A Model of Electrodiffusion and Osmotic Water Flow and its Energetic Structure

0.48 MB·English

by  Yoichiro Mori

A Model of Electrodiffusion and Osmotic Water Flow
and its Energetic Structure
Yoichiro Moria,∗, Chun Liub, Robert S. Eisenbergc
1
aSchool of Mathematics, Universityof Minnesota, Minneapolis, MN 55414, U.S.A.
1
bDepartment of Mathematics, Pennsylvania State University,UniversityPark, PA 16802,
0
U.S.A.
2
cDepartment of Molecular Biophysics and Physiology, Rush University Medical Center,

n Chicago, IL 60612, U.S.A.
a
J

7
2
Abstract
]
B
We introduce a modelfor ionic electrodiffusionand osmotic water flow through
C
cells and tissues. The model consists of a system of partial differential equa-
. tionsforionicconcentrationandfluidflowwithinterfaceconditionsatdeforming
o
i membrane boundaries. The model satisfies a natural energy equality, in which
b
the sum of the entropic, elastic and electrostatic free energies are dissipated
-
q throughviscous, electrodiffusive and osmotic flows. We discuss limiting models
[ when certain dimensionless parametersare small. Finally, we develop a numer-

1 ical scheme for the one-dimensional case and present some simple applications
v of our model to cell volume control.
3
9
1
1. Introduction
5
.
1 Systemsinwithimportantelectrodiffusionandosmoticwaterflowarefound
0 throughout life [1, 2, 3]. Such systems include brain ionic homeostasis [4, 5],
1
fluidsecretionbyepithelialsystems[6],electrolyteregulationinthekidney[7,8],
1
fluid circulation in ocular systems [9, 10], and water uptake by plants [11].
:
v Mathematicalmodelsofelectrodiffusionand/orosmosishavebeenproposed
i
X and used in many physiological contexts, and have formed a central topic in
biology for a very long time [1, 12, 13]. Some are simple models using ordinary
r
a differentialequationswhileothersaremoredetailedinthatthey includepartial
differential equations (PDEs) describing the spatial variation of the concentra-
tionandflowfields[14,15,16,17,18,19,20]. Inthispaper,weproposeasystem
of PDEs that describes ionic electrodiffusion and osmotic water flow at the cel-
lularlevel. Tothebestoftheauthors’knowledge,thisisthefirstmodelinwhich
osmoticwaterflowandelectrodiffusionhavebeentreatedwithinaunifiedframe-
work including cells with deformable and capacitance-carrying membranes. A
∗CorrespondingAuthor
Email addresses: [email protected] (YoichiroMori),[email protected] (ChunLiu),
[email protected] (RobertS.Eisenberg)
Preprint submitted toElsevier January 28, 2011
salientfeature ofourmodel is thatitpossessesa naturalthermodynamic struc-
ture;itsatisfiesafreeenergyequality. Assuch,thepresentworkmaybeviewed
as a generalization of the classical treatment of osmosis and electrodiffusion in
irreversible thermodynamics to spatially extended systems [21, 22, 23].
To introduce our approach, we first focus attention on uncharged systems.
In Section 2, we treat the case in which the diffusing chemical species carry no
electriccharge. Wewritedownequationsthataresatisfiedbythewatervelocity
fieldu,thechemicalconcentrationsc ,k =1,··· ,N andthemembraneposition
k
X. Themodelisshowntosatisfyafreeenergyequalityinwhichthesumofthe
entropicfreeenergyandtheelasticenergyofthemembraneisdissipatedthrough
viscous water flow, bulk diffusion, transmembrane chemical fluxes and osmotic
water flow. One interesting consequence of this analysis is that the classical
van t’Hoff law of osmotic pressure arises naturally from the requirement that
osmoticwaterflowbedissipative. Wenotethatmodelswiththesimilarpurpose
ofdescribingdiffusingnon-electrolytesandtheirinteractionwithosmoticwater
flowacrossmovingmembranes,havebeenproposedintheliterature[19,24,25]
(in the Appendix Appendix A.2, we discuss the relationshipof our model with
that of [19, 24]).
In Section 3, we extend the model of Section 2 to treat the case of ionic
electrodiffusion. We introduce the electrostatic potential φ which satisfies the
Poisson equation. The membrane now carries capacitance, which can result in
a jump in the electrostatic potential across the membrane. We shall see that
this model also satisfies a free energy equality. The free energy now includes
an electrostatic contribution. The verification of the free energy equality in
this case is not as straightforwardas in the non-electrolyte case, and requires a
careful examination of surface terms.
In Section 4, we discuss simplifications of our model. We make the system
dimensionless and assess the relative magnitudes of the terms in the equations.
An important simplification is obtained when we take the electroneutral limit.
Inthiscase,theelectrostaticpotentialbecomesaLagrangemultiplierthathelps
to enforce the electroneutrality condition.
In Section 5, we develop a computational scheme to simulate the limiting
system obtained in the electroneutral limit, when the geometry of the cell is
assumed spherical. As an application, we treat animal cell volume control.
2. Diffusion of Non-electrolytes and Osmotic Water Flow
2.1. Model Formulation
Consider a bounded domain Ω ⊂ R3 and a smooth closed surface Γ ⊂ Ω.
This closed surface divides Ω into two domains. Let Ω ⊂ Ω be the region
i
boundedbyΓ,andletΩ =Ω\(Ω ∪Γ). Inthecontextofcellbiology,Ω maybe
e i i
identified with the intracellular space and Ω the extracellularspace. Although
e
cell physiologicalsystems of biologicalcells serve as our primary motivation for
formulating the models of this paper, this identification is not necessary.
2
In this section we formulate a system of PDEs that governs the diffusion of
non-electrolytes and osmotic flow of water in the presence of membranes. In
Section 3, we shall build upon this model to treat the electrolyte case.
WeconsiderN non-electrolytechemicalspecieswhoseconcentrationswecall
c ,k=1,··· ,N. Let ω be the entropic part of the free energy per unit volume
k
of this solution. The following expression for ω is the most standard choice:
N
ω = k Tc lnc . (2.1)
0 B k k
k=1
X
This expression is valid when the ionic solution is sufficiently dilute and lead
to linear diffusion of solute. Our calculations, however, do not depend on this
choiceofω. Ifthe solutioninquestiondeviatessignificantlyfromideality,other
expressions for ω may be used in place of ω .
0
Given ω, the chemicalpotential µ ofthe k−thchemicalspecies is givenas:
k
∂ω
µ =σ , σ ≡ . (2.2)
k k k
∂c
k
We have introduced two symbols µ and σ in anticipation of the discussion of
k k
the electrolyte case, where µ and σ are different. For water, it is convenient
k k
toconsiderthewaterpotentialψ ,thefreeenergyperunitvolume,ratherthan
w
the µ , the chemical potential (free energy per molecule). The water potential
w
andwaterchemicalpotentialarethus relatedbythe relationv ψ =µ where
w w w
v is the volume of water per molecule. We define:
w
N N
∂ω
ψ ≡π +p, π = ω− c σ = ω− c (2.3)
w w w k k k
∂c
! k!
k=1 k=1
X X
where p is the pressure. As we shall see, p will be determined in our model via
the equations of fluid flow (Eq. (2.8)). The entropic part of ψ (or the osmotic
w
pressure), π ,is givenasthe negativemultiple ofthe (semi)Legendretransform
w
ofω with respectto allthe ionic concentrationsc . This expressionfor osmotic
k
pressure can be found, for example, in [26]. As we shall see in the proof of
Theorem 2.1, the above definition of π is forced upon us if we insist that our
w
model satisfy a free energy dissipation principle.
We begin by writing down the equations of ionic concentration dynamics.
At any point in Ω or Ω
i e
∂c D
k +∇·(uc )=∇· c k ∇µ (2.4)
k k k
∂t k T
(cid:18) B (cid:19)
where D is the diffusion coefficient and u is the fluid velocity field. Ions thus
k
diffusedownthechemicalpotentialgradientandareadvectedwiththelocalfluid
velocity. We have assumed here that cross-diffusion (concentration gradient of
one species driving the diffusion of another species) is negligible.
3
We must supplement these equations with boundary conditions. Most for-
mulations of non-equilibrium thermodynamic processes seem to be confined ei-
ther to the bulk or to the interface between two bulk phases [21, 27, 23]. Here
we must couple the equations in the bulk and with boundary conditions at the
interface, which as a whole give us a consistent thermodynamic treatment of
diffusion and osmosis.
On the outer boundary Γ =∂Ω, for simplicity, we impose no-flux bound-
out
ary conditions. Let us now consider the interfacial boundary conditions on the
membraneΓ. Sincewewanttoaccountforosmoticwaterflow,themembraneΓ
will deform in time. Sometimes, we shall use the notation Γ to make this time
t
dependenceexplicit. LetΓ betherestingorreferenceconfigurationofΓ. The
ref
membranewillthenbeasmoothdeformationofthis referencesurface. We may
take some (local) coordinate system θ on Γ , which would serve as a material
ref
coordinatefor Γ . Thetrajectoryofa pointthatcorrespondsto θ =θ is given
t 0
by X(θ ,t)∈R3. For fixed t, X(·,t) gives us the shape of the membrane Γ .
0 t
Consider a point x=X(θ,t) on the membrane. Let n be the outward unit
normalonΓatthispoint. Theboundaryconditionssatisfiedontheintracellular
and extracellular faces of the membrane are given by:
D ∂X
c u− k ∇µ ·n=c ·n+j +a on Γ or Γ . (2.5)
k k k k k i e
k T ∂t
(cid:18) B (cid:19)
The expression “on Γ ” indicates that the quantities are to be evaluated on
i,e
the intracellular and extracellular faces of Γ respectively. The term j is the
k
passive chemical flux that passes through the membrane and a is the active
k
flux. Fluxes going from Ω to Ω is taken to be positive. Equation (2.5) is just
i e
astatementofconservationofionsatthemovingmembrane. Itiseasytocheck
that (2.4) together with (2.5) implies conservation of each species.
The flux j is in general a function of concentrations of all chemical species
k
on both sides of the membrane. The chemicals are usually carried by channels
and transporters, and the functional form of j describe the kinetic features of
k
these carriers. As we shall see in Section 2.3, j can also be a function of the
k
difference in water chemical potential across the membrane.
The passive nature of the j is expressed by the following inequality:
k
[µ ]j ≥0, [µ ]= µ | − µ | (2.6)
k k k k Γi k Γe
where ·| expresses evaluation of quantities at the intracellular and extra-
Γi,e
cellular faces of the membrane Γ respectively. We shall see that this condi-
tion is consistent with the free energy identity (2.20). For any quantity w,
[w]= w| − w| will henceforth always denote the difference between w eval-
Γi Γe
uated on the intracellular and extracellular faces of Γ. A simple example of j
k
occurs when j is a function only of [µ ] and satisfies the following conditions:
k k
∂j
k
j =j ([µ ]), j (0)=0, ≥0. (2.7)
k k k k
∂[µ ]
k
4
It is easily checked that j in this case satisfies (2.6). We shall see concrete
k
examplesinSection3. Condition(2.6)issomewhatrestrictiveinthe sensethat
it is possible to have a “passive” flux that does not satisfy (2.6) when multiple
speciesflowandinteract. We shalldiscussthis briefly inSection2.3. Condition
(2.6)needstobe relaxedtodescribesystemsinwhichdifferentchemicalspecies
flowthroughonechannelor(passive)transporter. Thosesystemsusuallycouple
fluxes of different chemical species. They often couple (unidirectional) influx
and efflux of the same species (symporters and antiporters) [28, 12, 29, 2, 3].
The active flux a is typically due to ionic pump currents often driven by ATP
k
[29, 2, 3].
We now discuss force balance. We shall treat the cytosol as a viscous fluid
and the cell membrane as an elastic surface. The cell membrane itself is just a
lipidbilayer,andcannotsupportalargemechanicalload. Thecellmembraneis
oftenmechanicallyreinforcedbyanunderlyingactincortexandoverlyingsystem
ofconnectivetissue,andinthecaseofplantcells,byanoverlyingcellwall. Ifwe
view these structures as partof the membrane, our treatmentof the membrane
being elastic may be a useful simplification. We could employ a more complete
model of cell mechanics incorporating, in particular, the mechanical properties
of the cytoskeleton and extracellular lamina. However,our emphasis here is on
demonstratinghowosmosiscanbeseamlesslycombinedwithmechanics,andwe
intentionally keep the mechanical model simple to clarify the underlying ideas.
Consider the equations of fluid flow. The flow field u satisfies the Stokes
equation at any point in Ω or Ω :
i e
ν∆u−∇p=0, ∇·u=0 (2.8)
whereνistheviscosityoftheelectrolytesolution. Notethattheaboveequations
can also be written as follows:
∇·Σ (u,p)=0, ∇·u=0, Σ (u,p)=ν(∇u+(∇u)T)−pI (2.9)
m m
where I is the 3×3 identity matrix and (∇u)T is the transpose of ∇u. Here,
Σ is the mechanical stress tensor. It is possible to carry out much of the
m
calculationsto followevenif weretaininertialtermsandworkwiththe Navier-
Stokes equations or use other constitutive relations for the mechanical stress.
In particular, such modifications will not destroy the free energy identity to be
discussed below. We do note, however, that incompressibility is important for
our computations.
We now turn to boundary conditions. We let u =0 on the outer boundary
Γ for simplicity. On the cell membrane Γ, we have the following conditions.
out
Take a point x = X(θ,t) on the boundary Γ, and let n be the unit outward
normal on Γ at this point. First, by force balance, we have:
[Σ (u,p)]=F . (2.10)
m elas
Here, F is the elastic force per unit area of membrane.
elas
We make some assumptions about the form of the elastic force. We assume
5
that the membrane is a hyperelastic material in the sense that the elastic force
canbe derivedfrom an elastic energy functional E that is a function only of
elas
the configuration X:
E (X)= E(X)dm (2.11)
elas Γref
ZΓref
where m is the surface measure of Γ and E is the elastic energy density
Γref ref
measured with respect to this measure. It is possible that E is a function of
spatial derivatives of X. The elastic force F (x) satisfies the relation:
elas
d
E(X(θ)+sY(θ))dm =− F (x)·Y(X−1(x))dm (2.12)
ds Γref elas Γ
(cid:12)s=0ZΓref ZΓ
(cid:12)
wher(cid:12)(cid:12)e Y is an arbitrary vector field defined on Γref and mΓ is the natural
measure on the surface Γ and is related to m by dm = Qdm where Q
Γref Γ Γref
is the Jacobiandeterminantrelating Γ to the reference configurationΓ . The
t ref
expression X−1(x) is the inverse of the map x = X(θ). Thus, F is given
elas
as the variational derivative of the elastic energy up to the Jacobian factor Q.
Consequently, we have the following relation:
d ∂X
E (X)=− F · dm . (2.13)
elas elas Γ
dt ∂t
ZΓ
Intheabove, ∂X shouldbethoughtofasafunctionofx,i.e., ∂X = ∂X(X−1(x)).
∂t ∂t ∂t
Weshallhenceforthabusenotationandlet ∂X beafunctionofxorθdepending
∂t
on the context of the expression.
In addition to the force balance condition (2.10), we need a continuity con-
ditionon the interface Γ. Since we are allowingfor osmoticwater flow, we have
a slip between the movement of the membrane and the flow field. At a point
x=X(θ,t) on the boundary Γ we have:
∂X
u− =j n (2.14)
w
∂t
wherej iswaterfluxthroughthemembrane. Wearethusassumingthatwater
w
flow is always normal to the membrane and that there is no slip between the
fluid and the membrane in the direction tangent to the membrane. Given that
n is the outward normal, j is positive when water is flowing out of the cell.
w
Like j in (2.5), we let j be a passive flux in the following sense. We let:
k w
j =j ([ψ ]), ψ = π | − ((Σ (u,p))·n)| (2.15)
w w w w Γi,e w Γi,e m Γi,e
(cid:12)
c c(cid:12)
where π was the entropic con(cid:12)tribution to the water potential defined in (2.3).
w
Ingeneral,j canbeafunctionofothervariables(seeSection2.3). Thefunction
w
j satisfies the condition, analogous to (2.6):
w
[ψ ]j ≥0. (2.16)
w w
c
6
This condition is clearly satisfied if:
∂j
w
j =j ([ψ ]), j (0)=0, >0. (2.17)
w w w w
∂[ψ ]
w
c
Waterflow acrossthe membrane is thus drivenby the difference inthe entropic
c
contribution to the chemical potential as well as the jump in the mechanical
force across the cell membrane. When the flow field u is equal to 0, the above
expression for ψ reduces to:
w
c ψ = π | + p| = ψ | . (2.18)
w Γi,e w Γi,e Γi,e w Γi,e
(cid:12)
c(cid:12)
We may thus view ψ a(cid:12)s a modification of ψ to take into account dynamic
w w
flow effects. Let ω = ω given in (2.1) in the definition (2.3) of π . Then, we
0 w
have, under zero flowcconditions,
N
[ψ ]= p−k T c . (2.19)
w B k
" #
k=1
X
c
Wethusreproducethestandardstatementthatwaterflowacrossthemembrane
isdrivenbythedifferenceinosmoticandmechanicalpressure,wheretheosmotic
pressure, π , is given by the van t’Hoff law.
w
2.2. Free Energy Identity
We now show that the system described above satisfies the following free
energy identity.
Theorem 2.1. Suppose c ,u,p be smooth functions that satisfy (2.4), (2.8),
k
and in Ω and Ω and satisfy boundary conditions (2.5), (2.10), (2.14) on the
i e
membrane Γ. Suppose further that c and satisfy no-flux boundary conditions
k
andu=0 on the outer boundaryΓ . Then, c ,u,pand φ satisfy the following
out k
free energy identity.
d
(G +E )=−I −J −J
S elas p p a
dt
G = ωdx
S
ZΩi∪Ωe
N
D
I = ν|∇u|2+ c k |∇µ |2 dx
p k k
ZΩi∪Ωe k=1 kBT ! (2.20)
X
N
J = [ψ ]j + [µ ]j dm
p w w k k Γ
ZΓ k=1 !
X
Nc
J = [µ ]a dm .
a k k Γ
ZΓ k=1 !
X
7
Here, E was given in (2.11) and |∇u| is the Frobenius norm of the 3×3 rate
elas
of deformation matrix ∇u.
If a ≡0, then the free energy is monotone decreasing.
k
The free energy is given as a sum of the entropic contribution G and the
S
membrane elasticity term E . This free energy is dissipated through bulk
elas
currentsI andmembranecurrentsJ . Dissipationinthebulkcomesfromionic
p p
electrodiffusion and viscous dissipation. Dissipation at the membrane comes
fromionicchannelcurrentsandtransmembranewaterflow. Ifactivemembrane
currents are present, they may contribute to an increase in the free energy
through the term J . It is sometimes useful to rewrite J +I as:
a p p
J +I =F +F ,
p p w c
F = ν|∇u|2dx+ [ψ ]j dm ,
w w w Γ
ZΩi∪Ωe ZΓ (2.21)
N D c N
F = c k |∇µ |2dx+ [µ ]j dm ,
c k k k k Γ
k T
ZΩi∪Ωek=1 B ZΓk=1
X X
where F and F are the dissipations due to water flow and solute diffusion
w c
respectively. In the statement of the Theorem, it is important that (2.6) and
(2.16) are used only to conclude that J be positive. Identity (2.20) should be
p
seen as giving us the definition of what a passive current should be.
We now prove Theorem 3.1. An interesting point about the calculation
to follow is how dissipation through transmembrane water flow comes from
two different sources, equations for ionic concentration dynamics and the fluid
equations. The former contributes the osmotic term π term, and the latter
w
contributesthemechanicaltermp,whichtogetheradduptothewaterpotential
ψ .
w
Proof of Theorem 2.1. First, multiply (2.4) with µ in (3.1) and integrate over
k
Ω and sum in k:
i
N N
∂c D
µ k +∇·(uc ) dx= µ ∇· c k ∇µ dx. (2.22)
k k k k k
∂t k T
k=1ZΩi (cid:18) (cid:19) k=1ZΩi (cid:18) B (cid:19)
X X
The summand in the right hand side becomes:
D D
µ ∇· c k ∇µ dx= µ c k ∇µ ·n dm
k k k k k k Γ
k T k T
ZΩi (cid:18) B (cid:19) ZΓi(cid:18) B (cid:19) (2.23)
D
− c k |∇µ |2 dx
k k
k T
ZΩi(cid:18) B (cid:19)
8
where n is the outward normal on Γ. Consider the left hand side of (2.22).
N N
∂c ∂ω ∂c ∂ω
µ k +∇·(uc ) = k +u·∇c = +∇·(uω), (2.24)
k k k
∂t ∂c ∂t ∂t
k=1 (cid:18) (cid:19) k=1 k (cid:18) (cid:19)
X X
where we used (2.2) and the incompressibility condition in (2.8). Integrating
the above over Ω , we have:
i
∂ω ∂ω
+∇·(uω) dx= dx+ ωu·ndm
Γ
∂t ∂t
ZΩi(cid:18) (cid:19) ZΩi ZΓi (2.25)
d ∂X
= ωdx+ ω u− ·ndm
Γ
dt ∂t
ZΩi ZΓi (cid:18) (cid:19)
where we used the fact that u is divergence free in the first equality. The term
involving ∂X comes from the fact that the membrane Γ is moving in time.
∂t
Performing similar calculations on Ω , and adding this to the above, we find:
e
d
ωdx+ [ω]j dm
w Γ
dt
ZΩi∪Ωe ZΓ
(2.26)
N N
D D
= µ c k ∇µ ·n dm − c k |∇µ |2 dx.
k k k Γ k k
k T k T
k=1ZΓ(cid:20) B (cid:21) k=1ZΩi∪Ωe(cid:18) B (cid:19)
X X
whereweused(2.14). Using(2.5)and(2.14),wemayrewritethesecondbound-
ary integral as follows:
D
µ c k ∇µ ·n dm = ([µ c ]j −[µ ](j +a ))dm . (2.27)
k k k Γ k k w k k k Γ
k T
ZΓ(cid:20) B (cid:21) ZΓ
We now turn to equation (2.8). Multiply this by u and integrate over Ω :
i
u·(ν∆u−∇p)dx= (Σ (u,p)n)·udm − ν|∇u|2dx=0 (2.28)
m Γ
ZΩi ZΓi ZΩi
Performing a similar calculation on Ω and adding this to the above, we have:
e
[(Σ (u,p)n)]·udm − ν|∇u|2dx=0 (2.29)
m Γ
ZΓ ZΩi∪Ωe
We may use (2.10), (2.13) and (2.14) to find
d
E (X)= [(Σ (u,p)n)·n]j dm − ν|∇u|2dx (2.30)
elas m w Γ
dt
ZΓ ZΩi∪Ωe
9
Combining (2.26), (2.27) and (2.30), we have:
d
ωdx+E (X)
elas
dt
(cid:18)ZΩi∪Ωe (cid:19)
N
D
=− c k |∇µ |2 dx− ν|∇u|2dx (2.31)
k k
k T
k=1ZΩi∪Ωe(cid:18) B (cid:19) ZΩi∪Ωe
X
− [µ ](j +a )dm − [ω−c σ −(Σ (u,p)n)·n]j dm
k k k Γ k k m w Γ
ZΓ ZΓ
Recalling the definition of ψ in (2.15), we obtain the desired equality. In
w
the absence of active currents a , is decreasing given that j and j satisfy
k k w
conditions (2.6) and (2.16) recspectively.
In the last line of the above proof, note that the expression for ψ arises
w
naturally as a result of integrating by parts. In this sense, we may say that
osmotic water flow arises as a natural consequence of requiring thatcthe free
energy be decreasing in time.
2.3. Cross Coefficients and Solvent Drag
As can be seen from (2.20) or (3.15) the only condition we need to impose
forthe free energyto decreasewith time in the absenceofactive currentsis the
following:
N
[ψ ]j + [µ ]j ≥0. (2.32)
w w k k
k=1
X
c
This condition is weaker than conditions (2.6) and (2.16) being satisfied sep-
arately by j and j . We now discuss an important case in which j and j
k w k w
maynotindividually satisfy (2.6)and(2.16)but (2.32)is satisfiednevertheless.
This ariseswhenever fluxes arecoupledas is usually the case for fluxes through
transporters or (single filing) channels [28, 12, 29, 2, 3]. We note that such
cross-diffusion can be relevant even in bulk solution [30, 31, 32, 33, 34].
If [µ ] and [ψ ] remain small, the dissipation J in (2.20) may be approxi-
k w p
mated by a quadratic form in the jumps:
c
J = [µ]·jdm = [µ]·(L[µ])dm ,
p Γ Γ
ZΓ ZΓ (2.33)
µ=(µ ,··· ,µ ,ψ )T,j=(j ,··· ,j ,j )T,
1 N w 1 N w
where L is a symmetric (N +1)×c(N +1) matrix. Requiring that the free
energy be decreasing implies that L must be positive definite. The maximum
dissipation principle requires that j be given as variational derivatives of J /2
p
with respect to [µ]:
j=L[µ]. (2.34)
10