Brownian particles in periodic potentials coarse-graining versus ne structure Lucianno Defaveri1 Eli Barkai2 and David A. Kessler1 1Department of Physics Bar-Ilan University Ramat Gan 52900 Israel and
2025-04-30
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Brownian particles in periodic potentials: coarse-graining versus fine structure
Lucianno Defaveri1, Eli Barkai2, and David A. Kessler1
1Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel and
2Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan 52900, Israel
We study the motion of an overdamped particle connected to a thermal heat bath in the presence
of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle
positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading
particles, all starting from a common origin, converges to a normal distribution centered at the
origin with a mean-squared displacement that grows as 2D∗t, with an effective diffusion constant
that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-
grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs
(BG) factor e−V(x)/kBT, the latter being non-normalizable. We explain this result and construct a
theory of observables using the Fokker-Planck equation. These observables are classified as those
that are related to the BG fine structure, like the energy or occupation times, while others, like
the positional moments, for long times, converge to those of the large-scale description. Entropy
falls into a special category as it has a coarse-grained and a fine structure description. The basic
thermodynamic formula F=T S −Eis extended to this far-from-equilibrium system. The ergodic
properties are also studied using tools from infinite ergodic theory.
I. INTRODUCTION
Problems involving diffusion of atoms and molecules
on surfaces, lattices, and general periodic potentials have
been studied for decades [1–11] due to their applicability
to a wide range of systems such as diffusion of adatoms
[4,12], of proteins on a membrane [13] and in one dimen-
sional corrugated channels [14–22]. Brownian particles
in a one-dimensional periodic potential landscape V(x),
stretching across all space (−∞,∞), cannot reach a state
of equilibrium since, due to the nonbinding nature of the
potential, the equilibrium distribution is not normalized
as R∞
−∞ e−V(x)/kBTdx → ∞, where kBis the Boltzmann
constant and Tthe temperature of the environment. For
short times, particles moving in a periodic lattice be-
come stuck in attractive regions, or wells, of the poten-
tial. Eventually, however, the particles will experience
an environmental fluctuation large enough to overcome
the finite potential barrier and will reach a neighboring
well [23,24]. A schematic representation of this model is
shown in Fig. 1. This macroscopic motion is character-
ized by an effective diffusion constant D∗which is always
smaller than the free diffusion constant D[1].
Despite these types of systems being unable to reach
a state of true equilibrium and therefore not obeying the
ergodic hypothesis, the Boltzmann-Gibbs factor, though
non-normalizable, can still be used to study the prop-
erties of the system, as is the case with other non-
confining potentials [25–27], logarithmic potentials used
in subrecoil-laser-cooled gases [28], diffusion processes
with heterogeneous diffusion fields [29,30] and random
potentials used in Sinai diffusion [31]. We show here
that this non-normalizable state, i.e., the Boltzmann-
Gibbs factor, gives the fine structure of the probabil-
ity packet, and discuss the consequences of this. This
non-normalized state was foreseen by Sivan and Farago
[32,33]. By fine structure, we mean the density fluctu-
ations on the scale of the period of the potential, which,
in the long time limit, is of course much smaller than the
scale associated with diffusion √2D∗t.
x
a
V0
FIG. 1. A schematic representation of the Brownian motion of
non-interacting particles, using the potential in Eq. (1). The
lattice period is a, the height of the potential barrier between
wells is V0and the system is at temperature T.
Experimental advances in optical lattices [34,35] which
allow experimentalists to probe the fine-grained nature
of systems, motivate us to ask: how does the interplay
between the fine structure and more coarse-grained de-
scriptions, which are both present in the probability den-
sity function (PDF), affect the properties of observables?
What are their ergodic properties? Our aim in this pa-
per is to answer those questions. It should be noted that
one may observe the density of the spreading packet of
particles either at a coarse-grained level or by paying at-
tention to the fine structure. That it is say, when we
observe the concentration of many non-interacting parti-
cles in the periodic potential, we may bin the data with
bin sizes either smaller or greater than the period of the
lattice. The latter case, which we call coarse-graining,
will lead to the loss of information, though it is some-
times needed, since in the long time limit, within a small
bin, we may not find a statistically sufficient number of
particles. The coarse-graining issue is then translated to
other observables, like entropy. As explained below, it
can generally result in widely different points of view on
arXiv:2210.10935v2 [cond-mat.stat-mech] 14 Feb 2023
2
the system if compared to a fine-scale observation.
We note that considerable attention was devoted in the
literature to the coarse-graining problem, in a thermody-
namical setting [36–42], here, however, we deal with a
new domain, that of infinite ergodic theory [25–27,43–
46]. As we explain below, the time-invariant infinite den-
sity in our system is the Boltzmann-Gibbs factor, which,
as we mentioned, is non-normalizable.
The manuscript is organized as follows. In Section II
we describe the potential and the basic concepts and tools
of our model. In Section III we present, using intuitive
arguments, the long-time PDF of the Brownian particle.
We discuss the different types of observables with respect
to their ensemble averages in Section IV, and with respect
to their time, together with their ergodic properties, in
Section V. In Section VI we calculate the entropy for both
coarse-grained and fine structure descriptions. In Section
VII we provide a rigorous derivation of the PDF using
an eigenfunction expansion. Finally, in Section VIII we
present our concluding remarks.
II. MODEL
We consider the one-dimensional overdamped motion
of a Brownian particle in a thermal environment of tem-
perature Twhich is also subjected to the external peri-
odic potential V(x) = V(x+a), consisting of attractive
well regions (local minima) separated by potential barri-
ers (local maxima) of height V0. A potential that fulfills
these characteristics is given by
V(x) = −V0
2cos(2πx/a),(1)
where ais the lattice spacing. In Fig. 1we show a
schematic representation of the model. The probability
density Pt(x) of the particle at time tis described by the
Fokker-Planck equation (FPE) [47]
∂Pt(x)
∂t =D∂2Pt(x)
∂x2+1
kBT
∂
∂x ∂V
∂x Pt(x),(2)
where Dis the bare diffusion constant and Tis the tem-
perature of the environment.
Equivalently, we could describe the system at the
level of individual trajectories, or realizations, using the
Langevin equation
γ˙x=−∂V
∂x +p2γkBT η(t),(3)
where γis the damping constant, which obeys Ein-
stein’s relation D=kBT/γ, with η(t) being a stochas-
tic Gaussian white noise with zero mean and variance
hη(t)η(t0)i=δ(t−t0). For each realization, we would
have a stochastic trajectory xη, so that
Pt(x) = hδ(x−xη)iη,(4)
where the brackets h...iηrepresent averages taken over
an ensemble of trajectories xη. We will later use the
Langevin equation to numerically compute the time av-
erages of physical observables, while in the first part of
the manuscript we will use the Fokker-Planck equation.
An important dimensionless control parameter for
studying the system is the ratio between the height of
the potential barrier and the typical energy from ther-
mal fluctuations V0/kBT. Our main results are valid in
all temperature ranges, provided that the time is large
enough.
III. ASYMPTOTIC SOLUTION
In Section VII we present a derivation, using an eigen-
function expansion, of the asymptotic solution for the
PDF Pt(x) governed by Eq. (2). For the moment, we
will rely on the more physically transparent ansatz-based
derivation of Sivan and Farago [32,33] which we herein
recapitulate in order to make the current work self-
contained. For long times, the mean squared displace-
ment, which is equivalent to the second positional mo-
ment, follows the expression hx2i ∼ 2D∗t, where D∗is
the effective diffusion constant, which can be calculated,
as shown by Lifson and Jackson [1], as
D∗=D
De
V(x)
kBTEaDe−V(x)
kBTEa
,(5)
where we define the average over a lattice period as
hfia= (1/a)Ra/2
−a/2f(x)dx. For the specific case of the
potential in Eq. (1) we have D∗=D/I2
0(V0/2kBT), with
I0(...) being the 0-th modified Bessel function of the first
kind. This allows us to define the effective diffusive
lengthscale √2D∗t.
Any periodic potential ¯
V(x) can be shifted by a con-
stant value δV ≡(kBT /2) ln e−V /kBTa/eV /kBTa,
giving a new potential V(x) = ¯
V(x) + δV . For this new
potential, we have that eV/kBTa=e−V /kBTa. For
simplicity, we will use this convention and study poten-
tials that obey this equality, as the force field is clearly in-
variant under the above-mentioned transformation, and
therefore Eqs. (2) and (3) are unchanged.
For long times and a range of positions much less than
the diffusive lengthscale, x√2D∗t, the PDF becomes
proportional to the BF distribution as
Pt(x)∝e−V(x)
kBT
tα.(6)
where α > 0. We see that this is a solution by plugging
Eq. (6) in the FPE (2), the right-hand side is identically
zero and the left-hand side is ∂tPt(x)≈αPt(x)/t, and in
the limit of large twe have ∂tPt(x)→0. For large length-
scales, x∼√2D∗t, the fine structure of the PDF can be
neglected, leading to a free particle-like description, with
an effective diffusion constant D∗, that is,
Pt(x)≈e−x2
4D∗t
√4πD∗t.(7)
3
We compare Eq. (6) and Eq. (7), to conclude that
α= 1/2. By matching both limits we obtain a uniform
approximation as
Pt(x)≈const e−V(x)
kBTe−x2
4D∗t
√4πD∗t.(8)
The constant is calculated by imposing the normalization
of the PDF,
const Z∞
−∞
e−V(x)
kBTe−x2
4D∗t
√4πD∗tdx ≈1.(9)
We perform a change of variables to y≡x/√t,
const Z∞
−∞
e−V(y√t)
kBTe−y2
4D∗
√4πD∗dy ≈1,(10)
where we see that the Boltzmann-Gibbs factor
e−V(y√t)/kBToscillates rapidly allowing it to be replaced
by its average value in a period, that is,
const De−V
kBTEaZ∞
−∞
e−y2
4D∗
√4πD∗dy ≈1,(11)
where the integral is clearly unity, and const =
1/e−V/kBTa. The uniform approximation becomes
Pt(x)≈e−V(x)
kBTe−x2
4D∗t
Zt
,(12)
where we define the normalizing term
Zt≡De−V/kBTEa
√4πD∗t=√4πDt . (13)
Using this uniform approximation, it is possible to obtain
an time-invariant infinite density of the system as
lim
t→∞ ZtPt(x) = e−V(x)
kBT,(14)
a result that is known for asymptotically flat potentials
[25,26], which is here seen to also be valid in the case of
periodic potentials. For finite long times, Eq. (14) holds
for x√2D∗t, that is, xmuch smaller than the dif-
fusive lengthscale. This expression, which is valid re-
gardless of initial conditions, shows that the system re-
laxes to a state closely related to thermal equilibrium de-
scribed by the Boltzmann-Gibbs factor, even if the latter
is non-normalized, with the time-dependent Ztdefined in
Eq. (13) replacing the usual normalizing partition func-
tion. In panel (a) of Fig 2we show the relaxation of
Pt(x) to the Boltzmann-Gibbs factor using a numerical
integration of the FPE (2).
The uniform approximation can be improved by con-
sidering additional long-time corrections. In Section VII
we present a rigorous eigenfunction derivation, while in
this section we will follow the same principle used by
012345
x/a
0.0
2.5
5.0
7.5
ZtPt(x)
(a)
Zt=√4πDt e−V(x)/kBT
0 2 4 6 8 10 12 14
x/a
0.00
0.25
0.50
0.75
1.00
ZtPt(x)eV(x)/kBT
(b)
Dt
a2=20.0
Dt
a2=50.0
Dt
a2=200.0
FIG. 2. Panel (a): numerical results for ZtPt(x) (solid lines),
for the three different times shown in the legend of panel
(b). The black dashed line represents the Boltzmann-Gibbs
factor e−V(x)/kBT. We can observe that, for longer times
and for x√4D∗t,ZtPt≈e−V(x)/kBT, as expected from
Eq. (14). Panel (b): numerical results for ZtPt(x) divided by
the Boltzmann-Gibbs factor (solid lines) for three different
times shown in the legend. The black dashed lines represent
the prediction in Eq. (24). In both panels V0/kBT= 4 and
D∗/D ≈0.19.
Sivan and Farago in [32,33] and propose a solution in
the form
Pt(x) = e−V(x)
kBT−x2
4D∗t
Zt1−τ(x)
2t,(15)
where τ(x) is an ansatz. We plug the proposed solution
in Eq. (15) into the FPE (2), and limit ourselves to long
time contributions up to O(t−3/2). The left-hand side of
the FPE, in this limit, becomes,
∂Pt(x)
∂t ≈ −e−V(x)
kBT
2tZt
,(16)
and the right-hand side of the FPE,
D∂2Pt(x)
∂x2+V0(x)
kBT
∂Pt(x)
∂x =−D
D∗
e−V(x)
kBT
2tZt
(D∗τ00(x)
+1 −V0(x)
kBT[x+D∗τ0(x)],(17)
4
leading to a differential equation for the ansatz as
τ00(x)−1
D∗
V0(x)
kBTx+D∗τ0(x)=1
D−1
D∗.(18)
This equation can be solved as
τ(x) = 1
DZx
0
e
V(y1)
kBTZy1
0
e−V(y2)
kBTdy2dy1+
−x2
2D∗+C0
DZx
0
e
V(y)
kBTdy +C1,(19)
where C1is a constant that ensures the normalization
of Pt(x) and C0ensures that there is no biased parti-
cle flow. For a symmetric potential and initial condition
at the potential minimal, we expect the PDF to be dis-
tributed in space symmetrically, therefore we must have
that τ(x) = τ(−x), which leads to C0= 0. For an asym-
metric potential, we must instead impose that there is
no macroscopic drift of particles, that is, C0is defined to
ensure that τ(a) = τ(−a),
C0=1
2Ra
0e
V(y)
kBTdy Za
0
e
V(y1)
kBTZa
y1
e−V(y2)
kBTdy2dy1
−Za
0
e
V(y1)
kBTZy1
0
e−V(y2)
kBTdy2dy1.(20)
We can manipulate the expression for τ(x) to write
that
τ(x) = x U1(x)
D∗+U2(x)
D∗,(21)
where U1(x) and U2(x) are a-periodic functions with
U1(x) = aRx
0e
V(y)
kBTdy
Ra
0e
V(y)
kBTdy −x . (22)
The initial conditions are present in U2(x), which we will
here omit giving the full expression. The scaling y≡
x/√trepresents the diffusive motion of the particles, it
reflects the Gaussian spreading of the PDF, and we use
this scaling to write
τ(x)
2t=yU1(x)
2D∗√t+U2(x)
2t,(23)
In this scale, we neglect terms of order O(t−1), leaving
us with the yU1(x) term, which contains contributions
to both coarse-grained and fine-grained structures. We
reach the final expression [32,33]
Pt(x)≈e−V(x)
kBTe−x2
4D∗t
Zt1−x U1(x)
2D∗t,(24)
where Zt, which was defined before, plays a similar role
as the partition function for regular Boltzmann-Gibbs
equilibrium. This final expression is valid regardless of
the symmetry properties of V(x) and can be used for
any initial condition x0simply by translating the x-axis
so that x0becomes the new origin.
In Fig. (2) we compare our results in Eqs. (14) and
(24) with the numerical integration of Eq. (2). In the
top panel (a), we show how the PDF Pt(x) multiplied by
Ztconverges in the long time limit to the Boltzmann-
Gibbs factor, akin to systems with perfectly normalized
BG states. In the lower panel, we plot the density divided
by the Boltzmann-Gibbs factor versus x. In the long time
limit, we expect a Gaussian propagator, similar to that
of a free particle, with an effective diffusion constant D∗,
however, at not too long times, the correction term in
Eq. (24) is clearly important.
IV. ENSEMBLE AVERAGES
In this section, we focus on the ensemble average of
a physical observable O(x) at a given time t, which we
label hOit, given by
hOit=Z∞
−∞ O(x)Pt(x)dx . (25)
We will now classify the different observables and their
dependence on the non-normalized Boltzmann-Gibbs
state in the long time limit. We will see that some observ-
ables are sensitive to the fine scale of the solution, namely,
to the Boltzmann-Gibbs factor, while others are con-
trolled by the coarse-grained description of Pt(x), which
amounts to a Gaussian.
A. Positional moments
It is possible to calculate the q-th moments of x,h|x|qit,
using the PDF in Eq. (12). The statistical properties of
these observables are controlled by the large-scale solu-
tion of the packet. For long times, their statistics follow
those of a free particle with the effective diffusion con-
stant D∗. As an example, we calculate the ensemble aver-
age of the second moment, the mean square displacement
(MSD), hx2it. We perform the same change variables to
y≡x/√tas we did to calculate the normalization in
Section III, to obtain the expression
hx2it≈tZ∞
−∞
y2e−V(y√t)
kBTe−y2
4D∗
√4πD dy . (26)
In the long-time limit, we see that e−V(y√t)/kBTwill os-
cillate rapidly, which allows us to replace its value for an
average in a period, that is,
hx2it≈tDe−V(x)
kBTEaZ∞
−∞
y2e−y2
4D∗
√4πD dy
≈trD
D∗Z∞
−∞
y2e−y2
4D∗
√4πD dy = 2D∗t , (27)
where we have used that he−V /kBTia=pD/D∗and ob-
tained the expected variance for normal diffusion with
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Brownianparticlesinperiodicpotentials:coarse-grainingversus nestructureLuciannoDefaveri1,EliBarkai2,andDavidA.Kessler11DepartmentofPhysics,Bar-IlanUniversity,RamatGan52900,Israeland2DepartmentofPhysics,InstituteofNanotechnologyandAdvancedMaterials,BarIlanUniversity,RamatGan52900,IsraelWestudythemoti...
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