Steepest-entropy-ascent quantum thermodynamic modeling of heat and mass diffusion in a far-from-equilibrium system based on a single particle ensemble

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by  Guanchen Li

Steepest-entropy-ascent quantum thermodynamic modeling of heat and mass diffusion
in a far-from-equilibrium system based on a single particle ensemble
Guanchen Li∗ and Michael R. von Spakovsky†
Center for Energy Systems Research,
Mechanical Engineering Department
Virginia Tech, Blacksburg, VA 24061
(Dated: January 13, 2016)
6
1 Thispaperpresentsanonequilibriumthermodynamicmodelfortherelaxationofalocal,isolated
0 system in nonequilibrium using the principle of steepest entropy ascent (SEA), which can be
2 expressed as a variational principle in thermodynamic state space. The model is able to arrive
at the Onsager relations for such a system. Since no assumption of local equilibrium is made, the
n
conjugate fluxes and forces, which result, are intrinsic to the subspaces of the system’s state space
a
andaredefinedusingtheconceptsofhypoequilibriumstateandnonequilibriumintensiveproperties,
J
which describe the non-mutual equilibrium status between subspaces of the thermodynamic state
2
space. The Onsager relations are shown to be a thermodynamic kinematic feature of the system
1
independent of the specific details of the micro-mechanical dynamics. Two kinds of relaxation

processes are studied with different constraints (i.e., conservation laws) corresponding to heat and
]
h mass diffusion. Linear behavior in the near-equilibrium region as well as nonlinear behavior in the
c far-from-equilibrium region are discussed. Thermodynamic relations in the equilibrium and near-
e equilibriumrealm, includingtheGibbsrelation, theClausius inequality,andtheOnsagerrelations,
m aregeneralizedtothefar-from-equilibriumrealm. Thevariationalprincipleinthespacespannedby
- theintrinsic conjugate fluxesand forces is expressed via thequadratic dissipation potential. Asan
t application, themodelisappliedtotheheatandmassdiffusion ofasystemrepresentedbyasingle
a
particleensemble,whichcanalsobeappliedtoasimplesystemofmanyparticles. Phenomenological
t
s transport coefficients are also derived in near-equilibrium realm.
.
t
a
m
I. INTRODUCTION [11], extended irreversible thermodynamics [12], etc.
- The thermodynamic features captured can be regarded
d
n The study of nonequilibrium relaxation processes as a coarse graining of the microscopic dynamics or
o including chemical kinetics, mass diffusion, and heat as a pattern in ensemble evolution [13, 14], which
c diffusion is typically accomplished using approaches computationally is more efficient. However, most
[ based on microscopic mechanics or thermodynamics. of these approaches have limited or no applicability
2 With the former, system state space is spanned by in the far-from-equilibrium realm, since the local or
v microstates and the governing equation is based on near-equilibrium assumption is needed or analytical
4 the dynamics of classical mechanics (e.g., molecule solutions are only available at steady state. In
4 dynamics [1] or kinetic theory [2]), quantum me- addition,theirgoverningequationsarephenomenological
3 chanics (e.g., nonequilibrium Green’s functions [3] or or stochastic in nature and, thus, do not have a first-
1
the quantum Boltzmann equation, i.e., the Uehling- principle basis. To address these issues and push the
0
. Unlenbeck-Boltzmann equation [4–6]) or a stochastic application of thermodynamic principles further into
1 process (e.g., Monte Carlo simulation of the Ising model the nonequilibrium realm, it is of great importance to
0 [7]). These approaches provide complete information find a general and simple description of nonequilibrium
6
of the microscopic process such as individual particle state corresponding to a thermodynamic pattern of the
1
: collisions or quantum state scattering. However, the microscopic description, to fundamentally define the
v large amount of very detailed information required macroscopicpropertiesofanythermodynamicstate(i.e.,
Xi inevitably results in large computational burdens, which extensive or intensive properties for both equilibrium
r limit the applicability of these approaches. andnonequilibriumstates),andtouseathermodynamic
a Approaches based on thermodynamics, on the other governing equation based on first principles.
hand,arenotsimilarlyburdenedandareabletogenerally Steepest-entropy-ascent quantum thermodynamics
capturethefeaturesoftherelaxationprocessisofinterest (SEAQT), which is a first-principle, thermodynamic-
via, for example, the Onsager relation. Approaches ensemble based approach, addresses all of the issues
of this type include nonequilibrium thermodynamics raised above, providing a governing equation able to
[8, 9], linear response functions and the fluctuation- describe the nonequilibrium process from an entropy
dissipation theorem [10], stochastic thermodynamics generation viewpoint. The description of system
state is based on the density operator of quantum
mechanics or probability distribution in state space
of classical mechanics. The macroscopic properties
∗ [email protected] of entropy, energy, and particle number, which are
† [email protected] well defined for any state of any system [15], are
2
used to develop the governing equation and describe thermodynamic concepts. To begin with, the equation
system state evolution. Recently, this description has of motion for such a system is derived in Sec. II.A from
been simplified via the concept of hypoequilibrium the geometric principle of steepest entropy ascent. This
state [16], which captures the global features of the is followed in Sec. II.B by a discussion of the kinetics
microscopic description for the relaxation process. anddynamicsoftherelaxationprocess,whichenablethe
In addition, the concept of nonequilibrium intensive study of the thermodynamic trajectory regardless of the
properties introduced in [16] based on the concept of details of the microscopic interactions. A description of
hypoequilibrium state enables a complete description of nonequilibrium state and its evolution for the relaxation
the nonequilibrium evolution of state when combined process is then given in Sec. II.C using the concepts
with the set of nonequilibrium extensive properties. of hypoequilibrium state and nonequilibrium intensive
Unlike the intensive property definitions of other properties. Finally, in Sec. II.D, a presentation of the
nonequilibrium thermodynamic approaches (definitions variational principle of steepest entropy ascent is used
which require local-equilibrium, near-equilibrium, to derive the SEAQT equation of motion for the local
or steady state assumption or a phenomenological system.
basis), the definitions in the SEAQT framework are
fundamental and available to all nonequilibrium states
and are especially suitable for the description of the A. SEAQT equation of motion for an isolated
evolution in state of relaxation processes. Both of these system
concepts enables the generalization of many equilibrium
(or near-equilibrium) thermodynamic relations such as Based on the discussion by Grmela [13, 14, 17] and
the Gibb’s relation, the Clausius inequality, and the
Beretta [18, 19], the general form of a nonequilibrium
Onsager relations into the far-from-equilibrium realm.
framework is a combination of both irreversible relax-
To describe the SEAQT framework, the paper starts ation and reversible symplectic dynamics. If written in
with a derivation in Secs. II.A and B of the SEAQT the generalized form of the Ginzburg-Landau equation
relaxation dynamics starting from the geometry of [13, 19],the equationofmotiontakesthe followingform:
system state space. Some useful mathematical features
ofthe relaxationprocessarethen presentedin Sec. II.C, d
γ(t)=XH +YH (1)
enablingthedefinitionoftheconceptsofhypoequilibrium dt γ(t) γ(t)
state and nonequilibrium intensive properties. As
an alternative to the geometric derivation from state where γ(t) represents the state evolution trajectory,
space, the relaxation dynamics of SEAQT can also be XH and YH are functions of the system state γ(t)
γ(t) γ(t)
derived using a variational principle in system state and represent the reversible symplectic dynamics and
space as is done in Sec. II.D. Section III then follows irreversible relaxation dynamics, respectively. In the
with a discussion of mass diffusion in a local, isolated SEAQT framework, the system state is represented by
system in nonequilibrium. Subsequently, the transport the density operator ρˆ, XH follows the Schr¨odinger
γ(t)
equations, the Gibbs relation, and the Onsager relations equation, and YH is derived from the SEA principle.
aregeneralizedforthenonequilibriumrelaxationprocess, γ(t)
Thus,
andthedissipationpotentialisgiventocompletethelink
from the variational principle in system state space to dρˆ 1 1
that for the conjugate fluxes and/or forces [9]. Section dt = i~[ρˆ,Hˆ]+ τ(ρˆ)Dˆ(ρˆ) (2)
IV discusses the process of heat diffusion by choosing
a set of system constraints different from those used where Dˆ is the dissipation operator determined via a
for mass diffusion. In Sec. V, the SEAQT model is
constrained gradient in Hilbert space. A metric tensor
applied to the study of the heat and mass diffusion
must be specified in the derivation of this dissipation
of a simple system consisting of ideal gas (hydrogen),
term, since it describes the geometric features of the
which can be represented by a single-particle ensemble.
Hilbert space [18]. τ is the relaxation time, which
Linear behavior in the near-equilibrium realm provides
representsthespeedofsystemevolutioninHilbertspace.
the phenomenological transport equations, and higher-
In the application here to the modeling of heat and
order, nonlinear behavior in the near-equilibrium realm
mass diffusion, the system is restricted to the class
is also studied. Finally, section VI concludes the paper
of dilute-Boltzmann-gas states in which the particles
with some final comments.
have no quantum correlation between eigenstates, and
are independently distributed [20]. Such states can be
represented by a single-particle density operator that
II. THEORY: SEA-QT EQUATION OF MOTION
is diagonal in the basis of the single-particle energy
eigenstates. The Hilbert space metric chosen here is
The basic framework of SEAQT is introduced in this the Fisher-Rao metric, which is uniform in different
section. This frameworkdescribesthe relaxationprocess dimensionsofHilbertspace. Under these conditions,the
of a local, isolated system in nonequilibrium based on symplectic Scho¨dinger term in the equation of motion
3
vanishes. Thus, the focus is this paper is on the The principle of SEA upon which the equation of
irreversible relaxation process only. motion is based is defined as the system state evolving
A group of energy eigenlevels ǫ ,i = 1,2,... is along the direction that at any instant of time has the
i
{ }
determined from the system Hamiltonian. The state largestentropygradientconsistentwiththe conservation
of the system can then be represented by a probability constraints. The equation of motion is given by
distribution among the energy eigenlevels p ,i =
{ i dx 1
1,2,... , which is the diagonal terms of the density = g (12)
} dt τ(x) S⊥L(gI,gE)
operator. Usingthe Fisher-Raometric ofthe probability
space pi,i = 1,2,... , one can define the distance where τ, which is a function of system state, is the
{ }
in probability space, which can be used as the state relaxation time which describes the speed at which the
distance. Equivalently,thesquarerootoftheprobability state evolves in state space in the direction of steepest
distribution{xi,i=1,2,...}canbeusedtorepresentthe entropyascent. L(gI,gE) is the manifoldspanned by gI
system state with the result that the Fisher-Rao metric andg ,andg istheperpendicularcomponent
ofthe probabilityspacebecomestheEuclideanmetricin of theEgradienSt⊥oLf(tghIe,gEen)tropy to the hyper-surface that
square root of the probability space of xi,i = 1,2,... . conserves the probability and energy. It takes the form
{ }
The later representation is used in the paper. Both are of a ratio of Gram determinants expressed as
expressed as follows:
g g g
S I E
State: {pi,i=1,2,...}, (cid:12) (gS,gI) (gI,gI) (gE,gI) (cid:12)
1 dlnp (cid:12)(gS,gE) (gI,gE) (gE,gE) (cid:12)
Distance: dl= 2sXi pi( dθ i)2dθ (3) gS⊥L(gI,gE) = (cid:12)(cid:12)(cid:12)(cid:12) ((ggI,,ggI)) ((ggE,,ggI)) (cid:12)(cid:12)(cid:12)(cid:12) (13)
I E E E
State: x ,i=1,2,... , (cid:12) (cid:12)
{ i } (cid:12) (cid:12)
where(...,...)denotes(cid:12)thescalarproducto(cid:12)ftwovectors
(cid:12) (cid:12)
Distance: dl= (dxi)2 (4) in state space. The explicit form of Eq. (13) for pj is,
{ }
s i thus, [20]
X
where dl is the distance between p(θ +dθ) and p(θ) or p lnp p ǫ p
j j j j j
x(θ+dθ) andx(θ), andθ is a continuousparameter. An − s 1 e
extensive property of the system can then be defined as dp 1(cid:12)(cid:12) hesi e he2i (cid:12)(cid:12)
a function of the state x such that j = (cid:12) h i h i h i (cid:12) (14)
{ i} dt τ (cid:12)(cid:12) 1 e (cid:12)(cid:12)
(cid:12) e he2i (cid:12)
I = x2i (5) (cid:12)h i h i(cid:12)
(cid:12) (cid:12)
i where (cid:12) (cid:12)
X (cid:12) (cid:12)
E = e = ǫ x2 (6)
h i i i i he2i= ǫ2ix2i, hesi= −ǫix2i ln(x2i) (15)
X i i
S = s = x2ln(x2) (7) X X
h i − i i The state representationandthe equationof motioncan
i
X be simplifiedbycombiningdegenerateenergyeigenlevels
where ... means the ensemble average. The von [16]. The system is defined by a group of different
h i
Neumann formula for entropy is used, because as energy eigenlevels ǫ ,i = 1,2,... and their degeneracy
i
{ }
shown in [21], it has all the properties required by n ,i=1,2,... . The state of the system is described by
i
{ }
thermodynamics. The gradient of a given property in a probability distribution among the energy eigenlevels
state space is then expressed by p ,i=1,2,... or square rootofthe probability x ,i=
i i
{ } {
1,2,... . As a result, the equation of motion changes to
∂I }
g = eˆ = 2x eˆ (8)
I i ∂xi i i i i −pjlnnpjj pj ǫjpj
X X s 1 e
∂E (cid:12) h i h i (cid:12)
gE =Xi ∂xieˆi =Xi 2ǫixieˆi (9) ddptj = τ1(cid:12)(cid:12)(cid:12)(cid:12) hesi1 heei he2i (cid:12)(cid:12)(cid:12)(cid:12) (16)
g = ∂Seˆ = [ 2x 2x ln(x2)]eˆ (10) (cid:12) e he2i (cid:12)
S ∂xi i − i− i i i (cid:12)(cid:12)h i h i(cid:12)(cid:12)
Xi Xi where the properties are de(cid:12)fined by (cid:12)
(cid:12) (cid:12)
where eˆi is the unit vector for each dimension. x2
Furthermore, for an isolated system, the system satisfies e = ǫ x2, s = x2ln( i)
h i i i h i − i n
the conservation laws for probability and energy, i.e., i i i
X X
x2
I = x2i =1, E = ǫix2i =constant (11) he2i= ǫ2ix2i, hesi= −ǫix2i ln(nii) (17)
i i
i i X X
X X
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B. Nonequilibrium evolution: Kinetics and paper,however,is on the thermodynamic features of the
Dynamics nonequilibrium relaxation so that τ is set equal to 1.
More discussion on how τ is chosen using the mechanics
In general, the equation of motion for a system with a andonthedynamicsofnonequilibriumisleftforafuture
given group of conservation laws has the form: paper.
dp 1
j = D (p) (18)
dt τ(p) j C. Nonequilibrium state and state evolution
description: Hypoequilibrium
whereD (p)iscalculatedfromtheconservationlawsand
j
the principle ofsteepestentropyascent[22]. Specifically,
The solution of the SEA-QT equation of motion
ittakestheformofEq. (14)foranisolatesystemyielding
exhibits some good properties, which allows for a
to mass and energy conservation.
complete description of nonequilibrium state and the
Since for a given initial state of the system, the
general definition of nonequilibrium temperature. More
nonequilibrium thermodynamic path of state evolution
discussion is presented in reference [16], and an example
is uniquely solvedfromthis equation, Eq. (18), the path
is provided below. The energy eigenlevels of the system
can be used to define a new parameter τ˜ given by
ǫ ,i = 1,2,3... with degeneracy n ,i = 1,2,3... can
i i
{ } { }
be divided into to M sets ǫK (degeneracy nK ) with
dτ˜= 1 dt, or τ˜= 1 dt′ =τ˜(t) i = 1,2,3,..., K = 1,2,...,{Mi,}so that the sta{teis}pace of
τ(p(t)) τ(p(t′))
Zpath the system (the Hilbert space) can be represented by
(19) H
the sum of M subspaces , with K =1,2,...,M, i.e.,
K
whereτ˜iscalledthedimensionlesstime. Withthistime, H
the independent variable for the equation of motion can M
be changed so that = (22)
K
H H
K=1
dp M
j =D (p) (20)
dτ˜ j To be complete, M can be infinite. The system state
can be represented by the distributions in M subspace
The solution of this equation is written as:
energy eigenlevels pK,K =1,...,M . If the probability
{ i }
distribution in one subspace, for example, the Kth
p =p (τ˜) (21)
j j
subspaceyieldstothecanonicaldistributionofparameter
Independent of how the relaxation time τ depends on βK, the temperature of the Kth subspace is defined to
the realtime t andthe state, the equationof motioncan be TK = 1/βK. Given a way to divide the energy
always be transformed to Eq. (20) with the parameter eigenlevels, if the system probability distributions in the
change defined by Eq. (19). Furthermore, the evolution M subspaces are all canonical distribution, the state
of system state will follow the same function [Eq. (21)] of the system is called a Mth-order hypoequilibrium
in τ˜. Physically, this means that the system follows state [16], which can be described uniquely by the
the same trajectory in state space. By using this total probability in each subspace ({pK = pKi ,K =
transformation,the kinetics and dynamics of the system 1,...,M ) and the temperature of each of the subspaces
} P
are separated. The former is found via Eqs. (20) and ( TK,K =1,...,M ). If the initial state of the systemis
{ }
(21) and result in the trajectory in state space based on a Mth-order hypoequilibrium state then
the parameter τ˜ or a constant relaxation time τ. The
pKnK
dynamics are found via Eq. (18) and the functional pK(t=0)= i e−βKǫKi ,i=1,2,3,... (23)
i ZK(βK)
dependence τ = τ(p) [Eq. (19)] and result in the
trajectory in state space based on the real time t.
where k is the Boltzmann constant, ZK(βK) is the
b
In the discussionon mass diffusion (Sec. III) and heat
partition function of subspace K at temperature TK. A
diffusion (Sec. IV), it is shown that the kinetics (or its
moregeneralformto representany nonequilibriumstate
associated trajectory) of the nonequilibrium relaxation
isgivenin[22]usingthelanguageofquantummechanics.
results in a generalized Gibbs relation and the Onsager
Li and von Spakovsky [16] have proven that the system
relations in the far-from-equilibrium realm and in linear
retains a Mth-order hypoequilibrium state throughout
phenomenological equations in the near-equilibrium
the nonequilibrium relaxation process if it initial starts
realm, which are independent of the dynamics. The
out in such a state. The solution to Eq. (18), thus,
kinetics appears as a system feature or pattern of the
becomes
thermodynamics in the sense of the GENERIC [13, 14,
17], which is an ensemble or group feature. Information pK(t)
about the mechanical details (e.g., how the particles pKi (t)= ZK(βK(t))nKi e−βK(t)ǫKi ,i=1,2,3,... (24)
interact in the system mechanically) can be included
in the dynamics (e.g., by how τ is chosen) when the As a result, each subspace has temperature defined
state evolution in time t is studied. The focus of this throughoutthe entirenonequilibriumrelaxationprocess.
5
This result applies to an isolate system with probability Summation over all energy eigenlevels in the
andenergyconservation. Forasystemwithadifferentset Kthsubspace yields the evolution of the probability
of conservation laws, a similar relation exists. However, in the Kth subspace, represented by pK, namely,
the generalproofis leftfor afuture paper. The prooffor
asystemwithheatdiffusiononlyisgivenintheAppendix dpK = 1( pKlnpK +pK s˜ K pKA2 +pK e˜ KA3)
A. dt τ − h i − A h i A
1 1
(28)
where .˜.. K isthespecificpropertyintheKthsubspace.
h i
D. Local variational principle in thermodynamic To calculate the specific property, the probability
state space distribution in the Kth subspace is found from
According to Beretta [20], the equation of motion can pK pK, p˜K pKj (29)
bederivedfromalocalvariationalprinciple,whichcanbe ≡ j j ≡ pK
j
regarded as the variational form of the steepest entropy X
ascent principle, i.e.,
where pK is the particle number in the Kth subspace.
Maximize: S˙(x˙)=(x˙,gS) subject to The specific property in the Kth subspace is then
expressed as
E˙ =(x˙,gE)=0, I˙=(x˙,gI)=0, (x˙, x˙)=ξ(x)
with δx˙ =0, δx=0 (25) e˜ K ǫKp˜K (30)
6 h i ≡ j j
j
The third constraint on x˙ indicates that only the X
p˜K
direction of x˙ is of interest. This variational principle s˜ K p˜Kln j (31)
is in microscopic state space, which contrasts with the h i ≡− j nK
j j
variational principle in the space spanned by conjugate X
fluxes and forces presented later for the Onsager
relations.
B. Particle number and temperature evolution
when the initial state is a hypoequilibrium state
III. THEORY: DIFFUSION IN A
Ifthe systemis inaMth-orderhypoequilibriumstate,
NONEQUILIBRIUM SYSTEM
the probability evolution yields Eq. (24). For simplicity,
the following definition is made:
In the next two sections, Secs. III and IV, two
interactions are studied with the SEAQT framework. αK =lnZK lnpK (32)
The theory for mass diffusion in a local, isolated system −
in nonequilibrium is presented in this section, Sec. III.
With this definition, the probability evolution of one
The Gibbs relation, the entropy generation for a non-
energy eigenlevel is given by
quasi-equilibrium process, and the Onsager relations are
derived based on the concept of hypoequilibrium state pK(t)
and intensive properties. The variationalprinciple using pKi (t)= ZK(βK(t))nKi e−βK(t)ǫKi
conjugate forces is given at the end of Sec. III.
=nKe−αK(t)−βK(t)ǫKi (33)
i
αK andβK arenonequilibriumintensivepropertiesofthe
A. Equation of motion for mass diffusion
Kth subspace,correspondingto the extensive properties
pK and EK pk e˜ K. Furthermore, by defining
The mass (or probability) diffusion across energy ≡ h i
eigenlevels (or across subspaces) can be studied for an A A
2 3
isolated system in nonequilibrium. Using Eq. (14), α= , β = (34)
A −A
1 1
one energy eigenlevel in the Kth subspace yields to the
following equation of motion: the particle number and energy evolution of the Kth
subspace can be acquired from Eq. (26), i.e.,
dpK 1 pK A A
j = ( pKln j pK 2 +ǫ pK 3) (26)
dt τ − j nKj − j A1 j j A1 dpK = 1pK(αK α)+ 1pK e˜ K(βK β) (35)
dt τ − τ h i −
where d e K 1 1
h i = pK e˜ K(αK α)+ pK e˜2 K(βK β)(36)
1 e s e s 1 dt τ h i − τ h i −
A1 =(cid:12) hei hhe2ii(cid:12),A2 =(cid:12)hhesii hhe2ii (cid:12),A3 =(cid:12) hhesii hei(cid:12) From Eqs. (24) and (26), the intensive properties αK
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (27(cid:12)) and βK obey the evolutions (see Appendix B for the
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
6
derivation) pK. The totaldifferentialentropychangeforthesystem,
which for a pure nonequilibrium relaxation process
dαK 1
= (αK α) (37) corresponds to the entropy generation, is
dt −τ −
dβK 1 dS = dSK = βKdEK + (αK 1)dpK
= (βK β) (38) −
dt −τ − K K K
X X X
The authors prove that α and β have the physical = (βK β)dEK + (αK α)dpK (48)
− −
meaning of intensive properties from measurements of K K
X X
a nonequilibrium state [23]. At stable equilibrium, the
wherebothenergy( EK =0)andprobability( pK =
intensive properties in any subsystem obey the following
0) conservations have been applied. The Casimer
relations: conditionholdsandPJK =dEK/dtandJK =dpKP/dtare
E p
α(t=teq)=αK(t=teq)=αeq (39) definedtobetheinternalfluxesofenergyandprobability
β(t=teq)=βK(t=teq)=βeq (40) insidethesystem,whileXpK =βK−β andXEK =αK−α
are the conjugate forces. The result is
dS
C. Gibbs relation, entropy generation for a σ(J,X)= dt = XEKJEK + XpKJpK (49)
non-quasi-equilibrium process, and the Onsager K K
X X
relations in the nonlinear realm
The Onsager relations are then acquired from Eqs. (35)
and(36)intheformofJ=ΛX,whereΛisasymmetric
Differential changes of the extensive properties in the
and positive definite operator. Thus,
Kth subspace are written as
1 1
dEK = d(ǫKj pKj ) (41) JpK = τpKXpK + τEKXEK (50)
i 1 1
X JK = EKXK + e2 KXK (51)
d pK pK dpK E τ p τh i E
dSK = ( pKln j )= ( ln j 1) j (42)
dt − j nK − nK − dt The quadratic dissipationpotentialusing force represen-
i j i j
X X tation [9, 24] is written as
whereEK andSK pK s˜ K aretheenergyandentropy
≡ h i 1 1
in the Kth subspace, respectively. Ξ(X,X)= X,ΛX = [pK(αK α)2
When a system is in a Mth-order hypoequilibrium 2h i 2τ −
K
state andundergoesapure relaxationprocess,arelation X
+2EK(αK α)(βK β)+ e2 K(βK β)2] (52)
for property evolution in one subspace is acquired by − − h i −
using relation of Eq. (33), namely, Using force representation, the variational principle is
given by
dSK dpK dEK dpK
= (ǫKβK+αK 1) j =βK +(αK 1)
dt j − dt dt − dt δ[σ(J,X) Ξ(X,X)]J =0, J=const, δJ=0, δX=0
− 6
X (43) (53)
The proof of this relation for one subspace applies to
any differential changes (not only the time derivative). where the σ(J,X) and Ξ(X,X) are given by Eqs. (49)
Thus, a generalization of the Gibbs relation to the Kth and (52). Furthermore, even though the following
subspace of a system in nonequilibrium, is expressed by constraints apply to the fluxes:
dSK =βKdEK +(αK 1)dpK (44) JK =0, JK =0 (54)
− p E
K K
Thus, from the Gibbs relation at stable equilibrium X X
written as The reciprocity seen in Eqs. (50) and (51) is
1 µ completely consistent with the Onsager theory since
dS = dE dN (45) according to Gyarmati [9], the validity of Onsager’s
T − T
reciprocal relations is not influenced by a linear
the physical meaning of βK and αK is shown to be homogeneous dependence valid amongst thefluxes. Thus,
the physical interpretation given here is fully consistent
∂SK 1
βK = = (46) with other investigations [23] and does not require a
∂EK Tk
(cid:18) (cid:19)pK reformulationintermsofindependentfluxeseventhough
∂SK µK ∂EK this could be done. In addition, it is from the gradient
αK 1= = , µK = (47) dynamics of the nonequilibrium relaxation process that
− ∂pK −Tk ∂pK
(cid:18) (cid:19)EK (cid:18) (cid:19)SK the entropy generation, the Onsager relations, and
where TK is subspace temperature and µK is subspace the quadratic dissipation potential of a local, isolated
chemical potential with respect to subspace probability system in nonequilibrium have been derived using the
7
geometric principle of SEA as well as the concepts where the contribution of each subspace to the total
of hypoequilibrium state and nonequilibrium intensive property is defined by
properties. Alternatively, the variational principle of
SEAin systemstate spacecanbe usedto arriveatthese hsi=hsia+hsib, hsia = paisai, hsib = pbisbi(58)
relationsasisdonein[22]usingthelanguageofquantum
e = e a+ s b, e a =X paǫa, e b =Xpbǫb(59)
mechanics. Of course, these relations also correspond h i h i h i h i i i h i i i
to the variational principle in the space spanned by X X
where
conjugate forces and fluxes [9].
pa(b) p˜a(b)
sa(b) = ln j =s˜a(b) lnpa(b), s˜a(b) ln j
j − na(b) j − j ≡− na(b)
j j
IV. THEORY: HEAT DIFFUSION IN A (60)
NONEQUILIBRIUM SYSTEM By defining
pa 0 e a s a 0 e a
A local, isolated system in nonequilibrium with heat h i h i h i
B = 0 pb e b ,Ba = s b pb e b
diffusion only is considered in this section. This 1 (cid:12) h i (cid:12) 2 (cid:12) h i h i (cid:12)
(cid:12) e a e b e2 (cid:12) (cid:12) es e b e2 (cid:12)
requires a model with a different set of constraints (i.e., (cid:12)h i h i h i (cid:12) (cid:12) h i h i h i (cid:12)
the probability redistribution is only allowed in each (cid:12)(cid:12) s b 0 e b(cid:12)(cid:12) (cid:12)(cid:12) s a pa 0 (cid:12)(cid:12)
subspace) then when only heat diffusion is considered. Bb =(cid:12) hsia pa heia(cid:12) ,B =(cid:12) hsib 0 pb (cid:12)(61)
2 (cid:12)h i h i (cid:12) 3 (cid:12) h i (cid:12)
The entropy change of the system and subspaces due to (cid:12) es e a e2 (cid:12) (cid:12) es e a e b (cid:12)
(cid:12) h i h i h i (cid:12) (cid:12) h i h i h i (cid:12)
heat diffusion for non-quasi-equilibrium process is given (cid:12) (cid:12) (cid:12) (cid:12)
Eq. (57) c(cid:12)an be simplified t(cid:12)o (cid:12) (cid:12)
and the relationship between the SEAQT equation of (cid:12) (cid:12) (cid:12) (cid:12)
motion and the phenomenological diffusion equation is
dpa 1 Ba B
presented. j = (pasa pa 2 ǫapa 3) (62)
dt τ j j − j B − j jB
1 1
Moreover, the equation of motion for the probability
distributioninonesubspacecanalsobewritteninterms
A. Equation of motion for heat diffusion
of the normalized probability by dividing both sides of
Eq. (62) by pa so that
Different from previous forms of the equation of
motion, that for pure heat diffusion yields to a different dp˜a 1 Ba B
j = (p˜asa p˜a 2 ǫap˜a 3) (63)
set of conservation equations. If the system is separated dt τ j j − j B − j jB
1 1
into M subspaces with energy flow but no probability
flow across the subspaces, there are M +1 conservation Furthermore, if the system is in a second-order
laws. System probability conservation is replaced by hypoequilibrium initially so that each subspace has a
that for M individual subspaces. In Appendix A, it is canonicaldistribution,Eq. (63)canbesimplifiedfurther
proven that the concept of hypoequilibrium state and to arrive at the form
nonequilibrium temperature are also well defined under
dp˜a 1
these new constraints given by j = p˜a[(s˜a s˜ a) β(ǫa e˜ a)] (64)
dt τ j j −h i − j −h i
IK = (xK)2 =pK, K =1,2,...M (55) where s˜a is defined by Eq. (60) and s˜ a and e˜ a are
i j h i h i
i defined by Eqs. (30) and (31). The parameterβ is given
X
E = ǫix2i =constant (56) B paA˜aβa+pbA˜bβb
Xi β≡ B31 = pa1A˜a1 +pbA˜b11 (65)
For simplicity, a second-order hypoequilibrium state is B =papb(paA˜a+pbA˜b) (66)
1 1 1
studied first. The system is separated into 2 subspaces B =papb(paβaA˜a+pbβbA˜b) (67)
(subspace a and subspace b) so that the equation of 3 1 1
motion takes the form is a weighted average of the inverse temperatures of the
subsystems relative to the mole fractions and the energy
pasa pa 0 ǫapa fluctuation (or nondimensional specific heat at constant
j j j j j
s a pa 0 e a volume) of the subspaces written as
(cid:12) h i h i (cid:12)
(cid:12) s b 0 pb e b (cid:12)
ddptaj = τ1(cid:12)(cid:12)(cid:12)(cid:12) hhesiipaheia0heibe ahhe2ii (cid:12)(cid:12)(cid:12)(cid:12) (57) A˜a1(b) =he˜2ia(b)−(he˜ia(b))2 =−∂∂hβe˜iaa((bb)) = (βCaVa((bb)))2(68)
(cid:12) (cid:12)
(cid:12) 0 pb heib (cid:12) 1 ∂ e˜ a(b)
(cid:12)(cid:12) e a e b he2i (cid:12)(cid:12) CVa(b) ≡ k ∂hTia(b) (69)
(cid:12)h i h i h i (cid:12) b
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
For the more general case of a Mth-order hypoequilib- V. MODEL: COMPOSITE SYSTEM IN A
rium state and the system separated into M subspaces, NONEQUILIBRIUM STATE
equation (63) remains the same but with
Interacting systems can form a composite nonequi-
β B3 = MK pKA˜K1 βK (70) librium system with the interaction resulting in the
≡ B1 P MK pKA˜K1 nonequilibrium relaxation process for the composite.
UsingtheSEAequationofmotion,thestateevolutionof
A given interaction type P(e.g., heat diffusion) results thecompositesystemcanbedetermined. Properdivision
in a given relaxation time τ (see Sec. II.B), while the of this composite allows the subspaces to be viewed as
ratio B3/B1 provides an average temperature based on the interacting subsystems within the composite. The
subspace mole fractions and energy fluctuations. At behavior of each subsystem, thus, can be studied by
stable equilibrium, β = βeq. Now, if one subspace R an analysis of the state evolution of each subspace.
is attached to a reservoir, the evolution of the other In particular, if two individual subsystems involved in
subspaces behave according to the equation of motion, an interaction both have canonical state distributions,
Eq. (63), with βR constant. For example, only part and each subsystems energy eigenvectors span one
of the energy eigenlevels can absorb energy from the subspace, the composite system is in a secondorder
environment. Mathematically, if subspace R yields to hypoequilibrium state. On the other hand, if each
one of two conditions, individual subsystem’s state cannot be described by a
canonical distribution, a higher-order hypoequilibrium
K =R,CR CK, pR pK (71)
∀ 6 V ≫ V ≫ state will be required.
the relation β = βR holds and subspace K(= R) yields In the following section, (Sec. V.A), the SEAQT
6 framework using single particle energy eigenlevels is
to the equation of motion.
applied to the study of a simple system. Section V.B
dp˜K 1 p˜K then explains the process of subspace division followed
dtj = τp˜Kj [(−lnnjK −hs˜iK)−βR(ǫKj −he˜iK)] (72) in Sec. V.C by a comparisonwith the phenomenological
j equations for mass and heat diffusion. In Sec. V.D, the
Notethatinthis equation,theonlyparameterrelatedto physicaldetailsofthesystemusedaredescribed. Finally,
subspace R is the reservoirtemperature βR. The energy in Sec. V.E, the coupling of mass and heat diffusion is
eigenstructure of subspace R plays no role! modeled and discussed.
B. Property of heat diffusion: A. Multi-particle classical simple system
non-quasi-equilibrium process and second law of
thermodynamics Theoretically and in general, the SEAQT equation of
motion is applicable to multi-particle systems provided
Based on Eq. (62), the total entropy and energy energy eigenstructure of the system is known [23].
evolution in one subspace can be determined via However, for a multi-particle classical simple system,
the energy eigenstructure of a single particle and its
dSK d s K d s˜ K 1
= h i =pK h i = pK(βK β)βKA˜K(73) associated equation of motion can be used to study the
dt dt dt τ − 1 system, since all of the particles (or particle groups)
dEdtK = dhdetiK =pKdhde˜tiK = τ1pK(βK −β)A˜K1 (74) hanavdedtehgeenesaramcey enne,rgiy=ei1g,e2n,l3e,v.e.l.s.{ǫTi,hius=, th1e,2s,y3s,t.e.m.}
i
{ }
state can be represented by the particle number (or
Dividing Eq. (73) by (74) yields
particlegroupnumber)ateachenergyeigenlevel m ,i=
i
{
dSK dSK dEK 1,2,3,... . The mole fraction of particles at the ith
= / =βK (75) energy e}igenlevel is given by y = m / m . The
dEK dt dt i i i
extensive property constraints of the system are then
The equation is a generalized form of the differential P
entropy transfer due to heat diffusion using the M = m =constant (77)
i
nonequilibrium temperature for each subspace, i.e.,
i
X
δQK E = ǫimi =constant (78)
dSK =βKdEK = (76)
T Xi
y
i
S = m ln (79)
Moreover, Eq. (76) can applied to all kinds of − i n
i
i
thermodynamic processes, and is not limited to quasi- X
equilibrium processes. This argument comes from Dividing the constraints by the total particle number
the universal definition of nonequilibrium temperature m , the system state can be represented by the mole
i
provided in this paper. fractions y ,i = 1,2,3,... , which is equivalent to the
i
P { }
9
single-particleprobabilitydistribution p ,i=1,2,3,... .
i
{ }
For an isolated system the constraints become
I = y = p =constant (80)
i i
i i
X X
E
e = = ǫ y = ǫ p =constant (81)
i i i i
h i M
i i
X X
S y p
i i
s = = y ln = p ln (82)
i i
h i M − n − n
i i
i i
X X FIG. 1. For mass diffusion, mass flow and energy flow are
... represents an average specific property. The both allowed across the partition. For heat diffusion, only
chonsitraints and equation of motion reduce to the single- energy flow is allowed.
particle case (Eqs. (5) to (7) and (14)) when y =p ,i=
i i
1,2,3,....
subsystem on the “left” and that on the “right”. This
In general, a systems energy eigenstructure and
wavelength represents the classical limit, i.e.,
extensive properties are a function of the total particle
number of each constituent. This is also true of its h h
p = 2mk T, λ= = , ∆L λ (83)
specific properties in the presence of a mass interaction v b p √2mk T ≫
v b
or chemical reaction if the system is partitioned and p
not simple [25], since partitioning influences each where pv is the expectation value of the particle
partition’s energy eigenstructure. Thus, for mass momentum for a system at temperature T.
diffusion, the framework outlined here requires an
invarianteigenstructureandasaconsequencethe simple
C. Phenomenological transport equation
system assumption. With this assumption, particle
numbernolongerinfluencesthe specific properties. This
same assumption, however,is not required in the case of With the assumption that the two subspaces of
heatdiffusionsincethetotalnumberofparticlesforeach the system are two subsystems at two positions, the
system partition (i.e., subsystem) does not change. phenomenological transport equations of mass and heat
diffusion can be derived.
B. Interacting systems
1. Mass diffusion
It is assumed that a group of observable operators
Fˆ commuting with the Hamiltonian operator Hˆ exists. Using Eq. (28), the equations of motion for two
subspaces are written as
The degenerate energy eigenlevels of the system can be
distinguished by eigenvalues of the observations of Fˆ, dlnpa 1 1 A A
= lnpa+ ( s˜ a 2 + e˜ a 3) (84)
1wen,h2eir,cg3hy,.h.e.aigveewnlivetahvleuldessecgaFenn1,ebFrea2cs,ye·p··arn,aFKtMe,di,i=nsotot1Mh,a2t,s3et.t.hs.e{,ǫsKiyws,htieerm=e ddlntpb =−τ1 lnpb+τ1(hs˜ib− AA12 +he˜ibAA31) (85)
K = 1,} ,M. In each se{t,ievery energy e}igenlevel dt −τ τ h i − A1 h i A1
represent·s·a·n eigenstate common to both Fˆ and Hˆ with When the two subspaces of the system have the same
the same eigenvalueofFˆ. Eigenstatesin eachofthe sets eigenstructure and temperature, subtracting Eq. (85)
canbespannedintoasubspaceofthesystemstatespace from Eq. (84) yields
and can be designed as a subsystem.
d 1
Practically, by choosing the observable operator, the (lnpa lnpb)= (lnpa lnpb) (86)
dt − −τ −
system can be view as an composite system whose
subsystems can be properly arranged to study specific Substituting the subspace probability(or molefraction),
phenomena. For the kinetics of a chemical reaction, F Eq. (87) into Eq. (86) results in
can be chosen to be an observable operator of species,
na nb
whose eigenvalues are “Reactant” and “Product”. In pa = pb = (87)
heat and mass diffusion, the observable operator F is na+nb na+nb
chosen to be the relative location to a partition, with d na 1 na
(ln )= (ln ) (88)
eigenvalue of “left” (left of the partition) and “right” dt nb −τ nb
(right of the partition). The partition allows mass
Iftheglobalmassdistributioniscontinuous,andposition
and heat diffusion (see Fig. 1).To be more precise, it
A and position B are close enough,
is assumed that the de Brogli wavelength λ is much
smaller than the distance ∆L between the center of the na =nb+∆n (89)
10
which transforms Eq. (88) into fluctuation (or nondimensional specific heat at constant
volume) of the subspaces, i.e.,
d ∆n 1∆n
( )= (90)
dt nb −τ nb κ′ = 1 C 1, κ= 1 δxC (95)
1 d na nb δx dn 2τ V A 2τ A V
Ja(b) (na nb)= − = (91)
≡ 2Adt − − 2τA −2τAdx
The above formulation is applicable for any kind
where the approximation ln(1 + x) x for small of interaction resulting in a flow of energy only.
x has been used and higher-order ter≃ms dropped to Furthermore, if the heat and mass diffusion are affected
arrive at Eq. (90). In Eq. (91), δx is the distance via the same kind of micro-mechanical interactions such
between two positions, A is the cross-sectional area of as the collision of particles, it can be assumed that the
the interacting surface, and Ja(b) is the flow of particle same τ is applicable when the system is in the near-
number equal to (dna/dt)/A or ( dnb/dt)/A (hence the equilibrium region near to the same stable equilibrium
division by 2 in Eq. (91)). Eq−. (91) recovers Fick’s point. In this case, κ = CVD. This last result is the
law with the diffusion coefficient (diffusivity) given by same as that found from classical transport theory and
D = δx . The specific form of D is directly related to isadirectconsequenceofthe thermodynamicfeaturesof
2τA
the form of τ, which contains the detailed mechanical the system minus any direct knowledge of the details of
information. The phenomenological linear equation can the micro-mechanicalinteractions taking place.
be derived without the form of τ, which is a pure
thermodynamic feature orpattern ofthe nonequilibrium
D. Mass and heat diffusion of hydrogen
relaxation process. In addition to results such as these
for the near-equilibrium realm, thermodynamic features
orpatternsinthe far-from-equilibriumrealmcanalsobe To model the mass and heat diffusion for a specific
studiedusingEqs. (35)and(36)providedtheinitialstate case, a composite system of hydrogen is set up with two
is a hypoequilibrium state. For the case when it is not, subspaces correspondingto subsystems on two sides of a
Eq. (14) can directly be used as is done, for example, in partition. The energy eigenlevels of the two subsystems
[16]. together form the energy eigenlevels for the composite
system as a whole. Denoting the state space of the
subsystem on the “left” by a and that on the “right”
2. Heat diffusion by b, the composite systeHm state space takes the
H H
form
Forsysteminwhichtheonlyinteractionisthatofheat
diffusion, Eq. (76) captures the energyflow betweentwo
= a b (96)
subsystems, i.e., H H ⊕H
dT 1 Theavailableenergyeigenvaluesforonesubspace(“left”
Q˙ =JEa(b) =−κ′δxAdx = τpa(βa−β)A˜a1 or “right”) are constructed from the energy eigenvalues
of each degree of freedom for translation and rotation,
1 papbA˜aA˜b
= 1 1 (βa βb) (92) i.e., from
τ paA˜a+pbA˜b −
1 1 ǫa(b) =ǫ +ǫ (97)
t,H2 r,H2
where Q˙ is the rate of energy transferred, Ja(b) is
E The translational energy eigenvalue ǫ uses the form of
the heat flux, A is the cross-sectional area of the t
the infinite potential well, while the rotational energy
interactingsurface,TaandTbaretemperatureofthetwo
eigenvalueǫ usesthe formofthe rigidmotor. Theseare
subsystems,andEq. (65)hasbeensubstitutedforβ. Eq. r
expressed as follow:
(92) recoversFourierslawofheatdiffusion(conduction).
Thethermalconductivityperunitlengthκ′ andthermal ~2 n2 n2 n2
conductivity κ are expressed as: ǫt(nx,ny,nz)= 8m Lx2 + Ly2 + Lz2 (98)
x y z!
1papbβaβbA˜aA˜b 1 j(j+1)~2
κ′ = τ paA˜a+pb1A˜b1A, κ=κ′δx (93) ǫr(j,m)= 2I (99)
1 1
where n , n , and n are the quantum numbers for
Inthe near-equilibriumregionwiththe sameconstituent x y z
the translational degrees of freedom, j and m are the
in the two subsystems,
quantum numbers for the rotational degrees of freedom,
1 I is the moment of inertia and Lx, Ly, and Lz are
pa =pb = 2, CVa =CVb =CV (94) chosen based on mean-free-path of the particles in
each subsystems (e.g., the dimension of subsystem is
and the thermal conductivity per unit length and the used for an ideal gas). The vibrational energy are
thermalconductivityareexpressedintermsoftheenergy not included since the temperature in the study is