Continuous stochastic theory of birth and death processes with long-range interaction. Application to electrolytes

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Continuous stochastic theory of birth and death processes with long-range interaction. Application to

electrolytes

V.R. Chechetkin, V.S. Lutovinov

To cite this version:

V.R. Chechetkin, V.S. Lutovinov. Continuous stochastic theory of birth and death processes with long-range interaction. Application to electrolytes. Journal de Physique, 1988, 49 (2), pp.159-165.

�10.1051/jphys:01988004902015900�. �jpa-00210681�

Continuous stochastic theory of birth and death processes with long-

range interaction. Application ^to electrolytes

V. R. Chechetkin and V. S. Lutovinov

Institute of Radioengineering, Electronics and Automation, 117454, Moscow, ^U.S.S.R.

(Requ ^{le 9} juin 1987, révisé le 25 septembre ^1987, accepté ^{le 25} septembre 1987)

Résumé.

L’article porte ^{sur la} description stochastique des processus de naissance ^et de mort dans des

systèmes spatialement distribués et sièges d’interactions à longue portée. ^{On montre comment on} peut formuler le problème à l’aide d’une équation ^maîtresse régissant l’évolution d’une fonctionnelle

probabilité ^».

La théorie est appliquée ^au calcul d’une

correction relaxationnelle » à la mobilité des ions dans un

électrolyte. ^{On montre en} particulier que les processus de dissociation ^et recombinaison s’ajoutent ^{à l’effet}

d’écran des ions mobiles (Debye screening). L’importance relative de ces deux phénomènes ^sur la mobilité est

examinée.

Abstract.

The problem of stochastic description ^{of the} spatially distributed birth and death processes with

long-range interaction is considered. It is shown how the problem ^{can be} formulated in terms of the functional master equation ^{for the} probability functional. The theory ^is applied ^to the calculation of the relaxational correction to the mobility of ions in the electrolytes. The influence of recombination

dissociation processes

on the Debye screening ^of moving ions and their relative contributions to the mobility ^are discussed. The

particular example ^{concerns the} system ^{of a} two-component electrolyte ^{in the} limit of small fluctuations of the concentrations (e2/03B503BBD kB T ~ 1, ^where 03BBD ^{is the} Debye radius and T is the absolute temperature).

Classification

Physics ^Abstracts

05.40

02.50

’

1. Introduction.

The dynamics ^{of the} physical systems with chemical transformations evolves always ^{on the} background

of intrinsic fluctuations. These fluctuations are due to the intrinsic occasional nature of elementary

reactions and spatial diffusion of discrete particles. ^If

the local relaxation in momentum space is fast,

which is a typical situation in dense gases and liquid solutions, ^{then the} stochastic evolution of a system

can be described in terms of birth and death processes with continuum realizations (see, e.g., [1, 2]). The mathematical description ^{of such a} problem

in a continuous medium approximation appears ^to be rather complicated. ^{The most advanced studies}

are based either on the discretised approximation ^of

the continuous medium [1, 3-6] ^{or on the Bose}

operator ^formalism [7-10] and direct stochastic treat- ment of the corresponding differential equations [2].

We have developed ⁱⁿ [11] the functional formalism for the continuous stochastic theory of birth and death processes and formulated the problem ⁱⁿ

terms of the functional ^master equation ^{for the} probability distribution functional. Then this

equation ^{can be} reformulated either in terms of the infinite set of coupled equations ^{for the} equal ^time

correlators [11] ^{or can} be studied directly ^{with the} corresponding regular functional perturbational

methods [12]. ^In previous investigations ^{one has}

considered systems ^{of the} ideal gas type ^with respect

to the mutual interactions between particles. ^{We will}

consider in this paper the generalization ^{of the}

functional theory ^{to a} system ^with long-range ^interac-

tion with particular application ^{to the} electrolytes.

We shall study ^the following physical situation. Since the classical results by Debye ^and Huckel, ^and Onsager (see, e.g., [13]) it is well known that the

mobility ^{of ions} in solutions is determined by ^the

interaction with solvent and by ^{the Coulomb interac-}

tion with other ions in solution. The last effect leads

to the screening ^{of external electric field and causes}

the effective diminishing ^{of the} mobility. ^{We are}

interested in how such an effect is influenced by

recombination-dissociation processes.

The paper is organized ^as follows. In section 2 we

describe the general formalism. This formalism is

applied in section 3 first to a system ^of charged particles ^{without chemical reactions and then in}

section 4 the recombination-dissociation _processes

are included in the scheme. A particular example

concerns the system ^{of a} two-component electrolyte

in the limit of small fluctuations of concentrations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004902015900

160

(e2/ eÀ D kB ^T 1 ). The results are discussed in the final section 5.

2. Master equation ^for probability distribution func- tional.

For the convenience of the reader we summarize here the main results of the functional formalism

[11]. It is assumed that the deterministic evolution of

elementary processes,

is given by ^{the set of} equations :

where X (r, ^{t )} is the number of the j-th ^molecules

per unit volume (as ^{usual the same} symbol Xj ^{will be}

used for the notation of the j-th element in the chemical reactions and the number of the j-th particles per unit volume), {v j p} and {JL j p} ^{are the}

stechiometric coefficients, the force F (r, t ) ^corres- ponds ^{to a fixed} given ^{external field, and the}

diffusion coefficients {Dj} ^{are related to the mobili-}

ty coefficients {b j } ^by the Einstein relationship :

where T is the absolute temperature and kB ^{is the}

Boltzmann constant.

The stochastic evolution of a system ^{is described} in terms of evolution of the probability distribution functional S ( Xi (r’ ),

XN (r’ ) ; t } , ^where {Xl (r’),

XN (r’ ) ; t } corresponds ^{to the} proba- bility of detection of the local densities {Xj (r’ )} ^{at a}

moment t, ^{and is} given by ^{the master} equation

Here {S / SX(r)} ^are the functional derivatives and Vr’ F (r, t ) ^{is the} operator ^{whose action is} equivalent ^to

the multiplication by ^{a vector function} F (r, t ) ^and subsequent taking ^of divergence ^{from a whole} expression.

The operators exp(± 5l5Xj(r)) ^are the local birth and death operators. Their action on the arbitrary functional CP {Xl (r’), ... , XN (r’ )} ^{is given by}

Thus, it is easy ^{to see} that the second sum on the right ^{hand side of} equation (2.4) corresponds ^{to a} typical probabilistic balance with « income » and ^« outcome ». We assumed for simplicity ^{that the} characteristic reaction correlation radii for the processes (2.1) ^{are small in} comparison ^{with the other} characteristic distances (e.g., ^with Debye radius for the electrolytes) and used the local form for the reaction rates in

equations (2.2), (2.4) (cf. discussion in [11]).

The diffusion terms with the fluctuation corrections may easily be derived by comparison with the discrete

jump ^model [2] ^{and then} by returning ^{back to the} continuous medium limit (a ^{detailed account can be found}

in [14]).

Equation (2.4) ^can be transformed into an evolution equation for the characteristic functional [11],

Since the equal time correlators ^are determined according ^to

the corresponding ^evolution equations ^{for the cor-}

relators are derived by the proper functional dif- ferentiation of both sides of equations (2.7).

3. Fluctuations in diffusion fluxes of particles ^with long-range interaction.

The evolution of the electrolyte system is determined

by (i) long-range self-consistent Coulomb interaction between ions, (ii) recombination-dissociation _pro-

cesses (Eqs. (2.1)) ^and by (iii) external field (if ^{it is} applied). The first and third effects are considered

usually ^{in the framework} of the standard BBGKY scheme (see, e.g., [15, 16]). We should also re-

member that birth and death formalism is applicable only ^{for times} greater than the elastic collision times

[1]. Before the formal application ^of equations (2.4)

and (2.7) ^{to the} electrolytes ^it is useful to compare the results with the BBGKY theory ^{as a} preliminary step before the inclusion of recombination-dissocia- tion effects. Thus, ^we first discuss the stochastic evolution of the charged particles in the absence of chemical reactions (i.e., {A:p}

^{0 in equation (2.4))}

and, ^for simplicity, in the absence of external field.

As is seen from the structure of equation (2.4) ^the

evident modification to a system ^of particles ^with long-range interaction consists in the proper redefini- tion of the forces F (r, t ) ^{which must now} include the effects of self-consistent Coulomb interaction. The

particles ^{will then move in a} collective self-consistent internal field. The corresponding evolution equa- tions have the form :

where ^E is the dielectric constant of a medium without charged particles (in ^the problem concerned it

corresponds ^to the dielectric constant of the solvent), _ezj ^{is the} charge ^of the j-th ^ions (it may be either

negative ^or positive). It is also assumed that the system ^is electroneutral, i.e., that its total charge ^is equal ^to

zero :

The electric field E in equation (3.1) ^must be understood in the exact (unaveraged) ^{sense. We will} only

consider the limit of an infinite three-dimensional system ^where boundary ^{effects can be} neglected. ^{Then the}

internal electric field is given by

After the substitution of this expression ^into equation (3.2) ^one obtains the following equation ^{for the}

characteristic functional :

This gives ^the following ^{set of} equations ^{for the} equal time correlators :

162

etc. The comparison with the discrete master

equation ^shows (see [14]) that the differentiations with functions must always be understood in the sense, e.g.,

etc. Equations (3.6), (3.7) and others correspond ^to

the infinite set of the coupled equations and any subset of these equations ^{is not} closed. For this

reason one must use various approximations ^to

disentangle ^the system ^of equations. ^{We will below}

restrict ourselves to the simplest ^closure precedure

which is valid in the limit of small fluctuations of concentrations.

In steady-state spatially homogeneous ^conditions

one may ^assume

where

Then one obtains according ^to definitions (3.9) ^and (3.10) :

In the limit of small fluctuations the term

(åXj(rl) ^{AXk (r2)} åXm(r3» ^may approximately ^be neglected ^{and the} system (3.6), (3.7) ^becomes

closed. We should note that since the system ^of

equations ^is coupled, ^{the term}

(âXj(rl) ^{OXk (rz )} âXm(r3» ^{is expressed through}

the terms

etc., ^{and is} generally ^not equal ^{to zero as} might ^be expected from the naive parity argumentation. ^{It is}

also useful to extract separately the Poissonian contribution from the pair correlator (cf. [11, 15]) :

Substituting expressions (3.9)-(3.13) ^into equations (3.6), (3.7) ^and taking ^{into account the} electroneutrality ^condition (3.3), ^{the Einstein re-}

lationship (2.3), ^the identity

and the fact that the integrals ^{of the} type

do not depend ^{on r} it is easy ^to prove that

equation (3.6) ^{is satisfied} identically, ^while

equation (3.7) gives

or

Thus, ^{the infinite set of} equations ^{for the} equal ^time

correlators appears ^to be in fact equivalent ^{to the}

BBGKY scheme (cf., e.g., [15, 16]).

The amount of fluctuations can be estimated by

the ratio :

So the approximation

is valid only ^when

4. Mobility of ions in electrolytes.

We now include recombination-dissociation _proces-

ses and apply the formalism to the following physical problem. ^{Let the} mobility ^{of an} isolated ion in the solvent be equal ^to b a (0). Our aim consists in the calculation of the correction to the mobility b a (0) ^due

to the mutual Coulomb interaction between ions and the recombination-dissociation processes. For simp- licity ^{we restrict ourselves to a} two-component electrolyte ^{and to} the range of parameters ^{in which} the inequality (3.20) is satisfied. The recombination- dissociation processes ^are defined according ^{to the}

scheme :

In steady-state conditions ^one obtains, neglecting

fluctuations in the lowest order :

Here X(+) and X (- ) denote the numbers of positive

and negative ions per unit volume and N corresponds

to that of neutrals. The indices a, b, ^c will denote either negative ^or positive ions, while index n will

invariably be attributed to the neutrals.

The total field acting ^on the a-th ion is equal ^to

where E is ^a static spatially homogeneous ^external

electric field and the contribution AE is related to the self-consistent Coulomb interaction with the other ions. Since the perturbation ^{of the electric} potential ^{at a} point rl ^created by ^{the whole} system ^of ions, ^{with the condition} that the a-th ion is placed ^at point r2, is given by (see ^also [13])

where K,,, (rl, rz) ^{is the} pair correlation function (see Eqs. (3.11) ^and (3.13)), ^{one obtains}

For the relatively ^small external fields E the pair

correlation function Kca(rl’ r2 ) will contain the addi- tional contribution Kjj ^{which is} mainly proportional

to E (generally there is the expansion ^with respect ^to degrees ^of E). ^This gives respectively the linear contribution to AE and, thus, the total field acting ^on

the a-th ion will be equal ^to

The renormalization of the effective external field leads to the effective redefinition of the observable

mobility :

where b a (0) ^{is the} mobility of the isolated a-th ion in the solvent and the correction 8 is related to the interaction effects. It is important ^{that correction}

K§j (and 5) ^{will be} determined by ^both long-range

Coulomb interaction effects and recombination-dis- sociation processes. Thus, ^the problem is reduced to the calculation of Kab(rl, r2). ^{We will use below the} approximation (åXj(rl) ^{OXk (r2 )} åXm(r3) = ^{0 (or}

Eq. (3.20)) ^and equations (4.2), (4.3).

Combining equations (2.7), (2.8) ^and (3.5) ^one

derives analogously ^to section 3 the following

steady-state equations (cf. ^{also § 4 in} [11]) :

164

Here indices a, b, ^c denote either negative ^or positive ions, while index n is attributed to the neutrals, ^{E is a} static spatially homogeneous external electric field, the correlators Kjm (rl - rz) ^{are defined} according ^to equation (3.11).

Analogously ^to [13] ^the system (4.9)-(4.11) ^{can be solved} by expanding ^with respect ^to external field E.

In the lowest order (E

0 ) ^{one obtains}

where Q (rl - r2 ) is determined by equations (3.17) ^and (3.18). ^{Then this} gives in the first order with respect

to E the folloving equations :

The equation ^for KlJ(rl - r2) is obtained from equation (4.17) by replacement of indices « + » - « - »

We simplify partially the situation and consider below only ^the symmetric ^{case when}

Relating q;:1) ^and K§j(1) according ^to equation (4.5) ^and solving equations (4.15)-(4.21) by ^Fourier

transformation one finally ^obtains (see ^also Eqs. (4.7), (4.8)) :

The correction 5 is equal for both the positive ^and negative ions. It is always negative.

Since the conductivity ^{of the} electrolyte ^is given by [13] :

the correction to b will correspond ^{to that of cr.}

We considered here only the so-called relaxational correction ^to mobility because the electrophoretic

correction to mobility will remain unchanged ^{in this} approximation (cf. [13]).

5. Discussion.

As is seen from equations (4.22)-(4.27) ^{the main}

dimensionless parameters ^{of the} theory ^{are the}

ratios e2z 2/ E’k D kB T, k2 X(O)/k, A D k2 X(O)/ D ^and AD kl/D. ^{The parameter} e2 Z2 / E AD kB T ^charac-

terizes the strength ^of screening. ^The larger ^{values of}

this parameter correspond ^to larger effects of

screening. ^{The ratio} kz X(O)/k1 characterizes the relative rates of recombination and dissociation. For the strong electrolytes ^{this ratio is} usually small,

while for the weak electrolytes ^the inequality ^is typically opposite ^{even at} rather moderate concent- rations X(o). The time À 5/ D describes the Coulomb relaxation of charge unbalance. Thus, ^{the ratios}

À6 kzX(O)/D and A 2 k,ID ^describe the relative rates of the recombination and dissociation and the relaxation of charge.

In the limit k2 X(O),k 2 ID ^{1, k1} ’k 2 ID ..c ^{1 the}

recombination and dissociation can be neglected ^and

one obtains

which corresponds ^to the classical result by Debye

and Hfckel, ^and Onsager [13]. ^{It can be} proved ^that

the recombination and dissociation processes will

always ^cause the reduction of À eff ^{and thus enhance}

the mobility ^with respect ^to (5.1) (see, ^however, below). ^{In the limit} k2 X(O),k 2 ID > ^{1, k1} a D/D > ¹

the Coulomb charge relaxation can be neglected,

which gives

If moreover the inequality k2 X(O)Ikl .c. 1 ^holds,

then Aeff ^will again ^be equal approximately ^to (5.1).

In the opposite case k2 X(O)Ikl > ^{1 one obtains}

A eff - À D kl/ k2 X(O) (if D - D (n). ^{In the inter-}

mediate range k1 - k2 X(O) ^{the answer} (5.2) ^differs

from (5.1) by ^a numerical factor of the order of

unity.

Finally, ^{we should note} that the results can easily

be generalized ^{to the case of} alternating spatially homogeneous ^external electric field by ^{the mere} replacement E -+ £ (úJ ) (except ^{for the} expression ^of

the zeroth order with respect to E), Å 12 -+ Å 12 +

i co /2 D, ’k22 ’k22 + ito I(D + D (n)).

Acknowledgments.

The authors are thankful to A. A. Vedenov, ^{A. L.}

Chernjakov, ^{A. M.} Kamchatnov and E. B. Levchen- ko for helpful discussions and remarks.

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