Irreversible dynamics of a massive intruder in dense granular fluids

Abstract

A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a massive intruder in a dense granular fluid. The model reproduces numerical correlation and response functions, violating the equilibrium Fluctuation Dissipation relations. The source of memory is identified in the coupling of the tracer velocity $V$ with a spontaneous local velocity field $U$ in the surrounding fluid. Such identification allows us to measure the intruder's fluctuating entropy production as a function of $V$ and $U$, obtaining a neat verification of the Fluctuation Relation. Comment: 5 pages, 3 figures accepted for publication in EPL

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arXiv:1007.0364v1 [cond-mat.stat-mech] 2 Jul 2010
Memory effects and entropy production in granular fluids
Alessandro Sarracino,1,2Dario Villamaina,1,2Giacomo Gradenigo,1,2and Andrea Puglisi1,2
1CNR (ISC) - p.le A. Moro 2, 00185, Roma Italy
2Dipartimento di Fisica, Universit` a Sapienza, p.le A. Moro 2, 00185, Roma Italy
A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a
massive intruder in a dense granular fluid. The model reproduces numerical correlation and response
functions, violating the equilibrium Fluctuation Dissipation relations. The source of memory is
identified in the coupling of the tracer velocity V with a spontaneous local velocity field U in
the surrounding fluid. Such identification allows us to measure the intruder’s fluctuating entropy
production as a function of V and U, obtaining a neat verification of the Fluctuation Relation.
PACS numbers: 05.70.Ln,45.70.Mg,05.10.Gg
Models of granular fluids are a natural framework
where the issues of non-equilibrium can be addressed [1].
Due to dissipative interactions among the microscopic
constituents, energy is not conserved and external sources
are necessary in order to maintain a stationary state.
Heat fluxes and currents continuously pass through the
system, time reversal invariance is broken and conse-
quently, properties such as the Equilibrium Fluctuation-
Dissipation relation (EFDR) do not hold. In recent years,
a rather complete theory, at least in the dilute limit, has
been developed and numerous aspects have been clarified,
in good agreement with numerical simulations [2, 3]. On
the other hand, a general understanding of dense gran-
ular fluids is still lacking. A common approach is the
so-called Enskog correction [2], which reduces the break-
down of Molecular Chaos (MC) to a renormalization of
the collision frequency. This is not appropriate to de-
scribe dynamical effects such as violations of the Einstein
relation at large packing fractions [4].
In this letter, we consider a massive tracer moving in
a gas of smaller granular particles, both coupled to an
external bath, and propose a model for its dynamics. In
particular, starting from the dilute limit, where the sys-
tem has a closed analytical description [5], and taking
such limit as reference point, we suggest a Generalized
Langevin Equation (GLE) with an exponential memory
kernel as first approximation also capable of describing
the dense case. Here, the main features are: i) the de-
cay of correlation and response functions is not simple
exponential and shows backscattering [6, 7] and ii) the
EFDR [8, 9] of the first and second kind do not hold.
In the model we propose detailed balance is not neces-
sarily satisfied, non-equilibrium effects can be taken into
account and the correct behavior of correlation and re-
sponse functions is predicted. Furthermore, the model
has a remarkable property: it can be mapped onto a two-
variable Markovian system, i.e. two coupled Langevin
equations with simple white noises. The dilute limit is
then naturally recovered by putting to zero the coupling
constant between the original variable and the auxiliary
one. The auxiliary variable can be identified in the local
velocity field spontaneously appearing in the surrounding
fluid. The introduction of such a field in terms of observ-
able quantities allows us to address other interesting is-
sues, such as the computation of the fluctuating entropy
production [10] and a fair verification of the Fluctuation
Relation [9, 11, 12]. This appears as a remarkable result
if compared with unsuccessful past attempts [13, 14].
We consider an “intruder” disc of mass m0= M and
radius R, moving in a gas of N granular discs with mass
mi = m (i > 0) and radius r, in a two dimensional
box of area A = L2. We denote by n = N/A the num-
ber density of the gas and by φ the occupied volume
fraction, i.e.φ = π(Nr2+ R2)/A and we denote by
V (or v0) and v (or vi with i > 0) the velocity vec-
tor of the tracer and of the gas particles, respectively.
Interaction among the particles are hard-core binary in-
stantaneous inelastic collisions, such that particle i, af-
ter a collision with particle j, comes out with a velocity
v′
i= vi−(1+α)
vector joining the particles’ centers of mass and α ∈ [0,1]
is the restitution coefficient (α = 1 is the elastic case).
The mean free path of the intruder is proportional to
l0= 1/(n(r +R)) and we denote by τcits mean collision
time. Two kinetic temperatures can be introduced for the
two species: the gas granular temperature Tg= m?v2?/2
and the tracer temperature Ttr= M?V2?/2.
In order to maintain a granular medium in a fluidized
state, an external energy source is coupled to every par-
ticle in the form of a thermal bath [15]. The motion
of a particle is then described by the following stochastic
equation (from hereafter, exploiting isotropy, we consider
only one component of the velocities)
mj
mi+mj[(vi−vj)· ˆ n]ˆ n where ˆ n is the unit
mi˙ vi(t) = −γbvi(t) + fi(t) + ξb(t). (1)
Here fi(t) is the force taking into account the collisions
of particle i with other particles, and ξb(t) is a white
noise (different for all particles), with ?ξb(t)? = 0 and
?ξb(t)ξb(t′)? = 2Tbγbδ(t − t′). The effect of the external
energy source balances the energy lost in the collisions
and a stationary state is attained with mi?v2
For low packing fractions, φ ? 0.1, and in the large
mass limit, m/M ≪ 1, using the Enskog approximation
i? ≤ Tb.

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it has been shown [5] that the dynamics of the intruder
is described by the Langevin equation
M˙V = −ΓEV + EE, (2)
where ΓE= γb+γE
(g2(r + R) being the pair correlation function for a gas
particle and the intruder at contact), and EE is a white
noise with ?EE(t)EE(t′)? = 2TE
γE
g
function shows a simple exponential decay, with char-
acteristic time τE ∼ M/ΓE, and the EFDR is trivially
satisfied: time-reversal invariance, which is very weakly
modified for uniform dilute granular gases [16, 17], be-
comes perfectly satisfied for a massive intruder.
As the packing fraction is increased, the Enskog ap-
proximation is less and less effective in predicting the
dynamical properties of the system. In particular, ve-
locity autocorrelation C(t) = ?V (t)V (0)?/?V2? and lin-
ear response functions R(t) = δV (t)/δV (0) (for small
δV (0)) present an exponential decay modulated in am-
plitude by oscillating functions [7]. Moreover violations
of the EFDR C(t) = R(t) (Einstein relation) are observed
for α < 1 [4, 18]. By means of molecular dynamics sim-
ulations, we have measured C(t) and R(t), for several
different values of the parameters α and φ. In Fig. 1
g, with γE
g=g2(r+R)
l0
?2πmTg(1+α)
trΓEδ(t−t′), TE
tr= (γbTb+
1+α
2Tg)/ΓE. In this case the velocity autocorrelation
0 10 2030 40
5060
70 8090 100
t/τc
0.01
0.1
1
C(t)
0 100 200300
t/τc
0
0.2
C(t)
φ=0.33
φ=0.01φ=0.1φ=0.2 φ=0.33
α=0.6
FIG. 1: (Color online). Semi-log plot of C(t) (symbols) for
different values of φ = 0.01,0.1,0.2,0.33, and α = 0.6. Times
are rescaled by τc. The continuous lines are the best fit curves
obtained with Eq. (6). Inset: C(t) and the best fit are shown
in linear scale for φ = 0.33 and α = 0.6.
symbols correspond to the velocity correlation functions
measured in the inelastic case, α = 0.6, for different val-
ues of the packing fraction φ. The other parameters are
fixed: N = 2500, m = 1, M = 25, r = 0.005, R = 0.025,
Tb= 1, γb= 200. Times are rescaled by the mean colli-
sion times τc, as measured in the different cases. Numer-
ical data are averaged over ∼ 105realizations.
In order to describe such a phenomenology, a model
with more than a characteristic time is needed. In par-
ticular, due to the high density of the system, stronger
correlations arise and memory effects have to be taken
into account. Notice that the Enskog approximation [2]
cannot predict the observed functional forms, because it
only modifies by a constant factor the collision frequency.
As a first extension of Eq. (2), we consider a Langevin
equation with a single exponential memory kernel [19]
M˙V (t) = −
?t
−∞
dt′Γ(t − t′)V (t′) + E′(t),(3)
where Γ(t) = 2γ0δ(t) + γ1/τ1e−t/τ1and E′(t) = E0(t) +
E1(t) with ?E0(t)E0(t′)? = 2T0γ0δ(t − t′), ?E1(t)E1(t′)? =
T1γ1/τ1e−(t−t′)/τ1and ?E1(t)E0(t′)? = 0.
α → 1, the parameter T1is meant to tend to T0in order
to fulfill the EFDR of the 2nd kind ?E′(t)E′(t′)? = T0Γ(t−
t′). Within this model the dilute limit is recovered if
γ1 → 0.In this case, the parameters γ0 and T0 are
expected to coincide with ΓEand TE
The exponential form of the memory kernel can be jus-
tified within the mode-coupling approximation scheme.
In this framework [20], it can be written as a sum of two
contributions: Γ(t − t′) = αδ(t − t′) + β˜Γ(t − t′), where
α and β are model dependent coefficients, and
In the limit
trof Eq. (2).
˜Γ(t − t′) =
?
dq qd−1p(q)e−(ν+D)q2(t−t′). (4)
The function p(q) weights the coupling of the tracer ac-
celeration ˙V with modulations of the tracer displace-
ment and of the velocity field of the surrounding fluid
at different wave numbers q.
e−(ν+D)q2(t−t′), with D and ν respectively the diffusion
coefficient and the kinematic viscosity of the fluid, follows
from the assumption that the velocity field and the tracer
displacement both evolve according to simple diffusional
equations [21].As further approximation, we assume
that - for not too high packing fractions - memory arises
due to re-collisions within a limited region at distance
∼ λ1 around the tracer and that this can be modeled
by an effective p(q) which is peaked around q1= 2π/λ1,
i.e. a single mode contributes to the integral in Eq. (4),
yielding˜Γ(t − t′) ∼ e−(ν+D)q2
The exponential term
1(t−t′)and then
τ1= λ2
1(2π)−2(ν + D)−1∼ τg
c(λ1/lg
0)2, (5)
with τg
path respectively. Eq. (5) relates the time-scale τ1, char-
acterizing the tail of the memory kernel, with a typical
length-scale λ1present in the system. This length-scale
will turn out to play a central role in the following.
The model (3) predicts C = fC(t) and R = fR(t) with
cand lg
0the fluid mean free time and mean free
fC(R)= e−gt[cos(ωt) + aC(R)sin(ωt)].(6)
The four parameters g, ω, aCand aR, together with the
variance ?V2?, are known functions of γ0, T0, γ1, τ1and
T1. In the elastic (T1→ T0) as well as in the dilute limit
(γ1 → 0), one gets aC = aR and recovers the EFDR
C(t) = R(t). A multi-branch fit of measured C and R

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3
against Eqs. (6), together with a measure of ?V2?, yields
five independent equations to determine the five param-
eters entering the model. We have explored the range
α ∈ [0.6,1] and φ ∈ [0.01,0.33] and in Fig. 1 we show
the best fit curves, in good agreement with the numerical
data. In Table I the best fit results are reported, together
with the predictions given by the Enskog approximation
(last four columns). We used the parameters mentioned
before, changing α or the area A (to change φ): this
makes the limit φ → 0 equivalent to γg ∼ 1/l0 → 0
(“super-dilute” limit). The last row reports about the
true dilute limit: i.e. R is reduced, at fixed l0(equal to
the value of the previous case φ = 0.2), in order to get
φ = 0.01 and γg> 0. We notice that γ0is in good agree-
ment with the drag coefficient calculated under the En-
skog approximation, ΓE, even in the dense cases. Rough
empirical identification can also be made for the other
parameters: the most stable is T1∼ Tb; a fair similitude
is also found between T0 and Ttr. In the most dense
cases it appears that γ1∼ γE
the “super-dilute” limit, but cannot hold in the dilute
one, where γ1→ 0 ≪ γE
increases as l0increases, which - at constant diameters -
is equivalent to decreasing with the packing fraction; it
also weakly grows with inelasticity. Note that, at large φ,
Ttr> TE
collisions dissipate less energy.
A fundamental feature of this model is its ability to
reproduce violations of EFDR. In Fig. 2, we plot the
correlation and response functions in a dense case (elas-
tic and inelastic). In the inelastic case, deviations from
EFDR R(t) = C(t) are clearly observed. In the inset of
Fig. 2 the ratio R(t)/C(t) is also reported.
Looking for an insight of the relevant physical mecha-
nisms of the model, it is useful to map it onto a Marko-
vian equivalent model with an auxiliary field [22]:
g∝ φ: this is confirmed in
g. Finally, the coupling time τ1
tr, which is coherent with the idea that correlated
M˙V = −γ0(V − U) +
?
2T0γ0EV
?
γ2
˙U = −U
τ1
−
γ1
γ0τ1V +
2T1γ1
0τ2
1
EU,(7)
where EV and EU are uncorrelated white noises. The
variable U(t) ∝ γ1/(τ1γ0)?t
−∞e−t−t′
τ1 [V (t′)+E1(t′)]dt′is
TABLE I: Parameters of model (3) fitting the numerical data.
αφTtr
Tg
γ0
M
T0
T1
γ1
M
τ1
τc
ΓE
M
55
48
42
24
15
γE
g
M
47 1.00 1.00
40 0.84 0.89
34 0.73 0.83
16 0.82 0.91
7 0.89 0.96
TE
tr
TE
g
1.0 0.33 1.00 1.00 55 0.99 1.0 44 67
0.8 0.33 0.92 0.90 47 0.91 1.0 42 68
0.6 0.33 0.86 0.84 44 0.82 1.1 43 89
0.6 0.20 0.92 0.91 27 0.90 1.0 26 54
0.6 0.10 0.95 0.96 17 0.95 0.99 12 29
0.6 0.01 0.99 1.00 9.6 0.99
0.6 0.01∗0.88 0.94 21 0.88
/
/
0 2.8 8.6 0.6 0.98 0.99
0 21 2012 0.85 0.93
30 40
50 60
70 80
t/τc
0.001
0.01
0.1
0 1020 30 40
50
0.9
1
R(t)/C(t)
α=0.6
α=1.0
R(t) & C(t)
FIG. 2: (Color online). C(t) (black circles) and R(t) (red
squares) for α = 1 and α = 0.6, at φ = 0.33. The continuous
lines are the best fit curves obtained with Eqs. (6). Inset: the
ratio R(t)/C(t) is reported in the same cases.
determined up to a constant multiplicative factor. We
chose the definition leading to the system (7), because,
in the dilute limit, such a form is expected to hold (see
Appendix of [5]) if U is the local velocity field of the gas
surrounding the tracer [23]. More specifically, we fix a
distance l and average the velocity of the gas particles
within a circle Clof radius l + R centered on the tracer.
In this way we define Ul= 1/Nl
number of particles in Cl, and then compute the correla-
tions ?V Ul? and ?U2
is the region where the tracer is really correlated with the
surrounding particles. A first guess is provided by the
estimate obtained from Eq. (5). Indeed, by measuring
the diffusion coefficient D and the kinematic viscosity ν
and using the value τ1 from the best fit, an estimate of
the length-scale l∗of the correlated region around the
tracer can be obtained. Alternatively, once all the pa-
rameters are fixed by the best fit results, the model pro-
vides precise values of the correlation functions ?U2? and
?V U?. Then the distance l∗such that ?V Ul∗? ∼ ?V U?
and ?U2
correlated region. Remarkably, the two estimates give
compatible results and identify a narrow range of values
for l∗.
An important independent assessment of the effective-
ness of model (3), comes from the measurement of the
fluctuating entropy production [10] which quantifies the
deviation from detailed balance in a trajectory. Given
the trajectory {V (s)}t
in Ref. [24] it has been shown that the entropy production
for the model (3) takes the form
?
i∈Clvi, where Nlis the
l?. It is difficult to estimate how large
l∗? ∼ ?U2? can be identified as the length of the
0and its time-reversed {IV (s)}t
0,
Σt= log
P({V (s)}t
P({IV (s)}t
0)
0)≈ γ0
?1
T0
−
1
T1
??t
0
ds V (s)U(s).
(8)
Boundary terms - in the stationary state - are sublead-
ing for large t and have been neglected. This functional
vanishes exactly in the elastic case, α = 1, where equipar-
tition holds and T1 = T0, and is zero on average in

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the dilute limit, where ?V U? = 0, since V and U are
uncorrelated.Formula (8) has the simple meaning of
quantifying the energy transferred by the “force” γ0U on
the tracer, weighting it with the difference between the
temperatures of the two “thermostats” coupled in sys-
tem (7). Following the procedure described above, in the
-4-3-2 -101234
Σt/<Σt>
-4
-2
0
2
4
log[P(Σt)/P(-Σt)]/<Σt>
t/τc=300
t/τc=600
t/τc=2400
t/τc=4800
y=x
-0.4-0.20 0.2 0.4
Σt/t
0
0.5
1
-log[P(Σt)]/t
FIG. 3: (Color online). Check of the fluctuation relation (9)
in the system with α = 0.6 and φ = 0.33. Inset: collapse of
the rescaled probability distributions of Σt at large times.
case φ = 0.33 and α = 0.6, we obtain the estimate for the
correlation length l∗∼ 9r ∼ 6l0. Then, measuring Ul∗
and V Ul∗ along a trajectory up to time t, Eq. (8) allows
us to compute the probability P(Σt= x) and compare it
to P(Σt= −x), in order to verify the Fluctuation Rela-
tion
log
P(Σt= x)
P(Σt= −x)= x.(9)
In Fig. 3 we report our numerical results: the inset shows
how logP(Σt)/t converges to the large deviation rate
function, and curves for different times collapse. The
main frame confirms that at large times the Fluctuation
Relation (9) is well verified. Note also that formula (8)
does not contain parameters, i.e. the slope of the graph
is exactly 1, provided that one has identified the cor-
rect entropy production rate: this is achieved through
a completely independent procedure which allows us to
determine the prefactor (1/T0− 1/T1) ≈ (1/Ttr− 1/Tb)
and the “force field” γ0U(t).
In conclusion, we designed a first granular dynami-
cal theory describing non-equilibrium correlators and re-
sponses for a massive tracer.
posal is to offer a significant insight into the mecha-
nisms of re-collision and dynamical memory and their
unexplored relation with the breakdown of time-reversal,
which brings to EFDR violations and non-zero entropy
production. Small non-Gaussian corrections [17], always
present in granular fluids, are neglected here in favor of
the largest contribution given by memory terms to viola-
tions of EFDR and entropy production. For most of the
parameters in the theory (γ0∼ ΓE, γ1∼ γE
The value of this pro-
g, T0∼ Ttr
and T1∼ Tb) we have empirical estimates and reasonable
arguments, while τ1, related to the coupling length-scale
λ1, deserves further investigations. A kinetic theory is
required to give close analytical predictions of all param-
eters and, eventually, deduce possible extensions to the
case M ∼ m, larger densities, as well as to hard spheres.
We thank A. Vulpiani for a careful reading of the
manuscript. The work of the authors is supported by the
“Granular-Chaos” project, funded by the Italian MIUR
under the FIRB-IDEAS grant number RBID08Z9JE.
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