ISSN 1023�1935, Russian Journal of Electrochemistry, 2010, Vol. 46, No. 8, pp. 948–951. © Pleiades Publishing, Ltd., 2010.Original Russian Text © B.M. Grafov, 2010, published in Elektrokhimiya, 2010, Vol. 46, No. 8, pp. 1009–1012.

948

INTRODUCTION

Passing of electrochemical reactions is supportedby continuously restored Brown fluctuation motion ofmedium in close proximity of the electrode/electro�lyte interface [1–4]. Two general theoreticalapproaches exist at the present time, which allow link�ing the rate and the fluctuation characteristics of elec�trochemical reactions. They are: the Bochkov–Kuzovlev fluctuation theorem [5, 6] and its generali�zation in the form of the Jarzynski fluctuation theorem[7], on the one hand; the Stratonovich relations fornonlinear nonequilibrium thermodynamics [8, 9], onthe other hand.

In this work we aimed at the analyzing of the appli�cability of the Bochkov–Kuzovlev–Jarzynski andStratonovich fluctuation theorems to the theory ofslow discharge. We shall discuss the passing of electro�chemical reactions in galvanostatic regime.

FLUCTUATION THEOREMS OF BOCHKOV–KUZOVLEV AND JARZYNSKI

The fluctuation theorem of Bochkov–Kuzovlevwas formulated in works [5, 6]. The Bochkov–Kuzov�lev fluctuation relation was generalized in [7] underthe name of Jarzynski fluctuation theorem. Its electro�chemical version is as follows:

(1)

where β is the reciprocal of the product of temperatureand Boltzmann constant. The indexes A and B denotetwo states of ideally polarizable (or perfectly polariz�

βWAB–( )exp⟨ ⟩ β σA σB–( )S{ },exp=

able) electrode [10], whose surface area is S, the sur�face tension values are σA and σB. In the case of solidelectrodes, σA and σB are the works of formation ofunit electrode surface during a reversible thermody�namic process. During a real process transferring anideally polarizable electrode from a reversible state Ato a reversible state B, the electrode must be impartedwith certain charge QAB; some work WAB must also beexecuted. The work WAB is but in part executed overthe electrode/electrolyte interface. Its dissipation partW(diss) is spent up on the formation of irreversibleentropy and is converted to heat. The amount of thedissipated energy W(diss) depends on the fluctuationproperties of the electrical circuit that supplies theelectric charge QAB. Therefore, the dissipated energy isa random variable and possesses different values eachtime when an ideally polarizable electrode is trans�ferred from the state A to the state B. The brackets inequation (1) imply the operation of averaging over anensemble of realizations of the electrode transfer fromthe state A to the state B.

Let the capacitance C of the electrical double layeris charged in two steps. In the first stage, constantanodic current Ia is passed for the time ta. In the secondstage, constant cathodic current Ic is passed for thetime tc. We assumed that the anodic charge Qa = Iataequals the cathodic charge Qc = Ictc, that is,

QAB = Qa+ Qc = 0. (2)

Eventually, the state B of the ideally polarizable elec�trode coincides with the state A; the surface tension ofthe ideally polarizable electrode is kept constant.

On the Applying of Fluctuation Theorems to the Theory of Slow Discharge in Galvanostatic Regime

B. M. Grafovz

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

Received May 12, 2009

Abstract—Expressions for the equation of slow discharge in galvanostatic regime are drawn, as power expan�sion with respect to overvoltage, in terms of the Bochkov–Kuzovlev and Jarzynski fluctuation theorems com�bined with the Stratonovich relations for nonlinear nonequilibrium thermodynamics. Coefficients of theexpansion are linear functions of the intensities of cumulant correlation functions that characterize the elec�trode potential fluctuations.

Key words: slow discharge, galvanostatic regime, electrical noise, fluctuation–dissipation relations, Boch�kov–Kuzovlev theorem, Jarzynski theorem, Stratonovich theory

DOI: 10.1134/S1023193510080136

z Corresponding author: bmg@elchem.ac.ru (B.M. Grafov).

SHORT COMMUNICATIONS

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 46 No. 8 2010

ON THE APPLYING OF FLUCTUATION THEOREMS 949

Under these conditions the fluctuation theorem ofJarzynski passes into that of Bochkov–Kuzovlev:

(3)The Jarzynski theorem [7] revealed the condition

of applicability of the Bochkov–Kuzovlev theorem[5, 6] (fulfillment of the condition (2) [11]). There�fore, expression (3) can well be called the Bochkov–Kuzovlev–Jarzynski fluctuation theorem.

CUMULANT REPRESENTATION OF THE BOCHKOV–KUZOVLEV–JARZYNSKI

THEOREM

The left�hand part of equation (3) can be thought ofas a characteristic function of the irreversible workW(diss) that is a random variable. This means that thecoefficients of expansion of the left�hand part of expres�sion (3) into Maclaurin series in terms of (–β) mustcoincide with the moments of the irreversible workW(diss). It is known [12, 13] that logarithm of any char�acteristic function of random variable is a generatingfunction for cumulants of the same random variable.After taking the logarithm of equation (3) we have

(4)whence we obtain

(5)

Here W(k) is the kth cumulant of the irreversible workW(diss).

SLOW DISCCHARGE

We now assume that the electrical double layercapacitance C is charged both in anodic and cathodicdirections via electrochemical reaction whose rate isdetermined by slow discharge. We write the discharge�current (I) dependence of the overvoltage E as a cur�rent�power expansion:

(6)

βW diss( )–( )exp⟨ ⟩ 1.=

βW diss( )–( )exp⟨ ⟩ln 0.=

W diss( )⟨ ⟩β W 2( )

β2/2 W 3( )

β3/3!– W 4( )

β4/4!+=

– W 5( )β

5/5! W 6( )

β6/6! ….–+

E I( ) R0I R1I2/2 R2I3

/3! R3I4/4!+ + +=

+ R4I5/5! R5I6

/6! …,+ +

where R0 is the slow�discharge resistance at zero over�voltage, R1, R2, … are the first�, second�, etc. orderderivatives with respect to current of the slow�dis�charge resistance, which are also taken at zero over�voltage.

In galvanostatic regime, the dissipating energy isdetermined not only by the overvoltage E(I) but also bythe fluctuation e.m.f. e(t), which characterizes theslow discharge. Therefore, the dissipating energyW(diss) can be represented by a sum of its average value�W(diss)� and fluctuation value ΔW(diss):

W(diss) = �W(diss)� + ΔW(diss), (7)

where

�W(diss)� = E(Ia)Qa + E(Ic)Qc, (8)

ΔW(diss) = Ia (t)dt + Ic (t)dt. (9)

We emphasize that equation (9) is written under theassumption that the fluctuation e.m.f. e(t) impedes thepassing of electrical current I. Figure a illustratesthe situation. Arrows in the figure show the positivedirection of the direct electrical current I and the pos�itive direction of action of the fluctuation e.m.f. e(t).

The condition (2) can be always fulfilled by appro�priate choice of the duration of the anodic (ta) andcathodic (tc) steps of the electrical double layer charg�ing; hence, the possibility to independently change theanodic (Ia) and cathodic (Ic) direct currents exists. Thecumulant equation (5) eventually parts into two simi�lar cumulant equations for the anodic and cathodicsteps of charging of ideally polarizable electrode:

(10)

(11)

In equations (10) and (11) the random energy dissi�pated during the anodic step of charging of the ideally

ea

0

ta

∫ ec

0

tc

∫

E Ia( )Qaβ Wa2( )β

2/2 Wa

3( )β

3/3!– Wa

4( )β

4/4!+=

– Wa5( )β

5/5! Wa

6( )β

6/6! …,–+

E Ic( )Qcβ Wc2( )β

2/2 Wc

3( )β

3/3!– Wc

4( )β

4/4!+=

– Wc5( )β

5/5! Wc

6( )β

6/6! ….–+

C

E = E(I)e(t)I

(b)

IC

E = E(I)e(t)I

(a)

I

Electrical circuit for galvanostatic charging of the electrical double layer capacitance С according to Bochkov–Kuzovlev–Jarzyn�ski under the conditions when the positive direction of external current I and that of fluctuation e.m.f. (a) do not correspond or(b) correspond.

950

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 46 No. 8 2010

GRAFOV

polarizable electrode is characterized by the cumu�lants denoted by index a:

(12)

that dissipated during the cathodic step of charging ofthe ideally polarizable electrode, by the cumulantsdenoted by index c:

(13)

Because equations (12) and (13) are identical, the furtheranalysis will be carried out basing on equation (12).

THE CUMULANTS OF DISSIPATED ENERGY

According to equation (12), the random dissipatedenergy and the fluctuation e.m.f. characterizing theslow discharge in galvanostatic regime are linked via alinear integral expression. Therefore, the set of thecumulants of dissipated energy is unambiguouslydetermined by a set of cumulant coefficients corre�sponding to random slow discharge [13]. We omit theindex a and obtain, according to equation (12), that

W(2) = QIe(2), (14)

W(3) = QI2e(3), (15)

W(k) = QIk – 1e(k). (16)The cumulant coefficients e(k) by definition are theintensities of the corresponding cumulant correlationfunctions of the fluctuation voltage:

�e(0), e(t)� = e(2)δ(t), (17)

�e(0), e(t1), e(t2)� = e(3)δ(t1)δ(t2), (18)

�e(0), e(t1), e(t2), …, e(tk)� = e(k + 1)δ(t1)δ(t2)…δ(tk),(19)

where we used the notation proposed by Malakhov[13] for the cumulant correlation functions; the sym�bol δ(t) denotes the Dirac delta�function.

The cumulant coefficients e(k) depend on the elec�trical current:

(20)

Expression (20) is the Maclaurin power series in termsof electrical current.

THE SLOW DISCHARGE RESISTANCEAND ITS DERIVATIVES

Equations (5), (10), (14)–(16) in aggregate allowobtaining the following equation for the slow dis�charge in galvanostatic regime:

(21)

ΔWadiss( ) Ia ea t( ) t,d

0

ta

∫=

ΔWcdiss( ) Ic ec t( ) t.d

0

tc

∫=

e k( )

= e0k( ) e1

k( )I e2k( )I2

/2 e3k( )I3

/3! e4k( )I4

/4! ….+ + + + +

E I( ) e 2( )Iβ/2 e 3( )I2β

2/3! e 4( )I3

β3/4!+–=

– e 5( )I4β

4/5! e 6( )I5

β5/6! e 7( )I6

β6/7!– ….+ +

Basing on equation (21), by equating the summands pro�portional to the same power of electrical current, wefound the coefficients of the expansion in equation (6):

(22)

(23)

(24)

(25)

(26)

(27)

The sequence (22)–(27) can be continued in the formof expressions for the derivatives from the slow dis�charge resistance of sixth, seventh, and higher orders.This means that the slow discharge resistance and itsderivatives of any order with respect to electrical cur�rent, related to the equilibrium state, are linear func�tions of the intensities of the cumulant functions ofovervoltage fluctuations that occur in the galvanostaticregime of the slow discharge. We emphasize that thisfundamental result directly follows from the Boch�kov–Kuzovlev–Jarzynski fluctuation theorem.

Let us reverse the direction of positive action of theslow�discharge fluctuation e.m.f. e(t) as shown inFig. b. Then we obtained, instead of equation (9), thefollowing equation:

(28)

When the condition (28) is fulfilled, the set of equa�tions (22)–(27) is replaced by the following set ofequations:

(29)

(30)

(31)

(32)

(33)

(34)

Unlike the set of equations (22)–(27), no sign (plus–minus) alternating occurs in the set (29)–(34).

R0 e02( )β/2,=

R1/2 e03( )β

2/3!– e1

2( )β/2,+=

R2/3! e04( )β

3/4! e1

3( )β

2/3!– e2

2( )β/4,+=

R3/4! e05( )β

4/5!– e1

4( )β

3/4!+=

– e23( )β

2/ 3!2!( ) e3

2( )β/ 2!3!( ),+

R4/5! e06( )β

5/6! e1

5( )β

4/5!– e2

4( )β

3/ 4!2!( )+=

– e33( )β

2/ 3!3!( ) e4

2( )β/ 2!4!( ),+

R5/6! e07( )β

6/7!– e1

6( )β

5/6! e2

5( )β

4/ 5!2!( )–+=

+ e34( )β

3/ 4!3!( ) e4

3β

2/ 3!4!( )– e5

2( )β/ 2!5!( ).+

ΔW diss( ) Ia 1–( )ea t( ) td

0

ta

∫ Ic 1–( )ec t( ) t.d

0

tc

∫+=

R0 e02( )β/2,=

R1/2 e03( )β

2/3! e1

2( )β/2,+=

R2/3! e04( )β

3/4! e1

3( )β

2/3! e2

2( )β/4,+ +=

R3/4! e05( )β

4/5! e1

4( )β

3/4!+=

+ e23( )β

2/ 3!2!( ) e3

2( )β/ 2!3!( ),+

R4/5! e06( )β

5/6! e1

5( )β

4/5! e2

4( )β

3/ 4!2!( )+ +=

+ e33( )β

2/ 3!3!( ) e4

2( )β/ 2!4!( ),+

R5/6! e07( )β

6/7! e1

6( )β

5/6! e2

5( )β

4/ 5!2!( )+ +=

+ e34( )β

3/ 4!3!( ) e4

3β

2/ 3!4!( ) e5

2( )β/ 2!5!( ).+ +

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 46 No. 8 2010

ON THE APPLYING OF FLUCTUATION THEOREMS 951

THE STRATONOVICH FLUCTUATION APPROACH

In accordance with the Stratonovich irreversiblethermodynamics [8, 9], the slow discharge in galvano�static regime can be described by the following equa�tion:

(35)

The equation (35) is nothing but the equation (10.2)from work [8] and at the same time the equation(4.1.4) from [9]. The structure of equation (35) is sim�ilar to that of equation (21). However, in equation (21)the signs (plus–minus) do alternate, unlike equation(35). The reason is that the positive direction of actionof the fluctuation voltage, adopted in the Stratonovichtheory, corresponds to Fig. b, rather than Fig. a. It isthe set of expressions (29)–(34) that corresponds toequation (35). We see that in the galvanostatic regimeof current passing the Stratonovich theory is consis�tent with the Bochkov–Kuzovlev–Jarzynski theory.

CONCLUSIONS

The applying of the fluctuation approaches ofBochkov–Kuzovlev–Jarzynski [5–7] and of Stra�tonovich [8, 9] to the slow discharge theory showedthat in galvanostatic regime the slow discharge resis�tance and its derivatives of any order with respect toelectrical current are unambiguously determined bythe intensities of cumulant correlation functions of thefluctuation overvoltage. The corresponding expres�sions, up to the fifth�order derivative with respect tothe discharge resistance, are given by equations (22)–(27) and (29)–(34).

The Jarzynski theorem [7] was comprehensivelydeveloped as applied both to classical systems [14–16]and quantum ones [17]. Therefore, the investigation ofthe consistency of the Jarzynski theorem with the the�ory of elementary act is of basic significance not onlyfor the theory of slow discharge but also for the mod�ern theory of noise and fluctuations in general.

ACKNOWLEDGMENTS

Author is grateful to A.D. Davydov and S.F. Tima�shev for their stimulating discussion of differentaspects of the applying of fluctuation approach to the�ory of slow discharge.

REFERENCES

1. Kuznetsov, A.M. and Ulstrup, J., Electrochim. Acta,2001, vol. 46, p. 3325.

2. Marcus, R.A., J. Electroanal. Chem., 2000, vol. 483,p. 2.

3. Krishtalik, L.I., Elektrodnye reaktsii. Mekhanizm ele�mentarnogo akta (Electrode Reactions. Mechanism ofElementary Act), Moscow: Nauka, 1982.

4. Dogonadze, R.R. and Chizmadzhev, Yu.A., Dokl. ANSSSR, Fiz. Khimiya, 1962, vol. 144, p. 1077.

5. Bochkov, G.N. and Kuzovlev, Yu.E., Zh. Eksp. Teor.Fiz., 1977, vol. 72, p. 238.

6. Bochkov, G.N., Fluktuatsii v neravnovesnykhradiofizicheskikh sistemakh (Fluctuations in Nonequi�librium Radiophysical Systems), Gorky: Gorky Gos.Univ., 1981.

7. Jarzynski, C., Phys. Rev. Lett., 1997, vol. 78, p. 2690.8. Stratonovich, R.L., Nelineinaya neravnovesnaya termo�

dinamika (Nonlinear Nonequilibrium Thermodynamics),Moscow: Nauka, 1985.

9. Stratonovich, R.L., Nonlinear Nonequilibrium Thermo�dynamics I. Linear and Nonlinear Fluctuation�Dissipa�tion Theorems, Berlin: Springer, 1992.

10. Frumkin, A.N., Potentsialy Nulevogo Zaryada (Zero�Charge Potentials), Moscow: Nauka, 1979.

11. Jarzynski, C., J. Stat. Phys., 2000, vol. 98, p. 77.12. Zwanzig, R., J. Chem. Phys., 1954, vol. 22, p. 1420.13. Malakhov, A.N., Kumulyantnyi analiz sluchainykh

negaussovykh protsessov i ikh preobrazovanii (CumulantAnalysis of Stochastic Processes and Their Transforma�tions), Moscow: Sov. Radio, 1978.

14. Crooks, G., J. Stat. Phys., 1998, vol. 90, p. 1481.15. Sughiyama, Yu. and Abe, S., J. Stat. Mech, 2008,

p. 05008.16. Talkner, P., Lutz, E., and Hanggi, P., Phys. Rev. E:,

2007, vol. 75, p. 050102.17. Talkner, P., Campisi, M., and Hanggi, P., J. Stat. Mech,

2009, p. 02025.

0 E I( )βI e 2( )I2β

2/2– e 3( )I3

β3/3!– e 4( )I4

β4/4!–=

– e 5( )I5β

5/5! e 6( )I6

β6/6!– e 7( )I7

β7/7!– ….+