Validity of path thermodynamic description of reactive systems Microscopic simulations F. Barasa Alejandro L. Garciab and M. Malek Mansourc

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Validity of path thermodynamic description of

reactive systems: Microscopic simulations

F. Baras a, Alejandro L. Garcia b, and M. Malek Mansour c

(a) Laboratoire Interdisciplinaire Carnot de Bourgogne,

UMR 6303 CNRS-Université Bourgogne Franche-Comté,

9 Avenue A. Savary, BP 47 870,

F-21078 Dijon Cedex, France

(b) Dept. Physics and Astronomy,

San Jose State University,

San Jose, California, 95192 USA

B-1050 Brussels, Belgium

Abstract

Traditional stochastic modeling of reactive systems limits the domain of applicability of

the associated path thermodynamics to systems involving a single elementary reaction at

the origin of each observed change in composition. An alternative stochastic modeling has

recently been proposed to overcome this limitation. These two ways of modeling reactive sys-

tems are in principle incompatible. The question thus arises about choosing the appropriate

type of modeling to be used in practical situations. In the absence of suﬃciently accurate

experimental results, one way to address this issue is through the microscopic simulation

of reactive ﬂuids, usually based on hard-sphere dynamics in the Boltzmann limit. In this

paper, we show that results obtained through such simulations unambiguously conﬁrm the

predictions of traditional stochastic modeling, invalidating a recently proposed alternative.

1 Introduction

Thermodynamic description of non-equilibrium systems is based on the concept of entropy pro-

duction, the correct evaluation of which necessarily requires full knowledge of each elementary

process therein [1,2]. This prerequisite plays an important role in the corresponding statisti-

cal formulation, commonly modeled by means of an appropriate Markovian stochastic process.

In previous papers, we proved that the thermodynamic formulation of homogeneous reactive

systems based on their sample path, widely known as “path thermodynamics”, will lead to er-

roneous results whenever the system involves more than one elementary reaction leading to the

same change in composition [3,4]. Similar observations were reported in [5] and [6] (see also [7]).

This result can also be appreciated from a physical perspective. Let us consider a perfectly

homogeneous (no local ﬂuctuations) isothermal reactive system, such as can be produced ex-

perimentally in a "continuously stirred tank reactor" (CSTR). The state of such a system is

entirely characterized by its composition, which is indeed the only pertinent quantity accessible

to experimental investigations. Suppose now that we have at our disposal an ideal experimental

device allowing us to measure the precise number of each chemical species over an arbitrarily

long interval of time. Using this device, we may gain access to the exact state trajectory of

the system, traditionally referred to as its "sample path" within the framework of stochastic

processes. The question is whether such a state trajectory encompasses suﬃcient information to

arXiv:2210.01952v1 [cond-mat.stat-mech] 4 Oct 2022

allow us to determine the thermodynamic properties of that system. The key issue here is obvi-

ously the fact that entropy production in reactive systems is the sum of the entropy production

that is associated with each individual elementary reaction (cf. Section 9.5 in [1]). The answer

to this question will thus be positive if and only if each of the measured changes in composition

can be attributed to a speciﬁc elementary reaction. Such an attribution becomes impossible if

the reactive system includes more than one elementary reaction leading to the same change in

composition. No matter what method we use, the path thermodynamic properties of such a

reactive system can never be determined from its state trajectory.

Despite the undeniability of this result, from both the mathematical and the physical point of

view, its validity was recently contested by Gaspard in a Comment article [8]. Yet this very

issue had already been raised by that same author over a decade ago [9]. In short, extending

the work of Lebowitz and Spohn [10], Gaspard initiated the path thermodynamic formulation of

reactive systems in 2004 [11,12]. The theory is sound but, like all scientiﬁc theories, its domain

of applicability has limits; speciﬁc situations will exist outside the scope of the theory. Indeed,

three years later that theory was found to encounter certain inconsistencies when applied to the

Schnakenberg graph formulation of "current" ﬂuctuations in reactive systems involving more

than one elementary reaction leading to the same change in composition [9]. A remedy was

proposed in the speciﬁc case of "current" ﬂuctuations but, as we have shown in [3], this remedy

is inapplicable to traditional modeling of reactive systems based on jump Markov processes.

In short, a pure jump process, χ(t), is entirely determined by the concept of "transition rates",

W(X|X0), deﬁned by Kolmogorov as [13]

P(X, t + ∆t|X0, t) = W(X|X0) ∆t+ o(∆t),∀X6=X0(1)

The function P(X, t + ∆t|X0, t)represents the conditional probability to have χ(t+ ∆t) = X,

given that χ(t) = X0. The description of jump Markov processes in terms of their sample path

is based on this fundamental Kolmogorov equality [14]. Consider now a reactive system that

involves nelementary reactions R1· · · Rnleading the same change in composition X→X0, and

denote by Wρ(X|X0)the transition rate associated with the reaction Rρ. It was claimed in [8,9]

that we can write the sample path of χ(t)in terms of any individual transition rate Wρ(X|X0).

This claim, however, contradicts the fundamental property of jump processes.

Recall that the probability associated with a random event is unique. This property results

from the basic deﬁnition of the concept of probability, ﬁrst stated by Pascal, and reﬁned over

the years by various mathematicians to Kolmogorov’s axiomatic formulation [15,16]. A tran-

sition X0→Xis clearly a random event. The probability that this event occurs in the time

interval [t, t + ∆t]is precisely P(X, t + ∆t|X0, t). Accordingly, this probability cannot assume

several values simultaneously, that is, for any given X, X0, t, and ∆t, the conditional prob-

ability distribution P(X, t + ∆t|X0, t)is unique. The Kolmogorov equality (1) then implies

that this is also the case for the transition rate W(X|X0). In particular, if a reactive system

involves several elementary reactions leading to the same change in composition, then the result-

ing transition rate is necessarily the sum of the transition rates associated with each of them,

i.e., W(X|X0) = PρWρ(X|X0). Consequently, expressing the sample path of χ(t)in terms of

individual transition rates Wρ(X|X0)does not apply to jump stochastic processes.

Given this limitation, Gaspard proposed in his Comment a brand new type of stochastic modeling

of reactive systems that extends the domain of validity of the associated path thermodynamics

to the controversial situation of reactive systems involving more than one elementary reaction

leading to the same change in composition [8]. The resulting stochastic process proves to be

quite diﬀerent from that associated with the traditional stochastic modeling of reactive systems.

They don’t even share the same state space. The state space of the stochastic process associated

with the traditional modeling of an ncomponent isothermal homogeneous (CSTR) reactive

system is simply Zn, in one-to-one correspondence with quantities that can actually be measured

in laboratory experiments (Zrepresents the set of non-negative integers). First proposed by

McQuarrie in 1967 [17], traditional stochastic modeling was then reﬁned over the years by Van

Kampen [15], Haken [18], Nicolis and Prigogine [19], Kurtz [20], Gardiner [16], and many others.

Everything we currently know about the statistical properties of non-equilibrium reactive systems

was established in this way, including the stochastic formulation of path thermodynamics by

Seifert [21], Lebowitz and Spohn [10], and Gaspard [11,12].

The state space of the stochastic process associated with the new type of modeling proposed in [8]

is diﬀerent. In addition to the number of particles of chemically active molecules, it also includes

an extra variable designed to select the precise elementary reaction at the origin of an observed

change in composition. As clearly stated by the author, the main consequence of this modeling

is that it allows us to deﬁne several Markov processes associated with a given reactive system

(Section III, in [8]). Among them, there will undoubtedly be found a process guaranteeing the

validity of the associated path thermodynamics in controversial situations.

One may argue that in this new formulation the model has been changed simply to shoehorn

it to ﬁt the theory. As they have diﬀerent state spaces, it is impossible to use the framework

of one of these stochastic processes to conﬁrm or deny the validity of the other. Conversely, it

is impossible to discuss the validity of the proposed new modeling within the framework of the

traditional stochastic modeling.

Nevertheless, the main purpose of the two approaches is to provide a theoretical description of a

"real-world" system: a system with properties that can be observed in laboratory experiments.

In this respect, we know that reactive processes result from local interactions between chemically

active molecules (reactive collisions). If the reactive system is well stirred (CSTR), it will be

impossible to determine experimentally which reaction led to an observed change in composition,

unless that reaction is unique. In other words, determining the precise state trajectory of a well-

stirred reactive system, as required by the new modeling [8], is beyond the reach of real-life

laboratory experiments.

On the other hand, the new stochastic modeling of reactive systems, however strange, may

nevertheless represent reality. The only way to investigate this possibility is through laboratory

experiments. Unfortunately, the accuracy of the available experimental data are insuﬃcient to

address this issue. The alternative option is to perform microscopic simulations of reactive ﬂuids,

usually based on Newtonian hard sphere dynamics. Introduced in the mid-seventies [22,23],

this technique provides quite useful information on the relevance and accuracy of theoretical

developments in non-equilibrium reactive systems [24–28] (see [29] for a review).

In Section 2, we consider the microscopic simulation of Schlögl-like reactive systems [30], often

used to illustrate some peculiar aspects of path thermodynamic properties of reactive systems

[11,12]. The results obtained are in perfect agreement with the traditional stochastic modeling

of reactive systems, thus calling into question the alternative modeling proposed by Gaspard.

Conclusions and perspectives are presented in Section 3; algorithmic details of the simulations

are in an appendix.

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Validityofpaththermodynamicdescriptionofreactivesystems:MicroscopicsimulationsF.Barasa,AlejandroL.Garciab,andM.MalekMansourc(a)LaboratoireInterdisciplinaireCarnotdeBourgogne,UMR6303CNRS-UniversitéBourgogneFranche-Comté,9AvenueA.Savary,BP47870,F-21078DijonCedex,France(b)Dept.PhysicsandAstronomy,SanJo...

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Validity of path thermodynamic description of reactive systems Microscopic simulations F. Barasa Alejandro L. Garciab and M. Malek Mansourc

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