Shape-Dependent Thermodynamics and Non-Local Hydrodynamics in a Non-Gibbsian Steady-State of a Drift-Diffusion System

5 Pages·0.13 MB·English

by  Francis J. Alexander

Shape-Dependent Thermodynamics and Non-Local Hydrodynamics in a
Non-Gibbsian Steady-State of a Drift-Diffusion System
Francis J. Alexander
Center for Computational Science, Boston University, Boston, MA 02215 and
8 Gregory L. Eyink
9 Department of Mathematics, University of Arizona, Tucson, AZ 85721
9
(February 7, 2008)
1

n
a
themselves. Theyareexpectedtooccurinphysicaldrift-
J
Shape-dependent thermodynamics and non-local hydro-
diffusion systems, e.g. electrolytes and semiconductors.
5 dynamics are argued to occur in dissipative steady states of
Correlations of power-law type are predicted in such
2
driven diffusive systems. These predictions are confirmed by
systems by linear fluctuating hydrodynamics [1,2]. In a
numerical simulations. Unlike power-law correlations, these
] DDS with one species of (unit-charged) particle at con-
h phenomena cannot be explained by a hypothesis of “critical-
stantmeandensityρ,incontactwithaheatbathattem-
c ity”. Instead, they require the effective Hamiltonian of the
e system to contain very long-range potentials, making the in- peratureT andinanappliedelectricfieldE,theequation
m variant probability measures formally “non-Gibbsian”. in Fourier space for density fluctuations δρ(k,t) is
-
t PACS Numbers: 05.70.Ln, 66.30.Hs, 82.20.Mj, 64.60.Lx δρ˙(k,t)=[ic ·k−k·D ·k]δρ(k,t)−ikb·δJ(k,t). (1)
a E E
t
.s HerbecE(ρ)= ddjρE(ρ)isthedriftbvelocity,adenbsityderiva-
t Power-law decay of correlations are generic in dissi- tive of the conduction current. The latter is given, at
a
m pative steady-states of open, driven systems with con- low density ρ and small field-strength E, by Ohm’s law,
- servation laws [1,2]. In equilibrium systems such power jE = σ·E = ρµ·E, with σ the conductivity tensor and
d laws can be due to two different mechanisms: the inter- µ the mobility tensor. DE(ρ) is the diffusion tensor in
n action potentials in the system Hamiltonian can them- theconductingstate,whileδJ(k,t)istheFouriercompo-
o
selves be long-range power-laws (e.g. dipolar) or else nent of the current noise with zero mean and covariance
c b
[ the potentials can be short-ranged but the system may hδJ(k,t)δJ(k′,t′)i = 2Ls(ρ)δ3(k+k′)δ(t −t′). In the
E
be at a critical point [3]. The latter circumstance has limit E → 0, LE(ρ) coincides with the Onsager matrix
1 b b
prompted the view that dissipative, nonequilibrium sys- L=σk T. ItiseasytosolveEq.(1)forthe staticstruc-
v B
8 tems are attracted without any tuning of parameters to ture function SE(k)≡limt→∞hδρ(k,t)δρ(−k,t)i:
5 a critical state, or exhibit so-called “self-organized crit-
2 icality” (SOC) [4]. We shall show here that such an b S (k)= k·LE·bk. b (2)
1 interpretation, taken literally, is not true in an impor- E k·DE·k
0 tant class of such systems. As we shall explain, an effec- b
8 See [1]. Inverse Fourier transforming gives a long-range
9 tive Hamiltonian may be introduced to characterize the density-density correlation S (r) ∼ r−d in dimension d,
E
t/ nonequilibrium statistics. We then exhibit phenomena when LE and DE are not proportional. This will gener-
a that can be explained only by the presence there of ef-
ally be true for E 6=0. Suchpower-lawcorrelationshave
m fective power-law potentials of very long range. Such an
been verified in numerical simulations of simple driven
- explanation for power-law correlations was earlier pro- latticegas(DLG)models[9,10]. Correlationsofthistype
d
posed in driven diffusive systems [5] and, more recently,
n are expected more generally in locally driven/damped
in a shaken granular flow [6]. Because the underlying
o conservative systems without detailed balance. For ex-
c dynamics are local in these systems, this mechanism ample, r−d correlations of similiar origin have recently
: may be justly termed “self-organizedlong-range interac-
v been predicted and verified computationally in the ho-
i tions”(SOLRI).Weshalldemonstrateintheclassofsys- mogeneouscoolingstateofrapidgranularflows[11]. Our
X
tems considered two closely related phenomena: shape-
results have implications for all such systems.
r dependentthermodynamicsandnon-localhydrodynamics.
a We illustrate our points with the DLG models [9,12].
These phenomena couldnot occur if the effective Hamil-
These were originally introduced as models of solid elec-
tonian were short-ranged but critical. In fact, the in-
trolytes, or superionic conductors [13,14]. However, un-
duced potentials must have such an extreme long-range,
like physical drift-diffusion systems of charged particles,
many-body character that the nonequilibrium measures
these models have only short-ranged dynamical interac-
are formally non-Gibbsian. (For an excellent review of
tions and are thus perfect for our theoretical objectives.
the relevant notions, see [7].) The indicators of the
We shallcomment later how the results carryoverto re-
non-Gibbsian nature, shape-dependent thermodynamics
alistic DDS with Coulombic interactions. The particles
and non-local hydrodynamics, have great importance in of the DLG model live on a cubic lattice Zd. Assuming
1
hard-core exclusion, the occupancy η ∈{0,1} for each An analogy was already remarked a few years ago
x,t
site x∈Zd and time t≥0. The evolution of the config- between the DDS and dipolar systems [5]. Of course,
uration is via a Kawasaki exchange dynamics, specified dipole spin-spin correlations are also ∝ r−d even at
bytheratec (x,y;η)forexchangeofoccupancyofnear- high-temperature. This is consistent with our point of
E
est neighbor sites x,y in the configuration η, i.e. for the view,becausethedipolepotentialjustmissesbeingabso-
transition η → ηxy. In addition to assuming that rates lutely summable (and is thus “non-Gibbsian” according
arefunctions ofoccupanciesatsites withinafinite range to ourcriterion!) An importantconsequenceofthis non-
of {x,y}, the main assumption is local detailed balance: summability was early recognized [17], namely, that the
thermodynamics of dipolar systems is shape-dependent.
c (x,y;η)=c (x,y;ηxy)exp[−β(H(ηxy)−H(η)
E E This situation arises because the dipole potential energy
+E·(x−y)(η −η ))], (3)
x y sums are only conditionally convergent and hence may
leadtodifferentvaluesdependingupontheorderofsum-
for some short-rangedlattice-gasHamiltonianH(η), e.g.
mation, i.e. the shape, at least at nonzero field [18]. In
anIsingmodel. Thecondition(3)encouragesparticlesto
uniformly-magnetized (=high field), ellipsoidally-shaped
hopinthe directionofthe electricfieldEand,ininfinite
samples of dipolar materials the shape dependence of
volume, sets up an irreversiblesteady-state with a mean
thermodynamic free-energy functions is simply param-
current. In fact, these models have space-ergodic, ho-
eterized [19], in good agreement with experiment [20].
mogeneous, and time-invariant measures µ for each
ρ,E,β
It was argued in [8] that a similiar shape-dependence
density ρ∈[0,1], expected to be unique at small β. The
occursinDDS.Thethermodynamicfunctionsofinterest
questionariseswhetherthesemeasuresininfinitevolume
are the “pressure” p and the “Helmholtz free-energy”
are “Gibbsian” for an effective Hamiltonian E
f . The former is defined by the thermodynamic limit
E
H (η)= Φ (η) (4)
eff A 1
AX⊂Zd pE(µ,β|ρ∗)=Λl→imZd β|Λ|logheβµNΛiρ∗,β,E, (6)
with some set of many-body potentials Φ depending
A where Λ is a sequence of lattice volumes converging to
on spins ηx at sites x ∈ A ⊂ Zd. If it exists, this will Zd, N (η) is the number of particles within Λ for the
Λ
generally not be the same as the short-ranged Hamilto-
configurationη, and the averageh·i is with respect
ρ∗,β,E
nian H(η) used in defining the dynamics. It turns out
to the invariant measure µ of the DLG for refer-
that almost any reasonable measure (with local densi- ρ∗,β,E
ence density ρ . The “Helmholtz free energy” f is
∗ E
ties)is“Gibbsian”ifonepermits extremelylong-ranged,
thenintroducedbytheLegendretransformf (ρ,β|ρ )=
E ∗
many-body potentials and, indeed, there are a myriad
sup [µρ−p (µ,β|ρ )]. We include the ρ as a reminder
µ E ∗ ∗
of physically inequivalent such Hamiltonians! (E.g. see
ofthereferencedensityandtheEtoindicatethestrength
[15], Theorem V.2.2(a)). To guarantee uniqueness and
of the applied electric field. Since the measures here
other standard properties of usual Gibbs measures, the
are for irreversible steady-states with Ohmic dissipa-
condition of absolute summability is required:
tion, these are not usual free energies. They coincide
with the equilibrium free-energies in the limit E → 0.
kΦ k <∞ (5)
A ∞ The physical interpretation of f (ρ,β|ρ ) is as an “ex-
AX∋x E ∗
cess dissipation function,” i.e. as the total energy dis-
for all x ∈ Zd, where kΦ k ≡ sup |Φ (η)|. Following sipated per volume by an external field to change the
A ∞ η A
rather common practice [7], we shall agree here to call density to ρ from its reference value ρ∗ (in addition to
only measures with the latter property “Gibbsian”. It the Ohmic dissipation intrinsic to the reference state)
is a rigorous theorem of Asselah [16] (or Appendix B of [8]. The argument for shape-dependence is that the
[8])that the following alternativeholds: either all space- “susceptibility”(essentially,the isothermalcompressibil-
ergodic,invariantmeasuresoftheDLGareGibbsianwith ity) may be written both in terms of the free-energy,
absolutely summable potential or else none of them are. χ (ρ,β) = β∂2fE(ρ,β) −1, and also in terms of the
It may also be proved that in the domain of analyticity, E h ∂ρ2 i
noabsolutelysummablepower-lawpotentialcanproduce structure-function, via the limit χE = limk→0SE(k).
a correlation ∝ r−d (Theorem 1 of [3]), such as is ob- However,thelatterlimitisindeterminatewhenthestruc-
b
servedinthe DLG.Thus,underthefirstalternative,one ture function has the form in Eq.(2) and depends upon
isledtoconcludethattheGibbsmeasuremustbecritical the wavenumbervectordirectionkalongwhichthelimit
to account for the observed correlation decay. Alterna- is taken. Thus, the free-energy itself must be shape-
b
tive #1 for our purposes may thus be termed SOC. On dependent, by the same argument as for dipole systems.
the otherhand, inalternative#2 the potentialsarenon- To test this prediction we have performed a Monte
summable(=long-ranged),sothatthiscasecorresponds Carlo simulation of the DLG on a periodic square S×S
to the SOLRI scenario. To support the latter, we make lattice with Ising Hamiltonian H = −21 hx,yiηxηy
some key comparisons with long-rangedsystems. wherehx,yidenotesnearestneighborsites,foPrwhichthe
2
(inverse) critical temperature is βc ≈ 0.31 [12]. To stay Simple computation then gives δδµP(Er) =ρ∗ and
well within the single phase (high temperature) regime, (cid:12)µ=0
(cid:12)
weusedβ =0.2,E =10.0andreferencedensityρ =0.5. (cid:12)
∗ δ2P
We have determined the thermodynamic functions for δµ(r)δµE(r′)(cid:12) =βǫl→im0ǫ−d[hη[ǫ−1r]η[ǫ−1r′]iρ∗,E−ρ2∗]
rectangular subblocks Λ of the S ×S system, in which (cid:12)µ=0
(cid:12) ≡βS (r−r′). (9)
various aspect ratios of the sides of the rectangles were (cid:12) E
chosen. The pressure was evaluated by a double limit.
It follows that P [µ] = drρ µ(r)+ β dr dr′S (r−
First the infinite volume limit was obtained by a linear E ∗ 2 E
r′)µ(r)µ(r′)+O(µ3), anRd the LegendreRtranRsform yields
extrapolation in 1/S → 0 on the Monte Carlo average
heβµNΛiS in the steady-state with S = 64,128,256,512. 1
The thermodynamic limit in Eq.(6) for the pressure was FE[ρ]= 2β Z drZ dr′SE−1(r−r′)δρ(r)δρ(r′)+O(δρ3)
then evaluated by a second linear extrapolation in the
(10)
inverse volume 1/|Λ| of the subblock going to zero. The
largest subblock edge in this second extrapolation had
where δρ(r) ≡ ρ(r) − ρ and S−1 is the operator in-
length18. TheLegendretransformtothefree-energywas ∗ E
then carried out. The results are shown in Figure 1 for verse of SE. In other words, SE−1(k)=k·DE·k/k·LE·k.
subblockswithaspectratiosof3:1and1:3forsidelengths Hence, the inverse kernel S−1(r) is ∝ r−d for large r,
Ed
parallel and perpendicular to the field, respectively. Er- too. We seethatthehydrodynamicequationoftheDDS
rorbarsreflectbothstatisticaldeviationsinindependent at finite field strengths E thus must have an explicit,
runs and the double extrapolation procedure. The two severelynon-local form. [H. Spohn has emphasizedto us
functionsareclearlydistinct. WeseethattheDLG,con- that Eq.(7) linearized about the homogeneous state of
sidered as a model of a current-carrying electrochemical density ρ∗ will still be local if Eq.(2) holds, since then
cell,haswell-definedfree-energiesbuttheresultsdepend LE(ρ∗):∇∇ dr′SE−1(r−r′)δρ(r′)=DE(ρ∗):∇∇δρ(r).]
upon the shape of the cell! However, witRh the free-energy functional in (10) replac-
Thisshape-dependenceleads,however,toaratherseri- ing the local expression, all results of [8] remain valid:
ouspuzzleaboutthehydrodynamicbehavioroftheDDS. the H-theorem, the fluctuation-dissipation theorem, etc.
Thegeneralproblemistodescribehowaninitialsmooth Such nonlocal hydrodynamic behavior should also be
densityprofilerelaxestoaconstantdensityinthedriven presentin equilibrium systems with long-rangedinterac-
steady-state. In [8] a nonlinear hydrodynamic equation tions, e.g. dipolar hard sphere systems or ferrofluids. In
wasderivedbythe formalmethodofnonequilibriumdis- one such system, the Kawasaki lattice gas with a long-
tributions to describe this irreversible process. Density ranged Kac pair-potential, there is a rigorous result [21]
fields varying on a length-scale of the order of ǫ−1 com- that a hydrodynamic equation of the same form as (7)
pared with the lattice distance were formally shown to holds, with a similar nonlocal expression for the free-
evolve by a drift-diffusion equation, energy as in (10), simply replacing S−1(r) by the kernel
E
J(r) of the Kac potential.We see again a very striking
δF
ρ˙(r,t)=−∇· ǫ−1j (ρ)−βL (ρ)·∇ E , (7) and fruitful analogy between dissipative, driven systems
E E
(cid:20) (cid:18) δρ (cid:19)(cid:21) and equilibrium systems with long-range interactions.
over times of order ǫ−2. This equation has the “On- Our results verify that the SOLRI scenario holds in
our model, and not SOC. For Gibbsian measures with
sager form”, with j (ρ) the conduction current, L (ρ)
E E
summable potentials there is no shape-dependence of
theOnsagercoefficientmatrix,andF [ρ]thefree-energy
E
thermodynamics, such as we observe here, even at the
functional. Explicit analytical formulae were given in
critical point ( [15], Theorem I.2.5). Boundary condi-
[8] for each of these quantities, e.g. a Green-Kubo for-
tions play a role below the critical point in the phase co-
mula for the Onsager matrix. These formulae are ex-
existence regionin determining which of multiple phases
act even at high field strengths E, although they may
will occur, but even then the free energies are indepen-
be difficult to evaluate concretely. The free-energy func-
dent of the phase. The ordinary thermodynamic limit,
tional in [8] was nominally given by the local expression
with no shape-dependence, remains valid directly at the
F [ρ]= drf (ρ(r)). However, as observed there, such
E E
critical point. Likewise, the dynamics of short-ranged
a form isRindeterminate. Since the free-energy depends
systems in the coexistence region is expected to be de-
upon the limiting shape, which value is to be used?
scribedbythe Cahn-Hilliarddynamics,whichislocal. It
We can now resolve this issue. The free-energy func-
is actually a little perplexing how to interpret the SOC
tional actually shown in [8] to be relevant to hydro-
point of view that nonequilibrium steady-states are al-
dynamics is given by a Legendre transform F [ρ] =
E
ways “critical”, when these are observed themselves to
drρ(r)µ(r)−P [µ] of the pressure functional
E
undergo continuous phase-transitions at sharp values of
R
P [µ]= 1 limǫdloghexp[ βµ(ǫx)η ]i . (8) temperatureand/ordensity. Suchtransitionsoccurboth
E β ǫ→0 x ρ∗,β,E in the DLG [12] and in granular flow [6], not to mention
Xx
3
dipolesystems[22]. Awayfromthetransitionpointboth Phys.72 879 (1993).
DDSanddipolesystemshaveafinitecorrelationlengthξ [8] G. L. Eyink,J. L. Lebowitz, and H.Spohn,J. Stat.Phys.
which characterizes the crossover from a critical power- 83 385 (1996).
law ∝ r−(d−2+η) at intermediate range r ≪ ξ into the [9] S. Katz, J. L. Lebowitz, and H. Spohn, J. Stat. Phys. 34
asymptoticpower-law∝r−d atlong-ranger≫ξ [12,22]. 497 (1984).
[10] M. Q. Zhanget al., J. Stat. Phys. 52 1461 (1988).
Itismostinterestingtoconsidertheimplicationsofour
[11] J. A. G. Orza et al. preprint cond-mat/9702029; T. P. C.
model calculation for real systems. We expect that the
van Noije et al. preprint cond-mat/9706079.
mainconclusionsconcerningshape-dependenceandnon-
[12] B.SchmittmannandR.K.P.Zia,inPhaseTransitionsand
localitywillholdforphysicaldrift-diffusionsystems,such
Critical Phenomena, vol.17, C. Domb and J. L. Lebowitz,
as fluid or solid electrolytes, semiconductors, and, at a eds. (Academic Press, London,1995).
more mesoscopic level, colloidal suspensions [23]. In real [13] H.Sato and R.Kikuchi,J. Chem. Phys. 55 677 (1971).
charged-particlesystemsthere isthe addedcomplication [14] W. Dieterich, P. Fulde, and I. Peschel, Adv.Phys. 29 527
of a dynamics which is itself long-ranged,via Coulombic (1980).
interactions. However, these are expected to be Debye- [15] R. B. Israel, Convexity in the Theory of Lattice Gases.
screenedandeffectivelyshort-ranged. Thedrift-diffusion (Princeton University Press, Princeton, NJ, 1979).
[16] A.Asselah, Ann.Appl.Prob., submitted.
equationswerejustified longagofor nonequilibriumpro-
[17] J.H.vanVleck,J.Chem.Phys.5556(1937); L.Onsager,
cesses in electrolyte solutions at low density and small
J.Phys.Chem.43189 (1939); J.A.Sauer,Phys.Rev.57
fields within Debye-Hu¨ckel theory [24]. The equations
142 (1940).
are of the same form as those we have considered, sim- [18] R.B. Griffiths, Phys. Rev. 176 655 (1968).
ply generalized to multiple ionic species. The effects [19] P. M. Levy, Phys. Rev. 170 595 (1968); H. Horner, Phys.
of the Coulomb interactions are calculable and can all Rev.172 535 (1968).
be incorporated into an effective Onsager matrix, with [20] P. M. Levy and D. P. Landau, J. Appl. Phys. 39 1128
cross-speciestermsduetoionic-clouddistortionandelec- (1968).
trophoresis. The effects considered in our work corre- [21] G.GiacominandJ.L.Lebowitz,Phys.Rev.Lett.761094
spond to contributions to the invariant measures and (1997); J. Stat.Phys. 87 37 (1997).
thermodynamic potentials in at least the E2 power of [22] A.AharonyandM.E.Fisher,Phys.Rev.B83323(1973).
[23] D.N.PetsevandN.D.Denkov,PhysicaA183462(1992).
the field strength. It would be interesting to make a
[24] L. Onsager and R. Fuoss, J. Phys.Chem. 36 2689 (1932).
theoreticalestimateoftheorderofmagnitudeofsuchef-
[25] J.R.Dorfman,T.R.Kirkpatrick,andJ.V.Sengers,Ann.
fectsinelectrolytesolutions. Perhapsthemostaccessible
Rev.Phys. Chem. 45 213 (1994).
predictions are the long-ranged correlations themselves,
which could be observed in light-scattering experiments FIGURE CAPTION
similiarto those carriedoutonsimple fluids subjectto a
temperature gradient(see [25] for a recent review). Also Figure (1.) Free energy as a function of density for
ofpossiblepracticalinterestaretheimplicationsofanon- E =10, β =0.2, ρ =0.5 for aspect ratio 3:1 (triangles)
∗
local hydrodynamics for granular flow. and 1:3 (squares).
Acknowledgements: The authorswishtothankB.M.
Boghosian,H.Gould,W.Klein,J.L.Lebowitz,Y.Oono
andH.Spohnforhelpfuldiscussions. FJAwasfundedin
partbyNSFGrantDMR9633385andbyAFOSRGrant
#F49620-95-1-0285.
[1] P. Garrido et al., Phys. Rev.A 42 1954 (1990).
[2] G. Grinstein, D.-H.Lee, and S.Sachdev,Phys.Rev.Lett.
64 1927 (1990).
[3] M. Duneau and B. Souillard, Commun. Math. Phys. 47
155 (1976).
[4] P.Bak,C. Tang, andK.Wiesenfeld, Phys.Rev.A 38364
(1988).
[5] R.K. P.Zia and B. Schmittmann,Zeit. f. Phys.B 97 327
(1995).
[6] D. R.M. Williams, Physica A 233 718 (1996).
[7] A.C.D.vanEnter,R.Fern´andez,andA.D.Sokal,J.Stat.
4
0.080
0.070
0.060
0.050
0.040
f
0.030
0.020
0.010
0.000
0.35 0.45 0.55 0.65
ρ