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PHYSICAL REVIEW E 93, 012137 (2016)

Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolatedchemically reactive systems using density of states and the concept of hypoequilibrium state

Guanchen Li* and Michael R. von Spakovsky†

Center for Energy Systems Research, Mechanical Engineering Department, Virginia Tech, Blacksburg, Virginia 24061(Received 21 April 2015; revised manuscript received 10 September 2015; published 22 January 2016)

This paper presents a study of the nonequilibrium relaxation process of chemically reactive systems usingsteepest-entropy-ascent quantum thermodynamics (SEAQT). The trajectory of the chemical reaction, i.e., theaccessible intermediate states, is predicted and discussed. The prediction is made using a thermodynamic-ensemble approach, which does not require detailed information about the particle mechanics involved (e.g.,the collision of particles). Instead, modeling the kinetics and dynamics of the relaxation process is based onthe principle of steepest-entropy ascent (SEA) or maximum-entropy production, which suggests a constrainedgradient dynamics in state space. The SEAQT framework is based on general definitions for energy and entropyand at least theoretically enables the prediction of the nonequilibrium relaxation of system state at all temporaland spatial scales. However, to make this not just theoretically but computationally possible, the concept ofdensity of states is introduced to simplify the application of the relaxation model, which in effect extends theapplication of the SEAQT framework even to infinite energy eigenlevel systems. The energy eigenstructure ofthe reactive system considered here consists of an extremely large number of such levels (on the order of 10130)and yields to the quasicontinuous assumption. The principle of SEA results in a unique trajectory of systemthermodynamic state evolution in Hilbert space in the nonequilibrium realm, even far from equilibrium. Todescribe this trajectory, the concepts of subsystem hypoequilibrium state and temperature are introduced andused to characterize each system-level, nonequilibrium state. This definition of temperature is fundamental ratherthan phenomenological and is a generalization of the temperature defined at stable equilibrium. In addition, todeal with the large number of energy eigenlevels, the equation of motion is formulated on the basis of the densityof states and a set of associated degeneracies. Their significance for the nonequilibrium evolution of system stateis discussed. For the application presented, the numerical method used is described and is based on the densityof states, which is specifically developed to solve the SEAQT equation of motion. Results for different kinds ofinitial nonequilibrium conditions, i.e., those for gamma and Maxwellian distributions, are studied. The advantageof the concept of hypoequilibrium state in studying nonequilibrium trajectories is discussed.

DOI: 10.1103/PhysRevE.93.012137

I. INTRODUCTION

There are many modeling approaches in nonequilibriumthermodynamics. From a practical standpoint, models devel-oped from any one of these is typically applicable to a givenset of spatial and temporal scales so that in the vein of Grmelaand Ottinger [1,2], each model can be classified as being eithermore macroscopic or less microscopic (or vice versa). Thus,when the nonequilibrium phenomena studied cross differentspatial and temporal scales, some form of multiscale modelmust be employed. This is typically done via multiscalecomputational techniques, which are able to pass systemproperties from one scale to another [3–5]. For example,parameters or phenomenological coefficients used in a moremacroscopic model are calculated via a more microscopicmodel, and feedback is used to update them after a time (ordistance) interval that is between the characteristic time (ordistance) length of the two different models. Such multiscaleapproaches, nonetheless, present a number of significantdrawbacks, not the least of which are computational.

Of course, a general rigorous theoretical framework thatpermits the study of irreversible phenomena across spatial andtemporal scales is of great significance. Two such frameworksexist. The first developed by Grmela and Ottinger [1,2]

*[email protected]†[email protected]

provides a general equation for nonequilibrium reversible-irreversible coupling (GENERIC) able to couple models fromtwo different scales or levels, i.e., one more microscopic (alevel 1 model) and the other more macroscopic (a level 2model). In general, level 2 models are simpler and rely more onobservation and phenomenological descriptions, while thoseat level 1 are more fundamental and require a significantlyhigher level of complexity. Grmela and Ottinger suggest thata specific GENERIC equation of motion, which is based onmodels from two different levels of description, should takethe form that satisfies the compatibility of the two models. Theanalysis of this compatibility, i.e., the passing from a more to aless detailed level, involves a pattern recognition process [1,2],which corresponds to a kind of coarse graining [6,7] that resultsin what looks like dissipation even if the level 1 dynamics isreversible as in fact they are in mechanics. In this way, the twomodels and levels are linked via a single equation of motion.Clearly, this approach is more rigorous and fundamental thanthe multiscale computational technique described earlier. Itallows the study of the far-from-equilibrium realm where near-equilibrium parameters no longer hold and the link betweentwo different scale models ensures the compatibility of the twolevels in a way that simple parameter delivery cannot.

The second general, rigorous, theoretical framework isthat developed by Beretta [6,8–11], which provides a generalnonequilibrium dynamics for relaxation crossing differentscales. Unlike GENERIC, which bases its approach on

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http://dx.doi.org/10.1103/PhysRevE.93.012137

GUANCHEN LI AND MICHAEL R. VON SPAKOVSKY PHYSICAL REVIEW E 93, 012137 (2016)

the linking of models across scales into a single equationof motion, Beretta explains irreversible phenomena atdifferent scales using a single thermodynamic model anda single equation of motion, doing so on the basis of thegeneral principle of steepest-entropy ascent (SEA) [8,9,12]or equivalently maximum-entropy production (MEP) (e.g.,[13]). In this framework, the irreversible part of the equationof motion is built on a geometrical explanation of relaxation,which results in a gradient dynamics of entropy in statespace constrained by the energy, particle number, etc. Sincethermodynamic rules and the properties used (specifically,extensive thermodynamic properties such as the energy andthe entropy) are applicable across all spatial and temporalscales [14,15], the equation of motion of SEA is welldefined and rigorous at all scales. In addition, in Ref. [6],Beretta demonstrates that all of the well-known classicaland quantum nonequilibrium frameworks can be formulatedwithin his more general nonequilibrium thermodynamic SEAframework, which is applicable even far from equilibrium.

Although the GENERIC and SEA frameworks approachnonequilibrium thermodynamics from different viewpoints,Montefusco, Consonni, and Beretta [16] show that the dissi-pative components of the two theories are closely related andin some cases essentially mathematically equivalent, providedthat the choice of kinematics is the same, i.e., that bothhave a common starting point. Furthermore, both provide ageometrical foundation for their dynamics [6,17]. However,differences exist since the SEA framework is a local theory,which starts from local balance equations and implements theprinciple of maximum local entropy generation compatiblewith the local conservation constraints, while the GENERICframework is global, implementing an entropy gradient dy-namics compatible with the global conservation constraints.

To date, the SEA framework, which extends a first prin-ciples thermodynamic and ensemble representation into thenonequilibrium realm, even that far from equilibrium, hassuccessfully been applied to very microscopic systems such asthe state evolution of quantum systems (e.g., [18–23]) as wellas to a single-particle classical system [24] whose availablemicrostates are uniformly distributed in phase space. However,its application to high-dimensional state spaces and, thus, toinfinite energy eigenlevel (i.e., more macroscopic) systems hasnecessarily been limited.

To address this limitation, the GENERIC concept of pat-terns is used here. To begin with, one may view the irreversibleterm in the SEA framework as resulting from a pattern in amore microscopic model. In a manner similar to the way atstable equilibrium one can view the “Maxwellian distribution”as an invariant pattern even though the mechanical detailsof the individual particle states constantly change, steepest-entropy ascent can also be viewed as a changing global patternor effect of the details of the mechanics in relaxation. Thus,without including all of the details of the more microscopicmodel (i.e., of the mechanics), the pattern of this modelserves as a modification of the more macroscopic model(i.e., of the nonequilibrium thermodynamics). This, of course,greatly simplifies the computational complexity, while theclear physical meaning and geometrical description of theSEA landscape facilitates the discovery of general but uniquepatterns of relaxation in the nonequilibrium realm at any scale.

In view of this, two nonequilibrium relaxation patternspresent themselves, which offer two methods for practi-cally (i.e., computationally) extending the application of thesteepest-entropy-ascent quantum thermodynamics (SEAQT)framework to infinite-dimensional state spaces, even those inthe macroscopic continuous spectrum, and for extending anequilibrium-type description to nonequilibrium states. This ex-tension focuses on the irreversible process of thermodynamicmixture states in which the Hamiltonian dynamics vanishesand the state evolution is due only to thermodynamic irre-versibilities. Both methods originate from physical concepts.The discussion begins in Sec. II with a brief description of theSEAQT framework and focuses on the trajectory of systemthermodynamic state evolution and not its time evolutionsince it is the trajectory which reveals the geometric featuresof interest. The concept of hypoequilibrium state is thenintroduced, which permits temperature to be defined for allnonequilibrium states and implies that the energy eigenlevelsin mutual equilibrium evolve as a group. This is followed inSec. III by a description of our density of state method, whichallows the macroscopic-level SEAQT equation of motion tobe solved. It is based on the idea that similar eigenlevels withsimilar initial conditions evolve similarly. Finally, in Sec. IV,the thermodynamic-ensemble based approach developed andpresented in the previous sections is applied within the SEAQTframework to the study of an isolated, chemically reactive,macroscopic system undergoing a nonequilibrium evolutionin state. This approach, in fact, represents a computationallysimpler, alternative global method for predicting the chemicalkinetics of systems, even those far from equilibrium, anddoes so without the need for the detailed particle mechanics(e.g., that of particle collisions) of conventional approaches orfor such limiting assumptions as local or global equilibrium.Topics illustrated and discussed include the features of thenonequilibrium trajectories predicted, the density of states ofthe chemical reaction process, and the influence of the initialnonequilibrium states on the trajectories. Two generalizationsto the nonequilibrium realm of stable equilibrium concepts arealso physically illustrated and discussed, i.e., that of hypoe-quilibrium state and that of nonequilibrium temperature. Theformer leads to the definition of the latter and to the possibilityof representing a very large class of nonequilibrium states andof approximating an even larger class of other nonequilibriumstates in the study of nonequilibrium trajectories.

II. THEORY: STEEPEST-ENTROPY-ASCENTEQUATION OF MOTION

A. Nonequilibrium evolution framework

Based on the discussion by Grmela [1,2,17] and Beretta[6,16], the general form of a nonequilibrium framework isa combination of both irreversible relaxation and reversiblesymplectic dynamics. If written in a generalized form ofthe Ginzburg-Landau equation [1,16], the equation of motiontakes the following form:

d

dtα(t) = XH

α(t) + YHα(t), (1)

where α(t) represents the state evolution trajectory, XHα(t) and

YHα(t) are functions of the system state α(t) and represent

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STEEPEST-ENTROPY-ASCENT QUANTUM THERMODYNAMIC . . . PHYSICAL REVIEW E 93, 012137 (2016)

the reversible symplectic dynamics and irreversible relaxationprocess, respectively. In the SEAQT framework, the systemstate is represented by the density operator ρ, XH

α(t) followsthe Schrodinger equation, and YH

α(t) is derived from the SEAprinciple. Thus,

dρ

dt= 1

i�[ρ,H ] + 1

τ (ρ)D(ρ). (2)

D is the dissipation operator determined via a constrainedgradient in Hilbert space. A metric tensor must be specifiedin the derivation of this dissipation term since it describesthe geometric features of the Hilbert space [6]. τ is therelaxation time, which represents the speed of system evolutionin Hilbert space. A general discussion of the SEA formulationof dissipation, using other forms of metric and symplecticterms, is given in Ref. [6] with examples of five nonequilibriumthermodynamic frameworks. Furthermore, a version of Eq. (2)for a general quantum system is used in Ref. [21] to predictand compare with experimental results the decorrelation anddecoherence of a system in which the Schrodinger dynamicsand relaxation process are coupled. In this paper, the versionof Eq. (2) considered is for the case when the symplectic partvanishes and only the relaxation part remains. For this case, theSEA equation of motion exhibits useful mathematical features(e.g., hypoequilibrium states) that enable a clear physicalrepresentation of nonequilibrium state evolution and lead toa fast and accurate computational solution.

An example of such an evolution, i.e., that of a pure relax-ation process, is the application here of the SEAQT frameworkto the modeling of an isolated chemically reactive system. Thesystem is restricted to the class of dilute-Boltzmann-gas statesin which the particles are independently distributed [10]. Suchstates can be represented by a single-particle density operatorthat is diagonal in the basis of the single-particle eigenstates.In addition, the Hilbert space metric chosen is the Fisher-Raometric, which is uniform in different dimensions of Hilbertspace. Under these conditions, the symplectic Schrodingerterm in the equation of motion vanishes, and the study is ableto focus on the irreversible relaxation process only.

B. System and state

Sections II B and II C provide a brief introduction tothe SEA equation of motion for dilute Boltzmann gases.More details can be found in Refs. [10,25]. The system studiedhas the Hamiltonian H , and the eigenvalues and eigenvectorsof the corresponding operator H take the form

H |φk〉 = εk|φk〉, k = 1,2, . . . (3)

where k is the index for the energy eigenlevels. Degeneracyin which an energy eigenlevel can have the same eigenvaluewith different eigenvectors is allowed. Equivalently, the systemcan also be defined by a group of energy eigenlevels andeigenvectors such that

H =∑

k

εk|φk〉〈φk|. (4)

The thermodynamic state of the system can be defined viaa probability distribution {pk} among the energy eigenlevels{εk}, which accounts for the diagonal term of the density

operator. An element of {εk} can share the same value in caseof degeneracy. As an example, the stable equilibrium stateof the system has the following canonical distribution, whichprovides the maximum entropy:

pk = 1

Ze− εk

kbT , (5)

where Z is the partition function Z = ∑e− εk

kbT and theequilibrium temperature is T . Any other state, represented bya probability distribution other than the canonical distribution,is a nonequilibrium state since the entropy is not a maximum.

The probability space {pk} is the state space for the system.The statistical distance dl between pk(θ ) and pk(θ + dθ ) canbe defined by the Fisher-Rao metric such that

dl = 1

2

√√√√∑k

pk

(d ln pk

dθ

)2

dθ. (6)

More discussion on choosing a metric can be found in Ref. [6].The parameter θ is continuous and can be chosen as the time t .Now, in order to simplify the representation of the developmentof the equation of motion in the next two sections (Secs. II Band II C), the square root of the probability xi = √

pi is definedso that the probability space can be represented by x = {xi}.The statistical distance then takes the form

dl = 1

2

√√√√∑k

1

pk

(dpk

dθ

)2

dθ =√∑

k

(dxk)2, (7)

where the distance between any two distributions xa and xb isthe angle

d(xa,xb) = cos−1

(∑k

xak xb

k

)= cos−1(xa · xb). (8)

According to the discussion in Ref. [26], this statisticaldistance is equivalent to an angle in Hilbert space and hasa precise physical meaning in quantum mechanics. Thus, thestate of the system as well as the distance in state space can berepresented by x and dl2 = ∑

k(dxk)2 = dx · dx.Now, if time t is chosen as the parameter, from Eq. (7) one

can arrive at

dl

dt=

√√√√∑k

(dxk

dt

)2

, (9)

which is the speed of the motion in probability or state space.

C. Property and the equation of motion

A property of the system can be defined as a function ofstate {xk} such that

I =∑

k

x2k , (10)

E = 〈e〉 =∑

k

εkx2k , (11)

S = 〈s〉 =∑

k

−x2k ln

(x2

k

), (12)

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where 〈. . .〉 means the ensemble average. The von Neumannformula for entropy is used because, as shown in Ref. [14], ithas all the characteristics required by thermodynamics.

The gradient of a given property in state space is thenexpressed by

gI =∑

k

∂I

∂xk

ek =∑

k

2xkek, (13)

gE =∑

k

∂E

∂xk

ek =∑

k

2εkxkek, (14)

gS =∑

k

∂S

∂xk

ek =∑

k

[−2xk − 2xk ln(x2

k

)]ek, (15)

where ek is the unit vector for each dimension. Furthermore,for an isolated system, the system satisfies the conservationlaws for probability and energy, i.e.,

I =∑

k

x2k = 1, (16)

E =∑

k

εkx2k = constant (17)

and the principle of SEA upon which the equation of motionis based is defined as the system state evolving along thedirection that at any instant of time has the largest entropygradient consistent with the conservation constraints. Sincethe reversible term vanishes for dilute Boltzmann gases in anisolated system, the equation of motion for the irreversiblerelaxation process is given by

dxdt

= 1

4τ (x)gS⊥L(gI ,gE ), (18)

where τ , which is a function of system state, is the relaxationtime that describes the speed at which the state evolves in statespace in the direction of steepest-entropy ascent. L(gI ,gE) isthe manifold spanned by gI and gE , and gS⊥L(gI ,gE ), whichlies parallel to the hypersurface that conserves the probabilityand the energy, is the perpendicular component of the gradientof the entropy to the manifold L(gI ,gE). It takes the form ofa ratio of Gram determinants such that

gS⊥L(gI ,gE ) =

∣∣∣∣∣∣gS gI gE

(gS,gI ) (gI ,gI ) (gE,gI )(gS,gE) (gI ,gE) (gE,gE)

∣∣∣∣∣∣∣∣∣∣(gI ,gI ) (gE,gI )(gI ,gE) (gE,gE)

∣∣∣∣, (19)

where (. . . , . . .) denotes the scalar product of two vectors instate space. The explicit form of Eq. (18) for {xk} is, thus,

dx2k

dt= 1

τ

∣∣∣∣∣∣−x2

k ln x2k x2

k εkx2k〈s〉 1 〈e〉

〈es〉〈e〉〈e2〉

∣∣∣∣∣∣∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣(20)

and for {pk} [10]

dpk

dt= 1

τ

∣∣∣∣∣∣−pk ln pk pk εkpk

〈s〉 1 〈e〉〈es〉〈e〉〈e2〉

∣∣∣∣∣∣∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣. (21)

D. Equation of motion with degeneracy

The energy eigenlevels {εk, k = 1,2, . . .} and {|φk〉, k =1,2, . . .} can be reordered and grouped into {εij , i =1,2, . . . , j = 1, . . . ,ni} and {|φij 〉, i = 1,2, . . . , j =1, . . . ,ni}, where {εij = εi, j = 1, . . . ,ni} are degenerateenergy eigenlevels with degeneracy ni . For the equation ofmotion (21), the degenerate eigenstates for a given eigenvaluewith the same initial probability are equivalent, and have thesame occupation probabilities at all times, which means pij =pik for any j,k. For example, the initial distribution amongthese eigenlevels is proportional to the canonical distributionthat gives equal initial probabilities for each degenerateeigenstate. This is the case when these eigenlevels come fromthe same subsystem of a system in a hypoequilibrium state(see Sec. II F). Thus, without loss of generality, the degenerateenergy eigenlevels with the same initial probabilities canbe combined, and the energy eigenlevels represented by themonotonically nondecreasing energy eigenvalue series {εi}with degeneracy {ni}. The probability {pi} at energy eigenlevel{εi} is given by

pi = x2i =

ni∑j=1

x2ij , (22)

x2ij = 1

ni

pi = 1

ni

x2i . (23)

The system state can, therefore, be represented by a newvector {xi} or {pi} in a new probability space formed by theprobability distribution among the energy eigenlevels {εi}. Theequality of statistical distance in the original probability spacewith that in the new probability space is shown by the followingdevelopment:

dl2 =∑

i

ni∑j=1

(dxij )2 =∑

i

ni∑j=1

(1√ni

dxi

)2

=∑

i

(dxi)2,

(24)

d(xa,xb) = cos−1

⎛⎝∑

i

ni∑j=1

xaij x

bij

⎞⎠

= cos−1

⎛⎝∑

i

ni∑j=1

1√ni

xai

1√ni

xbi

⎞⎠

= cos−1

(∑i

xai xb

i

)= d(xa,xb). (25)

In this new vector space, system properties such as I , E, andS are expressed as

I =∑

i

x2i , (26)

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E = 〈e〉 =∑

i

εi x2i , (27)

S = 〈s〉 =∑

i

−x2i ln

(x2

i

ni

), (28)

while the equation of motion changes to the following formsfor {xj } and {pj }:

dx2j

dt= 1

τ

∣∣∣∣∣∣∣−x2

j lnx2

j

njpj εj x

2j

〈s〉 1 〈e〉〈es〉〈e〉〈e2〉

∣∣∣∣∣∣∣∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣, (29)

dpj

dt= 1

τ

∣∣∣∣∣∣−pj ln p

njpj εj pj

〈s〉 1 〈e〉〈es〉〈e〉〈e2〉

∣∣∣∣∣∣∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣. (30)

This new equation of motion for degenerate energy eigenlevelsis a simplification of the equation of motion of Sec. II C for theassumption that pij = pik for any j,k. Equations (24)–(30) areacquired by substituting Eq. (22) into Eqs. (7), (8), (10)–(12),(20), and (21). Since the discussions in the next sections arebased on Eq. (30), the tilde used to designate the probabilitiesof the new probability space are dropped for simplicity.

E. Nonequilibrium evolution trajectory:Kinetics versus dynamics

In this section, the kinetics and dynamics of nonequilibriumstate evolution are introduced. The result of this section appliesto a system with a relaxation process only (i.e., without asymplectic term in the equation of motion) but is not limitedto a Fisher-Rao metric system. The equation of motion for adegenerate system [Eq. (30)] is first simplified by defining A1,A2, and A3 such that

A1 =∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣, A2 =∣∣∣∣ 〈s〉〈e〉〈es〉〈e2〉

∣∣∣∣, A3 =∣∣∣∣ 〈s〉 1〈es〉〈e〉

∣∣∣∣.(31)

The equation of motion then becomes

dpj

dt= 1

τ

(−pj ln

pj

nj

− pj

A2

A1+ εjpj

A3

A1

). (32)

The solution of this equation is

pj = pj (t), (33)

where the time evolution of the probability can be regarded asa parametric equation with parameter t . If t is the real time,the solution of Eq. (32) provides both the trajectory in statespace and the system state at any instant of time.

In general, the relaxation time τ can be a function of systemstate such that

τ = τ [ p(t)] (34)

since for a given initial state of the system, the nonequilibriumpath of state evolution is uniquely obtained from the equation

of motion (32). This path can be used to define a new parameterτ given by

dτ = 1

τ [ p(t)]dt or τ =

∫path

1

τ [ p(t ′)]dt ′ = τ (t), (35)

where τ is called the dimensionless time. With this time, theindependent variable for the equation of motion can be changedso that

dpj

dτ= −pj ln

pj

nj

− pj

A2

A1+ εjpj

A3

A1. (36)

The solution for this equation is written as

pj = pj (τ ). (37)

No matter how the relaxation time τ depends on the realtime t and the state, the equation of motion can always betransformed into Eq. (36) with the parameter change definedby Eq. (35). Furthermore, the evolution of system state followsthe same function [Eq. (37)] in τ . Physically, this meansthat the system follows the same trajectory in state space,while τ decides the speed with which the system’s statemoves along the trajectory. If the relaxation time is chosento be a constant, Eq. (33) gives the same parametric equation[Eq. (37)] with a parametric scaling in the relaxation time τ .Thus, the kinetics and dynamics of the system are separated.The former are found via Eqs. (36) and (37) and result in thetrajectory in state space based on the parameter τ or a constantrelaxation time τ . The dynamics is found via Eq. (32) andthe functional dependence τ = τ ( p) [Eq. (34)] and result inthe trajectory in state space based on the real time t . Recentnumerical results show that a uniform (Fisher-Rao) metricmay for some systems provide poor performance relative tothe time evolution [24] at least in the near-equilibrium limit.In such cases, the time evolution needs more information onthe dynamics or the function τ . For this reason, the resultspresented here are restricted to the kinetic evolution trajectoryand the intermediate states of the relaxation process, and, thus,to the purely geometrical features of the relaxation process.

F. Description of the trajectory: Subsystem hypoequilibriumstate and temperature

In this section, the concepts of subsystem hypoequilibriumstate and temperature for a system in a nonequilibrium state aredefined. These two concepts support the physical description ofthe evolution trajectory in state space rather than just its math-ematical description. However, this description is restrictedto systems with irreversible relaxations only. These conceptsoriginate from a generalization of the canonical distribution toa nonequilibrium distribution in Hilbert space with a uniformmetric (Fisher-Rao metric). However, such a generalizationunder any metric is an open question. Furthermore, to begeneral, the relaxation time in this section can be constant ora function of system state, which means that the conclusionsdrawn apply to both the kinetic and dynamic characteristics ofthe state space trajectory.

For a given system represented by an energy eigenlevelset = {εk}, the system can be divided into M subsystems(i.e., subspaces in state space) i = {εik}, = ∪ i , and i ∩j = ∅ for any i,j = 1, . . . ,M . If the probability distribution

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in each subsystem yields to a canonical distribution, the systemis designated as being in an Mth-order hypoequilibrium state.Based on this definition, any state of the system (strictlyspeaking, a diagonal density operator in an eigenstate basis) isa hypoequilibrium state with order M , where M is less than orequal to the number of system eigenlevels. A hypoequilibriumstate of order 1 corresponds to a state in stable equilibrium.The probability distribution of the Mth-order hypoequilibriumstate takes the form

∀ i = 1,2, . . . ,M, pik = αinike−βiεik , k = 1,2, . . . ,wi

(38)where αi and βi are parameters, and nik is the degeneracy ofεik . To be complete, βi = 0 if wi = 1, and wi can be infinite.

αiZi(βi) has the physical meaning of particle number whereZi is the partition function of the subsystem. The inverse ofthe temperature of each subsystem i is βi with a scale ofthe Boltzmann constant. This temperature is defined for eachsubsystem when the system is in a state of nonequilibrium. Itis proven in the following that if a system begins in an Mth-order hypoequilibrium state, it will remain in an Mth-orderhypoequilibrium state throughout the time evolution as willthe subsystem division. To show this, Eq. (32) is reformulatedsuch that

d

dtln

pj

nj

= 1

τ

(− ln

pj

nj

− A2

A1+ εj

A3

A1

), (39)

where it is noted that d(ln nj )/dt is zero and that A1, A2, andA3 are the same for all chosen energy eigenlevels pj and only afunction of the entire probability distribution at a given instantof time. Subtracting the equations of motion for the ith andkth energy eigenlevels results in

d

dt

(ln

pj

nj

− lnpk

nk

)= − 1

τ

(ln

pj

nj

− lnpk

nk

)

+ 1

τ

A3

A1(εj − εk). (40)

Defining a new variable

Wjk = 1

εj − εk

(ln

pj

nj

− lnpk

nk

), (41)

the time evolution of Wjk yields to the ordinary differentialequation

dx

dt= − 1

τx + 1

τ

A3

A1. (42)

If pj and pk are in the same subsystem for which the initialprobability distribution is a canonical one, i.e., if

pj (t = 0) = αpnje−εj βp , pk(t = 0) = αpnke

−εkβp , (43)

then

Wjk(t = 0) = 1

εj − εk

(ln

pj

nj

− lnpk

nk

)= −βp. (44)

For ∀ pj ,pk in the same subsystem p, the time evolution ofWjk yields to the same ordinary differential equation (ODE)with the same initial value, namely,

dx

dt= − 1

τx + 1

τ

A3

A1, x = Wjk(t = 0) = −βp (45)

so that the solution of Wjk is the same Wjk(t) = βp(t). There-fore, the probability distribution in this subsystem maintainsthe canonical distribution with the parameter βp(t) given by

pj (t) = αp(t)nje−εj βp(t). (46)

In addition, the temperature of the subsystem at time t isdefined by

Tp(t) = 1

kbβp(t). (47)

Thus, for a system in a nonequilibrium state, the hypoe-quilibrium temperature for each subsystem is defined. Thistemperature can be the same or different from that of any othersubsystem. If a system is in an Mth-order hypoequilibriumstate, it remains at least of order M throughout as wellas after the evolution, and the probability distribution ofeach subsystem remains canonical. More discussion on theevolution of temperature is given in Sec. V. In addition, theODE for βp [Eq. (45)] is independent of αp so that differentsubsystems with the same initial βp also keep the same βp(t) inthe evolution. This phenomenon is consistent with the idea thatenergy via a heat interaction cannot transfer between systemswith the same temperature to produce a temperature difference.Equal temperature subsystems maintain equal temperatures.

Moreover, if two subsystems A and B are in mutualequilibrium at time tme [αp(tme)A = αp(tme)B , βp(tme)A =βp(tme)B], the combination of these two subsystems yields asubsystem with a canonical distribution. This new subsystemmaintains a canonical distribution throughout its state evolu-tion, which means that the two original subsystems maintaintheir hypoequilibrium states as well as a state of mutualequilibrium with each other.

The results just demonstrated are summarized as follows:(i) the manner of subdividing the system is invariant withrespect to the irreversible relaxation trajectory; (ii) the proba-bility distribution in each subsystem remains canonical alongthe trajectory so that temperature can be defined based on aparameter of the canonical distribution; and (iii) equal temper-ature subsystems maintain equal temperatures, and subsystemsin mutual equilibrium remain in mutual equilibrium. With theconcept of subsystem hypoequilibrium state, the trajectoryfor system state evolution in state space is described by twofunctions αp(t) and βp(t). Physically, each instantaneous valueof αp(t)Zp[βp(t)] is the total probability (particle number) ofa given subsystem, while each instantaneous value of βp(t) isthe parameter of the canonical distribution and the inverse ofthe subsystem temperature. Zp[β(t)] is the partition function,which is a function of β. Thus, αp(t) describes the probabilitytransfer between subsystems, while βp(t) describes the tem-perature evolution and heat diffusion between subsystems. Nolonger is the canonical distribution simply a characteristic ofstable equilibrium but instead a characteristic of subsystemhypoequilibrium and maximum-entropy generation as well.This feature provides a convenient pattern for studying theevolution of a system’s state distribution during an irreversiblerelaxation process.

For a complete discussion on subspace temperature anda set of general nonequilibrium intensive properties, thereader is referred to [27–29]. Some conclusions from these

012137-6


references are given in Appendix B, including a clear physicalpicture, based on the concepts of hypoequilibrium state andnonequilibrium intensive property, of the generalization ofthe Onsager relations and the dissipation potential to thenonequilibrium realm. In addition, for a discussion of the rela-tionship of the general Onsager framework of nonequilibriumthermodynamics [30] to SEAQT and GENERIC, the reader isreferred to [6,25] for the SEAQT framework and to [31] forthe GENERIC.

III. MODEL: DENSITY OF STATE METHOD

In this section, the numerical method for solving theSEAQT equation of motion is introduced. Called the densityof state method, its purpose is to solve this equation for anextremely large number of energy eigenlevels, a number, infact, so large that even the state evolution of macroscopicsystems can be determined. This numerical method followsthe same idea as the derivation of the equation of motionfor a degenerate system, namely, that similar eigenlevels withsimilar initial states evolve similarly. An example of a systemwith a very large number of levels is that of an ideal gas ata temperature higher than the characteristic temperature fortranslation. A practical illustration is given in Sec. IV.

A. Probability cut

For an isolated system with unbounded energy, the proba-bility distribution is assumed to be limited to a bounded rangeof energy eigenlevels. This means that in the limK→∞ p(ε >

εK ) = 0, and the energy eigenlevels in the bounded rangecan be used to approximate the unbounded range of thesystem. For a given system {εi, i = 1,2, . . .} and initialstate {pi, i = 1,2, . . .}, K is chosen such that

∑i>K pK < δ

resulting in εcut = εK and the set of system bounded energyeigenvalues εi < εcut.

B. Evolution of probability in energy intervals

For a system with bounded energy eigenvalues, a subset ofthe energy eigenlevels can be chosen to be a subsystem or anentire system. The range of energy eigenlevels of this subsetis then separated into finite intervals such that

ei = εground + i

R(εcut − εground),

Interval: [ei−1,ei], i = 1, . . . ,R

Interval length: �E = ei − ei−1, i = 1, . . . ,R (48)

where R is the number of intervals. Integrating the equation ofmotion (32) over the ith interval [ei−1,ei] yields the equationof motion for the probability in the ith interval, namely,

dPi

dt= 1

τ

(〈s〉i − Pi

A2

A1+ 〈e〉i A3

A1

), (49)

where Pi = ∑εk∈[ei−1,ei ] pk is the probability distributed

over the energy levels in the interval [ei,ei−1], 〈s〉i =−∑

εk∈[ei−1,ei ] pk ln(pk/nk) is the contribution of this energyinterval to the total entropy, and 〈e〉i = ∑

εk∈[ei−1,ei ] εkpk isthe contribution of this energy interval to the total energy. Bysumming the property of every energy interval, the property

of the whole system (e.g., the total entropy and total energy)can be determined.

C. Energy spectrum agglomeration

For a given initial state, the energy eigenlevels of thesystem can be separated into M groups. Each group formsa subsystem, and the probability distribution of each sub-system is assumed to be a gamma distribution or its linearcombination, which can cover quite a large range of initialconditions. A subsystem K is chosen, and its range of energyeigenvalues divided into R intervals. It is assumed that there aremi energy eigenvalues {εi

j , j = 1, . . . ,mi} with degeneracies{ni

j , j = 1, . . . ,mi} in the interval [ei−1,ei] of subsystem K

where mi can be infinity.A pseudosubsystem with energy eigenvalues {Ei, i =

1, . . . ,R} and degeneracies {Ni, i = 1, . . . ,R} is then con-structed such that

Ni =mi∑

j=1

nij , (50)

Ei = 1

Ni

mi∑j=1

nij ε

ij . (51)

For any distribution, Pi is the distribution of the ith interval ofthe Kth subsystem expressed by

Pi =mi∑

j=1

pij . (52)

If the subsystem has a distribution given by

pij = Cni

j

(εij

)θe−εi

j β , (53)

where C is a constant, nij is the degeneracy of energy eigenlevel

εij . The pseudosubsystem distribution at the same temperature

is expressed as

Pi = CNi(Ei)θ e−Eiβ, (54)

where C is the same as in Eq. (53). In Appendix A, Pi isproven to be equal to Pi for most energy intervals under thequasicontinuous condition expressed as

1

β� |Ei+1 − Ei | >

∣∣εij − Ei

∣∣. (55)

Note that Pi becomes a probability distribution with normal-ization condition when the system as a whole is considered. Itis assumed that the quasicontinuous condition holds both forthe pseudosubsystem as well as the original subsystem. Now,for the original subsystem,

〈e〉i =∑

j

pij ε

ij , (56)

〈s〉i = −∑

j

pij ln

pij

nij

, (57)

〈e〉K =∑

i

∑j

pij ε

ij , (58)

012137-7


〈s〉K = −∑

i

∑j

pij ln

pij

nij

, (59)

〈es〉K = −∑

i

∑j

pij ε

ij ln

pij

nij

, (60)

〈e2〉K =∑

i

∑j

pij

(εij

)2, (61)

while for the pseudosubsystem,

〈e〉i = EiPi, (62)

〈s〉i = −Pi lnPi

Ni

, (63)

〈e〉K =∑

i

PiEi, (64)

〈s〉K = −∑

i

Pi lnPi

Ni

, (65)

〈es〉K = −∑

i

PiEi lnPi

Ni

, (66)

〈e2〉K =∑

i

PiE2i . (67)

Here, 〈. . .〉i stands for the contribution of the ith intervalof the pseudosubsystem or original subsystem to the systemproperty, while 〈. . .〉K stands for the contribution of the entireKth subsystem to the system property. The system propertiesare then found as is done in Appendix A by summing thesubsystem properties over the index K . In Appendix A,it is proven that under the quasicontinuous condition, apseudosubsystem property in a given energy interval is in mostcases equal to that of the original subsystem, and the associatedsubsystem and system property as well as A1, A2, A3, A1, A2,and A3 are equal to those of the original subsystem. The latterare written as

A1 =∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣, A2 =∣∣∣∣ 〈s〉〈e〉〈es〉〈e2〉

∣∣∣∣, A3 =∣∣∣∣ 〈s〉 1〈es〉〈e〉

∣∣∣∣,(68)

A1 =∣∣∣∣ 1 〈e〉〈e〉〈e2〉

∣∣∣∣, A2 =∣∣∣∣ 〈s〉〈e〉〈es〉〈e2〉

∣∣∣∣, A3 =∣∣∣∣ 〈s〉 1〈es〉〈e〉

∣∣∣∣.(69)

The combination of M pseudosubsystems can then be usedas a numerical approximation of the original system for eachenergy interval [Eq. (49)], i.e.,

Original system:dPi

dt= 1

τ

(〈s〉i − Pi

A2

A1+ 〈e〉i A3

A1

),

Pi(t = 0) = C∑

j

nij e

−εij β , (70)

Pseudosystem:dPi

dt= 1

τ

(〈s〉i − Pi

A2

A1+ 〈e〉i A3

A1

),

Pi(t = 0) = CNie−Eiβ . (71)

Using the time evolution of the pseudosystem, the propertyevolution of the system, such as that for temperature andentropy, can be determined.

IV. APPLICATION TO AN ISOLATED CHEMICALLYREACTIVE IDEAL GAS MIXTURE

A. System definition and state representation

1. Microstate

The state space of a chemically reactive system is composedof two subspaces, reactant and product. The energy eigenlevelsof the reactants and those of the products together form theenergy eigenlevels for the system as a whole. Denoting thestate space of the reactants by Hreactant and that of the productsby Hproduct, the system state space H takes the form

H = Hreactant ⊕ Hproduct. (72)

The simple yet well-studied chemical reaction mechanismconsidered here is

F + H2 ⇔ FH + H (73)

since it is well adapted to illustrating our general approach. Re-action mechanisms in the vein of the general Guldberg-Waagechemical relaxation equation are considered in Ref. [32]where the SEAQT framework is used to model coupledreaction mechanisms while in Refs. [33,34] complex, coupledreaction-diffusion pathways are used to predict the effects ofmicrostructural degradation and chromium oxide poisoningon the performance of a solid oxide fuel cell cathode.Predictions made are compared with experimental data. Astudy of chemical reaction rate is also given in Ref. [18].In addition, Grmela provides a study using GENERIC [35]of the entropy production that occurs in chemically reactivesystems represented by the general Guldberg-Waage chemicalrelaxation equation.

Now, the available energy eigenvalues for one subspace (re-actant or product) are constructed from the energy eigenvaluesof each degree of freedom, i.e.,

εreactant = εt,H2 + εr,H2 + εv,H2 + εt,F , (74)

εproduct = εt,FH + εr,FH + εv,FH + εt,H . (75)

The translational energy eigenvalue εt uses the form of theinfinite potential well, the rotational energy eigenvalue εr theform of the rigid motor, and the vibrational energy eigenvalueεv the form of the harmonic oscillator, i.e.,

εt (nx,ny,nz) = �2

8m

[(nx

Lx

)2

+(

ny

Ly

)2

+(

nz

Lz

)2],

(76)

εr (j,m) = j (j + 1)�2

2I= j (j + 1)�2

2μr2, (77)

εv(ν) =(

ν + 1

2

)�ω + Ed, (78)

where nx , ny , and nz are the quantum numbers for thetranslational degrees of freedom; j and m are the quantum

012137-8


numbers for the rotational degrees of freedom; and ν is thequantum number for the vibrational degrees of freedom. εd isthe disassociation energy of a given molecule (e.g., H2 andFH ). Each combination of quantum numbers correspondsto one energy eigenlevel of reactant or product withoutdegeneracy (provided the rotational state is distinguishedby the magnetic quantum number m). The system energyeigenlevels are formed by all the available energy eigenlevelsof reactants and products. The separation of the two subspaces{εreactant} and {εproduct} is kept to maintain their clear physicalmeaning. In the modeling, these two subspaces serve as twosubsystems in the discussion of system nonequilibrium state.

2. Density of state

For temperatures higher than the characteristic temper-atures of translation and rotation, the energy eigenlevels

distribute from the ground state to infinity with high density.These spectrums can be approximated to be continuous. Thedensities of states of translational and rotational energy forspecies A are given by

Dt,A(ε)dε = 2πV

h3(2mA)

32 ε

12 dε, (79)

Dr,A(ε)dε = 2IA

�2dε. (80)

In contrast, that for vibration is assumed to be discrete. Todescribe the available energy eigenlevels, density of statesinstead of individual energy eigenlevels are used because ofthe extremely large number of the latter. Similar to the way ofdealing with individual energy eigenlevels, the density of statesis calculated separately for reactants and products. The jointdensity of states from the densities of states of M independentenergy forms can be calculated from

D(E)dE =∫

∑Mi=1 ei=E

D1(e1)D2(e2) . . . DM (eM )de1de2 . . . deM

= dE

∫ E

Eg

1

de1

∫ E−e1

Eg

2

de2 . . .

∫ E−∑M−2i=1

Eg

M

deM−1D1(e1)D2(e2) . . . DM−1(eM−1)DM

(E −

M−1∑i=1

ei

), (81)

where Eg

i is the ground state for the ith density of states.Specifically, the density of states for one subsystem of the system studied can be calculated by

Dreac(E)dE =∫

εt,H2 +εr,H2 +εv,H2 +εt,F =E

Dt,H2 (εt,H2 )Dr,H2 (εr,H2 )Dv,H2 (εv,H2 )Dt,F (εt,F )dεt,H2dεr,H2dεv,H2dεt,F , (82)

Dprod(E)dE =∫

εt,FH +εr,FH +εv,FH +εt,H =E

Dt,FH (εt,FH )Dr,FH (εr,FH )Dv,FH (εv,FH )Dt,H (εt,H )dεt,FH dεr,FH dεv,FH dεt,H . (83)

The joint distribution of the density of states of translational and rotational energy eigenlevels takes the form

Dreact,r (E)dE =

∫εt,H2 +εr,H2 +εt,F =E

Dt,H2 (εt,H2 )Dr,H2 (εr,H2 )Dt,F (εt,F )dεt,H2dεr,H2dεt,F

= 2πV

h3

(2mH2

) 32

2πV

h3(2mF )

32

2IH2

�2

1

3B

(3

2,3

2

)dε, (84)

Dprodt,r (E)dE =

∫εt,FH +εr,FH +εt,H =E

Dt,FH (εt,FH )Dr,FH (εr,FH )Dt,H (εt,H )dεt,FH dεr,FH dεt,H

= 2πV

h3(2mFH )

32

2πV

h3(2mH )

32

2IFH

�2

1

3B

(3

2,3

2

)dε, (85)

where B(. . . , . . .) is the beta function. Generally, the energyeigenlevels for a subspace are constructed from t translationaland r rotational degrees of freedom, and the density of statesfor the subsystem built from these eigenlevels takes the form

Dt,r (E)dE ∝ E12 t+r−1dE. (86)

For temperatures not much greater than or less than or equalto the characteristic temperature of the vibrational degrees offreedom, the form of the density of state for each subsystemis, respectively,

Dreac(E − εd,H2 )dE = Dreact,r (E)dE, (87)

Dprod(E − εd,FH )dE = Dreact,r (E)dE, (88)

where εd,H2 and εd,FH are the ground energy of vibrationaldegrees of freedom of H2 and FH , which are approximatedby the disassociation energies of H2 and FH .

For temperatures much greater than the characteristictemperature of the vibrational degrees of freedom, the formof the density of state for each subsystem is, respectively,

Dreac(E)dE = dE∑

ν,εν,H2 <E

Dreact,r (E − εν,H2 ), (89)

Dprod(E)dE = dE∑

ν,εν,FH <E

Dprodt,r (E − εν,FH ). (90)

Finally, the energy eigenlevel information for reactant({εreactant

i } and {nreactanti }) and product ({εproduct

j } and {nproductj }) is

012137-9


obtained. The system has energy eigenlevels {εreactanti ,ε

productj }

and density of states {nreactanti ,n

productj }.

B. Numerical process

1. Cutoff energy

With the density of states for each subspace, the cutoffenergy for the system at a fixed energy corresponding toan initial state or that of stable equilibrium can easily becalculated. Given the density of states n(E) of a subsystemand a type of probability distribution with parameter θ whichtakes the form that reduces to the canonical distribution whenθ = 0, i.e.,

p(E) ∝ D(E)Eθe−βE, (91)

the cumulative distribution function becomes

F (E) ∝∫ E

Eground

D(E′)Eθe−βE′dE′. (92)

The cutoff energy is then the inverse of the cumulativedistribution function F where δ is a very small number, namely,

Ecut = F−1(1 − δ). (93)

As an example, take a system consisting of an ideal gasmixture that has a temperature lower than the characteristictemperature of vibration but higher than that of translation androtation. This is indicative of the conditions for a very largenumber of ideal gas applications (102 K ∼ 103 K). The energyeigenlevels for a subspace are constructed from t translationaland r rotational degrees of freedom, and the density of statesfor the subsystem built from these eigenlevels takes the form

Dt,r (E) = C0E12 t+r−1, (94)

where the procedure for determining C0 is given in previoussection [e.g., see Eqs. (84) and (85)]. If the probabilitydistribution in the subsystem is a gamma function withparameters β and θ , then

p(E) ∝ Dt,r (E)Eθe−βE ∝ Eα+θ−1e−βE ∝ �(α + θ,β).(95)

The distribution among the subsystem energy eigenlevelsyields to the gamma distribution �(α + θ,β) with the follow-ing parameter definitions:

α = 1

2t + r, β = 1

kbT. (96)

The distribution and cumulative distribution functions thentake the form

p(E) = Eα+θ−1e−βE∫ ∞0 E′α+θ−1e−βE′

dE′ = βα+θ

�(α + θ )Eα+θ−1e−βE,

(97)

F (E) =∫ E

≈0 E′α+θ−1e−βE′dE′∫ ∞

0 E′α+θ−1e−βE′dE′ = γ (α + θ,βE)

�(α + θ ), (98)

where �(α + θ ) is the gamma function evaluated at α + θ , andγ (α + θ,βE) is the lower incomplete gamma function. Theenergy of the ground state is approximately zero. The cutoff

energy is found from the inverse of the cumulative distributionfunction such that

Ecut = F−1(1 − δ; α + θ,β). (99)

2. Pseudosystem

Since there are two subspaces for the chemically reactivesystem considered here, i.e., one for reactants and the otherfor products, two cutoff energies are calculated; and the largerone is used to truncate the open interval of infinite energyeigenlevels into a closed one. To establish the pseudosystem,an energy interval �E is chosen according to the quasicon-tinuous condition (55). Two pseudosubsystems are then setup, one for the reactants and the other for the products. Thepseudosystem for the composite system is the combinationof the two pseudosubsystems. The energy eigenvalue anddegeneracy for one pseudosubsystem corresponding to theith interval are then determined by integrating Eqs. (50) and(51) over that interval where the summations are replaced byintegrals using the density of states D(E) so that

Ni =∫ ei+�E

ei

D(E)dE, (100)

Ei = 1

Ni

∫ ei+�E

ei

D(E)E dE. (101)

Using Eq. (86) and considering only translational and ro-tational degrees of freedom, the energy eigenvalue anddegeneracy for the ith energy interval is

Ni = C0

∫ ei+�E

ei

Eα−1dE = C0

α

[(ei + �E)α − eα

i

], (102)

Ei = C0

Ni

∫ ei+�E

ei

EαdE = α

α + 1

(ei + �E)α+1 − eα+1i

(ei + �E)α − eαi

,

(103)

where C0 and α are defined by Eqs. (94) and (96), respectively,and calculated via Eqs. (82) and (83). Once the pseudosystemhas been defined in this way, the evolution of the isolatedchemically reactive system can be determined.

C. Initial condition

We provide the analytical solution for an initial conditionwhen the vibrational energy is frozen.

1. Initial condition 1: Second-order hypoequilibrium

For a second-order hypoequilibrium state, it is assumedthat the reactant energy eigenlevels and product energyeigenlevels form subspaces, respectively. In either subspace,the distribution is proportional to the canonical distribution.Using Eqs. (84), (85), (87), and (88),

preac(E − εd,H2 )dE = preact,r (E)dE ∝ Dreac

t,r (E)e−βEdE

∝ E3e−βEdE, (104)

pprod(E − εd,FH )dE = pprodt,r (E)dE ∝ D

prodt,r (E)e−βEdE

∝ E3e−βEdE. (105)

012137-10


Both distributions are �(4,β) gamma distributions and bothnormalization constants are β4/�(4), where �(4) is calculatedfrom the gamma function.

2. Initial condition 2: The gamma distribution

If the initial condition for one subspace (reactant or product)takes the general form of a gamma distribution, then, forexample, the reactant subspace distribution is given by

preac(E − εd,H2 )dE ∝ E3+θ e−βEdE (106)

since the product has little probability initially. Equation (106)is the gamma distribution �(4 + θ,1/β). The mean andvariance of E take the form

Mean = 4 + θ

β, (107)

Var = 4 + θ

β2. (108)

By varying θ , the influence of the initial condition can bestudied. For instance, the effusion process can result in agamma distribution if the probability distribution of one ofthe reactant’s (e.g., F ) energy eigenlevels is

pF (E)dE = pFt (E)dE = β2Ee−βEdE. (109)

This is the energy distribution of an effusion particle, whichcan be acquired from the velocity distribution

F (v)dv ∝ vfM (v)dv ∝ v × v2e− mv2

2kbT dv, (110)

where fM is the Maxwellian velocity distribution. The otherreactant H2 has the Maxwellian probability distribution of itsenergy eigenlevels given by

pH2t,r (E)dE = D

H2t,r (E)e−βE

ZH2t,r (β)

dE = β5/2

�(5/2)E3/2e−βEdE,

(111)

where DH2t,r is the density of state for H2 with vibrational

degrees of freedom frozen, and ZH2t,r is the partition function.

This results in a gamma distribution for θ = 0.5 in Eq. (106).The initial reactant state is then

preac(E − εd,H2 )dE

= preact,r (E) =

∫ E

0pF (E − e1)pH2 (e1)de1

= β9/2

�(9/2)E7/2e−βEdE. (112)

V. RESULTS AND DISCUSSION

A. Second-order hypoequilibrium initial condition

The fundamental theoretical result for the SEA equation ofmotion is composed of two parts: the nonequilibrium kinetictrajectory in state space and the subsystem temperatures. SEApredicts a unique trajectory for nonequilibrium state evolution(Sec. II E), while the definition of subsystem and subsys-tem temperature (Sec. II F) provides a good framework forstudying the trajectory. Unlike phenomenological definitionsof nonequilibrium temperature, the subsystem temperature

defined here is fundamental and has a clear physical picture:the temperature is defined via the canonical distribution.Furthermore, the principle of SEA leads to a set of verywell-defined physical and mathematical features for subsystemtemperature, which allow subsystem hypoequilibrium stateand temperature to be reasonable generalizations of the exist-ing concepts of state and temperature at stable equilibrium. Inorder to illustrate the above ideas, the nonequilibrium behaviorof the isolated chemically reactive system introduced earlieris modeled using the SEAQT framework. A discussion ofresults for the state evolution trajectory in state space andon an energy-entropy (E-S) diagram is given first followed bya discussion of results for the total probability and temperatureof the subsystems.

The initial state of the system is chosen to be a second-order hypoequilibrium state, and the two subsystems are thereactant and product subspaces. The temperatures of the twosubsystems for the initial nonequilibrium state are selected tobe 300 K and 400 K, respectively. Although it would be morereasonable to choose the same temperature, the temperatureschosen here are to better illustrate subsystem temperatureevolution. The total probability in the reactant subsystemis 0.9999, and that in the product subsystem 0.0001. Thecutoff energy is 5.76 × 10−19 J, and the energy interval is8.84 × 10−23 J. As a comparison, the real energy eigenlevelinterval is 2.62 × 10−43 J, and the interval calculated withthe initial temperature is 1/β(t = 0) = 6.9 × 10−21 J. Thus,the quasicontinuous condition holds. There are 14 127 energyeigenlevels in the reactant subspace (or subsystem) and 16 666energy eigenlevels in the product subspace (or subsystem).At the initial state, the system has more than a 0.999 999probability of being distributed in the 3500 energy eigenlevelsbelow the cutoff energy of 5.76 × 10−19 J. Thus, a total of30 793 levels are used to represent with great accuracy (asdemonstrated below) the estimated 10130 levels of the actualsystem. The relaxation time is chosen to be 1 so that the kineticsof state evolution is studied using dimensionless time.

Figure 1(a) shows the system state evolution trajectoryon an energy-entropy diagram [36]. For any state of thesystem (nonequilibrium or equilibrium), the system state canbe mapped to one point on the energy-entropy diagram. Thebold solid concave curve is the stable equilibrium curve, andany point on the curve represents a stable equilibrium state.Because the stable equilibrium state has the maximum entropyat a given energy, composition, and volume, there is no systemstate available to the right of the stable equilibrium curve. PointA0 is the initial state in the chemically reactive model. If thesystem state evolution trajectory is mapped from state spaceto the energy-entropy diagram, the trajectory is a horizontalline (red line) at a given energy. Points A1 to A4 are fourintermediate nonequilibrium states on the evolution trajectory,and point B on the concave curve is the stable equilibriumstate.

Figure 1(b) shows the changes in the entropy and entropygeneration rate values as the system state evolves alongthe trajectory. The entropy generation rate is not constantalong the dimensionless time axis. According to SEA, theentropy generation rate is proportional to gS⊥L(gI ,gE ), whichlies parallel to the constant probability and constant energyhypersurface and is the perpendicular component of the

012137-11


1.6 1.7 1.8 1.9 2 2.1 2.2

x 10−21

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5x 10

−19(a)

Entropy (J/K)

En

erg

y (J

)

A0 A

1 A

2 A

3 A

4 B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12x 10

−21(b)

Dimensionless time

En

tro

py

(J/K

)

0 5 10 15−1

0

1

2

3

4

5

6

7x 10

−23

En

tro

py

gen

erat

ion

rat

e (J

K−1

no

n−d

im t

ime

un

it−1

)

FIG. 1. (a) System state evolution trajectory on an energy-entropydiagram; (b) entropy (black line) and entropy generation rate[green (light gray) line] evolutions as a function of dimensionlesstime.

entropy gradient to the manifold L(gI ,gE). Thus, the entropygeneration rate reveals the entropy gradient changes along thetrajectory in state space.

Figure 2(a) shows the probability distribution among theenergy eigenlevels when the system is at states A0 to A4 andB in Fig. 1(a). In order to better show each distribution’sevolution, each curve is normalized by its peak and the resultshown in Fig. 2(b). In Fig. 2(a), one can observe that thesystem probability transfers from the reactant subsystem tothe product subsystem. At the same time, the distributions ofboth subsystems evolve from narrow to wider distributions,indicating that the temperature is increasing. The distributionsremain canonical at all times. The distribution evolution foreach subsystem is more clearly seen in Fig. 2(b).

As discussed in Sec. II F, the nonequilibrium evolution ofstate can be described via the evolutions of αp(t)Zp[βp(t)]and βp(t), which correspond to the particle number Np(t) andtemperature Tp(t) = 1/kbβp(t). Zp is the partition function ofthe subsystem. Given the evolution of the particle number and

−10 −8 −6 −4 −2 0 2 4 6

x 10−19

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3(a)

Energy (J)

Pro

bab

ility

den

sity

fu

nct

ion

H2,FFH,H

−8 −7 −6 −5 −4 −3 −2 −1 0

x 10−19

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(b)

Energy (J)

No

rmal

ized

Pro

bab

ility

den

sity

fu

nct

ion

H2,FFH,H

FIG. 2. System trajectory in state space represented by (a) theevolution of the probability distribution among the energy eigenlevelsand by (b) the evolution of the normalized distributions. The fourdashed lines correspond to four of the nonequilibrium states alongthe trajectory (states A1 to A4 in Fig. 1), while the single narrow solidline is the distribution for the initial state (A0) and the single boldsolid line is that for the equilibrium state (B).

temperature, one can rebuild the probability distribution viaEq. (38). Figure 3(a) shows the particle number evolution forthe process in which reactant changes to product. Figure 3(b)shows the temperature evolution. The energy released by thechemical reaction heats up both the reactant and product, andthe manner in which the temperature increases follows theSEA trajectory. At the end, when the system reaches stableequilibrium, the temperatures of the subsystems are equal, andthe nonequilibrium temperatures converge to that for stableequilibrium. As already indicated in the discussion abovesurrounding Fig. 2(b), the physical meaning of this temperatureincrease is a broadening in the probability distribution amongthe energy eigenlevels.

012137-12


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(a)

Dimensionless time

Nu

mb

er o

f P

arti

cle

0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Rea

ctio

n r

ate

(no

n−d

im t

ime

un

it−1

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

500

1000

1500

2000

2500

3000

3500

4000(b)

Dimensionless time

Tem

per

atu

re (

K)

H2,FFH,H

FIG. 3. (a) Evolutions of particle number and reaction rate and(b) the evolution of temperature. For the line colors in (a), theblack line and the green (light gray) line represent the FH,H andH2,F , respectively, and the red (dark gray) line represents reactionrate.

B. Influence of the density of states

Using a comparison of system state evolution with andwithout vibrational energy eigenlevels included in the systemdescription, one can study how the density of states influencessystem behavior. Figure 4(a) shows the comparison of thecorresponding stable equilibrium curves and the evolutiontrajectory using the same initial subsystem temperatures.Because the stable equilibrium temperature is less than 4000 K[Fig. 3(b)] and the characteristic temperature of vibrationis about 6000 K, the stable equilibrium curves show somedifference, which is consistent with the result found fromequilibrium thermodynamics. Figure 4(b), in contrast, showsthe difference of the two systems in the nonequilibrium region.At the beginning, the two systems perform the same. As theentropy generation rate approaches the maximum, the twosystems perform differently and the system with vibrationalenergy eigenlevels exhibits a larger entropy and entropy gener-ation rate. This phenomenon can be explained using subsystemtemperature. The initial temperature is about one order ofmagnitude less than the characteristic temperature of vibration,

1.6 1.7 1.8 1.9 2 2.1 2.2

x 10−21

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5x 10

−19(a)

Entropy (J/K)

En

erg

y (J

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12x 10

−21(b)

Dimensionless time

En

tro

py

(J/K

)

0 5 10 15−1

0

1

2

3

4

5

6

7x 10

−23

En

tro

py

gen

erat

ion

rat

e (J

K−1

no

n−d

im t

ime

un

it−1

)

FIG. 4. Comparison of (a) the energy-entropy diagrams and (b)the entropy (black line) and entropy generation rate [green (lightgray) line] evolutions for a system with and without vibrationaldegrees of freedom. The solid lines are for the system with vibrationalenergy eigenlevels, while the dashed lines are for the system withoutvibrational levels.

which leads to the vibrational energy eigenlevels being frozen.Thus, the two systems perform similarly. However, as eachsubsystem temperature increases [Fig. 3(b)], the vibrationalenergy eigenlevels are activated, resulting in a difference inperformance between the two systems.

Figure 5(a) shows the difference in the system stableequilibrium distributions with and without vibrational en-ergies. Figure 5(b) shows the density of states difference.One can observe that although total properties such as theentropy change little, the probability distribution and eventhe equilibrium particle number changes are somewhat moresignificant. All the differences result from the density of stateschanging.

C. Influence of the initial condition on the trajectory

Another important study is for the case when the initialcondition is a very high order hypoequilibrium state. In that

012137-13


−10 −8 −6 −4 −2 0 2 4 6

x 10−19

0

0.5

1

1.5

2

2.5

3

3.5x 10

−4(a)

Energy (J)

Pro

bab

ility

den

sity

fu

nct

ion

H2,FFH,H

−10 −8 −6 −4 −2 0 2 4 6

x 10−19

115

120

125

130

135

140

145

150(b)

Energy (J)

log

10(D

ensi

ty o

f st

ate)

H2,FFH,H

FIG. 5. Comparison of (a) the stable equilibrium distributions and(b) the density of states for the systems with (solid line) and without(dashed line) vibrational energy eigenlevels.

case, although the theoretical discussion in Sec. II F is stillvalid, the number of subsystem divisions of the system maybe so high that characterizing each nonequilibrium state as anMth-order hypoequilibrium state may no longer be practical.However, the density of states method still permits the solutionof the equation of motion to very high accuracy. This sectionuses the general gamma distribution to study the influence ofinitial condition on the nonequilibrium trajectory by varying θ

in Eq. (106). The mean value of the energy for three cases withdifferent θ ’s is chosen to be the same in order to ensure thesame system total energy and the same final stable equilibriumstate. From Eqs. (111) and (112), the ratio of β in the threecases is β−2 : β0 : β2 = 1 : 2 : 3 and the ratio of the varianceof energy is Var−2 : Var0 : Var2 = 1 : 1/2 : 1/3. The initialdistribution for the reactant subspace is shown in Fig. 6, whichtakes 0.9999 of the total probability. It can be observed thatfor lower θ , there is more probability distributed in the higherenergy eigenlevels (energies greater than −6.4 × 10−19 J) andthe energy variance is larger.

−6.8 −6.6 −6.4 −6.2 −6 −5.8

x 10−19

0

1

2

3

4

5

6x 10

−3

Energy (J)

Pro

bab

ility

den

sity

fu

nct

ion

θ=0θ=−2θ=2

FIG. 6. Initial distribution in reactant subspace for θ = 0(solid line), θ = −2 (dashed line), and θ = 2 (dashed-dottedline).

The evolution in dimensionless time is studied in Fig 7. Inorder to facilitate the comparison, all the curves are shiftedsuch that time 0 is the dimensionless time when the maximumreaction rate is reached, and both Figs. 7(a) and 7(b) use thesame 0 time. In Fig. 7(a), it is observed that the beginning ofthe three cases is quite different, while the reaction processesafter time 0 are similar. Furthermore, the negative θ case takesless time to arrive at stable equilibrium than the zero θ case(Maxwellian distribution), and the positive θ case takes evenmore. In Fig. 7(b), the entropy evolution is also similar aftertime 0. However, the beginning parts of the evolution exhibitthree features. The first is that the initial entropy of the negativeand positive θ cases are both smaller than that for θ = 0 sincethe Maxwellian distribution provides the largest entropy forgiven energy. That also means that the initial entropy cannotdecide how fast the reaction process is, which instead isdecided by the value and the sign of θ . The second featureis that for negative θ , the entropy evolution is faster than thatfor the zero θ case, and the maximum of the entropy generationrate is larger than that for the zero θ case. The opposite is thecase for the positive θ case. Both Figs. 7(a) and 7(b) showthat the negative θ provides the faster reaction process. Onepossible explanation is that for lower θ , more probability isdistributed in the higher energy eigenlevels, which acceleratesthe reaction process. Another view is that the lower θ casehas lower β, which phenomenologically can be explained ashigher temperature. The final or third feature is that in theevolution after time 0, the probability distributions for thepositive and negative cases are not Maxwellian distributionsin the strict sense used in the next section even though theparticle number and entropy evolution are almost the same.However, as shown in Figs. 7 and 8, a Maxwellian distributionin each subspace, for a second-order hypoequilibrium state,can be a very good approximation for studying a reaction’sintermediate nonequilibrium states.

012137-14


−2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(a)

Dimesionless time

Nu

mb

er o

f P

arti

cle

−2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Rea

ctio

n r

ate

(no

n−d

im t

ime

un

it−1

)

−2 0 2 4 6 81.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12x 10

−21(b)

Dimensionless time

En

tro

py

(J/K

)

−2 0 2 4 6 8−2

0

2

4

6

8

10x 10

−23

En

tro

py

gen

erat

ion

rat

e (J

K−1

no

n−d

im t

ime

un

it−1

)

FIG. 7. (a) Evolutions of particle number and reaction rate. Forthe line colors in (a), the black line and green (light gray) line representthe FH,H and H2,F , respectively, and the red (dark gray) linerepresents reaction rate. (b) Evolutions of entropy (black line) andentropy generation rate [green (light gray) line]. For the line stylesin both figures, θ = 0 (solid line), θ = −2 (dashed line), and θ = 2(dashed-dotted line). The dimensionless time when the maximumreaction rate is reached is set to be 0.

In Fig. 8, the trajectory of the chemical reaction isillustrated using the product particle number and systementropy. The particle number of the product can be regardedas an equivalent variable to the reaction coordinate here. Byusing product particle number instead of dimensionless time,only the information of the intermediate states is kept. Itmust be pointed out here that in Fig. 8, any trajectory witha positive dS/dN , i.e., a monotonically increasing functionlinking the initial state and stable equilibrium, does notviolate the second law of thermodynamics. However, thetrajectories plotted are not just any trajectories but thoseuniquely predicted by SEA, the maximum-entropy-production(MEP) trajectories. It can be observed that the trajectoriesfor the three θ ’s chosen are very similar in Fig. 8(a) exceptat the very beginning [shown in Fig. 8(b)]. As mentionedbefore, even though the probability distributions do notbecome second-order hypoequilibrium distributions until very

FIG. 8. Trajectory representation using product particle numberand entropy. Range of product particle number (a) from 0 to 0.9 and(b) from 0 to 0.004. The black line and the green (light gray) linerepresent entropy (S) and dS/dN , respectively. For the line styles,θ = 0 (solid line), θ = −2 (dashed line), and θ = 2 (dashed-dottedline).

late (after the product particle number reaches 0.5), thesecond-order hypoequilibrium state can provide a very goodapproximation of the intermediate nonequilibrium states. Inaddition, in Fig. 8(b), one can observe that the zero θ case hasthe largest initial entropy and lowest dS/dN . The trajectoriesat the beginning can be separated into two parts by the statewhich the particle number reaches at 0.002. From 0 to 0.002,dS/dN for the nonzero θ case has a huge difference with thatfor the zero θ case. If the entropy is used as the measure of thedifference of the distributions, the larger difference results in agreater speed of the non-Maxwellian distribution approachingthe Maxwellian one. In this process, the degree of reactiononly changes a little when compared with the change in thetotal particle number of the product. After the particle numberreaches 0.002, there is little difference in dS/dN betweenthe three cases, even though the difference in entropy lastsuntil the system distribution reaches that for a second-orderhypoequilibrium state. However, this difference is small whencompared with the total system entropy. Recall from Fig. 7

012137-15


−10 −8 −6 −4 −2 0 2 4 6

x 10−19

−12

−10

−8

−6

−4

−2

0

Energy (J)

log

10(R

elat

ive

erro

r)

H2,FFH,H

FIG. 9. Relative error for the probability distribution.

that there is quite a big difference in the state evolutions forthe three cases until time 0, when the product particle numberreaches about 0.5. For the range of product particle numberfrom 0.002 to 0.5, the difference in entropy is negligiblefor the intermediate states of the trajectory but nonethelesshas a large influence on the evolution in dimensionlesstime.

D. Numerical error

In Fig. 9, the state of the system without vibrational energyeigenlevels is compared with the analytical solution of thestate, the gamma function, since the analytical solution for asystem with vibrational levels is difficult to acquire. Figure 9shows the relative difference of the numerical solution withthe analytical solution at states A0 to A4 and B. The relativeerrors of almost all the energy eigenlevels are less than 10−2.There are 14 out of the 30 793 eigenlevels where the relativeerror is larger than 10−2. However, the total probability forthese levels is less than 6.5 × 10−8. The error is larger forthese energy eigenlevels since they coincide with the regionwhere the density of states is lower and increasingly steep.Thus, this error can be explained by truncation error. It is alsoobserved that the error is larger at the initial state than that atstable equilibrium. In addition, the states at lower subsystemtemperatures have narrower distributions, whose accuracy islimited by the quasicontinuous condition. The energy intervalchosen is about 10−2 times lower than 1/β = kbT , and thataccuracy sets an upper limit on the relative error at the initialstate. In conclusion, the density of states method developedand used here provides a very accurate numerical solutionto the SEAQT equation of motion for a system with a verylarge number of energy eigenlevels. Furthermore, the criterionused in the previous section to check whether a distributionis Maxwellian or not is determined from the study here, i.e.,if a distribution of energy has its distribution across most ofthe energy eigenlevels (for example, 99% of them) very closeto the Maxwellian distribution (relative difference less than

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−22

−20

−18

−16

−14

−12

−10

−8

−6

Dimensionless time

lnΛ

Numerical SolutionAnalytical Solution

FIG. 10. Time evolution of �.

1%) with the same mean value of energy, the distribution isregarded as a Maxwellian distribution.

Finally, validation of the numerical accuracy of the timeevolution can be acquired through a value defined by

� = kb(βproduct − βreactant) = 1/T product − 1/T reactant.

(113)

According to Sec. II F, βproduct and βproduct yield to ODE(42) with different initial temperatures. Subtracting the ODEequations for βproduct and βproduct results in

d�

dt= − 1

τ�, �(0) = 1/300 − 1/500. (114)

Using the dimensionless time scale for τ = 1, the analyticalsolution of ln � yields

ln �(t) = ln(1/300 − 1/500) − t = −6.6201 − t. (115)

Plotting ln � as a function of dimensionless time as is donein Fig. 10 shows the accuracy of the numerical solution fordifferent times. The deviation starts from time 12 onward.At dimensionless time 12, �, which is the difference in theinverse of the reactant and product temperatures, has a valuesmaller than 10−8. Thus, this validation proves that the densityof states method can provide an accurate numerical solutionfor the equation of motion.

VI. CONCLUSIONS

In the preceding study, the nonequilibrium state evolutiontrajectory for a chemical reaction process is predicted usinga first-principles, thermodynamic-ensemble based approach,which provides a computationally simpler, alternative globalmethod for predicting the chemical kinetics of systems. This iswell illustrated via the definition of hypoequilibrium state andthe existence of canonical distributions outside the realm ofstable equilibrium. The nature of the SEA equation of motiondirectly leads to the existence of a unique nonequilibriumevolution trajectory in state space, which represents thekinetics. With the categorization of nonequilibrium statesby different ordered hypoequilibrium states, subsystem and

012137-16


subsystem temperatures serve as a good description of thetrajectory in high-dimensional state space, whose propertiesare ensured by the equation of motion or the principle of SEA.In addition, with the goal of being able to model systemswith a very large number of energy eigenlevels (for thesystem considered here, on the order of 10130), the concepts ofdegeneracy and density of states are utilized, and the equationof motion for the degenerate system is developed. The equationof motion for an energy interval is presented and the numericalprocess to solve it introduced.

The clear physical meaning of the evolution trajectory,subsystem, and subsystem temperature are shown in the resultsas are the chemical reaction process via particle numberevolution and subsystem heating via temperature evolutions.This work provides a reasonable generalization of the conceptof temperature at stable equilibrium and offers a framework todescribe and study nonequilibrium states and their relaxationprocess. In addition, even if the order of the hypoequilibriumstate is very high so that using this concept in such a casemay not be practical, the density of state method still permitssolution of the equation of motion so that the nonequilibriumthermodynamic trajectory, especially that for the intermediatestates of the system, can be determined with the SEAQTequation of motion. As to the influence of the initial state,it is shown that a second-order hypoequilibrium state isa good approximation for determining the nonequilibriumthermodynamic trajectory when the initial condition or state isthat of a gamma distribution for the two subsystems. It is worthnoting that although the results given in this paper are based ona two-subsystem division and second-order hypoequilibriumstate, the approach presented here can be easily applied toa system with a much higher number of subsystems. Whatorder of hypoequilibrium state is sufficient for arriving atan approximation is left as an open question for futurework.

Finally, different from other methods for studying nonequi-librium systems, which are phenomenological or basedon mechanics, this paper introduces a practical alterna-tive approach based on a first-principles thermodynamicframework for studying the relaxation of nonequilibriumstates.

ACKNOWLEDGMENT

Funding for this research was provided by the US Office ofNaval Research under Award No. N00014-11-1-0266.

APPENDIX A: PSEUDOSYSTEM

In this Appendix, a system property calculated by thedensity of states method developed here is proven to be a goodapproximation of the property’s true value. For the moment ofenergy in each interval of a given subspace (subsystem),

∑j

nij

(εij

)θ = Eθi

∑j

nij

(εij

)θ

Eθi

= Eθi

∑j

nij

(1 − εi

j − Ei

Ei

)θ

= Eθi

∑j

nij

(1 − θ

εij − Ei

Ei

)= NiE

θi , (A1)

∑j

nij

(εij

)θln

(εij

) = ln(Ei)∑

j

nij

(εij

)θ + nij

(εij

)θln

(εij

Ei

)

= ln(Ei)∑

j

nij

(εij

)θ

(−εi

j − Ei

Ei

)

= NiEθi ln(Ei). (A2)

The third equal sign of Eq. (A1) and the second of Eq. (A2)hold when

εij − Ei

Ei

� 1 (A3)

which is the case for the intervals other than the lowest ones. InFig. 9, it is shown that those intervals account for only a verysmall probability, if the probabilities of the original system andpseudosystem have the forms

pij = Cni

j

(εij

)θe−εi

j β , (A4)

Pi = CNi(Ei)θ e−Eiβ . (A5)

For the properties of each interval of a given subspace(subsystem),

Pi =∑

j

pij = C

∑j

nij

(εij

)θe−εi

j β

= C∑

j

nij

(εij

)θe−Eiβe−(εi

j −Ei )β = C∑

j

nij

(εij

)θe−Eiβ

= Ce−Eiβ∑

j

nij

(εij

)θ = CNiEθi e−Eiβ = Pi , (A6)

〈e〉i =∑

j

pij ε

ij = C

∑j

nij

(εij

)θ+1e−εi

j β

= C∑

j

nij

(εij

)θ+1e−Eiβe−(εi

j −Ei )β

= C∑

j

nij

(εij

)θ+1e−Eiβ = Ce−Eiβ

∑j

nij

(εij

)θ+1

= CNiEθ+1i e−Eiβ = 〈e〉i , (A7)

〈s〉i = −∑

j

pij ln

pij

nij

= −β〈e〉i + Pi ln C −∑

j

pij θ ln

(εij

)= −β〈e〉i + Pi ln C − Piθ ln(Ei) = 〈s〉i . (A8)

The fourth equal sign in Eqs. (A6) and (A7) results from thequasicontinuous condition given by

1

β� |Ei+1 − Ei | >

∣∣εij − Ei

∣∣, (A9)

e−(εij −Ei )β .= 1. (A10)

In addition, under the quasicontinuous condition and the as-sumption that the probability distribution pi

j or the distribution

Pi or Pi have the property of higher order moment convergenceat least to the second order in energy (e.g., as for the case ofan ideal gas), the summation of discrete energy eigenlevelscan be approximated by an integral over a continuousspectrum. Thus, properties for a given subsystem are found

012137-17

michaelvonspakovsky

Sticky Note

should be +Beta instead of -Beta

michaelvonspakovsky

Sticky Note

1 + and not 1 -

michaelvonspakovsky

Sticky Note

1 + and not 1 -

michaelvonspakovsky

Sticky Note

this is completely wrong; look at Ishan's derivation

michaelvonspakovsky

Sticky Note

- PilnC and not = PilnC


from

〈es〉K =∑

i

∑j

pij ε

ij ln

pij

nij

=∑

i

∑j

nij ε

ij e

−βεij

[−βεij + ln C − θ ln

(εij

)]

= C

∫n(E)E[−βE + ln C − θ ln(E)]e−βEdE,

(A11)

〈es〉K =∑

i

PiEi lnPi

Ni

=∑

i

NiEie−βEi [−βEi + ln C − θ ln(Ei)]

= C

∫n(E)E[−βE + ln C − θ ln(E)]e−βEdE

= 〈es〉K, (A12)

〈e2〉K =∑

i

∑j

pij

(εij

)2 =∫

E2p(E)dE

= C

∫n(E)Eθ+2e−βEdE, (A13)

〈e2〉K =∑

i

∑j

Pi(Ei)2 =

∫E2p(E)dE

= C

∫n(E)Eθ+2e−βEdE = 〈e2〉K. (A14)

Note that both discrete summations of the original system andthe pseudosystem can be regarded as the Riemann summationof the integral, and the difference of the Riemann summationand the integral is of second order or greater for the energyinterval. This means that the difference between both Riemannsummations (original system and pseudosystem) is of secondorder or greater when the quasicontinuous conditions hold.The properties for the system as a whole are then given by

〈e〉 =M∑K

(∑i

〈e〉i)

K

, 〈e〉 =M∑K

(∑i

〈e〉i)

K

, (A15)

〈s〉 =M∑K

(∑i

〈s〉i)

K

, 〈s〉 =M∑K

(∑i

〈s〉i)

K

, (A16)

〈es〉 =M∑K

〈es〉K, 〈es〉 =M∑K

〈es〉K, (A17)

〈e2〉 =M∑K

〈e2〉K, 〈e2〉 =M∑K

〈e2〉K, (A18)

where the summation is over all subsystems. With theseexpressions and the proof above linking the original andpseudosubsystem properties, it is also clear that 〈A1〉, 〈A2〉,and 〈A3〉 are equal to 〈A1〉, 〈A2〉, and 〈A3〉 under the quasicon-tinuous condition. Thus, the solution of the equation of motionfor the pseudosystem [Eq. (71)] can be used as a numericalapproximation of that for the original system Eq. (70).

APPENDIX B: ONSAGER INVESTIGATION

Some results of the Onsager investigation using the con-cepts of hypoequilibrium state and nonequilibrium intensiveproperties found in Refs. [27,28] are given in the following. Fora general discussion of the Onsager relations in SEAQT usingthe language of quantum mechanics, the reader is referred to[25]. If the system is in an Mth-order hypoequilibrium state,the probability evolution yields Eq. (32). For simplicity, thefollowing definition is made:

αK = ln ZK − ln pK. (B1)

Thus, the probability evolution of one energy eigenlevel isgiven by

pKi (t) = pK (t)

ZK [βK (t)]nK

i e−βK (t)εKi

= nKi e−αK (t)−βK (t)εK

i . (B2)

αK and βK are nonequilibrium intensive properties in the Kthsubspace, corresponding to the extensive properties of pK andEK . Furthermore, by defining

α = A2

A1, β = −A3

A1(B3)

the particle number and energy evolution of the Kth subspacecan be acquired from Eq. (36) by summation over onesubspace, i.e.,

dpK

dt= 1

τpK (αK − α) + 1

τEK (βK − β), (B4)

dEK

dt= 1

τEK (αK − α) + 1

τ〈e2〉K (βK − β), (B5)

where 〈e2〉K is defined by Eq. (61) and pK and EK are theprobability and energy in the Kth subspace. When a systemis in an Mth-order hypoequilibrium state and goes through apure relaxation process, a relation for the evolution of extensiveproperties evolution in one subspace exists and is expressed as

dSK

dt= βK dEK

dt+ (αK − 1)

dpK

dt, (B6)

where SK is the entropy in the Kth subspace. This is the Gibbsrelation for the subspace. The physical meaning of βK and αK

is then given by

βK =(

∂SK

∂EK

)pK

= 1

T k, (B7)

αK − 1 =(

∂SK

∂pK

)EK

= −μK

T k, μK =

(∂EK

∂pK

)SK

, (B8)

where T K is the subspace temperature and μK is its chemicalpotential with respect to the subspace probability pK . Thedifferential change in the total entropy, which for a purerelaxation process is equivalent to the entropy generation, isthen written as

dS =∑K

dSK =∑K

βKdEK +∑K

(αK − 1)dpK

=∑K

(βK − β)dEK +∑K

(αK − α)dpK, (B9)

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where both energy (∑

dEK = 0) and probability (∑

dpK =0) conservation have been applied. The Casimir conditionholds and JK

E = dEK/dt and JKp = dpK/dt are defined to be

the internal fluxes of energy and probability inside the system,while XK

p = βK − β and XKE = αK − α are the conjugate

forces. Thus,

dS

dt=

∑K

XKE JK

E +∑K

XKp JK

p . (B10)

The Onsager relations are acquired from Eqs. (B4) and (B5)in the form of J = �X, where � is symmetric and positivedefinite, so that

JKp = 1

τpKXK

p + 1

τEKXK

E , (B11)

JKE = 1

τEKXK

p + 1

τ〈e2〉KXK

E , (B12)

while the quadratic dissipation potential in force representation[30,31] is given by

�(X,X) = 1

2〈X,�X〉 = 1

2τ

∑K

[pK (αK − α)2

+ 2EK (αK − α)(βK − β) + 〈e2〉K (βK − β)2].

(B13)

Furthermore, even though the following constraints apply tothe fluxes ∑

K

JKp = 0,

∑K

JKE = 0, (B14)

the reciprocity seen in Eqs. (B11)–(B13) is fully consistentwith the Onsager theory since according to Gyarmati [30]the validity of Onsager’s reciprocal relations is not influ-enced by a linear homogeneous dependence valid amongstthe fluxes. Thus, the physical interpretation of Eqs. (B11)–(B13) in terms of hypoequilibrium state and nonequilib-rium intensive properties does not require a reformulationin terms of independent fluxes even though this could bedone.

Thus, from the entropy generation of a nonequilibriumisolated system derived from the relaxation gradient dynamics,which are based on the geometry of system state space, oneis able to arrive at the Onsager relations and the quadraticdissipation potential using the concepts of hypoequilibriumstate and nonequilibrium intensive properties. Alternatively,one can arrive at these relations and this potential usinga variational principle in system state space as is done inRef. [25]. Of course, the Onsager relations and quadraticdissipation potential also correspond to a variational principlein the space spanned by conjugate forces and fluxes [30].

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