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/Interperiodica”0113

Russian Journal of Electrochemistry, Vol. 41, No. 2, 2005, pp. 113–117. Translated from Elektrokhimiya, Vol. 41, No. 2, 2005, pp. 131–136.Original Russian Text Copyright © 2005 by Grafov.

INTRODUCTION

The emergence and development of micro- andnanoelectrochemistry [1, 2] objectively enhance therole and significance of fundamental research into elec-tric noise and fluctuations. Though the contribution ofnonlinear noise to the overall electrochemical noise issmall, knowledge of higher-order noise correlations isof exceeding importance for the building unified theoryof electrochemical-reaction rates and fluctuations,which unavoidably accompany (and facilitate) thesereactions.

The aim of this work is to analyze fluctuations of theequilibrium electrode potential and find the way thethird correlation moment (asymmetry) depends onparameters that define the slow discharge rate.

SECOND MOMENTS OF THERMODYNAMIC FLUCTUATIONS

Statistical thermodynamics experiences certain dif-ficulties when trying to describe fluctuations of inten-sive thermodynamic quantities [3]. The latter includethe electrode potential

E

as well. In the general case, theapproaches of Gibbs [4] and Einstein [5, 6] are at vari-ance with one another. The Nyquist fluctuation–dissi-pation theorem [7–9] describes dispersions of thermo-dynamic fluctuations of both extensive and intensivequantities through dynamic (varying with time

t

) fluc-tuations, i.e. through electric noise [10, 11]. Beforegaining knowledge on the behavior of the third correla-tion moments, it seems of methodological importanceto build an analysis of the second correlation moments(dispersions) on the basis of the Nyquist theorem.

Let us employ the Nyquist theorem to find the dis-persion

⟨

e

2

(

t

)

⟩

of fluctuations of the equilibrium elec-

trode potential in a sufficiently general case where theslow-discharge resistance

R

, which depends on electriccurrent

I

, is shunted by nonlinear capacitance

C

of theelectrical double layer, which depends on electrodepotential

E

(Fig. 1). Here and below, angle bracketsdenote an averaging over an ensemble of realizations ofa stochastic process. Equilibrium noise being station-ary, dispersion

⟨

e

2

(

t

)

⟩

of fluctuations of the electrodepotential is independent of time instant

t

.

Figure 2 presents the linearized equivalent noise cir-cuit of Thévenin’s theorem [12, 13]. The circuitincludes generator of noise emf

ε

(

t

)

, which reflects thefact that the anodic

I

a

and cathodic

I

c

electric currentsflowing towards one another do not coincide at eachgiven time instant. Only when averaged, these currentsequal the exchange current

I

0

of an electrochemicalreaction.

The equilibrium fluctuation emf

ε

(

t

)

, which corre-sponds to a non-distributed resistor, is in essence whitenoise [14]. Hence, correlation function

⟨ε

(0)

ε

(

t

)

⟩

forfluctuation emf

ε

(

t

)

may be expressed through the delta-function of time

⟨ε

(0)

ε

(

t

)

⟩

=

D

2

δ(

t

)

(1)

On the Asymmetry of Thermodynamic Fluctuations of the Electrode Potential

B. M. Grafov

z

Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119071 Russia

Received March 9, 2004

Abstract

—The asymmetry of thermodynamic fluctuations of the equilibrium electrode potential and the asym-metry of thermodynamic fluctuations of the electrical double layer charge is characterized quantitatively on thebasis of third-order noise spectra.

Key words

: thermodynamic fluctuations, electrode potential, electrical double layer charge

z

Author’s e-mail: vek@elchem.ac.ru

C

(

E

)

R

(

E

)

Fig. 1.

Equivalent electric circuit for the electrode/electro-lyte interface:

R

(

I

)

is the nonlinear resistance of the slowdischarge and

C

(

E

)

is the nonlinear capacitance of the elec-trical double layer; arrows mark the positive direction of thepassing electric currents.

114

RUSSIAN JOURNAL OF ELECTROCHEMISTRY

Vol. 41

No. 2

2005

GRAFOV

where

D

2

defines the fluctuation intensity. To within anumerical coefficient, quantity

D

2

coincides with the

spectral density of the noise emf, for it followsfrom the Wiener–Khintchine theorem that [6, 14]

(2)

where

ω

is the angular frequency and

j

= .

Spectral density of noise current

i

(

t

)

, whichpasses through differential capacitance

C

of the electri-cal double layer (Fig. 2) is equal to spectral density

of noise emf

ε

(

t

)

divided by the square of theabsolute value of impedance

Z

ω

of the circuit formed byan in-series connection of resistance

R

of the slow dis-charge and differential capacitance

C

of the electricaldouble layer

(3)

(4)

Spectral density of charge fluctuations

q

(

t

)

of the

electrical double layer is equal to spectral density of fluctuations of electric current

i

(

t

)

divided by thesquare of angular frequency

ω

(5)

And, finally, spectral density of fluctuations of

electrode potential

e

(

t

)

is equal to spectral density of fluctuations of charge of the electrical double layerdivided by the square of differential capacitance

C

ofthe electrical double layer

(6)

We can now employ the second equation of theWiener–Khintchine theorem and describe dispersion

εω2⟨ ⟩

εω2⟨ ⟩ 2 dt jωt–( )exp ε 0( )ε t( )⟨ ⟩

∞–

+∞

∫ 2D2,= =

1–

iω2⟨ ⟩

εω2⟨ ⟩

iω2⟨ ⟩ εω

2⟨ ⟩ / ZωZω*( ),=

Zω R 1/ jωC( ); Zω*+ R 1/ jωC–( ).+= =

qω2⟨ ⟩

iω2⟨ ⟩

qω2⟨ ⟩ iω

2⟨ ⟩ /ω2.=

eω2⟨ ⟩

qω2⟨ ⟩

eω2⟨ ⟩ qω

2⟨ ⟩ /C2.=

⟨e2(t)⟩ of fluctuations of electrode potential e(t) by thefollowing integral expression:

(7)

As shown in Appendix A, given (2)–(6), performingactual integration converts (7) into expression (8).

(8)

We see that the denominator of expression (8) includesresistance R of the slow discharge. At the same time,according to the Nyquist theorem, spectral density

of fluctuation emf ε(t) is proportional to the sameresistance R of the slow discharge and is defined by theequation

(9)

Here, k is Boltzmann’s constant and T is the tempera-ture of the electrochemical cell. Comparing relation-ships (9) and (2) leads to the conclusion that

D2 = 2kTR. (10)

Consequently, the numerator of expression (8) alsoincludes, as a factor, resistance R of the slow discharge,which cancels resistance R of the slow discharge in thedenumerator. As a result, we obtain two equations,namely, equations (11) and (12)

⟨e2(t)⟩ = kT/C, (11)

⟨q2(t)⟩ = kTC. (12)

The former describes dispersion ⟨e2(t)⟩ of fluctuationsof the electrode potential and the latter, dispersion⟨q2(t)⟩ of fluctuations of the charge of the electrical dou-ble layer.

Equation (11) demonstrates the well known fact[15] that the dispersion of thermodynamic fluctuationsof the voltage across a capacitor connected in parallelwith an ohmic resistor is independent of the resistanceof the resistor and is completely defined by the electriccapacitance of the capacitor and the temperature.

The situation is very remarkable, one even could callit paradoxical. The reason for the emergence of thermo-dynamic fluctuations is the stochastic nature of the slowdischarge. At the same time, neither the dispersion offluctuations of the electrode potential expressed by rela-tionship (11) nor the dispersion of fluctuations of thecharge of the electrical double layer described by rela-tionship (12) depends on any characteristic of the slowdischarge. Note that statistical thermodynamics [9, 10],which has nothing to do with notions of the type “resis-tance of a slow discharge,” leads to the same expression(12) for the dispersion of fluctuations of the electriccharge of a capacitor. But the situation dramatically alterswhen dealing with the third-order correlation moments.

e2 t( )⟨ ⟩ 14π------ dω eω

2⟨ ⟩ .

∞–

+∞

∫=

e2 t( )⟨ ⟩ D2/ 2RC( ).=

εω2⟨ ⟩

εω2⟨ ⟩ 4kTR.=

e(t)

ε(t) R

C

Fig. 2. Equivalent noise circuit for the electrode/electrolyteinterface: R is the resistance of the slow discharge at the equi-librium potential, C is the capacitance of the electrical doublelayer at the equilibrium potential, and e(t) is the fluctuationconstituent of the equilibrium electrode potential; arrowmarks the positive direction of the action of noise emf ε(t).

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 41 No. 2 2005

ON THE ASYMMETRY OF THERMODYNAMIC FLUCTUATIONS 115

THIRD MOMENTS OF THERMODYNAMIC FLUCTUATIONS

Of late, third-order correlations have been arousingan ever keener interest [16–18]. As shown in works[19–22] published recently, the applicability of theNyquist theorem for describing noise properties of bothlinear and nonlinear equilibrium systems leads to theexistence, in the time domain, of Langevin’s dual linearequations, which relate fluctuations of current and volt-age that are recorded, respectively, in a short-circuitmode and in an open-circuit mode. In the frequencydomain, a consequence of the existence of Langevin’sdual linear equations is the invariance of cumulantspectra of the above fluctuation currents and voltages.

We will now consider, on the basis of [19–22], non-linear noise in an electric circuit that corresponds to anequilibrium electrode with a slow faradaic process(Fig. 2). The physical reason for the fluctuation(Brownian) movement in the circuit depicted in Fig. 2is the presence, at any given time instant, of misbal-anced stochastic fluxes of elementary acts of dis-charge—anodic and cathodic. Third-order correlationfunction ⟨ε(0)ε(t2)ε(t3)⟩ of fluctuation emf ε(t) corre-sponds to nonlinear white noise. Hence, it can be writ-ten in the most general form [23, 24]

⟨ε(0)ε(t2)ε(t3)⟩ = D3δ(t2)δ(t3). (13)

Quantity D3 in (13) is a measure of intensity of non-linear fluctuations. To within a numerical coefficient,quantity D3 coincides with third-order spectral density

, which is defined by the followingequation:

(14)

Expressions for the third-order spectral densities of anoise current and the fluctuations of the charge of theelectrical double layer can be written through relevantcorrelation functions similarly to relation (14). On thebasis of (13) and (14), then

(15)

The third-order spectral density is quite frequentlycalled a bispectrum of a stochastic process [25, 26]. Fora bispectrum, we will use below the simplified formω1 = ω2 + ω3

The invariance property of bispectra of a fluctuationcurrent and a fluctuation emf [19–22] give us groundsto utilize the following expression for bispectrum

of fluctuation current i(t) flowing in the RC cir-cuit in Fig. 2:

εω2 ω3+* εω2εω3

⟨ ⟩

εω2 ω3+* εω2εω3

⟨ ⟩ 4 dt2 dt3

∞–

+∞

∫∞–

+∞

∫=

× jω2t2– jω3t3–( ) ε 0( )ε t2( )ε t3( )⟨ ⟩ .exp

4D3 εω2 ω3+* εω2εω3

⟨ ⟩ .=

εω2 ω3+* εω2εω3

⟨ ⟩ εω1* εω2

εω3⟨ ⟩ ε1*ε2ε3⟨ ⟩ .≡=

i1*i2i3⟨ ⟩

(16)

Here, as before, Z denotes the impedance of a circuitcomprising resistance R of the slow discharge and dif-ferential capacitance C of the electrical double layerconnected in series:

(17)

Note that expression (16), which links bispectra of afluctuation current and a fluctuation emf with oneanother, is completely analogous to equation (3), whichlinks spectral densities of a fluctuation current and afluctuation emf with one another. On the basis of (16),we successively obtain

(18)

(19)

for the bispectrum of fluctuations of the charge of theelectrical double layer and the bispectrum of

fluctuations of the electrode potential . As aresult, we obtain

(20)

for the bispectrum of fluctuations of the electrodepotential. Note that, by virtue of the properties of a Fou-rier transform [25], the third-order moment ⟨e3(t)⟩ of thefluctuations of the electrode potential may be deter-mined by integrating expression (20) over the entiretwo-dimensional frequency range

(21)

Performing actual integration (Appendix B) leads to thefollowing result:

(22)

Our next task is to link intensity D3 of nonlinear fluctua-tions with a nonlinear characteristic of the slow discharge.

APPLICATION OF THE FLUCTUATION–DISSIPATION RELATIONSHIP

OF STRATONOVICH

According to Stratonovich [23], a third-order corre-lation function, which characterizes the equilibriumnoise emf of non-distributed resistance R(I), is definedby the expression

(23)

i1*i2i3⟨ ⟩ε1*ε2ε3⟨ ⟩

Z1*Z2Z3

---------------------.=

Z1* R1

j ω2– ω3–( )C----------------------------------; Z2+ R

1jω2C-------------;+= =

Z3 R1

jω3C-------------.+=

q1*q2q3⟨ ⟩ i1*i2i3⟨ ⟩ / j ω2– ω3–( ) jω2ω3[ ]=

e1*e2e3⟨ ⟩ q1*q2q3⟨ ⟩ /C3=

q1*q2q3⟨ ⟩e1*e2e3⟨ ⟩

e1*e2e3⟨ ⟩ = 4D3/ j ω2– ω3–( ) jω2ω3Z1*Z2Z3C3[ ].

e3 t( )⟨ ⟩ 1

4π( )2------------- dω2 dω3 e1*e2e3⟨ ⟩ .

∞–

+∞

∫∞–

+∞

∫=

e3 t( )⟨ ⟩ D3/ 3R2C2( ).=

ε 0( )ε t2( )ε t3( )⟨ ⟩ 6 kT( )2 dR/dI( )I 0= δ t2( )δ t3( ),=

116

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 41 No. 2 2005

GRAFOV

whose right-hand part includes derivative (dR/dI)I = 0 ofresistance R(I) to direct current I at a zero current. Com-paring relationships (13) and (23) yields

D3 = 6(kT)2(dR/dI)I = 0, (24)

which finally leads to the expression

⟨e3(t)⟩ = 2(kT)2C–2R–2(dR/dI)I = 0 (25)

which defines a third-order correlation moment of fluc-tuations of the electrode potential, and the expression

⟨q3(t)⟩ = 2(kT)2CR–2(dR/dI)I = 0 (26)

which defines a third-order correlation moment of fluc-tuations of the charge of the electrical double layer.Equations (25) and (26) is the major result of this work.

COMPARISON WITH STATISTICAL THERMODYNAMICS EQUATIONS

As follows from expression (26), a third-order cor-relation moment of fluctuations of the charge of theelectrical double layer is defined by the differentialcapacitance of the electrical double layer and by thelevel of small-signal quadratic faradaic rectification.However, modern statistical thermodynamics leads to asubstantially different expression for a third-ordermoment of fluctuations of charge in an equilibrium RCcircuit, specifically [9, 10]

⟨q3(t)⟩ = (kT)2dC/dE. (27)

Here, quantity dC/dE is the derivative from differentialcapacitance C of the electrical double layer with respectto electrode potential E.

We believe that such a dramatic discrepancybetween relationships (26) and (27) unequivocally sug-gests that it is quite legitimate to analyze and calculatethe second correlation moments (dispersions) of fluctu-ations of extensive quantities in the framework of sta-tistical thermodynamics. As to calculation of the thirdand higher cumulant moments of thermodynamic fluc-tuations of both extensive and intensive quantities, it isbeyond statistical thermodynamics.

CONCLUSIONS

In this work we have managed to make the follow-ing points.

The third correlation moment, which characterizesthe asymmetry of equilibrium fluctuations of the elec-trode potential (relationship (25)) depends in a substan-tial manner on the nonlinear properties of slow dis-charge (on the low-level faradaic rectification) and isabsolutely independent of the nonlinear properties ofthe capacitance of the electrical double layer.

The third correlation moment, which characterizesthe asymmetry of fluctuations of the charge of the elec-trical double layer at equilibrium (relationship (26)),depends in a very direct fashion on the faradaic rectifi-cation. At the same time, the fact that the component of

the electrical double layer is dependent on the electrodepotential makes no difference whatsoever to the valuesof this third moment.

Statistical thermodynamics offers no correct methodfor calculating third-order correlation moments that char-acterize equilibrium thermodynamic fluctuations of bothextensive and intensive quantities. At the same time, wecould point out to the existence of a quite correct methodfor calculating the asymmetry of equilibrium thermody-namic fluctuations. Such a method is directly related totheory of dynamic fluctuations and noise.

ACKNOWLEDGMENTS

The author is indebted to A.M. Kuznetsov,S.F. Timashev, R.M. Yul’met’ev, Yu.K. Evdokimov,R.R. Nigmatulin, and A.M. Gulyaev for the events ofdiscussion of the problem of nonlinear noise and fluc-tuations.

This work was supported by the Russian Foundationfor Basic Research (project no. 02-03-32114).

APPENDIX A

We will calculate integral (7)

(7)

in a complex plane using a contour integral method andviewing angular frequency ω as a complex variable.

The function under the integral sign

(Ä1)

in the upper half-plane of complex variable ω has a sin-gle pole, at the point ω = j/(RC). The residue of thefunction under the integral sign at this point is

(Ä2)

whence in accordance with the known mathematicalrelationship of a contour integral method [27]

(Ä3)

we obtain

(Ä4)

This is precisely what gives us a chance to make use ofrelationship (8).

e2 t( )⟨ ⟩ 14π------ dω eω

2⟨ ⟩∞–

+∞

∫=

εω2⟨ ⟩

εω2⟨ ⟩

2D2

CR( )2 ω 1/ jCR( )–[ ] ω 1/ jCR( )+[ ]---------------------------------------------------------------------------------------=

res εω2⟨ ⟩ D2/ jCR( )=

dω eω2⟨ ⟩

∞–

+∞

∫ 2πjres eω2⟨ ⟩ ,=

e2 t( )⟨ ⟩2πjD2

4πjRC-----------------

D2

2RC-----------.= =

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 41 No. 2 2005

ON THE ASYMMETRY OF THERMODYNAMIC FLUCTUATIONS 117

APPENDIX B

To calculate the integral in (21)

(21)e3 t( )⟨ ⟩ 1

4π( )2------------- dω2 dω3 e1*e2e3⟨ ⟩

∞–

+∞

∫∞–

+∞

∫=

we will use the same contour integral method aswhen calculating the integral in (7), but with thedifference that, now, we will have to perform dou-ble integration, specifically, with respect to fre-quency ω3 and with respect to frequency ω2. Thebispectrum

(B1)e1*e2e3⟨ ⟩4D3–

jRC( )3 ω2 ω3 1/ jRC( )–+[ ] ω2 1/ jRC( )+[ ] ω3 1/ jRC( )+[ ]-------------------------------------------------------------------------------------------------------------------------------------------------=

of fluctuations of the electrode potential, which isviewed as a function of complex frequency ω3, has onlyone pole in the upper half-plane of complex variable ω3,specifically, at point ω3 = –1/(iCR). As the residue ofthe integrand at this point is

(B2)

we obtain

(B3)

Viewing the integral as a function of

complex variable ω2, we find that it has only one pole inthe upper half-plane, specifically, at point ω2 =−1/(jRC). As the residue at this point is

(B4)

we obtain

(B5)

This is precisely what gives us a chance to make use ofrelationship (22).

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res e1*e2e3⟨ ⟩

= 4D3–

jRC( )3 ω2 2/ jRC( )–[ ] ω2 1/ jRC( )+[ ]----------------------------------------------------------------------------------------------

dω3 e1*e2e3⟨ ⟩∞–

+∞

∫ 2πjres e1*e2e3⟨ ⟩=

= 4D3–

jRC( )3 ω2 2/ jRC( )–[ ] ω2 1/ jRC( )+[ ]----------------------------------------------------------------------------------------------.

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