ISSN 1023-1935, Russian Journal of Electrochemistry, 2006, Vol. 42, No. 10, pp. 1026–1029. © MAIK “Nauka /Interperiodica” (Russia), 2006.Original Russian Text © B.M. Grafov, 2006, published in Elektrokhimiya, 2006, Vol. 42, No. 10, pp. 1141–1145.

1026

INTRODUCTION

Electrical noise, generated by the slow-dischargecurrent, is classified with white shot noise. Hence, the-oretical descriptions of such noise usually employ [1–3] the Schottky formula for the power spectrum

S

ofshot noise [4, 5]

S

= 2

eI

. (1)

Here,

e

represents an elementary charge and

I

is theabsolute value of the electric current. The Schottky for-mula holds for both the cathodic and anodic constitu-ents of the current of electrochemical reactions. Theformula has quite a number of theoretically importantapplications, but the question of an analogue of theSchottky formula that would describe polyspectra ofwhite shot noise has not been discussed broadly.

The aim of this work is to develop a theory forcumulant description of nonlinear properties of electri-cal noise generated by slow discharge. The work isbased on theory of integral stationary random processeswith independent increments [6, 7] and on higher-orderfluctuation–dissipation relationships [8].

CUMULANT FUNCTION OF RANDOM HOMOGENEOUS FLUXES

Mathematical theory of random homogeneousfluxes with independent increments [6, 7] operates withsuch a concept as cumulative number

n

of randomevents that occur during sufficiently long time period

t

.Consider a one-electron single-step electrochemicalreaction that obeys the kinetic equation of Butler–Volmer [9]

I

=

I

a

–

I

c

. (2)

Here, the anodic and cathodic constituents

I

a

and

I

c

ofthe overall current

I

are defined by equations

(3)

(4)

where

I

0

is the exchange current;

e

the elementarycharge

(

e

> 0)

;

E

the electrode potential, which isreferred to the equilibrium value;

k

Boltzmann’s con-stant; and

T

is the temperature. Symmetry factors

α

and

β

characterize the anodic and cathodic elementary acts,respectively, and give unity when summed:

α

+

β

= 1.The anodic flux of elementary acts

V

a

is related to theanodic discharge current

I

a

through the simple relation-ship

(5)

A similar expression links the cathodic electric current

I

c

and the cathodic flux of elementary acts:

(6)

The equation that defines the magnitude of theexchange flux

V

0

of an electrochemical reaction is ofthe same type:

V

0

=

I

0

/

e

0

. (7)

Within theory of homogeneous stochastic fluxeswith independent increments, one can introduce cumu-lative number of anodic elementary acts

n

a

and cumula-tive number of cathodic elementary acts

n

c

, which tookplace in the course of sufficiently long observation timeperiod

t

. Quantities

n

a

and

n

c

are random and their aver-

Ia I0 βeEkT------⎝ ⎠

⎛ ⎞ ,exp=

Ic I0 αeEkT------–⎝ ⎠

⎛ ⎞ ,exp=

Va Ia/e V0 βeEkT------⎝ ⎠

⎛ ⎞ .exp= =

V c Ic/e V0 αeEkT------–⎝ ⎠

⎛ ⎞ .exp= =

On Nonlinear Structure of Electrical Noise Generated by Slow Discharge

B. M. Grafov

z

Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119991 Russia

Received January 30, 2006

Abstract

—Nonlinear structure of electrical noise of slow electrochemical discharge is analyzed within theoryof integral random homogeneous fluxes with independent increments. It is shown that the degree of deviationof a flux of elementary acts from the Poissonian flux is defined by the magnitude of the factor of symmetry ofslow discharge. An equation that links polyspectra of slow discharge of first, second, and third orders is derived.

DOI:

10.1134/S1023193506100065

Key words

: electrochemical discharge, shot noise, polyspectra

z

Author’s email: bmg@elchem.ac.ru

RUSSIAN JOURNAL OF ELECTROCHEMISTRY

Vol. 42

No. 10

2006

ON NONLINEAR STRUCTURE OF ELECTRICAL NOISE 1027

age values

⟨

n

a

⟩

and

⟨

n

c

⟩

are equal to the product of aver-age fluxes

V

a

and

V

c

and the observation time

t

:

⟨

n

a

⟩

=

V

a

t,

⟨

n

c

⟩

=

V

c

t

. (8)

A full description of any random quantity (in ourcase,

n

) is given by its cumulants [10]. Relationships

(9)

serve as definitions for the first three cumulants

⟨

n

⟩

(

c

)

,

⟨

n

2

⟩

(

c

)

, and

⟨

n

3

⟩

(

c

)

of random quantity

n

. Quantity

V

in(9) is the average number of elementary acts that occurin a unit time. The angle brackets denote the averagingover an ensemble of temporal realizations, i.e. the meanvalue.

Any cumulant of the mth order may be determinedby means of the m-fold differentiation of cumulantfunction K(χ) at the point χ = 0:

(10)

In the case of a Poissonian flux of elementary acts V, allthe cumulants are equal to one another, and the cumu-lant function assumes the form [10]

K(χ) = Vt(expχ – 1). (11)The cumulant function of the Poissonian flux is a one-parameter function. Its sole parameter coincides withthe average number of elementary events Vt that occurduring sufficiently long observation time t.

In theory of integral homogeneous fluxes with inde-pendent increments [6, 7], the cumulant function

(12)

of a generalized Poissonian flux serves as direct gener-alization of cumulant function (11) of a Poissonianflux. Apart from parameter Vt, cumulant function (12)contains additional parameter, P. The latter is a factorthat determines the degree (measure) to which a gener-alized Poissonian flux deviates from a purely Poisso-nian flux. Parameter P may be labeled a P factor (Pois-sonian factor) of shot noise. For purely Poissonian shotnoise, the P factor is equal to unity.

The first two cumulants out of those defined bycumulant function (12) of the generalized Poissonianflux coincide with the first two cumulants of a purelyPoissonian flux no matter the P factor. This implies thatan expression for the power spectral density of noisegenerated by electric current, which would correspondto cumulant function (12), is always (and not only in thecase of a purely Poissonian flux of random events)defined by the Schottky equation for shot noise (1). Inother words, the fact that the Schottky formula isobeyed does not necessarily imply that the noise underinvestigation is a purely Poissonian random event.According to (12), the P factor for a Gaussian flux ofelementary acts is equal to zero. Nevertheless, theSchottky formula is still valid.

n⟨ ⟩ c( ) n⟨ ⟩ Vt, n2⟨ ⟩ c( )n n⟨ ⟩–( )2⟨ ⟩ ,= = =

n3⟨ ⟩ c( )n n⟨ ⟩–( )3⟨ ⟩ ,=

nm⟨ ⟩ c( ) dm

dχm---------K χ( ) χ 0= .=

K χ( ) VtPχ( ) 1– Pχ–exp

P2-------------------------------------------- χ+ ,=

The anodic and cathodic fluxes are characterizedeach by its pair of parameters: Vat, Pa and Vct, Pc. Eitherflux is also characterized by its cumulant function, spe-cifically, Ka(χ) and Kc(χ):

(13)

(14)

where the P factors Pa and PÒ are, generally speaking,dependent on the electrode potential. In the generalcase, formulas (13) and (14) solve the problem con-cerning nonlinear structure of both the anodic andcathodic constituents of shot noise generated by slowdischarge.

Note that the theory of random homogeneous fluxeswith independent increments [6, 7] has recently beenresorted to by the authors of [11] for analyzing nonlin-ear structure of a Wiener random process with non-Gaussian noise sources.

POISSONIAN FLUX OF ELECTROCHEMICAL DISCHARGE

The above naturally raises the problem connectedwith electrochemical identification of the P factor for ageneralized Poissonian flux. For its solution, it is desir-able first of all to know what type of an electrochemicaldischarge a purely Poissonian flux corresponds to.When analyzing dual fluctuation-dissipation relation-ships for the bispectra of equilibrium fluctuations ofelectric current and electrode potential in [8], we putforth an idea that both the anodic and cathodic equilib-rium fluxes of elementary acts are purely Poissonianonly in the case of a symmetrical slow discharge. Theimplication is that both anodic and cathodic P factorsare equal to unity (Pa = PÒ = 1) solely in conditions of asymmetrical slow discharge. Moreover, both cumulantfunctions (13) and (14) convert into the cumulant func-tion for a purely Poissonian flux (11) solely under theseconditions. It should be noted that the idea, accordingto which anodic and cathodic Poissonian fluxes corre-spond to a symmetrical slow discharge, warrants exper-imental verification. Of course it is desirable that thisidea be verified in the framework of modern theory ofelementary act as well [12, 13].

FLUCTUATION–DISSIPATION RELATIONSHIPS

Of essence is that, when at equilibrium, the P factorfor a cumulant function of a generalized Poissonianflux may be derived from the fluctuation–dissipationrelationship linking bispectrum Bi of an equilibriumfluctuation current to the derivative of the power spec-trum of the equilibrium fluctuation current S withrespect to the electrode potential. According to [8],

Ka χ( ) V atPaχ( ) 1– Paχ–exp

Pa2

----------------------------------------------- χ+ ,=

Kc χ( ) V ctPcχ( ) 1– Pcχ–exp

Pc2

----------------------------------------------- χ+ ,=

1028

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 42 No. 10 2006

GRAFOV

(15)

For independent fluctuations of anodic and cathodicconstituents of the overall electric current,

Bi = Bic – Bia, (16)S = Sa +Sc, (17)

Consequently, fluctuation–dissipation relation (15)splits into two expressions for the anodic and cathodicnoise:

(18)

(19)

Equalities (18) and (19) entail the following relation-ships between the 2nd- and 3rd-order cumulants thatcharacterize, respectively, the anodic and cathodicfluxes of elementary acts:

(20)

(21)

Fluctuation–dissipation relations (20) and (21) allow usto determine the magnitude of the P factor for theexchange flux of elementary acts of discharge in boththe anodic and cathodic directions.

CUMULANT FUNCTIONS FOR A SLOW DISCHARGE

Given (8), (5), and (6), simple differentiation ofcumulant functions (13) and (14) yields

(22)

(23)

(24)

(25)

When performing differentiation, we assumed that, forthe model of the electrode process under consideration,the P factors of electrochemical noise are independentof the electrode potential at least near equilibrium. Sub-stituting (22) and (24) into (20), then

Pa|E = 0 = 2β. (26)Similarly, on the basis of (21), (23), and (25) we have

Pc|E = 0 = 2α. (27)Thus, the P factor for shot noise generated by a slow

discharge at an equilibrium potential is twice the sym-metry factor, i.e. 2β and 2α for, respectively, anodic andcathodic directions of the exchange current. The P fac-tor for shot noise generated by activationless discharge[14] is equal to zero. Hence, noise of activationless dis-charge is purely Gaussian noise. For shot noise gener-

BiE 0= 4

kTe

------ ddE-------S

E 0= .–=

Bia E 0=4

kTe

------ ddE-------Sa E 0=

,=

Bic E 0=–4

kTe

------ ddE-------Sc E 0=

.=

na3⟨ ⟩ c( )

E 0=2

kTe

------ ddE------- na

2⟨ ⟩ c( )

E 0=,=

nc3⟨ ⟩ c( )

E 0=–2

kTe

------ ddE------- nc

2⟨ ⟩ c( )

E 0=.=

na2⟨ ⟩ c( )

V at,=

nc2⟨ ⟩ c( )

V ct,=

na3⟨ ⟩ c( )

PaV at,=

nc3⟨ ⟩ c( )

PcV ct.=

ated by barrierless slow discharge [14], P = 2. Symme-try factors α and β for a slow discharge range from zeroto unity. Consequently, the equilibrium value of the Pfactor for a random flux of slow discharge lies betweenzero and two.

Symmetry factors α and β for the electrochemicalreaction under consideration are independent of theelectrode potential. This gives us certain grounds toassert that the P factors for the noise generated byanodic and cathodic discharge (Pa, Pc) are also indepen-dent of the electrode potential. If this is true, relations(18), (19) and (26), (27) are valid not only at equilib-rium but at any steady-state electrode potential as well.Whether or not this assertion is true may be determinedon the basis of either statistical theory of elementary act[12–15] or by studying polyspectra of noise generatedby slow discharge experimentally.

POLYSPECTRA OF NOISE GENERATED BY SLOW DISCHARGE

Equations (13) and (14) of theory of integral randomhomogeneous fluxes with independent incrementsmake it possible to formally determine all the polyspec-tra for the noise generated by slow discharge in steady-state conditions, i.e. at a direct current. The first threepolyspectra for white shot noise generated by slow dis-charge have the following form:

S = 2e(Ia + Ic), (28)Bi = (2e)2(PcIc – PaIa), (29)

Tr = (2e)3( Ic + Ia). (30)

Here, Tr denotes the trispectrum of white shot noise. Inthe general case, the P factors for anodic and cathodicdischarges (Pa, Pc) in (29) and (30) are potential-depen-dent and may differ from doubled symmetry factorsrelating to the electrochemical reaction at equilibrium.

Equation (28) has long since been known: it wasrepeatedly derived form the Schottky formula for whiteshot noise. Equations (29) and (30) for the bispectrumand trispectrum of white shot noise are published forthe first time ever. Note that trispectrum (30) in theoryof integral random homogeneous fluxes with indepen-dent increments is a positive definite quantity.

At high overvoltages, practically the entire noisecorresponds to either anodic or cathodic constituent ofelectric current of slow discharge. Hence, experimen-tally studying anodic (cathodic) polyspectra far fromequilibrium in principle allows us to determine the Pfactor for anodic (cathodic) noise of electrochemicaldischarge with the relationship

(31)

This formula may be used for experimental assessmentof the degree to which the slow discharge-generatednoise deviates from the Poissonian noise. It also leadsto a very important conclusion concerning properties

Pc2 Pa

2

2ePBiS

--------TrBi

--------.= =

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 42 No. 10 2006

ON NONLINEAR STRUCTURE OF ELECTRICAL NOISE 1029

inherent in polyspectra for anodic and cathodic shotnoise far from equilibrium. The square of the bispec-trum for anodic and cathodic shot noise equals thepower spectrum times the trispectrum:

Bi2 = STr. (32)When far from equilibrium, similar relationships

take place also for higher-order polyspectra of shotnoise generated by anodic and cathodic slow dis-charges. At low (zero) frequencies, polyspectra for anycolored noise exhibit properties of white shot noise.This assertion is valid for quantum noise as well.

CONCLUSIONS

The analysis we performed in the foregoing leads tothe following conclusions.

(1) Mathematical theory of integral random homo-geneous fluxes with independent increments happens tobe indispensable for studying non-Gaussian structureof noise generated by slow discharge.

(2) In the general case, the noise that is generated byanodic and cathodic discharges differs from the Poisso-nian noise and represents a generalized Poissoniannoise. The noise generated by a symmetrical single-step discharge is Poissonian. The noise generated by apurely activationless discharge is purely Gaussian.

(3) The cumulant functions of anodic and cathodicnoise that are described by relationships (13) and (14)depend on two parameters, specifically, anodic(cathodic) flux of elementary acts and an additionalparameter, which is the P factor (P). Deviation of thelatter from unity serves as a measure of the degree towhich noise of slow discharge deviates from the purelyPoissonian noise.

(4) When at equilibrium, the P factor of the noisethat is generated by an anodic (cathodic) discharge istwice the symmetry factor of the anodic (cathodic) dis-charge process, see formulas (26) and (27).

(5) When at equilibrium, the sum of the P factors forthe anodic and cathodic slow discharges is equal to two.

(6) When far from equilibrium, knowing the ratiobetween the bispectrum and the power spectrum or theratio of the trispectrum to the bispectrum allows one tocompute the P factor for the slow discharge-generatednoise with expression (31). By this means one canexperimentally assess the degree to which the slow dis-charge-generated noise deviates from the purely Pois-sonian noise.

(7) When far from equilibrium, the square of thebispectrum for noise that is generated by a slow dis-

charge is equal to the product of the power spectrumand the trispectrum, see formula (32).

(8) Theory of integral homogeneous fluxes withindependent increments may be indispensable for stag-ing works concerning creation of unified theory of slowelectrochemical discharge and electrochemical noise.

ACKNOWLEDGMENTSThe author of this work expresses his deep appreci-

ation to A.M. Kuznetsov, S.F. Timashev, andA.A. Dubkov for useful discussion concerning theproblem of theoretical description of internal noisegenerated in nonlinear systems.

This work was supported by the Russian Foundationfor Basic Research, project no. 05-03-32 294.

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