We study the nonequlibrium state of heat conduction in a one-dimensional
system of hard point particles of unequal masses interacting through elastic
collisions. A BBGKY-type formulation is presented and some exact results are
obtained from it. Extensive numerical simulations for the two-mass problem
indicate that even for arbitrarily small mass differences, a nontrivial steady
state is obtained. This state exhibits local thermal equilibrium and has a
temperature profile in accordance with the predictions of kinetic theory. The
temperature jumps typically seen in such studies are shown to be finite-size
effects. The thermal conductivity appears to have a very slow divergence with
system size, different from that seen in most other systems.Comment: 5 pages, 4 figures, Accepted for publication in Phys. Rev. Let

Dhar, Abhishek

English

arXiv

We study the nonequilibrium state of heat conduction in a one-dimensional system of hard point particles of unequal masses interacting through elastic collisions. A BBGKY-type formulation is presented and some exact results are obtained from it. Extensive numerical simulations for the two-mass problem indicate that, even for arbitrarily small mass differences, a nontrivial steady state is obtained. This state exhibits local thermal equilibrium and has a temperature profile as predicted by kinetic theory. The temperature jumps typically seen in such studies are shown to be finite-size effects. The thermal conductivity appears to have a very slow divergence with system size, different from that seen in most other systems

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Physical Review Letters

Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses

Publications of the IAS Fellows

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Heat conduction in a one-dimensional gas of elastically colliding particles of unequal
masses
Abhishek Dhar †
Raman Research Institute, Bangalore 560080, India.
(February 1, 2008)
We study the nonequlibrium state of heat conduction in a one-dimensional system of hard point particles of unequal masses
interacting through elastic collisions. A BBGKY-type formulation is presented and some exact results are obtained from it.
Extensive numerical simulations for the two-mass problem indicate that even for arbitrarily small mass differences, a nontrivial
steady state is obtained. This state exhibits local thermal equilibrium and has a temperature profile in accordance with the
predictions of kinetic theory. The temperature jumps typically seen in such studies are shown to be finite-size effects. The
thermal conductivity appears to have a very slow divergence with system size, different from that seen in most other systems.
PACS numbers: 44.10.+i, 05.45.-a, 05.60.-k, 05.70.Ln
Introduction: The problem of finding a one-
dimensional system of interacting particles, evolving
through Newtonian dynamics, in which Fourier’s law of
heat conduction holds, has attracted considerable inter-
est in recent years [1]. One of the first studies in the field
was the work of Rieder et al [2] who obtained the exact
nonequilibrium steady state (NESS) of a chain of cou-
pled harmonic oscillators. They found a constant tem-
perature profile in the system and a heat current that
was independent of system size. This result was not
too surprising, considering that the harmonic chain is
integrable and there is no scattering of phonons. Since
then, a large number of studies have looked at the ef-
fects of introducing impurities, nonlinearity and external
potentials [3 − 8]. The specific question asked in most
of these studies is the following. Defining the thermal
conductivity K of a system through the linear response
formula j = −K(T )∇T , where j is the the current and T
the local temperature, is there a one-dimensional model
which gives a finite K ? Contrary to initial expectations,
it was found that adding impurities and/or nonlinearity
into a system did not result in finite K. The first model
in which a finite K was found was the so-called ding-a-
ling model [4] in which alternate particles on the line are
bound to harmonic springs and the other particles move
freely between pairs of the bound particles. Numerical
studies of several other models have also given finite con-
ductivity [9]. A common feature in these models is the
presence of external potentials which lead to momentum
nonconservation. Recently it has been proved rigorously
[10] that the conductivity as given by the Green-Kubo
formula always diverges in one-dimensional momentum
conserving systems with finite pressure. Two recent stud-
ies [11] have reported finite conductivity in momentum-
conserving systems but both of these have vanishing pres-
sure and so there is no contradiction.
While the proof of anamalous conductivity in momen-
tum conserving systems has been a significant progress,
there remain many issues that still need to be addressed.
First of all, the proof uses the Kubo formula and this
is not fully satisfactory since the validity of the Kubo
formula, even in the limit of arbitrarily small tempera-
ture gradients, has never been established rigorously. It
may be noted that any derivation of the Kubo formula
for thermal conductivity essentially makes the following
assumptions: (i) the NESS is in local thermal equilib-
rium (LTE) (ii) that Fourier’s law is valid and (iii) re-
gression of equilibrium fluctuations occur in the same
way as nonequilibrium processes [12]. These assump-
tions are physically motivated but may not be true in
all cases. Secondly we note that while the focus of most
of the work on heat conduction has been addressed to
obtaining Fourier’s law, the more general problem is one
of understanding the nonequilibrium energy current car-
rying state of a many-body system. For example one
important question is the existence or otherwise of LTE
in the steady state. Thus one would like to know if it
is possible to define a slowly varying temperature field
which determines all other local properties in the sys-
tem. This point has not attracted much attention, even
though it is quite crucial even for stating Fourier’s law
and using results of linear response theory. Naively one
might expect that if thermal currents in the system van-
ish in the thermodynamic limit, then LTE should hold
but this has been shown not to be true always [13]. An-
other interesting question is the determination of the
temperature profile itself. These questions are clearly
of interest and can be asked independently of whether or
not Fourier’s law holds. Finally we note that numeri-
cal studies of heat conduction are problematic for several
reasons. One needs long time numerical solution of non-
linear differential equations which is time-consuming and
not very accurate. This, in addition to long equilibration
times typically occuring in such systems, restricts one to
small system sizes. Also the treatment of boundary re-
lated problems, such as that of temperature jumps, is
1
not straightforward. Hence it has often been difficult to
arrive at correct conclusions from numerical studies and
it is desirable to have more accurate studies.
In this paper we study heat conduction in a system
of hard elastic particles of unequal masses moving on a
one-dimensional line. The only interaction between the
particles is through elastic collisions. This model was
first considered by Casati [14] as a possible candidate for
obtaining Fourier’s law but the numerical results were
insufficient to draw any definite conclusions. More re-
cently this model has been studied by Hatano [7] who
obtained a diverging conductivity. The equal-mass case
with dissipation was studied by Du et al [8] who obtained
a rather surprising NESS which implied a breakdown of
usual hydrodynamics.
Here we present extensive and accurate numerical work
on this model and also an analytic formulation of the
BBGKY-type. Our aim has been to give a more de-
tailed characterization of the NESS than has been pre-
viosly done. The present model is particularly suitable
for this purpose for two reasons: (i) simulations of this
model do not require a numerical solution of nonlinear
differential equations and it is possible to obtain very
accurate results, (ii) analytically the BBGKY hierarchy
has a comparatively simpler structure and some exact
statements can be made. The most interesting result
obtained is that the steady states for the case of equal
masses and the case with arbitrarily small mass differ-
ences are completely different. In the thermodynamic
limit the latter case exhibits LTE and the temperature
profile approaches a form predicted by kinetic theory for
a one-dimensional gas.This is surprising since kinetic the-
ory predicts a finite conductivity with a T 1/2 dependence
on temperature. On the other hand our model is momen-
tum conserving and the proof for diverging Kubo con-
ductivity holds. In our finite size studies we find a slow
divergence of the conductivity (∼ Lα with α < 0.2). Our
work also clarifies some of the problems related to bound-
ary effects. The jumps in temperature at the boundaries
are shown to be finite-size effects which are studied sys-
tematically.
Definition of model: We consider N point particles
numbered i = 1, ...N moving in a one-dimensional box
extending from 0 to L. The mass, position and velocity
of the ith particle are denoted by mi, xi and ui. The
only interaction between particles is through elastic col-
lisions. After a collision between particle i and (i+1), the
new velocities are obtained from momentum and energy
conservation and given by the linear equations:
u′i =
mi −mi+1
mi +mi+1
ui +
2mi+1
mi +mi+1
ui+1
u′i+1 =
2mi
mi +mi+1
ui − mi −mi+1
mi +mi+1
ui+1. (1)
Between collisions the particles travel with constant ve-
locity. The coupling to heat baths is implemented by
using Maxwell boundary conditions. Thus whenever a
particle of mass m collides with a wall at temperature
T , it is reflected back with a velocity chosen from the
distribution P (u) = (m | u | /T )exp(−mu2/(2T )). The
temperatures at the two ends are taken to be T1 and T2.
The case when all particles have equal masses can be
solved easily and behaves similarly to the ordered har-
monic springs case [2]. The “temperature” profile is flat
(with the value
√
T1T2), current is independent of sys-
tem size and there is no LTE. The equal-mass problem
is integrable and essentially reduces to a single-particle
problem and so the results are not surprising. As soon
as the masses are made different, the system becomes
nonintegrable and is expected to have good ergodicity
properties [15], and correspondingly the NESS should be
very different.
BBGKY-type equations: In principle, a complete so-
lution of the heat conduction problem could be ob-
tained from the steady-state solution of the master equa-
tion for evolution of the N -point distribution equation
ρ({xl, ul}, t). In practice this is difficult and a sim-
pler approach is to work with the so-called BBGKY
hierarchy which deals with reduced distribution func-
tions [16]. Let us first make the following definitions:
ρl(x, u, t) = 〈δ(x − xl)δ(u − ul)〉, ρl,l+1(x1, u1;x2, u2) =
〈δ(x1 − xl)δ(u1 − ul)δ(x2 − xl+1)δ(u2 − ul+1)〉,
where 〈A〉 = ∫ ρ({xl, ul}, t)A({xl, ul})∏l dxldul. Fur-
ther we define pl,l+1(x, u1, u2) as the number of collisions
per unit time occuring at x between the lth and (l+1)th
particles with respective velocities u1 and u2. Clearly
pl,l+1(x, u1, u2) = ρl,l+1(x, u1;x, u2)(u1 − u2)θ(u1 − u2).
In terms of these the BBGKY-type equations relating
one-point functions to two-point functions are the fol-
lowing:
∂ρl(x, u, t)/∂t+ u∂ρl(x, u, t)/∂x =∫
pl−1,l(x, u1, u2)δ(u
′
2 − u)du1du2 −
∫
pl−1,l(x, u1, u)du1 +
∫
pl,l+1(x, u1, u2)δ(u
′
1 − u)du1du2 −
∫
pl,l+1(x, u, u2)du2. (2)
These equations hold for x 6= 0, L. At the boundaries,
the distribution functions satisfy appropriate boundary
conditions.
The physical observables that we will be interested in
are the particle density n(x, t), the energy density ǫ(x, t)
and the energy current density j(x, t). These can be
expressed in terms of the one-point functions ρl(x, u, t).
Thus we have:
n(x, t) = 〈
∑
l
δ(x− xl)〉 =
∑
l
∫
ρl(x, u, t)du
ǫ(x, t) = 〈
∑
l
mlu
2
l
2
δ(x − xl)〉 =
∑
l
ml
2
∫
u2ρl(x, u, t)du
j(x, t) = 〈
∑
l
mlu
3
l
2
δ(x − xl)〉 =
∑
l
ml
2
∫
u3ρl(x, u, t)du (3)
2
The temperature field T (x) is defined as T (x, t) =
2ǫ(x, t)/n(x, t). Our first result is the following exact cur-
rent conservation equation: ∂ǫ(x, t)/∂t+∂j(x, t)/∂x = 0.
This result is easily obtained by multiplying Eq. (2)
by mlu
2/2, integrating over u, and summing over all
l. There is a pairwise cancellation of all terms on the
right-hand side. Similarly by multiplying Eq. (2) by
mlu/2, integrating and summing gives, in the steady
state: ∂ǫ(x)/∂x = 0. Thus for any choice of masses
{ml}, the energy density in the steady state is constant in
space. Physically the constancy of energy density follows
from the constancy of pressure and the linear relation
between the two quantites in an ideal gas. However this
does not imply a constant temperature profile since the
temperature also depends on the number density n(x)
which is not constant.
From the fact that the dynamics [Eq. (1)] is invariant
under a constant scaling of the masses, it follows that the
temperature profile does not change under mi → νmi.
Also from the boundary conditions it is easily shown that
T (νT1, νT2, x) = νT (T1, T2, x). From now on, we shall
consider the dimer case where we consider only particles
of two different masses m1 and m2 placed alternately on
the line. Because of the above scaling properties the only
independent variables are the ratios, m2/m1 and T2/T1,
and N . We will henceforth consider the case m1 = 1,
m2 = m, T1 = 2 and T2 = 8.
Numerical results: In our numerical simulations we let
the system evolve with the appropriate boundary condi-
tions and compute time-averages of various quantities in
the steady state. The time evolution does not require nu-
merical solution of differential equations since the exact
solution is essentially known. The system is evolved by
computing successive collision times and updating veloc-
ities using the collision rules Eq. (1). The only errors are
those due to round-off. We have verified that the simula-
tions reproduce all the exact known results both for the
equal and the unequal mass cases.
In our simulations we vary δ = m − 1 and N . The
mass values δ = 0.078, 0.11, 0.22, 0.44 were studied for
particle numbers N = 41, 81, 161, 321, 641 and 1281. The
size of the boxes were changed so that in all cases the
average density of particles was fixed at 2. The number
of particles is chosen to be odd so that at both ends the
particles in contact with the bath have the same mass.
The number of collisions over which the averaging is done
was between 109 − 1010. In all cases we checked that
increasing the time of averaging by a factor of 10 did not
significantly change the data.
In Fig. (1), we plot the steady state temperature pro-
files at different system sizes for δ = 0.22. The tem-
perature has a smooth and continuously varying profile.
There is a jump at the boundaries which decreases as the
system size is increased and is expected to vanish in the
thermodynamic limit when the current also becomes van-
ishingly small. We find that this is true for any non-zero
δ. For smaller δ one needs to go to larger system sizes
to get the same temperature profile. Infact for small δ
and large N the temperature-profile depends on δ and N
only through the scaling combination δ2N . This is illus-
trated in Fig. (2) where we plot data corresponding to
five different values of δ, each with a different N , chosen
such that δ2N is the same. Note that for δ = 0, the tem-
perature profile (which is flat and given by T =
√
T1T2)
and the energy current are both independent of system
size; thus δ = 0 is a singular point, while the steady state
in the limit δ → 0, N → ∞ with δ2N constant is quite
different and nontrivial.
For fixed value of δ, as we increase N the tempera-
ture profile approaches a limiting form. Quite amazingly
this limiting form seems to be exactly one that would be
predicted by kinetic theory. We recall that kinetic the-
ory for a one-dimensional gas predicts Fourier behaviour
with K ∼ T 1/2 and hence a nonlinear temperature pro-
file Tk(x) = [T
3/2
1 (1−x/L)+T 3/22 x/L]2/3. This has been
plotted in Fig. (1). We find that the following scaling
form gives a reasonable collapse of our data:
T (x,N, δ) = Tk(x) +
1
(δ2N)γ
g(x). (4)
The inset in Fig. (1) shows the collapse of data for δ =
0.22 obtained using the above scaling form with γ = 0.67.
We now look at the dependence of K on system size.
In Fig. (3) we plot j versus L for δ = 0.22. It is clear from
the data that the conductivity which is proportional to
Lj has a slow divergence given by K ∼ Lα with α < 0.2.
We note that this is significantly different from the sys-
tem size dependence of K ∼ L0.4 found by Hatano in the
same model [7] and also in other momentum-conserving
systems like FPU and the diatomic Toda [5,7].
Finally we have checked for LTE: to do so we compute
the steady state expectation u(4)(x) = 〈∑lmlu4l δ(x −
xl)〉. If there was LTE, this quantity would be deter-
mined by the local temperature T (x). In Fig. (4) we plot
u(4)(x), as determined directly by taking time averages
and also the value predicted from the local temperature
T (x). We see that at large system sizes LTE is indeed
achieved. We also find that for smaller values of δ, one
needs to go to larger system sizes to get LTE.
In summary we have studied heat conduction in the
unequal mass problem which appears to be the simplest
nontrivial deterministic system, in one dimension, for
which a very detailed investigation of the NESS can be
made. Our study shows that a meaningful hydrodynamic
description of the steady state is possible even in a situa-
tion where (presumably) K →∞ in the thermodynamic
limit. It is clear that further studies of this model can
throw much light on the difficult problem of transition
from the microscopic to macroscopic description in the
context of nonequilibrium phenomena.
I thank D. Dhar and M. Rao for many valuable sug-
gestions. I also thank J. Das, C. Dasgupta, R. Pandit, S.
3
Ramaswamy and B. S. Shastry for discussions.
† Also at the Poornaprajna Institute, Bangalore.
dabhi@rri.res.in
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ford, (1981).
[17] It is quite interesting to note that the T 1/2 dependence
of the conductivity, for arbitrary masses, can also be ob-
tained from the Kubo formula by simply scaling out the
temperature.
0.0 0.2 0.4 0.6 0.8
x/L
2.0
4.0
6.0
8.0
T
0.0 0.2 0.4 0.6 0.8
x/L
−40.0
−20.0
0.0
20.0
T s
c
FIG. 1. Temperature profiles for δ = 0.22 are plotted for
system sizes N = 21(indicated by o), 41, 81, 161, 321, 641 and
1281 (+). The solid line is the prediction of kinetic theory.
In the inset we have plotted Tsc = N
0.67(T (x) − Tk(x) [see
Eq. (4)] with the data for N = 161, 321, 641 and 1281.
0.0 0.2 0.4 0.6 0.8
x/L
2.5
3.5
4.5
5.5
6.5
T
δ=0.44;Ν=41
δ=0.22;Ν=161
δ=0.11;Ν=641
δ=0.078;Ν=1281
δ=0.055; Ν=2561
FIG. 2. Temperature profiles obtained with five different
sets of values for δ and N with δ2N constant.
4
100 1000
L
1
10
j
FIG. 3. Plot of j versus L for δ = 0.22 The straight line
shown corresponds to the decay j ∼ 1/L0.83.
0.0 0.2 0.4 0.6 0.8
x/L
0.0
50.0
100.0
150.0
200.0
u(
4)
NeqA; N=161
EqAv; N=161
NeqAv; N=641
EqAv; N=641
FIG. 4. Plot of u(4) as determined from a direct time
averaging (NeqAv) and from the local temperature (EqAv)
for two system sizes. For the bigger system, the curves almost
coincide.
5


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Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses

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