PHYSICAL REVIEW E 80, 011121 共2009兲
Nonequilibrium relation between potential and stationary distribution for driven diffusion
Christian Maes,1,* Karel Netočný,2,† and Bidzina M. Shergelashvili1
1
Instituut voor Theoretische Fysica, K.U. Leuven, B-3001 Belgium
2
Institute of Physics, AS CR, Prague, CZ-18221 Czech Republic
共Received 20 February 2009; revised manuscript received 11 May 2009; published 17 July 2009兲
We investigate the relation between an applied potential and the corresponding stationary-state occupation
for nonequilibrium and overdamped diffusion processes. This relation typically becomes long ranged resulting
in global changes for the relative density when the potential is locally perturbed, and inversely, we find that the
potential needs to be wholly rearranged for the purpose of creating a locally changed density. The direct
question, determining the density as a function of the potential, comes under the response theory out of
equilibrium. The inverse problem of determining the potential that produces a given stationary distribution
naturally arises in the study of dynamical fluctuations. This link to the fluctuation theory results in a variational
characterization of the stationary density upon a given potential and vice versa.
DOI: 10.1103/PhysRevE.80.011121
PACS number共s兲: 05.40.⫺a, 05.10.Gg
I. INTRODUCTION
Imagine independent colloidal particles in a potential field
and subject to friction and noise as imposed by a thermal
reservoir or background fluid. In thermal equilibrium at inverse temperature ␤, Prob共x兲 ⬀ exp关−␤V共x兲兴, where V is the
potential on the states x. Typical examples include the
Laplace barometric formula but also the distribution of particles in a fluid undergoing rigid rotation. We now add an
external forcing and we wait until a steady regime gets installed. The stationary statistics depends on the potential, but
most certainly and because of the forcing the resulting timeinvariant distribution of velocities and positions of the particles gets modified with respect to the Maxwell-Boltzmann
statistics. The relation between potential and stationary distribution is far from understood for generic nonequilibrium
systems, beyond its general specification as being for example a solution to the time-independent Fokker-PlanckSmoluchowski equation.
We show here that small local variations in the potential
can globally affect the relative density, provided a nongradient driving is present. That effect already arises in linear
order around equilibrium. For the inverse relation, we are
asking to reconstruct the potential which, under a known
driving, realizes a stationary distribution. In equilibrium, the
change in the potential needed to change the density ␳
→ ␳ exp共−A兲 is simply equal to A—not so in nonequilibrium.
It is then a genuine problem what potential field can produce
a given particle density at a fixed driving. Beyond obvious
applications in interpreting observational data, this question
also naturally emerges in dynamical fluctuation theory, cf.
关1兴. In particular, the large fluctuations of certain time averages around their stationary values are governed by a functional 共sometimes called an effective potential兲 that is given
in terms of the potential of the inverse problem. We will give
two possible approaches to finding the potential, one of
which is analytical and the other one is based on a variational
formula.
*[email protected]
†
[email protected]
1539-3755/2009/80共1兲/011121共10兲
The specific examples we treat in this paper and for which
the analysis can be applied are those of diffusing particles in
a background rotational velocity field. Diffusion in rotating
media is one of the central objects in geophysical and astrophysical applications. The question of nonlocal and irreversible effects is of particular interest for galactic dynamics,
where according to Chandrasekhar’s theory the huge number
of relatively small-size gravitational encounters gives rise to
an effective Brownian motion for test stars, cf. 关2兴; see also
关3兴 for an account in the context of galaxy formation. Arguably, also experimentally a most realistic scenario for maintaining a constant nonconservative force is via differential
rotation. One can think of concentric cylinders rotating at
different angular frequencies which are imposed on the fluid
by stick boundary conditions. Far from equilibrium and under a general angular 共possibly angle-dependent兲 driving one
can expect not only that currents are being maintained but
also that the time-symmetric aspects of diffusion can be essentially changed. It is known that the diffusion phenomenon
itself may be influenced by rotation, even by rigid rotation
关4兴. Related considerations on nonequilibrium diffusion also
apply to other scenarios, including, e.g., shear flow 关5–8兴 and
oscillatory flows 关9兴, and stochastic models of particle transport in turbulent media necessarily include discussions of
driven diffusion phenomena 关10兴. Colloidal particles in harmonic wells and driven by shear flow have been explicitly
treated for the violation of the fluctuation-dissipation relation
in 关11兴.
The relation between stationary density and potential is
generically nonlocal or long range but as our particles are
mutually independent, that obviously says nothing about the
generic long-range correlations in nonequilibrium systems.
As we will see however, the mathematical mechanisms are
not unrelated. Several of these nonlocal effects and generic
long-range correlations under nonequilibrium conditions
have been discussed in similar contexts of interacting particle systems, most recently from the point of fluctuation
theory 关12–14兴, in perturbation theory for the stationary density 关15兴, and as a result of the breaking of the fluctuationdissipation relation 关16–18兴. It was originally the modecoupling theory in hydrodynamic studies for a fluid not in
thermal equilibrium that revealed the 共macroscopic兲 long-
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©2009 The American Physical Society
PHYSICAL REVIEW E 80, 011121 共2009兲
MAES, NETOČNÝ, AND SHERGELASHVILI
range correlations between fluctuations. Light scattering experiments can reveal these correlations 关19–22兴. In contrast,
the present work analyzes the nonlocality on a mesoscopic
length scale and that does not result from particle interactions but as is already present in a single-particle distribution
共or, equivalently, in a density or an ensemble of independent
particles兲, due to the imposed nonequilibrium driving.
In Sec. II, we specify our dynamical model: independent
particles undergoing an overdamped diffusive motion in a
confining potential and under driving. The nonlocality in the
relation between potential and distribution is discussed
throughout Sec. III. The latter already speaks about the inverse problem of determining the potential for a given density, which is then elaborated in Sec. IV, in the context of
variational principles and for the purpose of the dynamical
fluctuation theory. In Appendixes A–C we add further details
about the McLennan’s interpretation of nonequilibrium distributions, about the Green’s functions encountered in the
response problem, and about the dynamical fluctuation origin
of the variational principles under consideration.
II. DIFFUSION IN A TWO-DIMENSIONAL ROTATIONAL
FLUID
Restricting ourselves for the moment to the twodimensional plane with points labeled by the polar coordinates x = 共r , ␪兲, we consider an ensemble of independent test
particles subject to a rotation-symmetric force with potential
U共r兲, sufficiently confining so that
Z = 2␲
冕
+⬁
e−Urdr ⬍ + ⬁.
Note that in the absence of a potential, U = ⌽ = 0, dynamics 共2.1兲 does continue to make sense, although a normalizable stationary distribution no longer exists. The transient
regime is still relevant and has been studied in detail 关23兴.
The stationary distribution for dynamics 共2.1兲 has density
␳ verifying the Fokker-Planck-Smoluchowski equation
ⵜ · J = 0,
J = ␳v − ␳ ⵜ 共U + ⌽兲 − ⵜ␳ .
共2.2兲
We refer also to 关8,24兴 for more thermodynamic and kinetic
gas considerations in the derivation of that nonequilibrium
dynamical equation. In polar coordinates the probability current J = 共Jr , J␪兲 takes the form
Jr = − ␳
⳵ 共U + ⌽兲 ⳵␳
− ,
⳵r
⳵r
J␪ = ␳v␪ −
␳ ⳵ ⌽ 1 ⳵␳
−
.
r ⳵␪ r ⳵␪
共2.3兲
By turning on the driving v␪, typically not only an angular
current J␪ is generated but also a nonzero radial component
Jr does get maintained. This is a priori not in contradiction
with the existence of a stationary distribution; its normalizability essentially depends on the imposed potential U + ⌽
and on the boundary conditions.
Our general aim is to analyze the relation between test
potentials ⌽ and stationary densities ␳, under a given confining rotation-symmetric potential U and as mediated by the
rotational field v␪. First, we examine the issue of spatial nonlocality.
0
The particles are suspended in a nonequilibrium fluid exerting an additional force that can have some conservative component with potential ⌽共r , ␪兲 and a nonconservative force
v = 共vr , v␪兲 for which we assume that the radial component
vr = 0 vanishes. The angular driving force v␪ can be associated with the local velocity of the background fluid which is
maintained in a differential rotation state. We assume a thermally homogeneous background 共setting the temperature to
one兲, modeled by a Gaussian and temporally white noise.
Given that the motion is noninertial, it satisfies the Langevin
equation
dxt = vdt − ⵜ共U + ⌽兲dt + 冑2dBt
共2.1兲
with Bt standard two-dimensional Brownian motion.
An implicit assumption in the above construction is the
smoothness of the functions U, ⌽, and v␪ on the domain
taken here as the entire two-dimensional plane. Yet, interesting modifications arise when the origin r = 0 is not accessible
and the particles can only move in the nonsimply connected
domain obtained by removing the latter. This allows for possible singularities when approaching the origin and the existence of a potential for the driving force v is no longer
equivalent to the condition ⵜ ⫻ v = 0; the rotational field of
the form v = 共0 , v␪兲, v␪ ⬀ 1 / r serving as example. The exclusion of the origin can be ensured, e.g., by an infinitely repellent potential therein or via suitable boundary conditions at
the origin.
III. LONG-RANGE RESPONSE TO CHANGING THE
POTENTIAL
Nonlocal features have been widely discussed in the nonequilibrium literature. Mostly however deals with the presence of long-range correlations, cf. 关13,16,19,25–27兴 for
time-separated viewpoints, or with models of self-organized
criticality, cf. 关28兴. In our case, we have independent particles; hence there are no correlations between the particles
and correlations between spatial points only appear because
of fixing the number of particles or by fixing the mass. We
think of the spatial dependence in the density as it is affected
by local changes in the external potential, and vice versa.
In the absence of driving, v = 0, stationarity equation 共2.1兲
has the usual equilibrium solution ␳ ⬀ e−U−⌽, J = 0, which is
manifestly a local functional of the test potentials ⌽ in the
sense that the response
冋
册
␦
␳共x兲
log
= ␦共z − y兲 − ␦共z − x兲
␦⌽共z兲
␳共y兲
is insensitive to perturbing ⌽ away from both points x and y.
One can ask to what extent this is an equilibrium property
but, in fact, it is easy to devise special nonequilibrium conditions where such a locality still holds. As a simple example, take ⌽ = 0 and let the angular driving velocity be
rotationally symmetric, v␪ = u共r兲. Then, the stationary density
and current become
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PHYSICAL REVIEW E 80, 011121 共2009兲
NONEQUILIBRIUM RELATION BETWEEN POTENTIAL AND…
冉
1
␳ = e−U,
Z
冊
u
J = 0, e−U .
Z
共3.1兲
Although the stationary density coincides with the equilibrium solution and is perfectly local in the potential U, it now
corresponds to a current-carrying steady state. Thus one cannot unambiguously decide just from the stationary distribution itself whether the system rests in equilibrium or whether
irreversible flows are present.
Yet, generic nonequilibrium distributions do get modified
due to driving and, as a result, they typically pick up some
nonlocality. Next comes a simple demonstration.
W共␪⬘, ␪兲 = ⌿共␪⬘兲 − ⌿共␪兲 +
1
p共r兲 = e−U共r兲 ,
Z
␳共r, ␪兲 = p共r兲q共␪兲,
j共r兲 =
共3.2兲
which is under the given assumptions the general form for a
density with everywhere vanishing radial current, Jr = 0. The
steady state decomposes into separated concentric motions
and the angular distribution q共␪兲 in Eq. 共3.2兲 is determined
from stationarity condition 共2.2兲, which reads J␪共r , ␪兲 = j共r兲
for some rotation-symmetric function j共r兲 that can be determined. Explicitly, from Eq. 共2.3兲,
rj共r兲
,
兵f共␪兲 − ⌿⬘共␪兲其q共␪兲 − q⬘共␪兲 =
p共r兲
共3.3兲
which decouples the polar coordinates and confirms ansatz
共3.2兲. The angular current j共r兲 is obtained by dividing Eq.
共3.3兲 by q and integrating over the angle variable, which
yields, always for v␪共r , ␪兲 = f共␪兲 / r,
冕
冕
p共r兲
j共r兲 =
r
2␲
0
2␲
f共␪兲d␪
.
共3.4兲
q 共␪兲d␪
−1
0
As expected, a nonzero steady current is maintained whenever the angular driving does not allow for a potential, i.e., if
the work performed over concentric circles is nonzero, w
= 兰20␲ fd␪ ⫽ 0. Using the normalization condition 兰20␲q共␪兲d␪
= 1, the solution of Eq. 共3.3兲 is obtained in the form
q共␪兲 =
⍀=
1
⍀
冕
0
with the work function
2␲
f d␰ .
␪⬘
p共r兲 w
共e − 1兲.
⍀r
共3.6兲
The equilibrium angular distribution is recovered for w = 0
共or j = 0兲, in which case the work function W共␪⬘ , ␪兲 derives
from a potential and formulas 共3.5兲 boils down to the
Boltzmann-Gibbs form.
For w ⫽ 0 the character of the stationary density becomes
modified, as can be read from the response to changes in the
test potential ⌽ = ⌿共␪兲 that we take to depend on the angle ␪
only. We take the functional derivative for changes in the
value of ⌿ at fixed angle ␩,
␦
␦⌿共␩兲
冋
log
册
␳共r, ␪兲
= Y共␩, ␪⬘兲 − Y共␩, ␪兲,
␳共r, ␪⬘兲
共3.7兲
where
Y共␩, ␪兲 = ␦共␩ − ␪兲 −
冖
eW共␩,␪兲
e
W共␩⬘,␪兲
.
d␩⬘
It is the second term that generates the nonlocality as its
reciprocal
冖
eW共␩⬘,␪兲−W共␩,␪兲d␩⬘ =
冕
␩⬘
e⌿共␩兲−⌿共␩⬘兲+养␩ 兵f−w␦共␰−␪兲其d␰d␩⬘
共3.8兲
still depends on ␪. Note also that since the work function
W共␩ , ␪兲 is discontinuous at ␪ = ␩, so is the nonlocal contribution in the response 关Eq. 共3.7兲兴 at ␩ = ␪ and ␩ = ␪⬘.
That nonlocal term is manifestly a correction of order
O共w兲; hence, some more explicit information can be obtained within the weak driving approximation, considering
the driving force f共␪兲 共or total work per cycle w兲 small. This
yields, up to O共w兲,
␦
␦⌿共␩兲
冋
log
册
␳共r, ␪兲
= ␦共␩ − ␪⬘兲 − ␦共␩ − ␪兲 + wqⴱ共␩兲
␳共r, ␪⬘兲
eW共␪⬘,␪兲d␪⬘ ,
0
冕 冕
2␲
2␲
␪
Here we have used the notation 养␪␪⬘ for the integral performed along the positively oriented path ␪⬘ → ␪ on the
circle, i.e., it coincides with 兰␪␪⬘ for ␪⬘ ⱕ ␪ whereas it equals
to 兰␪2⬘␲ + 兰0␪ otherwise. In the sequel we also employ the shorthand 养 for the integral 兰20␲.
Equivalently, the same solution to Eq. 共3.3兲 can also be
obtained in terms of the current j共r兲; this leads to the next
explicit expression for the latter,
A. Exactly solvable model
In the example above the radial currents are absent and
the steady state is rotation symmetric. A simple exactly solvable case where the rotation symmetry is broken can be obtained for an angular driving of the form v␪ = f共␪兲 / r and for
test potentials that are constant along radials, ⌽共r , ␪兲 = ⌿共␪兲.
The origin is excluded by the boundary condition Jr共0兲 = 0. In
this case, the stationary density is found to be of the form
冖
共3.5兲
⫻
eW共␪⬘,␪兲d␪⬘d␪
0
冦
冖
冖
␪
␪⬘
q ⴱd ␰ ,
␪⬘
␪
if ␩ 苸 共␪ → ␪⬘兲
− qⴱd␰ , if ␩ 苸 共␪⬘ → ␪兲,
where qⴱ共␩兲 stands for the auxiliary density
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PHYSICAL REVIEW E 80, 011121 共2009兲
MAES, NETOČNÝ, AND SHERGELASHVILI
q ⴱ共 ␩ 兲 =
e⌿共␩兲
冖
e ⌿d ␰
and ␩ 苸 共␪ → ␪⬘兲 indicates that ␩ belongs to the positively
oriented path from ␪ to ␪⬘. This explicitly confirms the nonlocal and discontinuous structure of the response: The relative density ␳共r , ␪兲 / ␳共r , ␪⬘兲 of our weakly driven system remains (strongly) sensitive to locally modifying the potential,
⌿共␪兲 → ⌿共␪兲 + ⑀␦共␪ − ␩兲, at an arbitrary angle ␩. The effect
disappears at equilibrium, w = 0.
Note that this special example is essentially one dimensional and we have so far only considered the response to an
r-constant change in the potential. The response to a strictly
local perturbation will be discussed in the next section by a
more general method.
The nonlocality of the inverse problem for this specific
example, potential as function of the density, is continued
around Eqs. 共3.13兲–共3.15兲.
B. General linear theory
Having observed that the nonlocal features of nonequilibrium states already emerge in the lowest order of the driving
strength, we can now follow the linear analysis more systematically. The method is by now well known; we refer to
earlier work of McLennan and others 关29,30兴. The analysis
starts from the overdamped form 关Eq. 共2.1兲兴, and again for
simplicity we keep in mind diffusion in the entire plane under natural boundary conditions 共decay at infinity兲.
One takes the reference equilibrium distribution 共for v
= 0兲 as
␳o共x兲 =
1 −V共x兲
e
,
Z
共3.9兲
where we combine V = U + ⌽ and Z is the normalization.
Assume now that the driving velocity field v is uniformly
small, 兩v兩 = O共␧兲.
The solution of the stationarity equation 共2.2兲 to linear
order in ␧ reads
log
1
␳
= 共ⵜ · v − v · ⵜV兲,
␳o LV
共3.10兲
where LV = −ⵜV · ⵜ+⌬, which is recognized as the generator
of a reversible diffusion in potential V; see Appendix A for
more details and for a physical interpretation in terms of the
McLennan’s theory 关29兴. Writing Eq. 共3.10兲 in the form
⌬␯ − ⵜV · ⵜ␯ = ⵜ · v − v · ⵜV
共3.11兲
equation for ␦␯ in which the Laplacian 共free diffusion兲 is
modified with a drift term in potential V. Note that since
LV = ␳10 ⵜ · 共␳0ⵜ兲 and ⵜ · J = 0, this problem is equivalent to
studying the electrostatic potential in an inhomogeneous dielectric environment as generated by a source with zero total
charge. Whereas the local component of the response to ␦V
is already hidden in the V dependence of the equilibrium
density ␳o, the nonlocal character of solutions to Eq. 共3.12兲
follows from general features of elliptic operators. The solution of the free Poisson equation 共with only the Laplacian
and natural boundary conditions at infinity兲 is explicit and
manifestly long range. The potential V or the finiteness of the
system introduces an extra confinement and the claim needs
to be refined. One expects that within the confinement region
where the density ␳o is approximately constant and near its
maximal value, the response ␦␯共x兲 at x to a local perturbation
␦V 共both localized within that region兲 can be well estimated
by replacing LV with the free Laplacian L0 = ⌬. This intuition
is indeed correct as we shortly explain in Appendix B. In that
region, the response ␦␯共x兲 to a local perturbation ␦V derives
from the Green’s function for d-dimensional Brownian motion, weighted by the local mean velocity J / ␳. In this sense
the nonlocal response is intrinsically a nonequilibrium feature.
Although the above linear analysis can formally be extended far from equilibrium and one obtains a generalization
of response formula 共3.12兲, the linear operator replacing LV
is no longer symmetric, in agreement with the breakdown of
Onsager reciprocity relations. We will see in the next section
that an inverse formulation of our linear-response question
does not suffer from the above complication and it also becomes remarkably easier to study outside the weak driving
regime.
C. Local response to nonlocal perturbation
In this section we discuss long-range aspects in the inverse problem, namely, how the test potential ⌽ that makes a
given density ␳ stationary is affected by a local change in ␳.
The relevance of this question and some other approaches
are considered in Sec. IV.
We start by revisiting the exactly solvable model of Sec.
III A. The confining potential U共r兲 and the driving v共r , ␪兲
= 关0 , f共␪兲 / r兴 are always considered fixed. For the class of
radial-angular uncorrelated densities, ␳共r , ␪兲 = p共r兲q共␪兲, the
stationary equation 共2.2兲 is satisfied for the test potential of
the form ⌽共r , ␪兲 = ⌽0共r兲 + ⌿共␪兲, with the radial part
⌽0 = − log p − U
with ␯ = log共␳ / ␳o兲, its variation along V → V + ␦V is
⌬␦␯ − ⵜV · ⵜ␦␯ − ⵜ␦V · ⵜ␯ = − v · ⵜ␦V.
and with the angular part equal to
In terms of the equilibrium generator LV and the stationary
current J = ␳共v − ⵜV兲 − ⵜ␳, this reads
J
L V␦ ␯ = − · ⵜ ␦ V
␳
共3.13兲
⌿ = − log q +
共3.12兲
always up to terms O共␧2兲; in the same order the density ␳ on
the right-hand side can be replaced by ␳o. This is a Poisson
冕冉 冊
␪
f−
␪0
C
d␰ .
q
共3.14兲
Here ␪0 is an arbitrary fixed angle and the constant C is
determined from
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NONEQUILIBRIUM RELATION BETWEEN POTENTIAL AND…
C=
冖
冖
f d␪
.
共3.15兲
q d␪
−1
The latter specifies the corresponding stationary current field
as J = 共0 , C / r兲. Away from equilibrium, i.e., for w = 养fd␪
⫽ 0, the angular component ⌿共␪兲 apparently becomes a nonlocal functional of the angular density q共␪兲, similar to what
we have observed for the functional dependence ␳关⌿兴.
In the general case, we are asked to find the potential ⌽
that solves the stationarity equation 共2.2兲 as a function of
density ␳ for given velocity field v and confining potential U,
say on d-dimensional space. By changing ␳ → ␳ + ␦␳ with
兰␦␳共x兲dx = 0, we get the linear-response equation for G = U
+ ⌽ + log ␳,
1
J
ⵜ · 共␳ ⵜ ␦G兲 = · ⵜ␦␯
␳
␳
共3.16兲
everywhere bounded from zero and for which all 共nonempty兲
equilevel lines, ␳共r , ␪兲 = const, are closed curves. We also
stick to trivial boundary conditions at infinity, as guaranteed
by a sufficient decay of all the fields ␳, U, and v. The auxiliary velocity
c = v − ⵜ共U + ⌽ + ␯兲,
for the function f = ␦G. The linear operator L␳ generates a
reversible diffusion in potential −log ␳, i.e., an equilibrium
process. The analysis is now exactly similar as from Eq.
共3.12兲, but with ␯ replacing −V, and now restricting to a
confinement region where ␳共x兲 is approximately constant and
maximal. The source is nonzero by the presence of J ⫽ 0 in
the right-hand side of Eq. 共3.16兲. We refer again to Appendix
B for the analysis, but the conclusion remains that a generic
local change in density ␦␳ in the confinement region requires
a nonlocal adjustment of the potential and the corresponding
response function derives from the Green’s function for the
d-dimensional Laplacian.
Remark that no weak driving or small current assumption
was employed in the above argument. In fact, the presence of
the 共auxiliary兲 reversible diffusion generated by L␳ suggests
that, even far from equilibrium, there are symmetries in the
response functions. This is indeed true; see Sec. B 3 of Appendix B for such a reciprocity relation.
共4.1兲
shares with the driving field v共r , ␪兲 an equal vorticity,
共4.2兲
ⵜ ⫻ c = ⵜ ⫻ v.
The probability current is J = ␳c, and stationarity condition
共2.2兲 reads
ⵜ · c + c · ⵜ␯ = 0.
共4.3兲
We only need to find the vector field c共r , ␪兲; then the potential ⌽共r , ␪兲 can be calculated as
with ␯ = log ␳ and J = ␳共v − ⵜG兲 the stationary current. We
recognize in the left-hand side of Eq. 共3.16兲 the action of the
generator
L␳ f共x兲 = ⌬f共x兲 + ⵜ log ␳ · ⵜf
␯ = log ␳
⌽共x兲 = − ␯共x兲 − U共x兲 +
冕
␥:x0x
共v − c兲 · dᐉ
共4.4兲
modulo a constant, where the integral is taken along an arbitrary curve connecting a fixed initial point x0 with x.
To determine c共r , ␪兲 solving Eqs. 共4.2兲 and 共4.3兲, we first
observe that it is unique by the Helmholtz decomposition
theorem when supplying the boundary condition that the difference c − v goes to zero at infinity. Still another boundary
condition has to be added in the case the origin is not accessible and excluded from the domain.
In the following we restrict ourselves again to the twodimensional plane. Equation 共4.3兲 is solved by any vector
field of the form
冉
c共r, ␪兲 = g关␯共r, ␪兲兴 −
冊
1 ⳵␯ ⳵␯
,
,
r ⳵␪ ⳵r
共4.5兲
with g an arbitrary function. The latter is fixed by condition
共4.2兲,
再
冉
1 ⳵␯ ⳵␯
,
r ⳵␪ ⳵r
ⵜ ⫻ g关␯共r, ␪兲兴 −
冊冎
= ⵜ ⫻ v,
共4.6兲
IV. MORE ABOUT THE INVERSE PROBLEM
One can ask how to actually construct the test potential ⌽
that makes a given density ␳ stationary, i.e., solving Eq.
共2.2兲. An immediate application is found in dynamical fluctuation theory; a brief review is left in Appendix C. In the
following we present a general procedure to solve the inverse
stationary problem for a class of densities. Next, a variational
formulation suitable for numerical implementation will be
given.
which after integration over the surface enclosed by any
equilateral curve of the density, ␯共r , ␪兲 = a, and using Stokes’
theorem yields
冖
v · dᐉ = g共a兲
␯=a
冖 冉
␯=a
−
冊
1 ⳵␯ ⳵␯
,
· dᐉ.
r ⳵␪ ⳵r
共4.7兲
Parametrizing the curve ␯共r , ␪兲 = a by its proper length so
that
A. General solution
We are back to the setup of Sec. II for two-dimensional
rotational diffusion. We restrict to those densities ␳ that are
011121-5
dᐉ =
冋 冉 冊 冉 冊册 冉
1 ⳵␯
r2 ⳵ ␪
2
+
⳵␯
⳵r
2 −1/2
−
冊
1 ⳵␯ ⳵␯
,
ds,
r ⳵␪ ⳵r
PHYSICAL REVIEW E 80, 011121 共2009兲
MAES, NETOČNÝ, AND SHERGELASHVILI
we finally get
冖
v · dᐉ
␯=a
冖 冋 冉 冊 冉 冊册
g共a兲 =
␯=a
1 ⳵␯
r2 ⳵ ␪
2
+
⳵␯
⳵r
.
2 1/2
共4.8兲
ds
I关␮兴 = inf G关␮, j兴
Formulas 共4.4兲, 共4.5兲, and 共4.8兲 together provide an explicit
solution for the test potential ⌽.
As a check we take the example Eq. 共3.1兲; there
c = 关0,u共r兲兴 = v
and the curves ␯ = a are concentric, corresponding to the
equipotential lines for U共r兲, assuming that it is monotone in
r. Therefore Eq. 共4.8兲 gives g共a兲 = −u共r兲 / U⬘共r兲 for a =
−U共r兲 − log Z.
Further, the more general example of Sec. III A has c
= 关0 , j共r兲 / q共␪兲兴 with nonvanishing divergence whenever q is
not a constant, in contrast with form 共4.5兲. The point is that
this example has a velocity field which is not defined at the
origin, it being excluded from the domain. Hence, Stokes’
theorem in the form of Eq. 共4.7兲 cannot be used and a modification is needed; we omit details.
The above solution, basically obtained by a suitable deformation of polar coordinates, provides a class of examples
with nonvanishing radial current. The latter is generally the
case whenever ␯ does not decompose into independent radial
and angular parts; compare with the model of Sec. III A.
B. Variational approach
Write now Fokker-Planck-Smoluchowski equation 共2.2兲,
considered again as the inverse stationary problem for the
test potential ⌽, in the form
ⵜ · 共J0 − ␳ⵜ⌽兲 = 0
defined for all normalized densities ␮ and all divergenceless
currents j, ⵜ · j = 0; see Appendix C for its meaning within the
dynamical fluctuations theory. This functional is manifestly
positive and zero only if ␮ and j coincide with the stationary
density, respectively, the stationary current 共for the case ⌽
= 0兲. It can be used to construct the variational functional
共4.9兲
j:ⵜ·j=0
with minimizer equal to the stationary density. This is a constrained variational problem that can be solved by Lagrange
multipliers; the solution reads
I关␮兴 =
共4.10兲
Recalling that it is an elliptic partial differential equation for
⌽, its solution coincides with the minimizer of the quadratic
functional
F␳关⌿兴 =
1
2
冕
␳ⵜ⌿ · ⵜ⌿dx +
冕
⌿ⵜ · J0共␳兲dx 共4.11兲
under the unchanged boundary conditions if present. This
formulation is suitable for numerical computations; see, e.g.,
via 关31兴.
There are other variational principles of physical importance that appear intimately related to our inverse stationary
problem. To explain those, consider the functional
G关␮, j兴 =
1
4
冕
␮−1关j − J0共␮兲兴 · 关j − J0共␮兲兴dx
共4.12兲
1
4
冕
␮ⵜ⌽ · ⵜ⌽dx
共4.14兲
with ⌽ the test potential that makes the density ␮ stationary,
cf. Eq. 共4.9兲,
ⵜ · 关J0共␮兲 − ␮ⵜ⌽兴 = 0.
共4.15兲
In this way, the solution to the inverse stationary problem is
an essential step in constructing the variational functional
I关␮兴 on densities.
Finally, recall that Eq. 共4.15兲 is equivalent to the variational problem F␮关⌿兴 = min with F␮ introduced in Eq.
共4.11兲. Combining with Eq. 共4.14兲 we have the relation
inf F␮关⌿兴 = F␮关⌽兴 = − 2I关␮兴
⌿
共4.16兲
that yields the next expression for the functional I关␮兴
共changing ⌿ → 2⌿ for convenience and integrating by
parts兲,
I关␮兴 = sup
⌿
冕
ⵜ⌿ · 关J0共␮兲 − ␮ⵜ⌿兴dx.
共4.17兲
This unconstrained variational formula is apparently more
useful for numerical computations than Eq. 共4.13兲 above.
with J0 = J0共␳兲 the ⌽-independent part of the probability current J,
J0共␳兲 = ␳共v − ⵜU兲 − ⵜ␳ .
共4.13兲
V. CONCLUSION
The relation between potential and stationary density in
mesoscopic 共stochastic兲 systems appears to be generically
long ranged whenever there is a true nonequilibrium driving.
That long-range effect is a priori distinct from the longrange correlations under conservative dynamics extensively
studied before, and it occurs already for free particles. In
models of overdamped diffusions considered in this paper,
we have linked that long-range effect to the slow spatial
decay of the Green’s function for a certain equilibrium diffusion process 共in the first order around equilibrium兲.
Vice versa, a similar nonlocal change of potential is generically needed to create a local change in the stationary
density. This issue appears relevant for the inverse stationary
problem that naturally emerges in the context of dynamical
fluctuations and nonequilibrium variational principles. We
have indicated how these specific issues become mutually
related, together with comparing some numerically feasible
schemes based on the dynamical fluctuation theory that
might be of use in applications.
011121-6
PHYSICAL REVIEW E 80, 011121 共2009兲
NONEQUILIBRIUM RELATION BETWEEN POTENTIAL AND…
ACKNOWLEDGMENTS
K.N. acknowledges the support from the Academy of
Sciences of the Czech Republic under Project No.
AV0Z10100520 and from the Grant Agency of the Czech
Republic under Grant No. 202/07/0404. C.M. benefits from
K.U. Leuven under Grant No. OT/07/034A.
APPENDIX A: McLENNAN THEORY OF STATIONARY
DISTRIBUTIONS
冕
1
d
␦␯t = LV␦␯t + ⵜ · 共␳ⵜ␦V兲,
dt
␳
we find
dt具v共xt兲 · ⵜV共xt兲 − ⵜ ·
␦␯t =
v共xt兲典ox ,
共B2兲
where the equilibrium condition has been used. As a result,
denoting with pt共x , y兲 the transition kernel,
␦␯t共x兲 = − ␦V共x兲 +
共v · ⵜV − ⵜ · v兲␮dx + O共␧2兲.
In other words, to linear order in the nonequilibrium background, −v共x兲 · ⵜV共x兲 + ⵜ · v共x兲 equals the mean dissipated
work per unit time provided the particle is at x. 共Incompressibility of the background fluid can be imposed by letting
ⵜ · v = 0.兲 That linear term is exactly what appears in the expectation in Eq. 共A1兲. Therefore, the linear nonequilibrium
correction to the Boltzmann distribution corresponds to the
total dissipated work under the equilibrium relaxation process as started from different initial configurations.
冕
dypt共x,y兲␦V共y兲
or
␦␯t共x兲
= − ␦共x − y兲 + pt共x,y兲.
␦V共y兲
v · 关共v − ⵜV兲␮ − ⵜ␮兴dx
as the expression between square brackets is the current profile at density ␮. Clearly, for small driving that equals
冕
esLVds LV␦V = 共etLV − 1兲␦V,
0
where 具 · 典ox denotes expectation over the equilibrium process
关that is process 共2.1兲 with v = 0兴 started from position x. 共For
a mathematical discussion on how to take the limits ␧ → 0,
T → +⬁, we refer to 关32兴.兲 It is interesting to recognize here
the linear part in the irreversible entropy flux. When the density of the test particles is ␮ then the instantaneous mean
work done by the background field v is
W=−
冉冕 冊
t
+⬁
共A1兲
冕
共B1兲
1
ⵜ · 共␳ⵜ␦V兲 = LV␦V,
␳
0
W=
␦␯0 = 0.
Using that the second term on the right-hand side equals
It is useful to see how the original McLennan reasoning
关29兴 can be used to provide some physical interpretation for
the stationary density of a weakly driven diffusion. We can
write formula 共3.10兲 in the equivalent form, always to linear
order in ⑀,
␳共x兲 = ␳o共x兲exp
Perturb the system by changing the potential at time t = 0 to
V + ␦V and let the system relax toward a new equilibrium.
The evolved distribution at time t be ␮t = ␳ + ␦␮t, corresponding to the effective time-dependent potential ␯ + ␦␯t, ␦␯t
= ␦␮t / ␳. By the linear-response theory,
共B3兲
Remember that ␦␯t is only determined up to a constant,
which explains why limt→⬁ ␦␯t = −␦V + 具␦V典␳ differs from the
“naturally expected” value −␦V; in the above the additive
constant has been fixed by the initial condition ␦␯0 = 0. The
nonlocal part in the linear response for fixed time t, thus,
exactly equals the transition-probability density pt共x , y兲. It is
nonlocal in the sense that over distances where ␳ is approximately constant and maximal (or around the minimum of V兲,
there is only slow decay in 兩x − y兩. As time grows larger, that
effect typically dies out, restoring a strictly local response in
the infinite-time limit.
As an example, the standard one-dimensional OrnsteinUhlenbeck process 共or, oscillator process兲 corresponding to
the potential V共x兲 = x2 / 2 has a response function with the
large-time asymptotics
␦
APPENDIX B: MORE TECHNICAL ASPECTS OF THE
NONLOCALITY
␦V共y兲
关␯t共x兲 − ␯t共x⬘兲兴 = ␦共x⬘ − y兲 − ␦共x − y兲
+ e−t共x⬘ − x兲y ␳共y兲 + O共e−2t兲,
1. Nonlocality in (equilibrium) transient distributions
The nonlocal features as discussed in the present paper
refer to fluctuations and responses in steady nonequilibria.
Nevertheless, their origin takes us to Poisson equations for
reversible dynamics; see Eqs. 共3.12兲 and 共3.16兲. We therefore
start here with a look at equilibrium dynamics but in the
transient regime.
Take a reversible 共v = 0兲 diffusion in a potential landscape
V at equilibrium, ␯ = log ␳ = −V 共up to an irrelevant constant兲.
共B4兲
the nonlocal component of which has a weight exponentially
damped in time.
2. Green’s function in confinement region
The response analysis in Secs. III B and III C reduces the
problem to finding the Green’s function,
011121-7
PHYSICAL REVIEW E 80, 011121 共2009兲
MAES, NETOČNÝ, AND SHERGELASHVILI
L␳共x兲G共x,y兲 = − ␦共x − y兲,
共B5兲
with L␳ = ␳1 ⵜ · 共␳ⵜ兲 generating diffusion in the potential
−log ␳. It has an explicit solution in terms of the transition
kernel 共or transition-probability density兲,
G共x,y兲 =
冕
+⬁
关pt共x,y兲 − ␳共y兲兴dt.
In a confinement region where ␳ is approximately constant
around its maximum, one expects that G共x , y兲 is essentially
determined by a free diffusion. To asses this conjecture, we
need to understand how the inhomogeneities in ␳ outside the
confinement region influence the Green’s function inside it
⬁
free
关pfree
t 共0,x兲 − pt 共0,x⬘兲兴dt =
0
冦
冋 冉 冊册
冋 冉 冊册
if d = 1
1
共log兩x⬘兩 − log兩x兩兲,
2␲
if d = 2
1 −d/2
␲ ⌫共d/2 − 1兲共兩x兩2−d − 兩x⬘兩2−d兲, if d ⱖ 3.
4
yi 1
+
L 2
e −␲
2n2/L2t
共B9兲
冕
⬁
0
关pt共x,y兲 − ␳共y兲兴dt =
共B7兲
冧
共B8兲
共B11兲
with the summation over the dual lattice, ki = . . . ,
−2␲ / L , 0 , 2␲ / L , . . .. Hence,
G共0,x兲 =
1
1 ik·x
e ,
d兺
L k⫽0 兩k兩2
共B12兲
which is to be compared with the free diffusion for which,
formally,
1
共2␲兲d
冕
dk ik·x
e .
兩k兩2
共B13兲
The latter is “infrared divergent” for d ⱕ 2 and the above
finite lattice version just provides its particular regularization
共above all it provides a cutoff of the neighborhood of k = 0兲.
Note that an alternative 共and more standard兲 way of regularizing the free Green’s function is to add a “positive mass:”
for d = 2 one obtains
1
L 1
− 兩y − x兩 + 共x2 + y 2兲.
12 2
2L
G⑀共0,x兲 =
共B10兲
=
Clearly, up to a correction O共1 / L兲 it coincides with the
Green’s function for free diffusion; moreover, the “infinite”
additive constant has been regularized and fixed by the
length of the region.
In general one has
,
1
兺 exp兵ik · x − 兩k兩2t其
Ld k
Gfree共0,x兲 =
For d = 1 the Green’s function can be obtained explicitly,
G共x,y兲 =
pt共0,x兲 =
1 2
xi 1
+ 兺 cos ␲n
+
L L nⱖ1
L 2
⫻cos ␲n
2/4t
the Green’s function Gfree共x兲 = 兰⬁0 pt共0 , x兲dt only exists in the
transient case, d ⱖ 3. For d = 1 , 2 the divergence cannot be
“renormalized” via a stationary density like in Eq. 共B6兲 as
the latter does not exist. Yet, its divergent part is in fact x
independent and one has in all dimensions well-defined
differences,
1
共兩x⬘兩 − 兩x兩兲,
2
Hence, the free diffusion Green’s function is well defined up
to a possibly infinite additive constant. However the latter
becomes irrelevant due to the “dipole” character of the
source term in Eq. 共3.12兲 and 共3.16兲; cf. also Eq. 共B20兲 below.
It remains to see in what sense the exterior of a confinement region enters the properties of the 共true兲 Green’s function. To simulate that, we consider the 共standard兲 diffusion in
a cube 关−L / 2 , L / 2兴d with reflexive boundary conditions. The
d
qt共xi , y i兲 with
transition kernel is pt共x , y兲 = 兿i=1
qt共xi,y i兲 =
−d/2 −兩x兩
pfree
e
t 共0,x兲 = 共4␲t兲
共B6兲
0
冕
and, more basically, how to make sense to the generally illdefined expression Eq. 共B6兲 for the free diffusion.
As well known 关33兴, for a free diffusion 共related to the
Brownian motion as 冑2Bt兲 in dimension d, with the transition
kernel
=
冕
冕
1
4␲2
1
2␲
⬁
0
dk
eik·x
兩k兩2 + ⑀2
uJ0共u兩x兩兲
du
u2 + ⑀2
1
K0共⑀兩x兩兲
2␲
共B14兲
with J0 and K0 the Bessel functions of the first and of the
011121-8
PHYSICAL REVIEW E 80, 011121 共2009兲
NONEQUILIBRIUM RELATION BETWEEN POTENTIAL AND…
second kind, respectively. Its short-distance asymptotics is
G⑀共0,x兲 =
log 2 − ␥
1
−
log共⑀兩x兩兲 + o共1兲
2␲
2␲
共B15兲
共␥ being the Euler constant兲.
These calculations indicate what the “boundary effects”
on the Green’s function in a region with approximatively
constant density profile are: the difference from the free
Green’s function becomes negligible on length scales much
smaller than size of the region. Although this comparison
includes the removal of an infrared divergence in low dimensions, the latter is well “renormalizable” in the above sense
of finite differences. Our conclusion is that one could for the
present purposes deal just with the free Green’s function 共although as such being ill defined兲.
As a specific example we consider the OrnsteinUhlenbeck process for the diffusion in a quadratic spherically symmetric potential. Its Green’s function can be found
by solving the Eq. 共B5兲 with ␳共x兲 ⬀ exp关−V共x兲兴
⬅ exp共−⑀2兩x兩2 / 2兲 共for simplicity we restrict here to a source
located at the origin兲. Again in two dimensions, this has a
solution
GOU共0,x兲 = −
1
2␲
冕
兩x兩
1/⑀
dr V共r兲
e
r
共B16兲
up to an arbitrary additive constant. For ⑀兩x兩 Ⰶ 1, what we
have called the confinement region, it reads
GOU共0,x兲 = −
1
log共⑀兩x兩兲 + o共1兲,
2␲
⌫共d/2兲
2␲d/2
冕
兩x兩
1/⑀
dr V共r兲
e ,
rd−1
␦G共x兲 =
共B18兲
yielding a power-law decay 兩x兩2−d if ⑀兩x兩 Ⰶ 1.
冕
dyⵜy
G共x,y兲
␳共y兲
· 共J␦␯兲共y兲
共B20兲
and therefore
J共x兲 · ⵜx
␦G
␦G
= J共y兲 · ⵜy
.
␦␯共y兲
␦␯共x兲
共B21兲
That relation vaguely resembles an Onsager reciprocity:
− ␦␯␦共y兲 ⵜxG can be interpreted as the extra 共gradient兲 force at x
needed to have a delta-function-like response at y.
APPENDIX C: DYNAMICAL FLUCTUATIONS
Variational principles often arise in the context of fluctuation theory. For example, the entropy and related thermodynamic potentials play a role both as important equilibrium
variational functionals 共second law兲 and as functionals governing the equilibrium fluctuations 共Einstein’s fluctuation
theory兲. Here we sketch how the functionals G and I introduced in Sec. IV B also fit such a scheme; a more technical
account of this problem for overdamped diffusions can be
found in 关1兴.
We consider the diffusion process 关Eq. 共2.1兲兴 with the test
potential ⌽ set to zero. For any random realization xt, t ⱖ 0,
of this process we introduce the empirical occupation density
共B17兲
in agreement with either of the two above regularization procedures. Similarly, for dimensions d ⬎ 2 one gets
GOU共0,x兲 = −
over that J is divergenceless and that ␦G is physically determined only modulo a constant, one can then write Eq. 共B19兲
as
¯␳T共z兲 =
1
T
冕
T
␦共xt − z兲dt
共C1兲
0
that counts the relative time spent at each point z. Apparently, ¯␳T共z兲 is the random density dependent on the realized
history. In the limit T → ⬁ it converges to the stationary density ␳, with probability one by the ergodic theorem. The main
result reads that for large but finite times T, the probability of
fluctuations of ¯␳T has the asymptotics given by the large
deviation law
3. “Reciprocity relations” in the inverse problem
P共¯␳T = ␮兲 ⬀ e−TI关␮兴 ,
In contrast to Sec. III B, the analysis of Sec. III C does not
make any use of close-to-equilibrium assumptions while all
the same, in Eq. 共3.16兲 and for the inverse problem, the linear response is given in terms of a reversible process. A
solution to Eq. 共3.16兲 can be written as
in which the functional I关␮兴 of Sec. IV B is recognized as an
exponential decay rate. Then, the variational inequality
I关␮兴 ⱖ I关␳兴 = 0 is a mere consequence of fluctuation law
共C2兲. As observed before, finding I关␮兴 amounts to solving
inverse stationary problems 共4.14兲 and 共4.15兲 or, equivalently, to evaluating variational expression 共4.17兲.
Similarly, functional 共4.12兲 reveals to be the exponential
decay rate in the large deviation asymptotics of the joint
probability law for the empirical occupation times and the
empirical current. For proofs and for more details see 关1兴.
The large deviation theory for stochastic systems has been
started and rigorously established by Donsker and Varadhan
关34,35兴. In the physics literature, these methods go back to
the seminal work of Onsager and Machlup 关36兴.
␦G共x兲 = −
冕
dyG共x,y兲
冉
冊
J
· ⵜ␦␯ 共y兲,
␳
共B19兲
where G共x , y兲 is the Green’s function, Eqs. 共B5兲 and 共B6兲.
Since the transition probabilities satisfy detailed balance, the
Green’s function exhibits the same symmetry: ␳共x兲G共x , y兲
= ␳共y兲G共y , x兲.
We now propose a partial integration which is allowed if
the current J decays sufficiently fast at infinity; using more-
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PHYSICAL REVIEW E 80, 011121 共2009兲
MAES, NETOČNÝ, AND SHERGELASHVILI
关1兴 C. Maes, K. Netočný, and B. Wynants, Physica A 387, 2675
共2008兲.
关2兴 S. Chandrasekhar, Rev. Mod. Phys. 21, 383 共1949兲.
关3兴 J. Binney and S. Tremaine, Galactic dynamics 共Princeton University Press, Princeton, NJ, 1987兲. See in particular Chap. 6
on disk dynamics and spiral structures.
关4兴 G. Ryskin, Phys. Rev. Lett. 61, 1442 共1988兲.
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