PHYSICAL REVIEW E 90, 042113 (2014)

Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementationof the maximum entropy production principle

Gian Paolo Beretta*

Universita di Brescia, via Branze 38, 25123 Brescia, Italy(Received 2 March 2014; revised manuscript received 21 May 2014; published 7 October 2014)

By suitable reformulations, we cast the mathematical frameworks of several well-known different approachesto the description of nonequilibrium dynamics into a unified formulation valid in all these contexts, which extendsto such frameworks the concept of steepest entropy ascent (SEA) dynamics introduced by the present author inprevious works on quantum thermodynamics. Actually, the present formulation constitutes a generalization alsofor the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a keyrole is played by the geometrical metric with respect to which to measure the length of a trajectory in state space.In the near-thermodynamic-equilibrium limit, the metric tensor is directly related to the Onsager’s generalizedresistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsagergeneralized resistance to the arbitrarily far-nonequilibrium domain, most of the existing theories of nonequilibriumthermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in statespace along the path of SEA compatible with the conservation constraints and the boundary conditions. Theresulting unified family of SEA dynamical models is intrinsically and strongly consistent with the second law ofthermodynamics. The non-negativity of the entropy production is a general and readily proved feature of SEAdynamics. In several of the different approaches to nonequilibrium description we consider here, the SEA concepthas not been investigated before. We believe it defines the precise meaning and the domain of general validity ofthe so-called maximum entropy production principle. Therefore, it is hoped that the present unifying approachmay prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models andtheoretical treatments of irreversible conservative relaxation towards equilibrium from far nonequilibrium states.The mathematical frameworks we consider are the following: (A) statistical or information-theoretic modelsof relaxation; (B) small-scale and rarefied gas dynamics (i.e., kinetic models for the Boltzmann equation);(C) rational extended thermodynamics, macroscopic nonequilibrium thermodynamics, and chemical kinetics;(D) mesoscopic nonequilibrium thermodynamics, continuum mechanics with fluctuations; and (E) quantumstatistical mechanics, quantum thermodynamics, mesoscopic nonequilibrium quantum thermodynamics, andintrinsic quantum thermodynamics.

DOI: 10.1103/PhysRevE.90.042113 PACS number(s): 05.70.Ln, 47.70.Nd, 05.60.Cd

I. INTRODUCTION

The problem of understanding entropy and irreversibilityhas been tackled by a large number of preeminent scientistsduring the past century. Schools of thought have formedand flourished around different perspectives of the problem.Several modeling approaches have been developed in variousframeworks to deal with the many facets of nonequilibrium.

In this paper, we show how to construct steepest entropyascent (SEA) models of nonequilibrium dynamics by adoptinga unified mathematical formulation that allows us to doit at once in several different well-known frameworks ofnonequilibrium description.

To avoid doing inevitable injustices to the many pioneersof all these approaches and to the many and growing fieldsof their application, here we skip a generic introduction andgiven no references nor a review of previous work. Rather,we dig immediately into the mathematical reformulations ofthe different frameworks in such a way that the constructionof the proposed SEA dynamics becomes formally a singlegeometrical problem that can be treated at once.

Our reformulations here not only allow a precise meaning,general implementation, and unified treatment of the so-called

*gianpaolo.beretta@unibs.it

maximum entropy production (MEP) principle (for a recentreview see Ref. [1]) in the various frameworks but also extendto all frameworks an observation that we have been developingin the quantum thermodynamics framework for the past threedecades [2–5]. In doing so, we also introduce an importantgeneralization for the quantum thermodynamics modelingframework.

The observation is that we cannot simply maximize theentropy production subject to a set of conservation constraintsor boundary conditions, but in order to identify a SEA path instate space we must equip the state space with a metric fieldwith respect to which to compute the distance traveled duringthe time evolution.

The generalization is as follows. In our previous work,we adopted the proper uniform metric for probability dis-tributions, namely the Fisher-Rao metric, because in quan-tum thermodynamics the state representative, the densityoperator, is essentially a generalized probability distribution.In other frameworks, however, the state representative notalways is a probability distribution. Moreover, the presentapplication to the framework of mesoscopic nonequilibriumthermodynamics [6,7] shows that standard results such asthe Fokker-Planck equation and Onsager theory emerge asstraightforward results of SEA dynamics with respect to ametric characterized by a generalized metric tensor that isdirectly related to the inverse of the generalized conductivity

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GIAN PAOLO BERETTA PHYSICAL REVIEW E 90, 042113 (2014)

tensor. Since the generalized conductivities represent, at leastin the near-equilibrium regime, the strength of the system’sreaction when pulled out of equilibrium, it appear that theirinverse, i.e., the generalized resistivity tensor, represents themetric with respect to which the time evolution, at least in thenear equilibrium, is locally SEA.

But the local SEA construction does much more, because itoffers a strongly thermodynamically consistent way to extendthe well-known near-equilibrium theories to the treatment ofnonequilibrium states.

The unified formulation of the local SEA variationalproblem is as follows and it is not restricted to near equilibrium:The time evolution of the local state is the result of a balancebetween the effects of transport or Hamiltonian dynamicsand the spontaneous and irreversible tendency to advancethe local state representative in the direction of maximalentropy production per unit of distance traveled in state spacecompatible with the conservation constraints.

Geometrically, the measure of distance traveled in statespace requires the choice of a local metric tensor. Physically,the local metric tensor contains the full information aboutthe relaxation kinetics of the material. The standard near-equilibrium results obtain when the local metric tensor isproportional to the inverse of the local matrix of generalizedconductivities, i.e., to the local generalized resistivity matrix.

The structure of the SEA geometrical construction for thedescription of highly nonequilibrium dissipative dynamicsin the nonlinear domain turns out to be closely related tothe formulation of dissipation in the well-known generalequation for the nonequilibrium reversible–irreversible cou-pling (GENERIC) [8–10]. The seeds of SEA and GENERICdeveloped independently in the early 1980s with differentmotivations and approaches, but the common general thrusthas been, and still is, to impose strong thermodynamic con-sistency in the dynamical modeling of systems far from stablethermodynamic equilibrium. SEA has focused exclusively onthe irreversible entropy-generation component of the timeevolution, while GENERIC has emphasized the coupling andinterplay between the reversible and irreversible componentsof the time evolution.

We will show elsewhere [11] that the main technicaldifferences are that (1) SEA chooses a (nondegenerate)Riemannian metric tensor as dissipative structure, whileGENERIC chooses two compatible degenerate structures(Poisson and degenerate co-Riemannian), and (2) in thedescription of a continuum, SEA uses the local entropydensity field as potential, while GENERIC uses the globalenergy and entropy functionals as potentials. Future workis needed to address also the relationships and establishdifferences and similarities between the SEA description offar from equilibrium dissipation and other closely relatedapproaches, such as the recent contact geometry of mesoscopicthermodynamics and dynamics [12–14], the general ideasof the rate-controlled constrained-equilibrium approach tofar-from-local-equilibrium thermodynamics [15,16] and of thequasiequilibrium approximation of invariant manifolds [17],as well as the works of Ziegler [18] and Edelen [19].

The question of what “the physical basis” is for the SEAscheme (or that for the GENERIC scheme) is tricky andphilosophically ill posed. It is as if one would ask what the

physical basis is for believing that a classical system shouldobey Hamilton’s equations or the equivalent minimum actionprinciple. The meaning of “physical reality” is well explainedin the classic book on this subject by Henry Margenau [20].There is a level of perceptions, the empirical world, that wetry to describe by defining concepts, their relations with theplane of perceptions (operational measurement procedures),and relations among concepts that we call laws or principles(often using the language of mathematics to express themefficiently). The farther the construction goes from the plane ofperceptions the more “abstract” it becomes, but the advantageis that more abstraction may allow us to encompass andregularize a broader set of less abstract theories, in short, tounify them. At any level of abstraction, what makes a theory“physical” are its links to the plane of perception, namely thefact that the theory allows us to model some empirical evidencewith some reasonable level of approximation.

Paraphrasing the words of Feynman, what makes a partic-ular law or principle “great,” such as the great conservationprinciples or the second law of thermodynamics, is the factthat they hold for whatever level of description of whateverempirical reality, provided the model has some basic structureand obeys some reasonable conditions, such as those thatgrant and give meaning to the concept of separability betweenthe object of study and its surroundings. The spirit of theSEA construction is precisely this. We consider a number offrameworks that have successfully modeled nonequilibriumsystems at some level of description, we focus on how thesesuccessful models of physical reality describe entropy produc-tion by irreversibility, and we cast them in a way that allowsus to see that they can all be encompassed and regularized bythe unifying geometrical SEA construction. The GENERICconstruction is even more ambitious in that it attempts tounify at once also the reversible and transport contributionsby recognizing their common Hamiltonian structure and theirrelations with the irreversible aspects of the dynamics.

Being more abstract (i.e., farther from Margenau’s planeof perceptions) than the various physical theories they unify,the SEA and GENERIC constructions emerge as generaldynamical principles which operate within the same domainof validity of the second law of thermodynamics and, hence,enjoy a similar level of “greatness”, by complementing it withthe additional essential and general features of nonequilibriumbehavior at all levels of description.

An important fraction of the greatness of the second lawof thermodynamics stems from the fact that it supports theoperational definition of entropy [21,22] as a property ofany well-defined system and in any of its equilibrium andnonequilibrium states. Other good fractions that have directimpact also on the near-equilibrium description of dynamicsderive from the stability and maximal entropy features of theequilibrium states.

An important fraction of the importance of the SEAprinciple stems from the fact that for any well-defined systemit supports the operational definition of the metric fieldG over its entire state space, which characterizes even inthe far-nonequilibrium domain all that can be said aboutthe spontaneous, irreversible, entropy-generating tendencytowards stable equilibrium. Another good fraction derivesfrom the fact that within the SEA construction the MEP

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principle acquires a precise and general validity whereby, inany well-defined model, the entropy-producing component ofthe dynamics effectively pulls the state of the system in thedirection of steepest entropy ascent compatible with the metricfield G and the imposed conservation laws.

The paper is structured as follows. In Sec. II we reformulateseveral of the well-known approaches for the description ofdissipation in nonequilibrium systems to express them all interms of a common geometrical formulation. In Sec. III wethen introduce our steepest-entropy-ascent unified variationalformulation of nonequilibrium dissipation and discuss its maingeneral features. In Sec. IV we give a pictorial representationof the same concepts and in Sec. V we draw our conclusions.

II. COMMON STRUCTURE OF THE DESCRIPTIONOF DISSIPATION IN SEVERAL NONEQUILIBRIUM

FRAMEWORKS

In this section we show that several well-known nonequi-librium frameworks at various levels of description can berecast in slightly nonstandard, but unifying, notation so thatthey all exhibit as common features the following essentialconditions:

C1: the state space, denoted by the symbol L , is a manifoldin a Hilbert space equipped with a suitable inner product (·|·);we denote its elements (the states) by γ or, alternatively, |γ );

C2: the system properties (energy, entropy, mass, momen-tum, etc.) are represented by real functionals A(γ ), B(γ ), . . .of γ such that their functional derivatives with respect to γ

are also elements of L ; we denote them by δA(γ )/δγ or,alternatively, by |δA(γ )/δγ );

C3: if the states are functions of time t only, γ = γ (t),their time evolution γ (t) obeys the equation of motion,

|dγ /dt) = |�γ ), (1)

where |�γ ) is also an element of L such that the rates ofchange of the entropy S(γ ) and of any conserved propertyCi(γ ), with i labeling a list of conserved properties, are

dS/dt = �S with �S = (�|�γ ) � 0, (2)

dCi/dt = �Ciwith �Ci

= (�i |�γ ) = 0, (3)

where �S and �Ciare the respective production rates and

|�) = |δS(γ )/δγ ) and |�i) = |δCi(γ )/δγ ) are shorthand fordenoting the variational derivatives with respect to γ of theentropy functional S(γ ) and the conserved functional Ci(γ ),respectively;

C3′: if the states are continuum fields γ = γ (t,x), where xis position and t is time, assume that the time evolution obeysthe equation of motion,

|∂γ /∂t) + Rγ |γ ) = |�γ ), (4)

where Rγ is an operator on L responsible for the descriptionof the local fluxes in the continuum and |�γ ) is an elementof L responsible for the description of the local productiondensities, such that the balance equations for entropy and anyconserved property are

∂S/∂t + ∇ · JS = �S with �S = (�|�γ ) � 0, (5)

∂Ci/∂t + ∇ · JCi= �Ci

with �Ci= (�|�γ ) = 0, (6)

where, of course, JS and JCiare the respective local Lagrangian

fluxes and �S and �Cithe respective local production

densities.In the next subsections we introduce the details of the

slightly nonstandard notations that allow us to reformulate inthe terms just outlined some of the approaches that have beendeveloped over the past several decades to provide thermody-namically consistent theories of nonequilibrium dissipation atvarious levels of description. This list of approaches is by nomeans exhaustive and their reformulations have no importantelements of novelty. Their presentation here is only intended toexplicitly substantiate the above common features in some ofthe most well-known nonequilibrium modeling frameworks.Perhaps the only major point is that in order to satisfy conditionC2 in frameworks (A), (B), (D), and (E), we will borrowfrom the formalism we originally developed for the quantumframework [2,3] (later introduced also in Refs. [4,23]): the useof square roots of probabilities (instead of the probabilitiesthemselves) as state representatives.

The reader who does not need to be convinced about suchdetails can skip the rest of this section and jump to Sec. III,where we provide the unified construction and implementationof the SEA concept, based only on the general assumptionsitemized above.

A. Framework A: Statistical or information-theoretic modelsof relaxation to equilibrium

Let L be the set of all n-vectors of real finite numbersA = vect(aj ), B = vect(bj ), . . . (n � ∞), equipped with theinner product (·|·) defined by

(A|B) = Tr(AB) =n∑

j=1

aj bj . (7)

In information theory [24], the probability assignment to aset of n events, pj being the probability of occurrence of thej -th event, can be represented by ρ = vect(pj ). In order toeasily impose the constraint of preservation of non-negativityof the probabilities during their time evolution and to obtaincondition C2 above, we adopt the description in terms of thesquare root of ρ that we denote by

γ = vect(γj = √pj ). (8)

Typically, we consider a set of conserved expectation valuesof the process

{Ci(γ )} = {H (γ ),N1(γ ), . . . ,Nr (γ ),I (γ )}, (9)

where H (γ ) = Tr(γ 2H ) = ∑nj=1γ

2j ej with H denoting the

constant vector H = vect(ej ), for i = 1, . . . ,r , Ni(γ ) =Tr(γ 2Ni) = ∑n

j=1γ2j nij with Ni denoting the constant vector

Ni = vect(nij ), and I (γ ) = Tr(γ 2I ) = ∑nj=1γ

2j = 1, provid-

ing the normalization condition, with I = vect(1). Notice thatthe variational derivatives δH/δγ = vect(2γj ej ), δNi/δγ =vect(2γjnij ), δI/δγ = 2γ are vectors in L , thus satisfyingcondition C2 above. We denote them collectively by

�i = δCi/δγ, (10)

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A time evolution of the square-root probability distribution,γ (t), is a solution of the rate equation

dγ

dt= �γ , (11)

where the term �γ must be such as to satisfy the constraintsof conservation of the expectation values Ci(γ ), i.e., such that

�Ci= dCi

dt= d

dtTr(γ 2Ci) = (�i |�γ ) = 0. (12)

The entropy in this context is represented by the Shannonfunctional

S(γ ) = −kTr(ρ ln ρ) = (−kγ ln γ 2|γ ) (13)

so the rate of entropy production is given by

�S = dS

dt= −k

d

dtTr(ρ ln ρ) = (�|�γ ), (14)

where � denotes its variational derivative with respect to γ ,

� = δS/δγ = vect(−2kγj − 2kγj ln γ 2

j

). (15)

It is noteworthy that an advantage of the state representationin terms of square-root probability distributions is that δS/δγ

is well defined and belongs to L for any distribution, even ifsome of the probabilities pi are equal to zero, whereas in suchcases δS/δρ is undefined and does not belong to L .

In Sec. III we present the SEA construction which in thisframework provides a model for the rate term �γ whereby �S

is maximal subject to the conservation constraints �Ci= 0

and the suitable additional constraint we discuss therein.An attempt along the same lines has been presented in

Ref. [25].

B. Framework B: Small-scale and rarefied gas dynamics

Let c be the classical one-particle velocity space, and Lthe set of real, square-integrable functions A,B, . . . on c,equipped with the inner product (·|·) defined by

(A|B) = Trc(AB) =∫

c

AB dc, (16)

where Trc(·) in this framework denotes∫c

· dc, with dc =dcx dcy dcz.

In the kinetic theory of rarefied gases and small-scale hy-drodynamics [26], the probability to find a particle (at positionx and time t) with velocity between c and c + dc [where, ofcourse, c = (cx,cy,cz)] is given by f (c; x,t) dc/

∫c

f dc,where f (c; x,t) is the local phase-density distribution whichfor every position x and time instant t is a function in L .

Also in this framework, in order to easily impose the con-straint of preservation of the non-negativity of f during its timeevolution and, perhaps more importantly, to obtain conditionC2, we introduce the local one-particle state representation notby f itself but by its square root, which we assume is also afunction in L that we denote by γ = γ (c; x,t). Therefore, wehave

f = γ 2 ,∂f

∂t= 2γ

∂γ

∂t,

(17)∂f

∂x= 2γ

∂γ

∂x,

∂f

∂c= 2γ

∂γ

∂c,

and for any functionals Af (f ) and Aγ (γ ) = Af (γ 2),

δAγ (γ )

δγ= 2γ

δAf (f )

δf

∣∣∣∣f =γ 2

(18)

Again, among the functionals that represent the one-particlephysical observables we focus on the conserved fields, i.e.,the collision invariants (mass, momentum, energy), which wedenote synthetically by the set

{Ci(γ )} = {m n(γ ),Mx(γ ),My(γ ),Mz(γ ),H (γ )}, (19)

where m is the single-particle mass, n(γ ) = Trc(γ 2) theparticle number density field, Mi(γ ) = mTrc(γ 2ci) the i-thcomponent of the momentum density field, and the total energydensity field H (γ ) = T (γ ) + U (γ ) is, in general, composedof a kinetic energy contribution T (γ ) = 1

2mTrc(γ 2c · c) and apotential energy contribution U (γ ), such that at position x andtime t the functional derivative δU (γ )/δγ = 2γ ϕγ (x,t) is afunction in L , thus obeying condition C2 above, where ϕγ (x,t)is the single-particle potential field. For example, for a uniformexternally applied field in the z direction, −∇ϕγ (x) = a =−a ∇z with a constant. Again, for the Vlasov-Poisson kinetictheory [27], U (γ ) = 1

2 Trc(γ 2ϕγ ), where ϕγ (x,t) is a nonlocalfunctional of γ , with ϕγ (x,t) = ∫

x′ dx ′∫c′

dc′V (|x −x′|) γ 2(c′; x′,t) representing a locally averaged mean-fieldsingle-particle potential due to the effects of the neighboringparticles via the interparticle potential V assumed to be afunction of particle distance only.

The dissipative time evolution of the distribution functionf is given by the Boltzmann equation or some equivalentsimplified kinetic model equation, which in terms of thesquare-root distribution may be written in the form

∂γ

∂t+ c · ∇γ − ∇ϕγ · ∂γ

∂c= �γ . (20)

In order to satisfy the constraints of mass, momentum, andenergy conservation, the collision term �γ must be such that

�Ci= (�i |�γ ) = 0, (21)

where

�i = δCi/δγ. (22)

The entropy density field in this context is represented by

S(x,t) = S(γ ) = −kTrc(γ 2 ln γ 2) = (−kγ ln γ 2|γ ), (23)

the rate of entropy production is

�S = (�|�γ ), (24)

where

� = δS/δγ, (25)

and the entropy balance equation is

−k∂Trc(f ln f )

∂t− k∇ · Trc(f c ln f ) = �S, (26)

where JS = −kTrc(f c ln f ) represents the entropy flux field.In Sec. III, we construct the family of models for the

collision term �γ such that �S is maximal subject to theconservation constraints �Ci

= 0 and the suitable additionalgeometrical constraint we discuss therein.

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The resulting family of SEA kinetic models of the collisionintegral in the Boltzmann equation is currently under investi-gation by comparing it with standard models such as the well-known Bhatnagar, Gross, and Krook (BGK) model as wellas with Monte Carlo simulations of the original Boltzmannequation for hard spheres [28]. In addition to the strongthermodynamic consistency even far from stable equilibrium,Ref. [28] gives a proof that in the near-equilibrium limit theSEA models reduce to the corresponding BGK models.

In a forthcoming paper [11], we work out the explicitrelation between SEA and GENERIC and we provide theexplicit SEA form of the full Boltzmann collision operatorby using its already-available GENERIC form [8,14,29,30].

C. Framework C: Rational extended thermodynamics,macroscopic nonequilibrium thermodynamics,

and chemical kinetics

Let L be the set of all n vectors of real numbers A =vect(aj ), B = vect(bj ), . . . (n � ∞), equipped with the innerproduct (·|·) defined by

(A|B) = Tr(AB) =n∑

j=1

aj bj . (27)

In rational extended thermodynamics [31], the local stateat position x and time t of the continuum under study isrepresented by an element γ in L , i.e.,

γ (x,t) = vect[αj (x,t)]. (28)

Thus, γ (x,t) represents a set of extensive underlying fieldsor internal variables αj (x,t) which represent the instantaneousspatial distributions within the continuum of the local densitiesthat define all its other local properties. In particular, for theconserved properties energy, momentum, and mass [32] it isassumed that their local densities and their local (Lagrangian)fluxes are all given by particular functions of γ that we denotesynthetically by

{Ci(γ )} = {E(γ ),Mx(γ ),My(γ ),Mz(γ ),mk(γ )}, (29)

{JCi(γ )} = {JE(γ ),JMx

(γ ),JMy(γ ),JMz

(γ ),Jmk(γ )}, (30)

so the energy, momentum, and mass balance equations takethe form

∂Ci

∂t+ ∇ · JCi

= �Ci= 0. (31)

Moreover, also for the local entropy density and the local(Lagrangian) entropy flux it is assumed that they are given byparticular functions of γ that we denote respectively by

S(γ ) and JS(γ ), (32)

so the entropy balance equation takes the form

∂S

∂t+ ∇ · JS = �S, (33)

where �S is the local production density.In general, the balance equation for each of the extensive

underlying field properties, i.e., the internal variables, is

∂αj

∂t+ ∇ · Jαj

= �αj, (34)

where Jαjand �αj

are the corresponding flux and productiondensity, respectively. Equivalently, this set of balance equa-tions may be written synthetically as

∂γ

∂t+ ∇ · Jγ = �γ , (35)

where Jγ = vect(Jαj) and �γ = vect(�αj

).It is then further assumed that there exist functions �αj

(γ )(Liu’s Lagrange multipliers [33]) that we denote here in vectorform by

� = vect(�αj) (36)

such that the local entropy production density can be written as

�S =n∑

j=1

�αj�αj

= (�|�γ ) (37)

and must be non-negative everywhere.For our development in this paper we additionally assume

that there also exist functions �i αj(γ ) that we denote in vector

form by

�i = vect(�i αj) (38)

such that the production density of each conserved propertyCi can be written as

�Ci=

n∑j=1

�i αj�αj

= (�i |�γ ). (39)

Typically, but not necessarily, the first 4 + nsp − nr underlyingfields αj (x,t) for j = 1, . . . ,4 + nsp − nr are convenientlychosen to coincide with the energy, momentum, and (inde-pendently conserved [32] linear combinations of the) massdensities, where nsp is the number of species and nr the numberof independent reactions, so Eqs. (34) for j = 1, . . . ,4 + nsp −nr coincide with Eqs. (31) because �αj

= 0 for this subset ofconserved fields.

The above framework reduces to the traditional On-sager theory of macroscopic nonequilibrium thermodynamics(NET) [6] if the αj ’s are taken to represent the local deviationsof the underlying fields from their equilibrium values. Inthis context, the usual notation calls the functions �αj

the “thermodynamic forces” and �αjthe “thermodynamic

currents.”In Sec. III we construct an equation of motion for γ such

that �S is maximal subject to the conservation constraints�Ci

= 0 plus a suitable additional constraint.The same framework reduces to the standard scheme of

chemical kinetics (CK) if the αj ’s include the local reactioncoordinates of each of the nr steps of the detailed kinetic mech-anism, the corresponding �αj

’s are the local rates of advance-

ment of the reactions, �αj= ∂S/∂αj = −∑nsp

�=1 νj

� μ�/T isthe entropic affinity of the j -th reaction step (equal to thede Donder affinity divided by the temperature, see, e.g.,Refs. [34,35], where μ� is the chemical potential of species�), and mk are the local values of the nsp − nr independentlyconserved linear combinations of the masses of the variousspecies (see Ref. [32] for the precise definition) so �mk

= 0are their local production densities.

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GIAN PAOLO BERETTA PHYSICAL REVIEW E 90, 042113 (2014)

In the CK framework, the SEA construction is closelyrelated to the gradient-dynamics formulations of theGuldberg-Waage mass action law as suggested in Refs. [36,37]and more recently in Refs. [38,39].

D. Framework D: Mesoscopic nonequilibrium thermodynamicsand continuum mechanics with fluctuations

Let L be the set of all n vectors A = vect(aj (ααα)),B = vect(bj (ααα)), . . . whose entries aj (ααα), bj (ααα), . . . are real,square-integrable functions of a set of mesoscopic (internal)variables denoted synthetically by the vector

ααα = vect(α1, . . . ,αm), (40)

whose m-dimensional range α is usually called the ααα space.Let L be equipped with the inner product (·|·) defined by

(A|B) =n∑

i=1

Trα(aibi) =n∑

i=1

∫α

ai(ααα)bi(ααα) dα, (41)

where Trα(·) in this framework denotes∫α

· dα , with dα =dα1 · · · dαm.

In mesoscopic nonequilibrium thermodynamics (MNET)(see, e.g., Refs. [6,40]) the αj ’s are the set of mesoscopic(coarse-grained) local extensive properties assumed to repre-sent the local nonequilibrium state of the portion of continuumunder study. Our mesoscopic description of the local state atposition x and time t is in terms of a square-root probabilitydensity on the ααα space α , which we denote by

γ (ααα; x,t)

such that γ 2(ααα; x,t) dα represents the probability that thevalues of the underlying fields are between ααα and ααα + dααα.In the standard formulation the state representative is theprobability distribution P (ααα; x,t) = [γ (ααα; x,t)]2.

It is assumed that the probability density γ obeys acontinuity equation that we may write as follows:

∂γ

∂t+ v · ∇γ = �γ with 2γ�γ = −∇α · �α, (42)

where v = v(ααα) is the particle velocity expressed in terms ofthe underlying fields (usually it is convenient to take the firstthree αj ’s to coincide with the velocity components), v · ∇γ =∇ · Jγ , where Jγ is the flux of square-root probability density,and

�α(ααα; x,t) = vect(�αj) , ∇α = vect

(∂

∂αj

), (43)

where the �αj’s are interpreted as probability weighted

components of a streaming flux in α , i.e., a current in thespace of mesoscopic coordinates.

The local densities Ci(x,t) of the conserved properties areassumed to have an associated underlying extensive propertywhich can be expressed in terms of the mesoscopic coordinatesas ci(ααα) such that

Ci(x,t) = Ci(γ ) =∫

α

ci(ααα) γ 2(ααα; x,t) dα, (44)

�i = δCi/δγ = 2γ ci . (45)

They obey the balance equation

∂Ci

∂t+ ∇ · JCi

= �Ci= 0, (46)

where the local Lagrangian flux JCi(x,t) and the local

production density �Ci(x,t) are defined as follows:

JCi(x,t) =

∫α

ci(ααα) v(ααα) γ 2(ααα; x,t) dα

�Ci(x,t) = (�i |�γ )

=∫

α

ci(ααα) 2γ (ααα; x,t)�γ (ααα; x,t) dα

= −∫

α

ci(ααα) ∇α · �α(ααα; x,t) dα

=∫

α

�α(ααα; x,t) · ∇αci(ααα) dα

= (ψi |�α), (47)

where in the next-to-last equation we integrated by parts andassumed that currents in ααα space decay sufficiently fast to zeroas the αj ’s → ∞, and we defined

ψi(ααα) = ∇αci(ααα). (48)

Also the condition of preservation of normalization is writtenin the same way, by setting c0(ααα) = 1 so �0 = 2γ andthe corresponding balance equation (46) with the condition�C0 (x,t) = 0 yields the following conditions on �γ and �α:

�C0 (x,t) = (�0|�γ ) =∫

α

2γ (ααα; x,t)�γ (ααα; x,t) dα

= −∫

α

∇α · �α(ααα; x,t) dα = 0. (49)

The local entropy density S(x,t) is expressed in terms ofthe local square-root probability density as

S(x,t) = S(γ ) = −k

∫γ

γ 2(ααα; x,t) ln γ 2(ααα; x,t) dα (50)

such that

� = δS/δγ = −2kγ (ααα; x,t) [1 + ln γ 2(ααα; x,t)], (51)

and the entropy balance equation takes the form

∂S

∂t+ ∇ · JS = �S, (52)

where the local Lagrangian flux JS(x,t) and the local produc-tion density �S(x,t) are defined as follows:

JS(x,t) = −k

∫α

γ 2(ααα; x,t) v(ααα) ln γ 2(ααα; x,t) dα

�S(x,t) = (�|�γ )

= −k

∫α

[1 + ln γ 2(ααα; x,t)]

× 2γ (ααα; x,t)�γ (ααα; x,t) dα

= k

∫α

ln γ 2(ααα; x,t) ∇α · �α(ααα; x,t) dα

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STEEPEST ENTROPY ASCENT MODEL FOR FAR- . . . PHYSICAL REVIEW E 90, 042113 (2014)

= −k

∫α

�α(ααα; x,t) · ∇α ln γ 2(ααα; x,t) dα

= (φ|�α), (53)

where we used the normalization condition (49) and again inthe next-to-last equation we integrated by parts and defined

φ(ααα; x,t) = −k∇α ln γ 2(ααα; x,t). (54)

In Sec. III, we construct an equation of motion for γ suchthat �S is maximal subject to the conservation constraints�Ci

= 0 and the suitable geometrical constraint we discusstherein. The result, when introduced in Eq. (42), will yieldthe Fokker-Planck equation for γ (ααα; x,t), which is also related(see, e.g., Ref. [41]) to the GENERIC structure [8–10]. Theformalism can also be readily extended to the family ofTsallis [42] entropies in the frameworks of nonextensivethermodynamic models [43].

E. Framework E: Quantum statistical mechanics, quantuminformation theory, quantum thermodynamics, mesoscopicnonequilibrium quantum thermodynamics, and intrinsic

quantum thermodynamics

Let H be the Hilbert space (dim H � ∞) associated withthe physical system and L the set of all linear operators A,B, . . . on H , equipped with the real inner product (·|·) definedby

(A|B) = Tr(A†B + B†A)/2, (55)

where A† denotes the adjoint of operator A and Tr(·) the usualtrace functional symbol.

In the quantum frameworks that we consider in this section,the state representative is the density operator ρ, i.e., a unit-trace, self-adjoint, and non-negative-definite element of L .

Instead, also here we will adopt the state representation interms of the generalized square root of the density operator thatwe developed in this context [2–5] in order to easily imposethe constraints of preservation of both the non-negativity andthe self-adjointness of ρ during its time evolution as wellas to satisfy Condition C2. Therefore, we assume that thestate representative is an element γ in L from which we cancompute the density operator as follows:

ρ = γ γ †. (56)

In other words, we adopt as state representative not the densityoperator ρ itself but its generalized square root γ . Therefore,we clearly have

dρ

dt= γ

dγ †dt

+ dγ

dtγ †. (57)

We then consider the set of operators corresponding to theconserved properties, denoted synthetically as

{Ci} = {H,Mx,My,Mz,N1, . . . ,Nr,I }. (58)

Here we assume that these are self-adjoint operators in L ,where each Mj and Ni commutes with H , i.e., HMj = MjH

for j = x,y,z and HNi = NiH for i = 1, . . . ,r , and that I isthe identity operator [44].

The semiempirical description of an irreversible relaxationprocess is done in this framework by assuming an evolution

equation for the state γ given by the equations

dγ

dt+ i

�Hγ = �γ , (59)

dγ †

dt− i

�γ †H = �γ † . (60)

As a result, it is easy to verify that for the density operator thedynamical equation is

dρ

dt+ i

�[H,ρ] = �γ γ † + γ �γ † , (61)

where [·,·] denotes the commutator. From this we see that inorder to preserve hermiticity of ρ the dissipative terms �γ and�γ † must satisfy the conditions

�γ † = �†γ and �γ = �

†γ † . (62)

In order to satisfy the constraints of conservation of theexpectation values Tr(ρCi), each Ci must commute with H ;moreover, the term �γ must be such that

�Ci= d

dtTr(ρCi) = Tr(Ci�γ γ † + γ �γ †Ci)

= (2Ciγ |�γ ) = 0. (63)

The entropy functional in this context is represented by

S(γ ) = −kTr(ρ ln ρ) = (−k(ln γ γ †) γ |γ ), (64)

so the rate of entropy production under a time evolution thatpreserves the normalization of ρ is given by

�S = −kd

dtTr(ρ ln ρ) = (−2k(ln γ γ †) γ |�γ ). (65)

In quantum statistical mechanics (QSM) and quantum infor-mation theory (QIT), ρ is the von Neumann statistical ordensity operator which represents the index of statistics from agenerally heterogeneous ensemble of identical systems (sameHilbert space H and operators {H,N1, . . . ,Nr}) distributedover a range of generally different quantum mechanicalstates. If each individual member of the ensemble is isolatedand uncorrelated from the rest of the universe, its state isdescribed according to quantum mechanics by an idempotentdensity operator (ρ2 = ρ = P|ψ〉 = |ψ〉〈ψ |

〈ψ |ψ〉 ), i.e., a projectionoperator onto the span of some vector |ψ〉 in H . If theensemble is heterogeneous, its individual member systemsmay be in different states, P|ψ1〉, P|ψ2〉, and so on, andthe ensemble statistics is captured by the von Neumannstatistical operator ρ = ∑

j wjP|ψj 〉. The entropy functionalhere represents a measure of the informational uncertainty asto which homogeneous subensemble the next system will bedrawn from, i.e., as to which will be the actual pure quantumstate among those present in the heterogeneous ensemble.

In this framework, unless the statistical weights wj changefor some extrinsic reason, the quantum evolution of theensemble is given by Eq. (61) with �γ = 0 so Eq. (61) reducesto von Neumann’s equation of quantum (reversible) Hamilto-nian evolution, corresponding to ρ(t) = ∑

j wjP|ψj (t)〉, wherethe underlying pure states |ψj (t)〉 evolve according to theSchrodinger equation d|ψj 〉/dt = −iH |ψj 〉/�.

In the framework of QSM and QIT, the SEA equationof motion we construct in the next Sec. III for γ and

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GIAN PAOLO BERETTA PHYSICAL REVIEW E 90, 042113 (2014)

hence, through Eq. (57), for ρ represents a model for therates of change of the statistical weights wj in such a waythat �S is maximal subject to the conservation constraints�Ci

= 0 (and a suitable additional constraint, see Sec. III).This essentially extends to the quantum landscape the samestatistical or information-theoretic nonequilibrium problem wedefined above as Framework A.

In quantum thermodynamics (QT), instead, the densityoperator takes on a more fundamental physical meaning. Itis no longer related to the heterogeneity of the ensemble, andit is no longer assumed that the individual member systems ofthe ensemble are in pure states.

The prevailing interpretation of QT (for a recent reviewsee Ref. [45]) is the so-called open-system model wherebythe quantum system under study (each individual system of ahomogeneous ensemble) is always viewed as in contact (weakor strong) with a thermal reservoir or “heat bath,” and its notbeing in a pure state is an indication of its being correlated withthe reservoir. The overall system-plus-reservoir composite isassumed to be a pure quantum mechanical state in H ⊗ HR

and reduces to the density operator ρ on the system’s spaceH when we partial trace the overall density operator over thereservoir’s space HR .

The semiempirical description of an irreversible relax-ation process is done in this framework by assuming for�ρ in Eq. (61) the Lindblad-Gorini-Kossakowski-Sudarshan(LGKS) form [46,47]

�ρ =∑

j

(VjρV

†j − 1

2{V †

j Vj ,ρ}+)

, (66)

where {·,·}+ denotes the anticommutator and operators Vj

are to be chosen to properly model the system-reservoirinteraction. The justification and modeling assumptions thatlead to the general form of Eq. (66) are well known.

In the framework of QT the SEA equation of motion weconstruct in the next section may be useful as an alternativemodel for �ρ (or for a term additional to the LGKS term) suchthat �S is maximal subject to the conservation constraints�Ci

= 0 (and the suitable additional constraint defined belowin Sec. III). In some cases this could be simpler than theLGKS model and it has the advantage of a strong built-inthermodynamics consistency. A similar attempt has beenrecently discussed in Ref. [48] as an application of theGENERIC scheme.

Mesoscopic nonequilibrium quantum thermodynamics(MNEQT) [7] starts from the formalism of QSM but attemptsto extend the Onsager NET theory and MNET to the quantumrealm. We will show elsewhere that the present SEA formu-lation reduces to MNEQT in the near-equilibrium limit andcan therefore be viewed as the natural extension of MNEQTto the far-nonequilibrium regime. The essential elements ofthis proof have actually already been given [4] but only for theparticular case corresponding to Eq. (70) below (Fisher-Raometric).

An even more fundamental physical meaning is assumedwithin the theory that we originally called quantum thermo-dynamics [2,3,49–53] but more recently renamed intrinsicquantum thermodynamics (IQT) to avoid confusion with

the more traditional theories of QT such as those justoutlined.

IQT assumes that the second law of thermodynamics shouldcomplement the laws of mechanics even at the single-particlelevel [49]. This can be done if we accept that the trueindividual quantum state of a system, even if fully isolatedand uncorrelated from the rest of the universe, requires densityoperators ρ that are not necessarily idempotent. Over the set ofidempotent ρ’s, QT coincides with quantum mechanics (QM),but it differs fundamentally from QM because it assumes abroader set of possible states, corresponding to the set ofnonidempotent ρ’s. This way, the entropy functional S(ρ)becomes in IQT an intrinsic fundamental property. In a senseIQT with its SEA dynamical law accomplishes the conceptualprogram, so intensely sought for also by Ilya Prigogine andcoworkers [54], of answering the following questions [2]:What if entropy, rather than a statistical, information-theoretic,macroscopic, or phenomenological concept, were an intrinsicproperty of matter in the same sense as energy is universallyunderstood to be an intrinsic property of matter? What ifirreversibility were an intrinsic feature of the fundamentaldynamical laws obeyed by all physical objects, macroscopicand microscopic, complex and simple, large and small? Whatif the second law of thermodynamics, in the hierarchy ofphysical laws, were at the same level as the fundamental lawsof mechanics, such as the great conservation principles? Whenviewed from such extreme perspective, the IQT conceptualscheme remains today as “adventurous” as it was acutelyjudged by John Maddox in 1985 [55].

In the framework of IQT the SEA equation of motion (61)for ρ which results from the expression for �γ we construct inthe next section represents a strong family of implementationsof the MEP principle at the fundamental quantum level whichcontains our original formulation as a special case.

Even the brief discussion above shows clearly that thedifferences among QSM, QIT, QT, IQT, and MNEQT areimportant on the interpretational and conceptual levels. Nev-ertheless, it is also clear that they all share the same basicmathematical framework. Hence, we believe that the SEAdynamical model, which we show here fits their commonmathematical basis, can find in the different theories differentphysical interpretations and applications.

III. STEEPEST-ENTROPY-ASCENT DYNAMICS:UNIFIED VARIATIONAL FORMULATION

OF NONEQUILIBRIUM DISSIPATION

In the preceding section we formulated the nonequilibriumproblem in various different frameworks in a unifying way thatallows us to represent their dissipative parts in a single formalway. In essence, as summarized by conditions C1–C4 above,the state is represented by an element γ of a suitable vectorspace L equipped with an inner product (·|·). The term in thedynamical equation for γ which is responsible for dissipativeirreversible relaxation and hence entropy generation is anotherelement �γ of L which, together with the variationalderivatives � and �i of the functionals S(γ ) and Ci(γ )representing, respectively, the entropy and the constants ofmotion, determines the rate of entropy production according

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STEEPEST ENTROPY ASCENT MODEL FOR FAR- . . . PHYSICAL REVIEW E 90, 042113 (2014)

to the relation

�S = (�|�γ ) (67)

and the rates of production of the conserved properties Ci

according to the relation

�Ci= (�i |�γ ). (68)

The formulations in terms of square roots of probabilitiesin Framework A, of the square root of the phase densityin Framework B, of the square-root probability density inFramework D, and of the generalized square root of the densityoperator in Framework E take care not only of the importantcondition that for the evolution law to be well defined it mustconserve the non-negativity of probabilities, phase densities,and density operators (which must also remain self adjoint)but also of condition C2, namely that functional derivatives ofthe entropy and the constants of the motion are also elementsof the vector space L .

We are now ready to formulate the SEA construction. Wedo this by assuming that the time evolution of the state γ

follows the path of steepest entropy ascent in L compatiblewith the constraints. So, for any given state γ , we must findthe �γ which maximizes the entropy production �S subjectto the constraints �Ci

= 0. But in order to identify the SEApath we are not interested in the unconditional increase in �S

that we can trivially obtain by simply increasing the “norm” of�γ while keeping its direction fixed. Rather, the SEA path isidentified by the direction of �γ which maximizes �S subjectto the constraints, regardless of the norm of �γ . Hence, wemust do the maximization at constant norm of �γ .

In the absence of Hamiltonian or transport contributions tothe time evolution of γ , the vector �γ is tangent to the pathγ (t). Therefore, the norm of �γ represents the square of thedistance d� traveled by γ in the state space L in the timeinterval dt , the square of the “length” of the infinitesimal bitof path traveled in state space in the interval dt . The variationalproblem that identifies the SEA direction at each state γ looksat all possible paths through γ , each characterized by a possiblechoice for �γ . Among all these paths it selects the one withthe highest entropy produced in the interval dt , �S dt per unitof distance d� traveled by γ .

It is therefore apparent that we cannot identify a SEA pathuntil we equip the space L with a metric field with respectto which to compute the distance d� traveled and the norm of�γ .

In our previous work [5], we selected the Fisher-Rao metricbased on the inner product (·|·) defined on L . Indeed, indealing with probability distributions it has been argued byseveral authors that the Fisher-Rao metric is the proper uniquemetric for the purpose of computing the distance between twoprobability distributions (see, e.g., Refs. [56–58]). Accordingto this metric, the distance between two states γ1 and γ2 isgiven by

d(γ1,γ2) =√

2 arccos(γ1|γ2), (69)

which implies that the distance traveled along a trajectory instate space is

d� = 2√

(�γ |�γ ) dt. (70)

As a result, for Framework E the SEA dynamics we haveoriginally proposed is most straightforward.

However, here we will not adopt a priori a specificmetric but rather assume a most general metric, which inFramework E generalizes our previous work and in the otherframeworks provides the most general formulation. We assumethe following expression for the distance traveled along a shortbit of trajectory in state space:

d� =√

(�γ | G(γ ) |�γ ) dt, (71)

where G(γ ) is a real, symmetric, and positive-definite operatoron L that we call the metric tensor field, (super)matrix, or(super)operator depending on the framework. In general, G(γ )may be a nonlinear function of γ . In Framework E, since L isthe space of operators on the Hilbert space H of the quantumsystem, G is a superoperator on H . For example, a simple caseis when G|A) = |[G†,[G,A]]) with G some operator in L .

We may now finally state the SEA variational problem andsolve it. The problem is to find the instantaneous “direction” of�γ which maximizes the entropy production rate �S subjectto the constraints �Ci

= 0. We solve it by maximizing theentropy production rate �S subject to the constraints �Ci

= 0and the additional constraint (d�/dt)2 = ε2 = prescribed. Thelast constraint keeps the norm of �γ constant as necessaryin order to maximize only with respect to its direction. FromEq. (71) it amounts to keeping fixed the value of (�γ | G |�γ )at some small positive constant ε2. The solution is easilyobtained by the method of Lagrange multipliers. We seekthe unconstrained maximum, with respect to �γ , of theLagrangian

ϒ = �S −∑

i

βi �Ci− τ

2(�γ | G |�γ ), (72)

where βi and τ/2 are the Lagrange multipliers. Like G, theymust be independent of �γ but can be functions of the stateγ . Using Eqs. (67) and (68), we rewrite (72) as follows:

ϒ = (�|�γ ) −∑

i

βi (�i |�γ ) − τ

2(�γ | G |�γ ). (73)

Taking the variational derivative of ϒ with respect to |�γ ) andsetting it equal to zero we obtain

δϒ

δ�γ

= |�) −∑

i

βi |�i) − τG|�γ ) = 0, (74)

where we used the identity (�γ | G = G |�γ ), which followsfrom the symmetry of G. Thus, we obtain the SEA generalevolution equation (the main result of this paper)

|�γ ) = L |� −∑

j

βj �j ), (75)

where we define for convenience

L = 1

τG−1. (76)

Since in the various frameworks L can be connected with thegeneralized Onsager conductivity (super)matrix in the nearequilibrium regime, we see here that τ L is the inverse of themetric (super)matrix G with respect to which the dynamics

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GIAN PAOLO BERETTA PHYSICAL REVIEW E 90, 042113 (2014)

is SEA. In other words, denoting the generalized Onsagerresistivity (super)matrix by R we have R = τ G. Since G ispositive definite and symmetric, so are L and R. In other words,the SEA assumption automatically entails Onsager reciprocitynear thermodynamic equilibrium.

Inserting Eq. (75) into the conservation constraints (68)yields the important system of equations which defines thevalues of the Lagrange multipliers βj ,∑

j

(�i | L |�j ) βj = (�i | L |�). (77)

This system can be readily solved for the βj ’s (for example,by Cramer’s rule) because the functionals (�i |L|�j ) and(�i |L|�) are readily computable for the current state γ .Notice that the determinant of the matrix [(�i | L |�j )] is aGram determinant and its being positive definite is equivalentto the condition of linear independence of the conservationconstraints. When Cramer’s rule is worked out explicitly, theSEA equation (75) takes the form of a ratio of determinantswith which we presented it in the IQT framework [5,50–53],namely,

|�γ ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

L� L�1 · · · L�n

(�1| L |�) (�1| L |�1) · · · (�1| L |�n)

......

. . ....

(�n| L |�) (�n| L |�1) · · · (�n| L |�n)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(�1| L |�1) · · · (�1| L |�n)

.... . .

...

(�n| L |�1) · · · (�n| L |�n)

∣∣∣∣∣∣∣∣∣∣∣

,

(78)

where the set of vectors L1/2 |�1), . . . ,L1/2 |�n) are linearlyindependent so the Gram determinant at the denominator isstrictly positive. These are all the vectors in the set {L1/2 |�i)}if they are linearly independent, otherwise they are a subset ofn of them which are linearly independent.

We can now immediately prove the general consistencewith the thermodynamic principle of entropy nondecrease (H -theorem in Framework B). Indeed, subtracting Eqs. (68), eachmultiplied by the corresponding βj from Eq. (67), and theninserting Eq. (75) yields the following explicit expression forthe rate of entropy production:

�S = (�|�γ ) = (� −∑

j

βj �j |�γ )

= (� −∑

i

βi �i | L |� −∑

j

βj �j ) � 0, (79)

which is clearly non-negative-definite by virtue, again, of thenon-negativity that must be assumed for a well defined metricsuperoperator G.

It is interesting to write the expression for the (prescribed)speed d�/dt at which the state γ evolves along the SEA path.This amounts to inserting Eq. (75) into the additional constraint

(d�/dt)2 = ε2 = prescribed. We readily find

d�2

dt2= (�γ | G |�γ )

= 1

τ 2

(� −

∑i

βi �i | G−1GG−1 |� −∑

j

βj �j

)(80)

= 1

τ�S = ε2 (81)

so we have the relations

τ =√(

� − ∑i βi �i | G−1 |� − ∑

j βj �j

)d�/dt

(82)

=(� − ∑

i βi �i | G−1 |� − ∑j βj �j

)�S

, (83)

from which we see that through the Lagrange multiplier τ wemay specify either the speed at which γ evolves along the SEAtrajectory in state space or the instantaneous rate of entropyproduction. Hence, using τ given by Eq. (83) the evolutionequation (75) will produce a SEA trajectory in state spacewith the prescribed entropy production �S . These relationsalso support the interpretation of τ as the “overall relaxationtime.” We see this as follows.

In general, we may interpret the vector

|�) = G−1/2 |� −∑

i

βi �i) (84)

as a vector of “nonequilibrium affinities” or, more precisely, of“generalized partial affinities.” In terms of this vector, Eq. (75)rewrites as

G1/2 |�γ ) = 1

τ|�). (85)

When only some of the partial affinities in the vector �

are zero, the state is partially equilibrated (equilibrated withrespect to the corresponding underlying components of thestate γ ). When the entries of the vector � are all zero, then andonly then do we have an equilibrium state or a nondissipativelimit cycle. In fact, it is when and only when the entropyproduction vanishes. (�|�), which, with respect to the metrictensor G, is the norm of the vector |� − ∑

j βj �j ), representsa measure of the “overall degree of disequilibrium” of thestate γ . It is important to note that this definition is valid nomatter how far the state is from the (maximum entropy) stableequilibrium state, i.e., also for highly nonequilibrium states.

We have proved in the IQT framework, and the resultcan be readily extended to all other frameworks, that amongthe equilibrium states only the maximum entropy one is notunstable (in the sense of Lyapunov [59]). As a result, themaximum entropy states emerge as the only stable equilibriumones and, therefore, we can assert that the SEA constructionimplements the Hatsopoulos-Keenan statement of the secondlaw [21,60] at the level of description of everyone of theframeworks we are considering.

Equation (83) rewrites as

�S = (�|�)

τ, (86)

which shows that the rate of entropy production is proportionalto the overall degree of disequilibrium. The relaxation time

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STEEPEST ENTROPY ASCENT MODEL FOR FAR- . . . PHYSICAL REVIEW E 90, 042113 (2014)

τ may be a state functional and needs not be constant, buteven if it is, the SEA principle provides a nontrivial nonlinearevolution equation that is well defined and reasonable, i.e.,thermodynamically consistent, even far from equilibrium.

We finally note that when the only contribution to theentropy change comes from the production term �S (forexample, in Framework B in the case of homogeneousrelaxation in the absence of entropy fluxes or in Framework Efor an isolated system), i.e., when the entropy balance equationreduces to dS/dt = �S , Eq. (81) may be rewritten as

d�

dt/τ= dS

d�, (87)

from which we see that when time t is measured in units of τ

the “speed” along the SEA trajectory is equal to the local rateof entropy increase along the trajectory.

If the state γ moves only due to the dissipative term �γ

(for example, in Framework E when [H,γ γ †] = 0), then theoverall length of the trajectory in state space traveled betweent = 0 and t is given by

�(t) =∫ t

0

√(�γ | G |�γ ) dt, (88)

and, correspondingly, we may also define the “nonequilibriumaction”

� = 1

2

∫ t

0(�γ | G |�γ ) dt = 1

2

∫ t

0

�S

τdt

= 1

2

∫ t

0

(�|�)

τ 2dt, (89)

where for the last two equalities we used Eq. (81) and Eq. (86),respectively.

IV. PICTORIAL REPRESENTATIONS

Let us give pictorial representations of the vectors that wedefined in the SEA construction. We consider first the simplestscenario of a uniform metric tensor G = I .

Figure 1 gives a pictorial representation of the linearmanifold spanned by the vectors |�i)’s and the orthogonalprojection of |�) which defines the Lagrange multipliers βi inthe case of uniform metric, i.e., the orthogonality conditions

FIG. 1. (Color online) Pictorial representation of the linear man-ifold spanned by the vectors |�i) and the orthogonal projection of|�) onto this manifold which defines the Lagrange multipliers βi

in the case of a uniform metric G = I . The construction definesalso the generalized affinity vector, which in this case is |�) =|� − ∑

i βi �i).

FIG. 2. (Color online) Pictorial representation of the SEA vari-ational construction in the case of a uniform metric G = I . Thecircle represents the condition (�γ |�γ ) = ε2. The vector |�γ ) mustbe orthogonal to the |�i)’s in order to satisfy the conservationconstraints �Ci

= (�i |�γ ) = 0. In order to maximize the scalarproduct (� − ∑

i βi �i |�γ ), |�γ ) must have the same direction as|� − ∑

i βi �i).

(�j |� − ∑i βi �i) = 0 for every j , which are Eqs. (77) with

L = I /τ . The construction defines also the generalized affinityvector, which in this case is |�) = |� − ∑

i βi �i) and isorthogonal to the linear manifold spanned by the vectors |�i)’s.

Figure 2 gives a pictorial representation of the subspaceorthogonal to the linear manifold spanned by the |�i)’s thathere we denote for simplicity by {|�i)}. The vector |�) isdecomposed into its component |∑i βi �i) which lies in{|�i)} and its component |� − ∑

i βi �i) which lies in theorthogonal subspace.

The circle in Fig. 2 represents the condition (�γ |�γ ) = ε2

corresponding in the uniform metric to the prescribed rate ofadvancement in state space, ε2 = (d�/dt)2. The compatibilitywith the conservation constraints �Ci

= (�i |�γ ) = 0 requiresthat |�γ ) lies in the subspace orthogonal to the |�i)’s. To takethe SEA direction, |�γ ) must maximize the scalar product(� − ∑

i βi �i |�γ ). This clearly happens when |�γ ) hasthe same direction as the vector |� − ∑

i βi �i) which inthe uniform metric coincides with the generalized affinityvector |�).

Next we consider the more general scenario of a nonuniformmetric tensor G. Figure 3 gives a pictorial representation ofthe linear manifold spanned by the vectors G−1/2 |�i) andthe orthogonal projection of G−1/2 |�) which defines the

FIG. 3. (Color online) Pictorial representation of the linear mani-fold spanned by the vectors G−1/2 �i and the orthogonal projection ofG−1/2 |�) onto this manifold which defines the Lagrange multipliersβi in the case of a nonuniform metric G. The construction definesalso the generalized affinity vector |�) = G−1/2 |� − ∑

i βi �i).

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GIAN PAOLO BERETTA PHYSICAL REVIEW E 90, 042113 (2014)

FIG. 4. (Color online) Pictorial representation of the SEA vari-ational construction in the case of a nonuniform metric G. Thecircle represents the condition (�γ | G |�γ ) = ε2, corresponding tothe norm of vector G1/2 |�γ ). This vector must be orthogonal tothe G−1/2 |�i)’s in order to satisfy the conservation constraints�Ci

= (�i |�γ ) = 0. In order to maximize the scalar product �S =(�|�γ ) = (� − ∑

i βi �i |�γ ), vector G1/2 |�γ ) must have the samedirection as |�) = G−1/2 |� − ∑

i βi �i).

Lagrange multipliers βi in the case of nonuniform metric G,where the orthogonality conditions that define the βi’s are(�j | G−1 |� − ∑

i βi �i) = 0 for every j , which are Eqs. (77).The construction defines also the generalized affinity vector|�) = G−1/2 |� − ∑

i βi �i) which is orthogonal to the linearmanifold spanned by the vectors G−1/2 |�i)’s.

Figure 4 gives a pictorial representation of the subspaceorthogonal to the linear manifold spanned by the G−1/2 |�i)’sthat here we denote for simplicity by {G−1/2 |�i)}. The vectorG−1/2 |�) is decomposed into its component G−1/2 | ∑i βi �i)which lies in {G−1/2 |�i)} and its component |�) =G−1/2 |� − ∑

i βi �i) which lies in the orthogonal subspace.The circle in Fig. 4 represents the more general condition

(�γ | G |�γ ) = ε2 corresponding in the nonuniform metricto the prescribed rate of advancement in state space, ε2 =(d�/dt)2. It is clear that the direction of G1/2 |�γ ), whichmaximizes the scalar product (� − ∑

i βi �i |�γ ), is when|�γ ) is in the direction of the point of tangency between theellipse and a line orthogonal to |� − ∑

i βi �i).The compatibility with the conservation constraints �Ci

=(�i |�γ ) = 0 requires that G1/2 |�γ ) lies in the subspaceorthogonal to the G−1/2 |�i)’s. To take the SEA direction,the vector G1/2 |�γ ) must maximize the scalar product(� − ∑

i βi �i |�γ ), which is equal to the entropy production�S = (�|�γ ) since (�i |�γ ) = 0. This clearly happens whenG1/2 |�γ ) has the same direction as the generalized affinityvector |�) = G−1/2 |� − ∑

i βi �i).

V. CONCLUSIONS

In this paper, we reformulate with a somewhat unusualnotation the essential mathematical elements of several differ-ent approaches to the description of nonequilibrium dynamicswith the purpose of presenting a unified formulation which,in all these contexts, allows us to implement the local SEAconcept whereby the dissipative, irreversible component ofthe time evolution of the local state is assumed to pull thestate along the path in state space which, with respect to an

underlying metric, is always tangent to the direction of max-imal entropy increase compatible with the local conservationconstraints.

The frameworks we consider are the following: (A)statistical or information-theoretic models of relaxation; (B)small-scale and rarefied gas dynamics (i.e., kinetic modelsfor the Boltzmann equation); (C) rational extended ther-modynamics, macroscopic nonequilibrium thermodynamics,and chemical kinetics; (D) mesoscopic irreversible thermo-dynamics and continuum mechanics with fluctuations; (E)quantum statistical mechanics, quantum thermodynamics,mesoscopic nonequilibrium quantum thermodynamics, andintrinsic quantum thermodynamics.

The present SEA unified formulation allows us to extendat once to all these frameworks the SEA concept which hasso far been considered only in the framework of quantumthermodynamics. However, a similar or at least closely relatedset of assumptions underlie the well-known GENERIC scheme[8–10], which developed independently.

In the present paper, we emphasize that, in the SEAconstruction, a key role is played by the geometrical metricwith respect to which to measure the length of a trajectory instate space. The metric tensor turns out to be directly relatedto the inverse of the Onsager’s generalized conductivitytensor. The SEA construction can be viewed as a preciselystructured implementation of the MEP principle. The formalrelation between the SEA metric tensor G and the GENERICdissipative tensor (usually denoted by M) can be establishedby means of a detailed technical analysis of the respective un-derlying mathematical landscapes. We present such discussionin a forthcoming paper, where we discuss the analogies anddifferences of the SEA and GENERIC approaches and showunder what conditions their descriptions of the dissipative partof the time evolution can be considered essentially equivalent.

The formulation discussed here constitutes a generalizationof our previous SEA construction in the quantum thermo-dynamics framework by acknowledging the need of morestructured and system-dependent metrics than the uniformFisher-Rao metric. It also constitutes a natural step towardsgeneralizing mesoscopic nonequilibrium quantum thermody-namics to the far-nonequilibrium nonlinear domain.

We conclude that in most of the existing theories ofnonequilibrium the time evolution of the local state representa-tive can be seen to actually follow in state space the path of SEAwith respect to a suitable metric connected with the generalizedconductivities. This is true in the near-equilibrium limit, wherein all frameworks it is possible to show that the traditionalassumption of linear relaxation coincides with the SEA result.Since the generalized conductivities represent, at least in thenear-equilibrium regime, the strength of the system’s reactionwhen pulled out of equilibrium, it appears that their inverse,i.e., the generalized resistivity tensor, represents the metricwith respect to which the time evolution, at least in the nearequilibrium regime, is SEA.

Far from equilibrium the resulting unified family of SEAdynamical models is a very fundamental as well as practicalstarting point because it features an intrinsic consistencywith the second law of thermodynamics which follows fromthe non-negativity of the local entropy production densityas well as the instability of the equilibrium states that do

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not have the maximum local entropy density for the givenlocal values of the densities of the conserved properties, ageneral and straightforward conclusion that holds regardlessof the details of the underlying metric tensor. In a varietyof fields of application, the present unifying approach mayprove useful in providing a new basis for effective numericaland theoretical models of irreversible, conservative relaxationtowards equilibrium from far nonequilibrium states.

ACKNOWLEDGMENTS

The author gratefully acknowledges the Cariplo–UniBS–MIT-MechE faculty exchange program cosponsored by theBrescia University (UniBS) and the CARIPLO Foundation,Italy, under Grant No. 2008-2290. This work is part of EOARD(European Office of Aerospace R&D) Grant No. FA8655-11-1-3068 and Italian Grant No. MIUR PRIN-2009-3JPM5Z-002.

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