Thermodynamics of Chemical Waves

Francesco Avanzini, Gianmaria Falasco, and Massimiliano EspositoComplex Systems and Statistical Mechanics, Physics and Materials

Science Research Unit, University of Luxembourg, L-1511 Luxembourg(Dated: April 19, 2019)

Chemical waves constitute a known class of dissipative structures emerging in reaction-diffusionsystems. They play a crucial role in biology, spreading information rapidly to synchronize and coor-dinate biological events. We develop a rigorous thermodynamic theory of reaction-diffusion systemsto characterize chemical waves. Our main result is the definition of the proper thermodynamicpotential of the local dynamics as a nonequilibrium free energy density and establishing its balanceequation. This enables us to identify the dynamics of the free energy, of the dissipation, and ofthe work spent to sustain the wave propagation. Two prototypical classes of chemical waves areexamined. From a thermodynamic perspective, the first is sustained by relaxation towards equi-librium and the second by nonconservative forces generated by chemostats. We analytically studystep-like waves, called wavefronts, using the Fisher-Kolmogorov equation as representative of thefirst class and oscillating waves in the Brusselator model as representative of the second. Given thefundamental role of chemical waves as message carriers in biosystems, our thermodynamic theoryconstitutes an important step toward an understanding of information transfers and processing inbiology.

I. INTRODUCTION

Chemical waves, or traveling waves, are complex con-centration patterns of chemical species moving in spacewith a constant velocity and without changes of shape [1].They are essential for communication in biosystems bothat the intracellular and intercellular level, and play a cru-cial role for the synchronization and coordination of bi-ological events. By using chemical reactions, chemicalwaves are able to spread signals more rapidly comparedto simple diffusion [2]. The information they carry is en-coded not only in the identity of the chemical species butalso in other features of the concentration patterns suchas the amplitude for step-like waves, called wavefronts, orthe wavenumber for periodic waves [3, 4]. Calcium waves,for instance, are highly versatile intracellular messengercreating different spatiotemporal patterns [5, 6] that reg-ulates several cellular activities over a wide range of timescales [7].

From a thermodynamic standpoint, chemical wavesare out of equilibrium processes requiring a contin-ual influx of energy and chemicals. They belong tothe broader class of dissipative structures occurring inreaction-diffusion systems [8, 9]. A lot of work hasbeen dedicated to investigating their origin and theirrelative stability, but mainly from a dynamical point ofview [10, 11]. However, a proper understanding of the en-ergetic cost needed to support chemical waves is missing.We fill this gap in this work by developing a thermody-namic theory of chemical waves.

We start in Sec. II by providing a local formulationof thermodynamics of reaction-diffusion systems. Ourtheory is based on recent advances in the thermody-namic description of chemical reaction networks [12, 13]and reaction-diffusion systems [14] inspired by stochas-tic thermodynamics [15, 16] and its links to informationtheory [17]. In Ref. [14], global thermodynamic quanti-

ties were defined in terms of the reaction-diffusion dy-namics to study the energetics of dissipative structures.However, to charaterize localized processes such as chem-ical waves, a local formulation is needed in the spiritof the phenomenological nonequilibrium thermodynam-ics developed in the last century [18, 19]. The theory wedevelop does so but without relying on any linear approx-imation of the thermodynamic currents in the forces. Itis systematically built on top of the dynamics and thusvalid arbitrarily far from equilibrium. The thermody-namic quantities are local and expressed in terms of den-sities. A crucial step is the identification of the properthermodynamic potential for the local dynamics, i.e., afree energy density. In a way which is reminiscent of equi-librium thermodynamics when passing from canonical togrand canonical ensembles, this potential is constructedfrom the Gibbs free energy density by eliminating theenergetic contribution due to the local diffusion of thechemical species and by making use of the conservationlaws of the chemical reaction network [13, 20, 21]. Itsbalance equation provides a local formulation of the sec-ond law and decomposes the evolution of the free energydensity into a flow of free energy caused by diffusion andthree source terms. The first two source terms are chem-ical works produced respectively by time dependent driv-ing and by nonconservative forces while the third one isa sink term due to entropy production. For undriven de-tailed balanced systems (i.e., in absence of work), thisfree energy is minimized as the system relaxes to equilib-rium.

We proceed in Sec. III by specializing our theory tostudy the thermodynamic cost of propagating and sus-taining chemical waves. We first identify the condi-tions needed for an open reaction diffusion system to al-low the existence of chemical waves. We then show ingeneral that while the free energy is exclusively chang-ing due to propagation, work and entropy production

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also quantify the cost for sustaining the waves. Wesubsequently consider in Sec. IV two prototypical wavepatterns which we characterize analytically, i.e., wave-fronts in the Fisher-Kolmogorov equation [1, 22] and pe-riodic/oscillating waves in the Brusselator model [23, 24].We find that when wavefronts are caused by relax-ation towards equilibrium in detailed balanced systems,as Fisher-Kolmogorov waves, they propagate efficientlyfrom an energetic standpoint because dissipation is solelylocalized at the front. Instead, oscillating waves, as in theBrusselator model, are significantly more energy costlyas they need to be sustained by nonconservative forces(non detailed balanced systems) which cause dissipationthroughout the system.

The implications of our work for information process-ing in biosystems as well as possible future developmentsare discussed in Sec. V.

II. REACTION-DIFFUSION SYSTEMS

A Reaction-Diffusion System (RDS) is considered hereas a dilute ideal mixture of chemical species Zα (α ∈ S)diffusing within a volume V with impermeable bound-aries ∂V and undergoing elementary reactions ρ ∈ R.Each chemical species α is classified either as a con-trolled/chemostatted species (αy ∈ Y ) if it is exchangedwith particular reservoirs called chemostats controllingits concentration or as internal species (αx ∈ X) other-wise [12]. It is characterized by its concentration fieldZα(r, t) which follows the reaction-diffusion equation:

∂tZα = Sαρ jρ −∇ · J

α + δααyIαy , (1)

where δααy is the Kronecker delta and repeated upper-lower indices imply the summation over all the allowedvalues of the indices in all the paper. jρ gives theconcentration variation due to the chemical reaction ρ,Sα+ρZαSα−ρZα. The stoichiometric coefficient Sα+ρ (Sα−ρ)specifies the number of molecules α involved in the for-ward reaction +ρ (backward reaction −ρ) whose chem-ical reaction current j+ρ (j−ρ) satisfy the mass-actionkinetics [18, 25],

j±ρ = k±ρ∏α

(Zα)Sα±ρ . (2)

The so-called stoichiometric matrix Sαρ = Sα−ρ−Sα+ρ givesthe full variation of the number of molecules α upon thereaction ρ, while jρ = j+ρ−j−ρ specifies the net reactioncurrent. The Fick’s diffusion currents

Jα = −Dα∇Zα, (3)

with the diffusion coefficient Dα for the species α, de-scribe the transport of chemical species within the vol-ume V . They vanish at the boundaries ∂V , i.e.,

∫V

dr ∇·Jα = 0 ∀α. If the RDS is open, the controlled speciesαy ∈ Y are exchanged with the chemostats according

to the external currents Iαy (r, t). These latter vanishinstead in a closed RDS.

The left null eigenvectors of the stoichiometric matrix,

lλαSαρ = 0 ∀ρ ∈ R, (4)

are the so-called conservation laws lλ [12, 20] definingthe quantities Lλ =

∫V

dr lλαZα that are conserved if the

RDS is closed [14]: dtLλ = 0. When the system is open,

the set of conservation laws lλ is split into two disjointsubsets: the unbroken conservation laws lλu and the bro-ken conservation laws lλb . The unbroken conservationlaws are left null eingenvectors of the submatrix of theinternal species, namely lλuαxS

αxρ = 0 and l

λuαy = 0, whereas

the broken conservation laws are not, namely lλbαxSαxρ 6= 0

for at least one ρ ∈ R. Therefore, the quantities

Lλu =

∫V

dr lλuαxZαx (5)

are conserved even if the RDS is open, dtLλu = 0, unlike

Lλb =∫V

dr lλbα Zα. They are called unbroken conserved

quantities. Notice that chemostating a species does notalways break a conservation law [12]. We thus distinguishthe set of controlled species αyb ∈ Yb ⊆ Y breaking allthe broken conservation laws from the others αyp ∈ Yp =Y \Yb. This allows us to introduce the so-called moieties

Mαyb := l̂αybλb

∫V

dr lλbα Zα (6)

where l̂αybλb

denotes the elements of the inverse of the

matrix whose entries are lλbαyb(see Ref. [14] for details).

The moieties represent the concentration of parts ofmolecules which are exchanged with the environmentthrough the chemostats since their time evolution is de-termined only by the external currents, dtM

αyb (t) =

l̂αybλb

∫V

dr lλbαyIαy (r, t).

The thermodynamic equilibrium is characterized byvanishing reaction currents jρ = 0, diffusion currentsJα = 0, and external currents Iαy = 0. The equilibriumconcentrations Zαeq are, consequently, homogeneously dis-tributed over the volume V . The existence of such equi-librium, together with mass-action kinetics, implies theso-called local detailed balance condition for the kineticconstants k±ρ of the chemical reactions,

k+ρ/k−ρ =∏α

(Zαeq)Sαρ . (7)

This constraint is assumed to be valid to ensure thatclosed systems relax to equilibrium and hence thermody-namic consistency [20]. Nonequilibrium conditions canbe created by chemostatting certain species. These maybe homogeneously distributed or not as well as constantin time or not. Our description of the reaction-diffusiondynamics presumes that all degrees of freedom otherthan concentrations in space to be equilibrated, as thetemperature T and the pressure of the solvent. In this

3

way, thermodynamic state functions can be specified bytheir equilibrium form but expressed in terms of nonequi-librium concentrations, like for the chemical potentialµα(r, t) = µ

◦α + RT ln(Zα(r, t)) of each species (µ

◦α is

the standard chemical potential). In this respect, thelocal detailed balance condition can be restated estab-lishing a correspondence between the kinetic constantsk±ρ, namely the dynamics of the RDSs, and the standardchemical potentials µ◦α, namely the thermodynamics:

k+ρk−ρ

= exp

(−Sαρµ◦αRT

). (8)

We consider now the nonequilibrium thermodynamiccharacterization of RDSs. First, we recapitulate theglobal thermodynamic theory developed in Ref. [14] andthen we generalize it to a local formulation. The globalsecond law of thermodynamics for open RDSs can bestated as the following balance equation

dtG = −T Σ̇ + ẆGdriv + Ẇnc, (9)

for the so-called semigrand Gibbs free energy G(t). T Σ̇is the non-negative entropy production rate accountingfor the energy dissipation, ẆGdriv is the driving work rateneeded to manipulate the concentration of the chemostat-ted species in a time dependent way and Ẇnc is the non-conservative work rate spent to prevent the system fromrelaxing towards equilibrium. We examine in detail eachterm in Eq. (9). Whenever possible, we provide the ex-pression of the thermodynamic quantities according totheir densities which will then be used in the local for-mulation of thermodynamics. With a slight abuse of no-tation, we will use the same name for the density andthe corresponding global quantities. For example, wewill refer to both T σ̇ and T Σ̇ =

∫V

dr T σ̇ as entropyproduction rate.

In analogy to equilibrium thermodynamics when pass-ing from canonical to grand canonical ensembles, thesemigrand Gibbs free energy

G := G− µrefαybMαyb (10)

is obtained from the Gibbs free energy G by eliminat-ing the energetic contributions of the matter exchangedwith the reservoirs. The latter amounts to the moietiesMαyb of Eq. (6), times the reference values of their chem-ical potentials µrefαyb

which are the values of chemical po-

tential fixed by the chemostats αyb . If a chemostat αybsets different values of the chemical potential µαyb (r) fordifferent points r in the volume V , then the referencechemical potential can be chosen arbitrarily among thesevalues. This is equivalent to set the minimum value ofG as it will become clear later. The Gibbs free energyG(t) =

∫V

dr g(r, t) is the integral over the volume Vof the Gibbs free energy density g(r, t) of ideal dilutesolutions

g = Zαµα −RTZS (11)

with RTZS = RT∑α Z

α accounting for the contribu-tion of the solvent [26, 27].

The total entropy production rate Σ̇ consists of twoparts Σ̇ = Σ̇rct + Σ̇diff, i.e., the reaction Σ̇rct and thediffusion Σ̇diff part:

Σ̇rct =

∫V

dr σ̇rct ≥ 0, Σ̇diff =∫V

dr σ̇diff ≥ 0. (12)

with

T σ̇rct = −(µαSαρ )jρ ≥ 0, T σ̇diff = −(∇µα) · Jα ≥ 0.

(13)

The free energy of reaction −(µαSαρ ) and the variationof the chemical potential across space −∇µα are ther-modynamic forces driving the system towards equilib-rium. The densities of the entropy production rates,as well as their global counterparts, are non-negativequantities and they vanish only at equilibrium. Indeed,the reaction entropy production rate can be written asT σ̇rct = RTj

ρ ln(j+ρ/j−ρ) ≥ 0, with the aid of the lo-cal detailed balance (8) and the mass-action kinetics (2).Similarly, the diffusion entropy production rate can bewritten as T σ̇diff = RTD

α|∇Zα|2/Zα ≥ 0. The drivingwork rate takes into account the time-dependent manip-ulation of µrefαyb

performed by the chemostats αyb

ẆGdriv = −dtµrefαyb

Mαyb . (14)

It obviously vanishes in autonomous systems. The non-conservative work rate

Ẇnc =

∫V

dr ẇnc, (15)

where the corresponding density is given by

ẇnc = FαyIαy , (16)

quantifies the energetic cost of sustaining fluxes of chem-ical species among the chemostats by means of the forces

Fαy = (µαy − µrefαyb l̂αybλb

lλbαy ). These forces have different

origin depending on whether we consider the chemostat-ted species αyb or αyp . Indeed, Fαyb (r) = µαyb (r)−µ

refαyb

is the difference of chemical potential of the same speciesimposed in different points of the space. It vanishesin case of homogeneous chemostatting. In contrast,

Fαyp (r) = µαyp (r) − µrefαyb

l̂αybλb

lλbαy is the difference of

chemical potentials imposed by different chemostats. Itvanishes if all the chemostats break a conservation law,i.e., Yb = Y .

In the absence of nonconservative forces Fαy = 0

∀αy ∈ Y , no nonconservative work is performed Ẇnc = 0and the system is said to be detailed balanced. This oc-curs when all the chemostats break a conservation lawYb = Y , and the chemostatted species are homogeneouslydistributed µαyb (r) = µ

refαyb∀r ∈ V . If a detailed bal-

anced system is also autonomous Ẇdriv = 0, it relaxes

4

towards equilibrium. This follows from the fact that thesemigrand Gibbs free energy approaches its equilibriumvalue Geq monotonously in time: i) the time derivative ofG(t) is always negative,

dtG = −T Σ̇ ≤ 0. (17)

and ii) G(t) is lower bounded by its equilibrium valueGeq. Indeed, G can be expressed in terms of its referenceequilibrium value Geq as

G = Geq +RT∫V

dr L({Zα(r, t)} ‖ {Zαeq}) (18)

by employing the relative entropy for the concentrationfields (i.e., non-normalized concentration distributions)

L({Zα} ‖ {Zαeq}) =∑α∈S

Zα ln

(Zα

Zαeq

)−(Zα − Zαeq

),

(19)which quantifies the difference between the two concen-tration distributions {Zα} and {Zαeq}. Since the relativeentropy is always positive, unless Zα = Zαeq, the semi-grand Gibbs free energy is greater than or equal to itsequilibrium counterpart G ≥ Geq. In other words, G(t)acts as a Lyapunov function. The reference equilibriumcondition

Zeqα = exp((µrefα − µ◦α)/RT ) (20)

is specified according to the reference chemical potentialsµrefα . We have already introduced the reference chemicalpotentials µrefαyb

for the controlled species αyb in Eq. (10).

Here, we call the chemical potentials µαx and µαyb evalu-ated at the equilibrium condition reference chemical po-tentials because they depend on µrefαyb

. A detailed dis-

cussion in Appendix A shows that, for the Yp controlled

species, µrefαyp is given by µrefαyp

= µrefαybl̂αybλb

lλbαy . This also

provides an interpretation of the thermodynamic force

Fαyp (r) = µαyp (r) − µrefαyb

l̂αybλb

lλbαy as the difference be-

tween the chemical potential of the species αyb imposedby the chemostat αyb and the corresponding referencechemical potential. In contrast, µrefαx is specified by both

the reference chemical potentials µrefαyband the unbroken

conserved quantities Lλu of Eq. (5) but an explicit ex-pression cannot be obtained. If all the conservation lawsare broken, there are no more conserved quantities and

also µrefαx is given by µrefαx = µ

refαyb

l̂αybλb

lλbαx . It is import to

understand that the choice of µrefαybof Eq. (10) set the

equilibrium condition {Zeqα } reached by the system if itwas detailed balanced. Consequently, the choice of µrefαybset Geq that is the minimum value of G(t).

We now generalize this description to a local ther-modynamic theory. To this aim, a fundamental step isthe identification of the proper thermodynamic potentialcharacterizing the local dynamics of RDSs as G does forthe global dynamics. Therefore, we seek a free energy

density that i) is lower bounded by its equilibrium valueand ii) approaches asymptotically its equilibrium valuefor undriven detailed balanced systems. One might won-der whether the semigrand Gibbs free energy density

g(r, t) := g(r, t)− µrefαyb l̂αybλb

lλbα Zα(r, t), (21)

such that G(t) =∫V

dr g(r, t), plays this role. Its balanceequation is specified as

∂tg = −T σ̇ + ẇgdriv + ẇnc −∇ · J

g , (22)

where T σ̇ is the entropy production rate of Eq. (13), ẇncis the nonconservative work rate of Eq. (16), and thedriving work rate is given by

ẇgdriv = −dtµ

refαyb

l̂αybλb

lλbα Zα. (23)

Compared to the evolution equation (9) for the semi-grand Gibbs free energy, a flow term

J g = (µα − µrefαyb l̂αybλb

lλbα )Jα (24)

arises at the local level. This term is absent at the globallevel because the diffusion currents vanish at the bound-aries ∂V and, consequently,

∫V

dr ∇ · J g = 0 .The semigrand Gibbs free energy density g(r, t) is not

the proper thermodynamic potential because it is notlower bounded by its equilibrium value geq: g(r, t) cannot be written as geq plus the relative entropy of Eq. (19).Consequently, g(r, t) is not minimized in undriven de-tailed balanced systems. This is a direct consequence ofthe definition (21) where g(r, t) is constructed from theGibbs free energy density g(r, t) by eliminating the en-ergetic contributions of the matter exchanged with the

chemostats. Indeed, the term l̂αybλb

lλbα Zα(r, t) in Eq. (21)

is the local representation of the moieties of Eq. (6). How-ever, at the local level diffusion allows all the species tobe exchanged with neighboring regions of space whichplay the same role as the chemostats from a local stand-point. This means that the proper thermodynamic po-tential must be specified as the following free energy den-sity

g(r, t) := g(r, t)− µrefα Zα(r, t), (25)

where the energetic contributions of all the species ex-changed through diffusion are removed from the Gibbsfree energy density g(r, t) of Eq. (11). The local equiva-lent to the term µrefαyb

Mαyb in Eq. (10) is precisely µrefα Zα.

The reference chemical potentials µrefα have the samemeaning as in Eq. (20). If all the conservation lawsare broken by the chemostats, the definition of the freeenergy density g(r, t) in Eq. (25) is equivalent to thedefinition of the semigrand Gibbs free energy density

g(r, t) in Eq. (21) since µrefαxZα = µrefαyb

l̂αybλb

lλbαxZαx and

µrefαypZα = µrefαyb

l̂αybλb

lλbαypZαyp .

5

We can now verify that g(r, t) can be expressed interms of the reference equilibrium free energy geq as

g(r, t) = geq +RTL({Zα(r, t)} ‖ {Zαeq}) (26)

by employing the relative entropy of Eq. (19). The freeenergy density is therefore always greater than or equalto the equilibrium counterpart g ≥ geq. The fact that therelative entropy of information theory [28] appears bothat the global and the local level supports the idea thatthermodynamics plays a fundamental role in the charac-terization of the information codified in complex patternsof chemical concentrations.

Furthermore, the free energy density g(r, t) satisfiesthe balance equation

∂tg = −T σ̇ + ẇgdriv + ẇnc −∇ · Jg (27)

which constitutes the local formulation of the second law.The non-negative entropy production T σ̇ and the non-conservative work rate ẇnc are the same as in Eq (13)and Eq. (16), respectively. However, the driving chemi-cal work rate

ẇgdriv = −dtµrefα Z

α (28)

now takes into account the time evolution of all µrefαthrough the time-dependent manipulation of µrefαyb

per-

formed by the chemostats αyb . Indeed, the referencechemical potentials µrefαyp and µ

refαx depend on µ

refαyb

’s (see

Appendix A). The flow of free energy,

Jg = (µα − µrefα )Jα (29)

describes the effects on the evolution of the free energydensity due to the local diffusion currents. Note that thediffusion entropy production rate T σ̇diff (resp. the freeenergy flow Jg) can be further split into a contributiondue to the internal species T σ̇Xdiff = −(∇µαx) ·J

αx (resp.JgX = (µαx − µrefαx)J

αx) and one due to the controlled

species T σ̇Ydiff = −(∇µαy ) · Jαy (resp. JgY = (µαy −

µrefαy )Jαy ). We will make use of this decomposition when

we examine the thermodynamic description of chemicalwaves. Equations (25) and (27) constitute the first majorresult of this paper.

For undriven detailed balanced systems g approachesgeq asymptotically. Indeed, the global free energy,

G(t) :=

∫V

dr g(r, t), (30)

is greater than its reference equilibrium value G ≥ Geq(because g ≥ geq) and its time derivative is always neg-ative

dtG = −T Σ̇ ≤ 0 (31)

since ẇgdriv = ẇnc = 0 and∫V

dr ∇ · Jg = 0. Inother words, G → Geq monotonously in time. Whilethis implies that g approaches geq asymptotically, i.e.,

g is minimized, it is not granted that its time derivative∂tg = −T σ̇ − ∇ · Jg is always negative because of thefree energy flow. Consider for example the case wherethe concentration for all the species except one is equalto its equilibrium value at one specific point r′ in space:Zα(r, t) = Zαeq ∀α 6= α′ if and only if r = r′. The con-centration of the species α′ differs instead from its equi-librium value for an arbitrarily small positive number,Zα′(r′, t) − Zα′eq = � > 0. Expanding the time deriva-

tive of the free energy ∂tg(r, t) for r = r′ in powers

of � and truncating the expansion at the lowest ordergives ∂tg(r

′, t) = �d�(T σ̇rct) + �Dα′∇2Zα′/Zα′eq , with the

derivative of the reaction entropy production rate eval-uated for � = 0. Therefore, the free energy increaseslocally (∂tg(r

′, t) > 0) if, for instance, the curvatureof the concentration field is very large at r = r′ (e.g.,

∇2Zα′ = Zα′eq/�2) since the contribution of the reactionentropy production rate becomes negligible.

III. CHEMICAL WAVES

We now examine the thermodynamic description ofchemical waves considered here as particular pattern so-lutions of RDSs also called traveling waves. The concen-tration of a species Zα(r, t) evolves as a traveling waveif it propagates in space with a constant velocity cα andwithout changing its shape:

Zα(r, t) = Zα(r − cαt). (32)

We label the spatial coordinate in the traveling wave αframe of reference as r̃α := r − cαt.

A. Dynamics of Traveling Waves

We first discuss a set of conditions that the reaction-diffusion equation (1) needs to satisfy to allow for travel-ing wave solutions. We will then employ these conditionsin Subs. III B to specialize the general thermodynamicdescription of Sec. II for traveling waves. We start bysplitting the chemical species into two disjoint subsets,i.e., the traveling waves αx ∈ X and the other speciesαy ∈ Y . The traveling waves are internal species byconstruction. We use the same notation for the nontrav-eling wave species as for the controlled species in Sec. IIsince we will show that these former must be undrivenchemostatted species.

Using Eq. (32), the reaction-diffusion equation (1) forthe traveling wave αx becomes

Dαx∇2Zαx + cαx · ∇Zαx + Sαxρ jρ = 0. (33)

In the frame of reference of the traveling wave αx, theterms ∇2Zαx and cαx · ∇Zαx are time-independent and,consequently, the above equation admits a solution if theconcentration variation due to the chemical reactions istime-independent as well: ∂t(Sαxρ jρ(r̃

αx + cαxt, t)) = 0.

6

This condition must be satisfied for every r̃αx and everyt justifying the hypothesis that each reaction current jρ

is time-independent in the frame of reference of the trav-eling wave αx, i.e., ∂t(j

ρ(r̃αx + cαxt, t)) = 0 ∀ρ ∈ R. Inother words, each reaction current evolves like the trav-eling wave αx with the propagation velocity c

αx . Sincethis must be true for every traveling species, the reac-tion currents jρ have to be time-independent in everyframe of reference r̃αx . Therefore, we conclude that allthe traveling waves evolve with the same velocity c andthe corresponding comoving coordinate is r̃ = r − ct.

This implies that the condition ∂t(jρ(r̃ + ct, t)) = 0,

using the mass-action kinetics Eq. (2) and ∂t(Zαx(r̃)) =

0, becomes

∑αy∈Y

∂jρ

∂Zαy[c · ∇Zαy + ∂tZαy ] = 0, (34)

where ∇Zαy and ∂tZαy are evaluated in frame of refer-ence r̃. The above equation must hold for every reactioncurrent jρ of every point r̃ and of every time t. We thusassume that each term between square bracket vanish in-dependently, c ·∇Zαy + ∂tZαy = 0. This means that thespecies Y are either traveling waves ∂tZ

αy = −c · ∇Zαyor undriven controlled species ∂tZ

αy = 0 with a con-centration field that is invariant under space translationalong the propagation direction c · ∇Zαy = 0. Since thespecies Y are not traveling waves by hypothesis, theyare undriven chemostatted species. In this respect, theirreaction-diffusion equation simplifies to

Sαyρ jρ −∇ · Jαy + Iαy = 0, (35)

with vanishing diffusion currents in the propagation di-rections c · Jαy = −Dαyc · ∇Zαy = 0.

In conclusion, to allow for traveling waves to possiblyarise, we consider only RDSs that satisfy the followingrequirements:

(i) the chemical species are split into two disjoint sub-sets, i.e., the traveling waves X (internal species)and the controlled species Y ;

(ii) the traveling waves are characterized by the samepropagation velocity c and they have the same co-moving coordinate r̃ = r − ct;

(iii) the concentration field of the controlled species Y isinvariant under space translation along the propa-gation direction c·∇Zαy = 0 and it is kept constantby the chemostats ∂tZ

αy = 0 (no driving work isperformed, ẇdriv = 0).

Although we do not expect it, we note that rigorouslyspeaking weaker conditions for traveling wave solutionsmay exist as our argumentation is not a mathematicalproof.

B. Thermodynamics of Traveling Waves

We now examine how the thermodynamic quantitiessuch as the local free energy, the entropy production rate,and the nonconservative work rate evolve when travelingwaves propagate in RDSs.

Using Eq. (32), the time evolution of the free energy isgiven by

∂tg = −(µαx − µrefαx)c · ∇Zαx , (36)

where one can recognize a drift contribution for everytraveling wave c · ∇Zαx . In other words, the free energyis “dragged” by the propagating traveling waves in the cdirection.

The nonconservative work rate and the entropy pro-duction rate account for the energetic cost of supportingthe wave propagation. Using Eq. (16) and Eq. (35), ẇncbecomes

ẇnc = −(µαy − µrefαy )Sαyρ j

ρ +∇ · JgY + T σ̇Ydiff (37)

The first term on the r.h.s., −(µαy − µrefαy )Sαyρ jρ, is the

energetic cost for balancing with the chemostats the con-centration variations of the Y species due to the chemi-cal reactions. On the other hand, the contribution to thefree energy flow JgY and to the diffusion entropy produc-tion rate T σ̇Ydiff resulting from the controlled species αytake into account the amount of work needed to preventmodifications to the pattern profiles Zαy (r) because ofdiffusion.

Finally, with the aid of Eqs. (27), (36) and (37), wefind that the entropy production rate reads

T σ̇ = −∂tg−∇ · JgX − (µαy − µrefαy )Sαyρ j

ρ + T σ̇Ydiff (38)

This general result highlights explicitly which phenomenaare responsible for the dissipation: i) a drift contributiondue to the propagation of the waves (µαx−µrefαx)c·∇Z

αx =−∂tg, ii) the free energy flow due to the wave speciesJgX = (µαx − µrefαx)J

αx , iii) the reaction consump-tion/production of the chemostatted species sustainingthe wave dynamics (µαy−µrefαy )S

αyρ jρ and iv) the diffusion

of the controlled species T σ̇Ydiff. The first two contribu-tions take into account the dynamics of the waves, whilethe others represent the dissipation due to the chemostat-ting of the controlled species. Equation (38) constitutesthe second major result of this paper.

The above expressions of the thermodynamic quanti-ties explicitly account for every process occurring whilewaves propagate. Furthermore, depending on the phys-ical properties of the system of interest, they can beeffectively simplified. For example, if the controlledspecies were homogeneously distributed Zσy (r) = Z

σy

∀r ∈ V , no work is required to prevent changes in thecontrolled species concentrations because of the diffusion.In this case, the nonconservative work rate simplifies toẇnc = −(µαy − µrefαy )S

αyρ jρ, while the diffusion entropy

production rate T σ̇Ydiff vanishes. In case the system is

7

detailed balanced, no nonconservative forces are applied,namely µαy = µ

refαy , and the dissipation is only due to the

propagation of the waves: T σ̇ depends on the drift andthe flow contribution, T σ̇ = (µαx−µrefαx)c·∇Z

αx−∇·JgX .In this case, the global dissipation is solely caused bythe transport mechanism of the wave dynamics T Σ̇ =∫V

dr (µαx − µrefαx)c · ∇Zαx .

IV. MODEL SYSTEMS

We apply here the thermodynamic description devel-oped in the previous section to study the properties ofthe traveling waves emerging in two model systems. Thefirst model, the Fisher-Kolmogorov equation, is detailedbalanced, and the relaxation towards equilibrium occursvia a step-like wave. On the other hand, for the Brussela-tor model two oscillating traveling waves are maintainedby nonconservative forces.

A. Fisher-Kolmogorov equation

A minimal RDS displaying traveling wave solutions isthe so-called Fisher-Kolmogorov equation in one infinitespatial dimension [1]. It can be used to describe an au-tocatalytic reaction between an internal species X anda controlled species Y according to the chemical equa-

tion X+Yk+1

k−12X. The system is detailed balanced

(no nonconservative forces) since one single species ischemostatted. The reaction-diffusion equation

∂tx = k+1xy − k−1x2 +D∂2rx, (39)

with y the homogeneous concentration of the controlledspecies and D the diffusion coefficient of the inter-nal species, specifies the dynamics of the concentrationZαx(r, t) = x(r, t) of the internal species. Waves solu-tions x(r, t) = x(r − ct) emerge for propagation veloci-ties c greater than or equal to a critical value c ≥ c =2√Dk+1y, and they are all characterized by a step-like

profile which formally means i) limr̃→+∞ x(r̃) = 0, ii)limr̃→−∞ x(r̃) = xeq and iii) ∂r̃x < 0 ∀r̃. Note thatr̃ denotes the single spatial coordinate in the frame ofreference of the traveling wave, r̃ = r − ct, while xeq la-bels the equilibrium concentration of the internal species,xeq = k+1y/k−1, corresponding to the amplitude of thewave. According to these general properties, the thermo-dynamic quantities of the global system, namely the en-tropy production rate T Σ̇ and the time derivative of thefree energy dtG (obtained by integrating the expressionsin Eq. (36) and (38) over the volume V ), can be directlyrelated to the features of the traveling wave, namely theamplitude xeq and the propagation velocity c. Indeed,by considering the infinite volume limit, one obtains

T Σ̇ = −dtG = RTcxeq (40)

FIG. 1. Thermodynamic quantities (entropy production rateT σ̇, free energy flow ∂rJ

g and time derivative of the freeenergy ∂tg) as a function of the comoving coordinate r̃ =r−ct for the wave solution of the Fisher-Kolmogoroff equationspecified in Eq. (41). We use RTk+1xeqy and

√D/k+1y as

units of measure for the thermodynamic quantities and thespatial coordinate, respectively. The profile of the travelingwave x(r̃) is scaled by a numerical factor 0.35 to fit in thefigure.

establishing a linear dependence between the thermody-namic quantities and the dynamical ones. The aboveequation, valid for every traveling wave solution, provesthat the higher the amplitude and/or the velocity, thegreater the dissipation of the global system during thepropagation.

Particular traveling wave solutions have to be takeninto account if one wants to investigate the local pro-file of the thermodynamic quantities. For the Fisher-Kolmogoroff equation, both asymptotic solutions and oneparticular analytical solution are available [1]. Since thetraveling waves share the same general behavior, we con-sider the analytical solution

x(r̃) =xeq(

1 + (√

2− 1)er̃√k+1y/6D

)2 (41)with propagation velocity c = (5/

√6)√Dk+1y, to cal-

culate the entropy production rate T σ̇, the free energyflow ∂rJ

g and the time derivative of the free energy ∂tgaccording to their definitions given in the Eqs. (38), (29)and (36). By inspecting Fig. 1 where these thermody-namic quantities are displayed, two main features can benoticed. First of all, the variation of the thermodynamicquantities occurs around the wavefront of the concentra-tion field as one would expect. Secondly, the entropyproduction rate is larger than the absolute value of thefree energy flow, T σ̇ > |∂rJg|, granting that the free en-ergy continuously approaches its equilibrium value evenlocally, i.e., ∂tg < 0 ∀r̃. In other words, the system is re-laxing continuously towards equilibrium for every point

8

of the traveling wave profile. In this model system, thepropagation dynamics of the traveling wave does not re-quire to be sustained by nonconservative forces and, con-sequently, it results to be particularly effective from thisspecific thermodynamic point of view.

B. Brusselator model

A simple nondetailed balanced system where oscillat-ing traveling waves emerge is the Brusselator model inone spatial dimension [23]. It describes the transforma-tion of two internal species, an activator X1 and an in-hibitor X2, according to the chemical equations

Y1k+1

k−1X1

X1+Y2k+2

k−2X2+Y3

2X1+X2k+3

k−33X1

X1k+4

k−4Y4

which involve four chemostatted species Y1, Y2, Y3, Y4.We choose Y1 and Y2 as the reference chemostattedspecies breaking the two conservation laws and, conse-quently, determining the equilibrium condition [14]. Thereaction-diffusion equation for the evolution in time andspace r ∈ V = [0, L] of the concentrations Zαx1 (r, t) =x1(r, t) and Z

αx2 (r, t) = x2(r, t) of the two internalspecies is specified as

∂tx1 = k+1y1 − k−1x1 − k+2x1y2 + k−2x2y3 + k+3x21x2 − k−3x31 − k+4x1 + k−4y4 +D1∂2rx1∂tx2 = k+2x1y2 − k−2x2y3 − k+3x21x2 + k−3x31 +D2∂2rx2

(42)

with y1, . . . , y4 the homogeneous concentrations of thecontrolled species, and D1 and D2 the diffusion coeffi-cients of the internal species. Equation (42) admits auniform steady-state solution (xss1 , x

ss2 ) that becomes un-

stable if the concentration y2 of the controlled species Y2,used here as a bifurcation parameter, exceeds a criticalvalue, y2 > y2. Depending on the critical point, differentphenomena emerge including chemical oscillations, Tur-ing patterns, and oscillating traveling waves [24, 29, 30].The bifurcation point leading to oscillating travelingwaves with velocity c and wavenumber ν = 2πn/L (withn ∈ N the number of oscillations in the volume V withperiodic boundary conditions) is determined by the con-centration y2 such that the matrix representing the lin-earized dynamical system (42) around the steady-stateacquires the pure imaginary eigenvalues ±ıνc (with c thecritical velocity).

In Ref. [24], J. F. G. Auchmuty and G. Nicolis em-ployed a perturbation expansion near the onset of insta-bility, y2 ' y2, for identifying the oscillating wave so-lutions for the concentration x1(r, t) and x2(r, t) aroundtheir steady-state values in the case of irreversible chem-ical reactions. We use here the same analysis (summa-rized in Appendix B) for the reversible model by as-suming that the backward reaction currents are almostnegligible j−ρ ' 0 ∀ρ ∈ R. This means that the ki-

netic constants of the forward reactions are greater thanthe ones of the backward reactions k+ρ � k−ρ, and theconcentrations y3 and y4 are significantly smaller thantheir equilibrium values, y3,eq = k+2k+3y2/k−2k−3 andy4,eq = k+1k+4y1/k−1k−4. In this case, the existenceof traveling wave solutions of Eq. (42) with wavenumberν = 2πn/L is constrained by the following condition

(D2ν

2)2 − (D1 −D2)k2+1k+3

k2+4(y1ν)

2 −k2+1k+3

k+4(y1)

2 ≤ 0

(43)granting the existence of the propagation velocity c. Thecorresponding critical concentration is specified as

y2(ν) =k+4k+2

+k2+1k+3

k+2k2+4(y1)

2 +(D1 +D2)

k+2ν2. (44)

Equation (43) and (44) mean that there is only a finite setof wavenumbers ν and corresponding critical concentra-tions y2(ν) such that the steady state becomes unstablewith respect to traveling wave solutions.

Similarly to what has been done for the Fisher-Kolmogorov equation discussed in Subs. IV A, we estab-lish a correspondence between a global thermodynamicquantity of the Brusselator model and the features ofthe traveling waves without considering specific solutions.In particular, the global nonconservative work rate Ẇnc,

9

i.e., the integral over the volume V of the work rate of

Eq. (37), Ẇnc = −∫ L

0dr (µαy − µrefαy )S

αyρ jρ, can be ex-

pressed as a function of the wave wavenumber ν. Indeed,by taking into account that i) the chemical potentials ofthe species Y1 and Y2 correspond to their reference po-tentials, µαy1 = µ

refαy1

and µαy2 = µrefαy2

, ii) the reaction

currents for ρ = 3, 4 can be specified as j3 ' k+2x1y2and j4 ' k+4x1, iii) the oscillation of each traveling waveis centered around the corresponding steady-state con-centration (xss1 , x

ss2 ) ' ((k+1y1/k+4), (k+2y2/k+3xss1 )),

the nonconservative work rate reads

Ẇnc = LRTk+1y1

[ln

(y3,eqy3

)k+2k+4

y2 + ln

(y4,eqy4

)].

(45)The concentration y2 can be approximated with its crit-ical value y2(ν) defined in Eq. (44) if wave solutionsnear the onset of the instability are considered. Sincey3,eq � y3, the global nonconservative work rate is amonotonically increasing function of the wavenumber:

dνẆnc = LRTk+1y1 ln

(y3,eqy3

)2(D1 +D2)

k+4ν > 0.

(46)This means that the energetic cost of sustaining oscillat-ing traveling waves for the Brusselator model increaseswith the wavenumber ν.

We consider now the specific traveling wave solutions,

x1(r̃) = xss1 + � cos(νr̃) + 2�

2φ1 cos(2νr̃ + θ1)

x2(r̃) = xss2 − �y2φ cos(νr̃ + θ) + �

2(d+ 2φ2 cos(2νr̃ + θ2))

(47)

whose derivation is discussed in Appendix B, to studythe local profile of the thermodynamic quantities definedin Eqs. (36), (37), (38) and (29), and displayed in Fig. 2for the case with n = 1. The particular values of thephysical quantities employed for the plot, like the con-centration of the chemostatted species, are reported inAppendix B. First of all, one can verify that all the ther-modynamic quantities share the same period as the trav-eling waves. Secondly, there is a significant differencebetween the nonconservative work rate and the entropyproduction rate, on the one hand, and the time deriva-tive of the free energy and the divergence of its flow, onthe other hand. The nonconservative work spent to sus-tain the propagation of the waves is mainly dissipated,ẇnc ∼ T σ̇, resulting in a limited variation of the free en-ergy in time ẇnc ∼ T σ̇ � |∂tg|. This behavior is causedby the need of keeping the system far from equilibrium toallow the emergence of oscillating traveling waves. Fromthis thermodynamic point of view, the propagation of os-cillating traveling waves in the Brusselator model is en-ergetically very expensive: it requires that a lot of workis performed.

A similar general behavior is observed if one consid-ers traveling waves emerging under different conditions,such as different concentrations of the controlled speciesand different wavenumber. In this regard, an interesting

FIG. 2. Thermodynamic quantities (nonconservative workrate ẇnc, entropy production rate T σ̇, free energy flow ∂rJ

g

and time derivative of the free energy ∂tg) as a function ofthe comoving coordinate r̃ = r−ct for the oscillating wave so-lutions of Brusselator model with ν = 2π/L. We use RT/k+2and

√D1/k+1 as units of measure for the thermodynamic

quantities and the spatial coordinate, respectively.

difference is the increase of the global nonconservativework rate with the wavenumbers according to Eq. (45).A similar trend can also be observed for the amplitude ofthe oscillating nonconservative work rate, as well as theamplitude of the free energy.

V. CONCLUSIONS

In this work, we established a local thermodynamictheory of reaction-diffusion systems valid arbitrarily farfrom equilibrium. We then used it to investigate thepropagation of chemical waves and its energetic cost. Tothis aim, the identification of the proper thermodynamicpotential as well as establishing its balance equationshave been the fundamental steps. We showed a clearcorrespondence between the thermodynamic quantitiesand the dynamical processes occurring while chemicalwaves propagate, both in general and for two model sys-tems. The global entropy production rate of the Fisher-Kolmogorov waves is linearly dependent on the veloc-ity and the wave amplitude. The global nonconservativework rate is a monotonically increasing function of thewavenumber for the periodic waves of the Brusselatormodel. The relationship between the features of the con-centration profiles of the chemical waves and the ther-modynamic quantities has also been analyzed in thesemodels at the local level.

We focused here on the propagation dynamics of chem-ical waves, but our thermodynamic description could beused to study any other reaction-diffusion processes. Adirect application might be the characterization of the en-ergetic cost of creating wave patterns from generic initial

10

conditions and of the response to perturbations in modelsystems like those studied in this work. Furthermore, wethink that our approach represents the proper frameworkto study the correspondence between the semantic infor-mation codified in complex chemical patterns like chem-ical waves and the syntactic information. The semanticinformation represents the information content relevantfor a specific system which depends on how the patternsare interpreted by for instance a receptor while the syn-tactic information represents the whole physical infor-mation content [31]. These topics are becoming moreimportant nowadays since they play a crucial role in thedevelopment of new ways of transferring and comput-ing information with chemical reactions [32–35]. Theyshould also shed light on how biosystems perform theseoperations with high efficiency [36–38].

Finally, our work is based on a deterministic descrip-tion of the reaction-diffusion dynamics, which emergesin the large size limit of a microscopic description interms of stochastically reacting and diffusing chemicals.A thermodynamic at that level is necessary to describeintracellular signaling because of the limited number ofmolecules. For instance, the importance of stochasticityin intracellular calcium signaling is still debated [39, 40].

VI. ACKNOWLEDGMENTS

This research was funded by the European ResearchCouncil project NanoThermo (ERC-2015-CoG Agree-ment No. 681456).

Appendix A: Reference Chemical Potentials

We examine here the relations between the referencechemical potentials of different species µrefα for openRDSs. Note that the classification of the species S asinternal species X or controlled species breaking the bro-ken conservation laws Yb or other controlled species Ypis particularly important in this framework. As alreadymentioned in the main text, the reference chemical poten-tials are the values of the chemical potentials at the equi-librium that would be reached if the system were detailedbalanced. For this reason and the local detailed balancedcondition k+ρ/k−ρ =

∏α(Z

αeq)

Sαρ = exp(−Sαρµ◦α/RT ),the set of reference chemical potentials is a left null vectorof the stoichiometric matrix,

µrefα Sαρ = 0, ∀ρ ∈ R. (A1)

This means that the reference chemical potentials µrefαcan be written as a linear combination of the closed sys-tem conservation laws lλα,

µrefα = ψλlλα (A2)

where ψλ are real coefficients. Because of Eq. (A2), thenumber of independent reference chemical potentials is

equal to the number of conservation laws |λ| of the closedRDS. The other chemical potentials are determined ac-cording to the constraints of Eq. (A1).

First of all, we show that the reference chemical po-tentials of the chemostatted species which break all thebroken conservation laws µrefαyb

set the value of the refer-

ence chemical potentials of the other controlled speciesµrefαyp . By definition of unbroken conservation laws, i.e.,

lλuαxSαxρ = 0 ∀ρ ∈ R and lλuαy = 0, the reference chem-

ical potentials µrefαyb(resp. µrefαyp ) are defined as a lin-

ear combination of only the broken conservation laws,µrefαyb

= ψλb lλbαyb

(resp. µrefαyp = ψλb lλbαyp

). The matrix

whose entries are lλbαybis square and nonsingular so that

it can be inverted. The elements of the inverse matrixare denoted by l̂

αybλb

, and the reference chemical potentials

µrefαyp can now be specified as

µrefαyp = µrefαyb

l̂αybλb

lλbαyp , (A3)

since ψλb = µrefαyb

l̂αybλb

.

Secondly, we show that the value of reference chem-ical potentials for the internal species µrefαx depends on

the reference chemical potentials µrefαyband the unbroken

conserved quantities Lλu =∫V

dr lλuαxZαx . Because of

ψλb = µrefαyb

l̂αybλb

,

µrefαx = ψλu lλuαx + µ

refαyb

l̂αybλb

lλbαx , (A4)

with the coefficients ψλu to be determined by employ-ing the unbroken conserved quantities evaluated for theequilibrium concentrations Lλu = V lλuαxZ

αxeq . Indeed, the

conserved quantities can be written as

Lλu = V∑αx∈X

lλuαx exp{

(ψλ′u lλ′uαx + µ

refαyb

l̂αybλb

lλbαx − µ◦αx)/RT

}(A5)

establishing |λu| nonlinear constrains for the |λu| coef-ficients ψλu . By solving the above system of equationsfor the coefficients ψλu , one obtains ψλu as a functionof {Lλu} and {µrefαyb}. Unfortunately, explicit expres-sions cannot be obtained from Eq. (A5). If there are no

conserved quantities, µrefαx = µrefαyb

l̂αybλb

lλbαx . In conclusion,

the reference chemical potentials µrefαyband the conserved

quantities Lλu set the values of all the other referencechemical potentials µrefαx , µ

refαyp

.

Appendix B: Traveling Waves in the BrusselatorModel

We summarize here the derivation of oscillating trav-eling wave solutions of the Brusselator model in one spa-tial dimension r ∈ [0, L] with irreversible chemical reac-tions proposed by J. F. G. Auchmuty and G. Nicolis [24].

11

Since the backward reaction currents are neglected, thereaction-diffusion equation (42) reads, in vector form,

∂t

(x1x2

)=

(k+1y1 − k+2x1y2 + k+3x21x2 − k+4x1 +D1∂2rx1

k+2x1y2 − k+3x21x2 +D2∂2rx2

).

(B1)

Oscillating traveling wave solutions of Eq. (B1) areperiodic functions of the only comoving coordinate:x1(r, t) = x1(r̃) = x1(r̃ ± L) and x2(r, t) = x2(r̃) =x2(r̃ ± L), with r̃ = r − ct. In this regard, the steady-state (xss1 , x

ss2 ) = ((k+1y1/k+4), (k+2y2/k+3x

ss1 )) can be

interpreted as a trivial traveling wave with null veloc-ity. By considering the linearization of the dynamicalsystem (B1) around the steady-state according to theoperator

Ĵ =(

k+2y2 − k+4 +D1∂2r k2+1k+3(y1)2/k2+4−k+2y2 −k2+1k+3(y1)2/k2+4 +D2∂2r

),

(B2)

one can identify the conditions where new wave solutionsmay bifurcate from the steady-state. Any such a solu-tion, written as a perturbation around the steady-state(xp1 , x

p2) = (x1 − xss1 , x2 − xss2 ), must have the following

form in the linear regime(xp1xp2

)=

(a1a2

)eıνr̃ (B3)

with (a1, a2) still unknown coefficients and ν = 2πn/L(n ∈ N), because of the periodic boundary conditions.Consequently, (xp1 , x

p2) of Eq. (B3) must be an eigenvector

of the linear operator Ĵ corresponding to the complex

eigenvalue −ıνc. To guarantee the existence of such asolution, the matrix representation of the operator Ĵ onthe basis element exp(ıνr̃)

J =(

k+2y2 − k+4 −D1ν2 k2+1k+3(y1)2/k2+4−k+2y2 −k2+1k+3(y1)2/k2+4 −D2ν2

)(B4)

must have purely complex eigenvalues meaning that itstrace vanishes T = 0, while its determinant is positiveD > 0. The condition T = 0 determines the criticalvalue y2 of the concentration of the chemostatted speciesY2, considered here as the bifurcation parameter, whichis specified in Eq. (44). On the other hand, the constraintD > 0 is equivalent to Eq. (43) when the critical concen-tration y2 is employed, and it ensures that the value ofthe critical velocity c(ν), namely the wave velocity eval-uated for y2 = y2, is a real number:

c(ν) = (D1 −D2)k2+1k+3k2+4

(y1)2 +

k2+1k+3k+4

(y1ν

)2− (D2ν)2 .

(B5)

When the concentration y2 exceeds its critical valuey2(ν), the steady-state becomes unstable compared tothe traveling waves with wavenumber ν. Waves solutionsof the reaction-diffusion equation close to the bifurcationpoint can be identified by expanding the dynamical sys-tem (B1) in powers of a small parameter � representingthe distance from the threshold. To this aim, Eq. (B1)can be written in such a way that i) the difference be-tween the actual concentration y2 (resp. velocity c) andthe critical one y2(ν) (resp. c(ν)) is made explicit, andii) the linear contributions to the dynamics are gatheredtogether and separated from the nonlinear one. In thisway, Eq. (B1) reads

−(c− c)(x1x2

)= (Ĵ + c)

(x1x2

)+ k2(y2 − y2)

(x1−x1

)+

(h(x1, x2)−h(x1, x2)

)(B6)

where Ĵ is the operator Ĵ evaluated at the critical con-centration y2(ν), while h(x1, x2) takes into account thenonlinear dynamics. We shall now expand every term inEq. (B6) in powers of �(

x1x2

)=

+∞∑q=1

�q(x(q)1

x(q)2

),

y2 − y2(ν) =+∞∑q=1

�qy(q),

c− c(ν) =+∞∑q=1

�qc(q),

(B7)

and then iteratively solve each equation obtained bygathering equal powers of �. Following this perturbation

procedure, one derives the traveling wave solutions withwavenumber ν of the Brusselator model. The details ofthe perturbation expansion in power of � are discussed inRef. [24], and they are not reported here.

The particular traveling wave solution in Eq. (47), aswell as any other quantities mentioned in the following, isspecified according to the set of unit measures reported inthe Table I. It has been obtained under the hypothesis ofkinetic constants k+ρ = 1 and stopping the perturbationexpansion at the second order. The explicit expressionsof the coefficients �, φ, φ1, φ2, θ, θ1, θ2 can be found inRef. [24]. For the plot in Fig. 2, we adopt the followingnumerical values of the physical quantities: y1 = 0.5,y2 = 1.01y2, D2 = 0.05, y3 = 10, y4 = 1 and k−ρ = 10

−4.Similar results to those discussed in the Subs. IV B are

12

TABLE I. Set of units measures adopted for the Brusselatormodel

Physical quantity Unit measures

time 1/k+1

space√D1/k+1

concentration k+1/k+2

obtained with different numerical values as long as theconcentrations of the chemostatted species Y3 and Y4

are much smaller than their equilibrium ones.Finally, it has to be mentioned that our thermody-

namic theory is based on the implicit assumption thata reference equilibrium condition is well defined. Thismeans that it can be applied only to the Brusselatormodel with reversible chemical reactions. On the otherhand, we employ the analytical traveling wave solutionsof the irreversible Brusselator as an approximate expres-sion of the traveling wave solutions for the case of negligi-ble, but not null, backward reaction currents. This choiceappears to be reasonable since the relations between thethermodynamic quantities, like ∂tg = −T σ̇+ẇnc−∂rJg,are still verified.

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Thermodynamics of Chemical WavesAbstractI IntroductionII Reaction-Diffusion SystemsIII Chemical wavesA Dynamics of Traveling WavesB Thermodynamics of Traveling Waves

IV Model SystemsA Fisher-Kolmogorov equationB Brusselator model

V ConclusionsVI AcknowledgmentsA Reference Chemical PotentialsB Traveling Waves in the Brusselator Model References