3 m1 3 v–2mv–n (v–m1 v–n)2 i lnNtcv–mNad

(

q(t) 2q

)

g(t , t ; v–m, v–n)b .

Equation (3.18) represents a small imaginary correction to the diffusion coefficient which, when inserted into the HJYR equation, produces a correction to the eigenvalues

of the order of the Lamb shift. It can be estimated by putting ln NvmN A ln Nv–nN, therefore (3.19) ad

(

_{q(t) 2q}

)

R

(

t , t0; ]k(s)(

)

b B B 2 ˇ_tc pm m , n

!

ad

(

q(t) 2q

)

g(t , t ; v – m, v–n)b i 2(v – m1 v–n) ln Ntcv–mN 4 4 2 ˇ_tc pm2ln NtcvNap 8(q) d

(

q(t) 2q

)

b ,

where v is an average oscillation frequency. Inserting this correction to the diffusion coefficient into the HJYR equation yields, since the contributions from dp(q) vanish on averaging, the energy shift

dE B

o



dD(q) p 9(q) dq

p

4 ˇ_tc

2 pm2ln NtcvNap 8(q) 2_{b .}

(3.20)

Adding the contributions from each separate spatial dimension, and using (3.8) (3.21) _{dE B} ˇtc 2 pmln

N

1 tcv

N»

¯2n ¯x2 1 ¯2n ¯y2 1 ¯2n ¯z2

«

4 ˇtc 2 pmln

N

1 tcv

N

a˘2_{n(x)b .}

For the harmonic oscillator in the ground state because p 9(q) vanishes, p 8(q) is a constant such that D0p 8(q) 4 (1O2) hv–, therefore

dE B ˇtc 2 pmln

N

1 tcv

N

3 mv2 (3.22)

and for a Coulombic potential n(x) 42 1ONxN

dE B ˇtc 2 pmln

N

1 tcv

N

4 pad(x)b (3.23)

and for an ns state (n principal quantum number)

dEns4 2 ˇtc pm

g

1 na

h

3 ln

N

1 tcv

N

, (3.24)

where a is Bohr radius. For a 1 s state

E1 s4 16.456 3 1091 1.085 lnN1O vN3109c.p.s. ,

(3.25)

where v is in atomic units. This result is numerically similar to the unrenormalized nonrelativistic calculation of Seke [11], including retardation effects.

Equation (3.24) has been included here so as to show that the present method is susceptible to yield consistent results, although, presently, there is not a proof avail-able for the applicability of the procedure to Coulombic systems (see, however, sect. 6).

1490

4. – Solution of the equations of the motion for the free particle

In this paragraph the solution of the equations of motion for a free particle subjected to the action of a ZPF, and eventually to a small static perturbing potential, is calculated. If the free particle is enclosed between two reflecting walls, then at each reflection the symmetric trajectory with respect to the wall must replace the original one, so that for this new trajectory no reflections whatever occur. This procedure must be repeated at each reflection, and finally yields a trajectory for a particle which is acted upon by the ZPF only, but is not confined between the two walls. The ZPF also undergoes the same reflection process, but this will not affect its statistical properties appreciably if the walls are sufficiently far apart, and therefore the reflections sufficiently separated in time, with respect to the correlation time of the field. It follows that reflections which are separated by intervals of time smaller than the correlation time of the ZPF are statistically insignificant. From the above argument it follows that many statistical properties will be the same for a free particle and for a particle enclosed between two walls sufficiently far apart, inasmuch as, for each trajectory enclosed between the walls, there exists an “equivalent” free trajectory ending at the same point. However, if reflections are taken into account, some among the formulae below may need to be modified accordingly. Let us therefore suppose that a particle whose HJYR equation is (5.1) obeys the equation of motion [3]

k

d dt 1 tc mn 9

(

q(t)

)

l

dq dt 4 2 1 mn 8

(

q(t)

)

1 b(t) , (4.1) therefore (4.2) q._{(t) 4} 1 m



t 80 t ds exp

y

₂tc m



s t n 9

(

q(a)

)

da

z

[

mb(s) 2n8

(

q(s)

)]

1 1 q.(t 80) exp

y

2 tc m



t 80 t n 9

(

q(a)

)

da

z

. Now, this system is unstable because dissipation is stronger when it has negative sign, since at the top of potential wells

(

n 9(q) E0

)

, the mean velocity is lower. Therefore the mean value in time

1 t 2t



t t n 9

(

q(a)

)

da (4.3)

is negative, in the absence of external force. However, this effect is second order in

n(q), because the difference in velocity is itself proportional to the excursion of the

potential n(q). Therefore, in order to stabilize the system, further effects must be invoked, like dipolar radiation due to the velocity inversion in collisions, which was evaluated in [1] in the low-frequency limit, following ref. [12]. This effect was discovered by Boyer [13].

In fact it can be shown that if stationary states with substantially imaginary drift velocity exist, which means low current, then the average value of n 9(q) must be positive, as can be guessed intuitively. The proof follows from (3.8). This confirms the

arguments displayed above on stabilization by velocity inversion and collisions. In fact, purely imaginary drift velocity on the real axis means zero current [1].

Therefore, it is assumed that it is allowable to put

lim t 80K 2Qq . (t 80) exp

y

2 tc m



t 80 t n 9

(

q(a)

)

da

z

_{4 0} (4.4) and (4.5) _{q(t) 4q}01



t0 t ds



2Q s ds e2b–(s 2s)

k

_{b(s) 2} 1 mn 8

(

q(s)

)

l

4 4 q01 1 2e 2b–(t 2t0) b –



2Q t0 ds e2b–(t02 s)

k

_{b(s) 2} 1 mn 8

(

q(s)

)

l

1 1



t0 t ds1 2e 2b–(t 2s) b –

k

b(s) 2 1 mn 8

(

q(s)

)

l

.

Equation (4.4) is justified if one assumes that the stationary state under study exists from t 42Q to the time of observation. This requires a specification of the initial distribution of coordinate probability density at time t0as well as its joint distribution of

the stochastic component of the velocity, because the system is not Markovian in the coordinate space. If the initial conditions are selected arbitrarily, a mechanism for damping must be specified in order that eq. (4.4) retains its validity.

5. – The spectrum of the response function

Supposing the leading terms of D×(q) to be bounded, which holds under the assumptions underlying eqs. (3.15) and (3.17), then the HJYR equation reads [1]

1 2 mp(q)

2

1 n(q) 1 D0p 8(q) 4E ,

(5.1)

where D0stands for the leading terms of the diffusion operator. Here the interest is in

approximate solutions of eq. (5.1) in the limits tcK 0 and n(q) K 0. These are

pg_{(q) 4} 2 imD0k(e ikq 2 ge2ikq) eikq_{1 ge}2ikq , (5.2) p 8g(q) 4 28 gmD0k2 (eikq 1 ge2ikq)2 (5.28) and so on.

The parameter k, which is a wave number multiplied by 2 p , must be determined so as to satisfy the boundary conditions, as discussed previously. g determines the average flux, as shown in [1]. A real number of modulus less than 1 will be assumed,

1492

without loss of generality. Then the response function g(t , s) can be written: (5.3) _{g(t , s) 4exp}

y

8 gD0k2



s t e22 ikq(a)_da

(

1 1ge22 ikq(a)

₎

2 2



s t b – (a) da

z

₄ 4 exp

y

8 gD0k2

!

l 40 Q (2g)l (l 11)



s t da e22 ik(l 1 1 ) q(a) 2



s t b – (a) da

z

. since Nge22 ikq(a)

N E 1 because q(a) is a real number by eq. (4.5). Now, exploiting the Gaussian property of q(a) inherited from the stochastic properties of b(t), it is possible to evaluate the average of (5.3) by a cumulant expansion. The only function which is needed is the autocorrelation function of the variable q:

aq(a) q(a 8)bc4 aq(a) q(a)b 2 aq(a)baq(a 8 )b .

(5.4)

This function, in the limit a 2t0, a 82t0K 1Q, has the following form, which is

obtained from (4.5) by putting n(q) 40, therefore q(a) Bq( 0 )(a): (5.5) aq( 0 )(a) q( 0 ) (a 8)b 4q2 01 1 b –₃

{

b – (a 2a8)



a 2a8 1Q djfb(j) 1b –

_

2Q a 2a8 dj jfb(j) 2 21 2e 2b–(a 2a8)



2Q a 2a8 dj eb–j_f b(j) 2 1 2 e b–(a 2a8)



2Q a 82a dj eb–j_f b(j)

}

2 1 b –₃



0 1Q dj(e2b–j 1 b–j) fb(j) . Therefore, (5.6) aq( 0 )(a)2 b 4q2 02 2 b –₃



0 1Q dj(e2b–j 1 b–j) fb(j) 4q021 2 ˇtc pm ln 1 tcb – , (5.7) lim a 2a8K1Qaq ( 0 )_{(a) q}( 0 ) (a 8)b 4q2 01 ˇtc pmln 1 tcb – .

The integrals appearing in eq. (5.5) are evaluated making use of eq. (1.8). Expanding the denominators in powers of t2c, yields the asymptotic values

(5.8)



z Q dj fb(j) 4 ˇ_tc pm



– 2Q 0 dh6h 2 e6hz 1 2t2 ch2 4 ˇ_tc pm 2 z3 1 O

g

1 z5

h

, DQ(z) E 0 , (5.88)



Q z dj jfb(j) ` 3 ˇtc pm



– 2Q 0 dh he 6hz 1 2t2ch2 4 2 ˇ_tc pm 3 z2 1 O

g

1 z4

h

, DQ(z) E 0 , (5.89) e2b–z



Q z dj eb–j_f b(j) 4 4 ˇ_tc pm



– 2Q 0 dh Zh 3_e6hz ( 1 2t2 ch2)(h 6b – ) 4 2 ˇ_tc pm 2 z3 1 O

g

1 z5

h

, DQ(z) E 0 .

a) Damped free particle

These asymptotic expansions can be verified by making use of (3.13). The evaluation of (5.3) therefore is obtained as follows:

(5.9) _{ag(t , s)b 4exp}

y

8 gD0k2

!

l 40 Q (2g)l (l 11)



s t

daae22 ik(l 1 1 ) q(a)_{b 1}

11 2( 8 gD0k 2₎2

!

l 40 Q

!

l 840 Q (2g)l 1l 8_{(l 11)(l 811)}



s t da



s t da 8Q

Q [ae22 ik(l 1 1 ) q(a)_e22 ik(l 8 1 1 ) q(a 8 )_{b 2 ae}22 ik(l 1 1 ) q(a)_bae22 ik(l 8 1 1 ) q(a 8 )_{b] 1}

1 1 3 !( 8 gD0k 2₎3

!

l 40 Q

!

l 840 Q

!

l 940 Q (2g)l 1l 81l 9_{(l 11)(l 811)(l 911)Q} Q



s t da



s t da 8



s t

da 9[ae22 ik(l 1 1 ) q(a) 2 2 ik(l 8 1 1 ) q(a 8 )_e22 ik(l 9 1 1 ) q(a 9 )_{b 2}

23ae22 ik(l 1 1 ) q(a)bae22 ik(l 8 1 1 ) q(a 8 ) 2 2 ik(l 9 1 1 ) q(a 9 )_{b 1}

12ae22 ik(l 1 1 ) q(a)bae22 ik(l 8 1 1 ) q(a 8 )_bae22 ik(l 9 1 1 ) q(a 9 )_{b] 1R2b}–_{(t 2s)}

l

_.

In eq. (5.9) the stochastic averages are computed over the realizations of the stochastic force k(t) and, eventually, over the given initial distribution of coordinate values in configuration space. If one assumes a normalized initial distribution of the form [1]

Pk(q0) 4

1

l( 1 1g2₎Ne

ikq0_{1 ge}2ikq0_N2_,

(5.10)

then (5.9) reduces to (3.11) and there results exactly

(

see c)

)

ag(t , s)b 4

g

1 2



s t iv–(a) da

h

e2b–(t 2s) (5.11) and (5.118) ad

(

q(t) 2q

)

g(t , s) g(t , s)b 4 4

g

1 2



s t iv–_{(a) da 2}



s t iv–(a) da

h

e2b–( 2 t 2s2s)_ad

₍

_{q(t) 2q}

₎

_{b ,}

since all the cumulants of n-th order reduce to products of n averages taken independently (uncorrelated averages). The diffusion coefficient can be evaluated

1494

using the response function (5.118) and yields, according to (3.5), (5.12) a D

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` `



t0 t dt



2Q t ds



2Q t ds

(

1 22iv–t 1iv–(s 1s)

)

e2b–(t 1t2s2s)_f b(s 2s)ad

(

q(t) 2q

)

b 4 4

y

iv – b –3



0 1Q dj e2b–j_f b(j) 1 1 b–2



0 t 2t0 dj(iv–_{j 21) f}b(j)

z

ad

(

q(t) 2q

)

b 1O

g

1 (t 2t0)2

h

4 4 2 ˇ_tc pmiv –_ln 1 tcb – ad

(

q(t) 2q

)

b 1O

g

1 (t 2t0)2

h

.

This is so because the argument leading to the result (3.15) retains its validity although the response functions are not bounded in the lower half-plane of the variable s, the contribution from the pole in this case being O(b–), consequently, the result is O(tc), as

follows from [10], and [1].

Equation (5.118) is based on the fact that the correlation function of the variable

p 8

(

q(t)

)

does not vanish in the limit of large-time difference in the arguments, Na 2 a 8N K 1Q. The mixing property [14, 15] does not hold for the variables q(t) and

p 8

(

q(t)

)

, unless the perturbation is taken into account.

The correlation function (5.5) has the same limiting behaviour along every direction of the complex plane as Na2a8NK1Q. Consequently, the term a R

(

t , t0; ]k(s)(

)

Q

d

(

q(t) 2q

)

b can be evaluated by following any path which joins the points 2Q with t, lying entirely inside the lower half-plane of the variable s.

b) Small perturbing potential

It is possible to examine how eq. (5.12) changes if a small static perturbing potential

n(q) is added to the system. The expression (5.118) inserted into (3.12) yields a pole

term in the variable s 2s, which contributes to the integral. There follows from (3.15) (5.13) a D

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 4 4 iˇtc 2 m



t0 t dt e2b–(t 2t)

o

_d

₍

_{q(t) 2q}

₎

k

₂ 1 m2p 8

(

q(t)

)

2 1 O(tc)

l

p

2 2a R(t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ,

where the first term on the r.h.s. comes from the pole in the variable s 2s, while the second one results from the integration over the variable s in the lower half-plane, so as to close the path clockwise from t to minus infinity. The pole term yields a contribution to the diffusion coefficient, which is O(tc), if n(q) is zero, as results from eq. (5.12).

Putting now n(q) finite, it is assumed that the averaged quantities are modified continuously for n(q) sufficiently close to zero. Consequently, the various terms in

(5.13) also vary continuously, so that n(q) can be selected so small as to make (5.14) _{Na R}n(t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 2

2a R0

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b NEeNad

(

q(t) 2q

)

b N ,

where e is an arbitrary positive quantity, and the subscripts refer to trajectories computed with putting n c 0 and n 40, respectively, the damping factor being left unchanged. Consequently, neglecting terms of higher order in tc,

(5.15) a D

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 4 iˇ 2 m[ 1 2e 2b–(t 2t0)_]ad

(

_{q(t) 2q}

)

_{b 1} 1iˇtc 2 m ¯ ¯q



t0 t dt e2b–(t 2t)_ad

₍

_{q(t) 2q}

₎

–_D

₍

_{t , t , t} 0; ]k(s)(

)

p 9

(

q(t)

)

b 2 2a Rn

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b ` iˇ 2 m[ 1 2e 2b–(t 2t0)_]ad

₍

_{q(t) 2q}

₎

_{b 2} 2a R0

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b 6eNad

(

q(t) 2q

)

b N .

Since e is arbitrary, from eq. (5.15) it follows that (5.16) lim n K0D×(t, t0) 4 iˇ 2 m[ 1 2e 2b–(t 2t0)_{] 2 a R} 0

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

bad

(

q(t) 2q

)

b21

for every t Dt0. In order to prove (5.14), it is sufficient to show that the expression

under the triple integral sign in (3.1) is a continuous function of n(q) in the lower half-plane of the variable s. The pole in the point s 4s2ie is, however, discontinuous in the limit t K1Q since the residue changes abruptly, as can be seen from (3.15).

c) The spectrum of response for a damped free particle

The following facts have therefore been established. The coordinate q(t), with

n(q) f 0 , is normally distributed, so that the two-point correlation function alone

determines all the averages. For large time, a , a 8K1Q, this function only depends upon the difference a 2a8, and moreover is analytical in its argument for Na 2 a 8N K 1Q, its asymptotic expression being deducible from (5.5), (5.7), (5.8), (5.88), (5.89). This expression has a limiting constant value which is independent of the direction in the complex plane. Consequently, it is possible to evaluate

(5.17) ¯ ¯s ag(t , s) 2_{g(s , s)b e}b–(s 2s)₄ 4 1 map 8

(

q(s)

)

g(t , s) 2_{g(s , s)b e}b–(s 2s)₄ 1 me 22 b–(t 2s)

!

l40 Q

!

m40 Q ₍₂₁₎l 1m l! m! 2 l_Q Q

»

p 8

(

q(s)

)

y

1 m



s t p 8

(

q(a)

)

da

z

l

y

1 m



s s p 8

(

q(a 8)

)

da 8

z

m

«

4

1496 4 e22 b – (t 2s)_{8 D} 0k2

!

l 40 Q (2g)l 11_{(l 11)}

»

exp [22ik(l11) q(s) ]

!

l40 Q

!

m 40 Q ₂l l! m! Q Q

y

!

l14 0 Q 28 D0k2(2g)l11 1(l11 1 )



s t

da exp [22ik(l11 1 ) q(a) ]

z

l Q Q

y

!

l 814 028 D 0k2(2g)l 811 1(l 811 1 )



s s

da 8 exp [22ik(l 811 1 ) q(a 8 ) ]

z

m

«

4 4 e22 b – (t 2s)_{8 D} 0k2

!

l 40 Q (2g)l 11_{(l 11)aexp [22ik(l11) q(s) ]b}

»

!

l 40 Q ₂l l!(28D0k 2₎l_Q Q

y

!

l140 Q (2g)l11 1_(l 111 )



s t

da exp

[

22 ik(l111 )[q(a)22 ik(l11 )aq(a) q(s)bc]

]

z

l

!

m40 Q 1 m!(28D0k 2₎m_Q Q

y

!

l 814 0 Q (2g)l 811 1_{(l 8} 11 1 )



s s

da 8 exp

[

22 ik(l 811 1 )[q(a 8 ) 2 2 ik(l 1 1 )aq(a 8 ) q(s)bc]

]

z

m

«

4 4 8 D0k2e22 b – (t 2s)

!

l 40 Q (2g)l 11_{(l 11)aexp [22ik(l11) q(s) ]b Q} Q

»

exp

y

₂ 2 m



s t p 8[q(a)22ik(l11)aq(a) q(s)bc] da 2 2 1 m



s s p 8[q(a8)22ik(l11)aq(a8) q(s)bc] da 8

z«

,

where the variable q(a) 22ik(l11)aq(a) q(s)bc is a Gaussian random variable which is

obtained from q(a) by the time-dependent complex-valued translation by the “length” 22 ik(l 1 1 )aq(a) q(s)bc

Dq(a) 42 2ik(l11)aq(a) q(s)bc.

(5.18)

It follows that the expansion leading to the result (5.17) can be resummed in the way indicated for every a which satisfies

Nexp [22 ik Dq(a) ] N 4 Nexp [24 k2(l 11)aq(a) q(s)bc] NG1 .

(5.19)

These inequalities are well verified for a Ks and in the limit a2sKQ, as one can find out by eqs. (5.6), (5.7). By putting, for 2QErE1Q,

(5.20) F_{(r , s) 4e}22 b–(t 2s)_Q Q

o

exp

y

₂ 2 m



s t p 8[q(a)22ikraq(a) q(s)bc] da 2 1 m



s s

there follows the system of homogeneous differential equations (5.21) ¯ ¯s F(r , s) C8D0k 2

!

l 40 Q (2g)l 11_{(l 11)Q}

Q aexp [22ik(l11) q(s) ]b exp [24k2

r(l 11)aq(s)2_b

c] F(r 1l11, s) ;

eqs. (5.21) follow from (5.17) for r F0, but can be retained as a mere definition for

r E0. Diagonalization of the matricial kernel of this system yields the eigenvalues

which are the time constants of the proper linear combinations of functions F(r , s) which decay exponentially. Redefining system (5.21) as

(5.22) ¯ ¯s F(r , s) 48D0k 2

!

m 4r11 Q (2g)m 2r_{(m 2r)Q} Q exp [22k2_(m2 2 r2)aq(s)2_b c2 2 ik(m 2 r)aq(s)b] F(m , s)

it is convenient to define, for the discrete variable r,

(5.23) F_{(r , s) 4} k 2 pe 2 k2_r2 aq(s)2bc



0 2 pOk dy C(y , s) e2 ikry, _{r 40, 61, 62, 63, R ,} C_{(y , s) 4}

!

r 42Q 1Q F(r , s) e22 ikry 2 2 k2r2aq(s)2bc_. (5.238)

By this substitution, system (5.22) becomes ¯

¯sC(y , s) 4

1

mp 8g

(

aq(s)b 2y

)

C(y , s) .

(5.24)

The proper initial conditions are taken into account by solving (5.238) with s4t0. Then,

in stationary conditions, one obtains the simple solution to (5.24) (5.25) F_{( 0 , s) 4 ag(t, s) g(t, s)b e}b–(s 2s)` ` k 2 p



0 2 pOk dy exp

y

₂ 1 m



s s

ds 8 p 8g

(

aq(s 8)b2y

)

z

C(y , s) .

In steady state, this expression must be subsequently averaged over the stationary probability density. The average value of the coordinate aq(s)b is calculated from eq. (4.5) and is sensibly a constant as long as n(q) is negligible and the effects of the walls are absent. Therefore one can put

(5.26) 1 mp 8g

(

aq(s)b 2y

)

4 1 mp 8g(q02 y) 4 4 28 gD0k 2 [ ( 1 1g2_{) cos 2 k(q} 02 y) 1 2 g 2 i( 1 2 g2) sin 2 k(q02 y) ] [ 2 g 1 (11g2_{) cos 2 k(q} 02 y) ]21 ( 1 2 g2)2sin22 k(q02 y) .

1498

The real part of this expression vanishes only in the limit NgNK1. If 0 EgE1 it can be seen that the modes such that 0 E2k(q02 y) E p are unstable, while those with the

sign of q02 y reversed are stable, the vice versa being true for 21 E g E 0 . This

requires some mechanism of equilibration, which can be found in the displacement of aq(s)b by collisions with the walls or with the perturbing static potential, and by an average drift velocity, in case of nonvanishing current. The perturbations make the system lean toward chaotic behaviour because of the effect of secular terms, which, as was shown in (3.15), add together in the course of time so as to enhance diffusion by a factor proportional to 1 Otcv–

(

see eq. (5.12)

)

. The inequalities

cos 2 k(q02 y) D 2 2 g 1 1g2 , g D0 , (5.27) cos 2 k(q02 y) E 2 2 g 1 1g2 , g E0 (5.28)

define the intervals in which the imaginary part of the eigenvalues (5.26) is negative, so that the eigenvectors oscillate with positive frequencies. The sign of the eigenvalues is uniform in the limit NgNK1. For NgNE1 the situation is found to be similar to that which was described in ref. [1], eq. (2.15), inasmuch as regions with different behaviour alternate, with respect to initial conditions. Notice that if q0is situated around a region

of maximum probability density, as described by (5.10), then inequalities (5.27) are satisfied everywhere except possibly around the wings of the frequency spectrum, which belong to modes which are linear combinations of functions of r with rapidly alternating signs. These disappear as soon as the probability density distribution has begun to smear out, owing to collisions. Averaging (5.25) with (5.10), eqs. (5.11), (5.118) result.

The stationarity condition (3.8) entails that the average of n 9(q) must have the sign opposite to that of p 8(q)2_{, which, for a real probability density distribution, is negative.}

This results to be consistent with the condition for the stability of the trajectories. Supposing (5.10) to be the steady solution, it follows that n(q) must have terms of the form

n(q) 4Nn0N cos ( 2 kq 1 p) 1 n1q 1Nn2Nq2, g D0 ,

(5.29)

n(q) 4Nn0N cos 2 kq 1 n1q 1Nn2N q2, g E0

(5.298)

in order that the condition of stationarity



0 2 pOk

n 9(q) Pk(q) dq D0 (5.30)

should be satisfied. Quite happily the conditions (5.27), (5.28) of compatibility are consistent with the stationarity condition (5.30), which forces the particle to spend most of the time in the environment of the minima of n(q).

6. – Summary

In a preceding paper [10], it was proved that the known property of a harmonic oscillator in a ZPF that is forced to obey a diffusion equation which results to be

equivalent to a Schroedinger equation [3], can be enlarged to comprise those systems interacting with a ZPF, whose response function is endowed with suitable regularity properties. Here, a simplified version of that proof is displayed, which is more intuitively understandable, being founded upon a simplified representation of the response function. The diffusion coefficient results therefore to be evaluated quite accurately, since it is shown by an elementary calculation that the first corrections in powers of tc lead naturally to frequency displacements of the same functional forms as

the Lamb shifts.

In the present work the following properties of diffusion in ZPF have been elucidated, according to the behaviour of the function of the variable s

(

see eq. (5.20)

)

:

ag(t , s) g(t , s)b 4F(0, s) e2b–(s 2s)

(6.1)

in the lower half-plane. The result (3.15) is exact to leading order, so that the problem is reconducted to the evaluation of the remainder a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b. If F( 0 , s)

is bounded, then the remainder is uniformly bounded by inequality (3.17), and is given by (3.18), in the limit t K1Q, plus bounded oscillating terms with zero mean [10].

The diffusion coefficient is calculated for the damped free particle, for which the function F( 0 , s) has the form (5.118). Since the diffusion coefficient is a constant, the stationary probability density distribution (5.10) is recovered self-consistently. This results from (5.9), (5.10), and from (5.25) as well. As n(q) vanishes, the contribution to the diffusion coefficient in this case mainly comes from the remainder, eq. (5.12).

The spectrum of the function F( 0 , s) is calculated for the damped free particle

(

eq. (5.30)

)

. Notice that, for a linear system, the constraint given by d

(

q(t) 2q

)

does not modify the Gaussian character of the trajectories, so that the arguments leading to (5.17) retain their validity. Subsequently, a small perturbing static potential is introduced, satisfying the conditions for positive average dissipation

(

eq. (5.30)

)

. The continuity of the remainder is assumed, whereas the pole term yields, in the limit

t 41Q, a definite invariant contribution to the result, however small the perturbation.

This assumption of continuity is based upon the proof that the function F( 0 , s) preserves the same form (5.25) if the coordinate distribution around each average trajectory originating from a single point, in a steady state, is Gaussian. This remark allows possibly to extend these concepts to a particle in an arbitrary static potential in the regime of validity of the semiclassical approximation.

The assumption of continuity of the remainder with respect to small perturbations is also based upon the remark that the rate of convergence of the integral over ds in (3.12) is essentially controlled by the parameters of the ZPF, thus being independent of the static perturbation, and of resonant effects, which are acting during long time intervals. In fact, a R

(

t , t0; ]k(s)(

)

d

(

q(t) 2q

)

b can be evaluated by an integration over

ds entirely over or in the proximity of the real axis, indenting below the pole at the point s , and taking account of inequality (3.14). Then



2Q t ds F( 0 , s) e2b–(s 2s)_f b(s 2s) (6.2)

1500 in the norm, lim n K0



2Q t NFn( 0 , s) 2F0( 0 , s) N2Ne22 b – (s 2s)_{NdsN 4 0 ,} (6.3)

where the subscripts here again denote quantities evaluated with n c 0 and n 40

(

see eq. (5.14)

)

. Notice that the remainder in (5.16) is considerably smaller than the resonant term for any conceivable static potential and interaction constant tc. In fact

the two terms on the r.h.s. equilibrate for

tcv–ln 1 tcb – A1 , (6.4) which requires an 9(q)b 4 m t2 c exp

y

₂ 1 tcv–

z

A 9 4( 137 ) 6_exp

k

23 2 ( 137 ) 3

l

_, (6.5)

which is an exceedingly small quantity.

Therefore, it can be inferred that for physical values of parameters the diffusion equation that has been deduced for a quasi-free particle in a ZPF is valid, and, a

fortiori, in the limit tcK 0 .

R E F E R E N C E S

[1] BATTEZZATIM., Nuovo Cimento B, 108 (1993) 559.

[2] DIRACP. A. M., Proc. R. Soc. London, Ser. A, 167 (1938) 148. [3] BATTEZZATIM., Can. J. Phys., 68 (1990) 508.

[4] MELETLIDOUE. and ICHTIAROGLUS., Phys. Lett. A, 188 (1994) 157. [5] FOXR. F., Phys. Rev. A, 33 (1986) 467.

[6] KUBO R., in Fluctuation, Relaxation and Resonance in Magnetic Systems, edited by D. HAAR(Oliver and Boyd, Edinburgh) 1962, p. 23.

[7] BATTEZZATIM, Nuovo Cimento B, 110 (1995) 1287. [8] EKSTEINH., Phys. Rev., 107 (1957) 333.

[9] FORDG. W., LEWISJ. T. and O’CONNELLR. F., Phys. Rev. A, 87 (1988) 4419. [10] BATTEZZATIM., Nuovo Cimento B, 107 (1992) 669.

[11] SEKEJ., Physica A, 203 (1994) 284.

[12] LANDAUL. and LIFSHITZE., Théorie des champs (Mir, Moscow) 1970, Chapt. IX. [13] BOYERT. H., Phys. Rev., 182 (1969) 1374.

[14] KHINCHIN A. I., Mathematical Foundations of Statistical Mechanics (Dover, New York) 1949.