THE JOURNAL OF CHEMICAL PHYSICS 124, 224103 共2006兲
Statistical mechanical theory for steady state systems.
V. Nonequilibrium probability density
Phil Attarda兲
School of Chemistry F11, University of Sydney, New South Wales 2006, Australia
共Received 15 March 2006; accepted 17 April 2006; published online 12 June 2006兲
The phase space probability density for steady heat flow is given. This generalizes the Boltzmann
distribution to a nonequilibrium system. The expression includes the nonequilibrium partition
function, which is a generating function for statistical averages and which can be related to a
nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the
linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the
probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown
to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the
Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the
generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent
mechanical work, and for nonequilibrium quantum statistical mechanics. © 2006 American Institute
of Physics. 关DOI: 10.1063/1.2203069兴
I. INTRODUCTION
II. HEAT FLOW
This paper is the culmination of a series on the theory of
steady state systems.1–4 The goal of the research has been to
develop a nonequilibrium phase space probability distribution. The earlier papers laid much of the ground work in
terms of general theory and specific application to the problem of heat flow. The present paper gives an explicit formula
for the probability density in the case of steady heat flow and
extends this formula to more general cases such as nonequilibrium work and nonequilibrium quantum statistical mechanics.
The focus on the nonequilibrium phase space probability
distribution is motivated by the status of the equilibrium
theory. The Boltzmann distribution is central to equilibrium
statistical mechanics, and the theory of equilibrium statistical
mechanics is entirely dependent on that result. Accordingly,
one can say that one has a theory for nonequilibrium statistical mechanics if, and only if, one can write down explicitly
the probability distribution.
The paper is set out as follows: Section II gives the
steady state probability distribution for heat flow and explores its mathematical properties. Section III gives the details of the nonequilibrium Metropolis Monte Carlo algorithm and presents results for the thermal conductivity of a
Lennard-Jones fluid. Section IV gives the probability distribution for hydrodynamic transport in general, for nonequilibrium mechanical work on a subsystem in contact with a
thermal reservoir, and for nonequilibrium quantum statistical
mechanics.
a兲
Electronic mail: [email protected]
0021-9606/2006/124共22兲/224103/13/$23.00
A. Steady state probability
In Paper I it was shown that the appropriate thermodynamic variables for the case of heat flow were the zeroth E0
and first E1 energy moments, and their respective thermodynamic conjugates, the zeroth T0 and first T1 temperature.1
The first temperature is related to the temperature gradient.
The idea that moments and gradients are conjugate is due to
Onsager.5
Consider a subsystem of length L in thermal contact with
two heat reservoirs of temperature T± located at z = ± L / 2.
The zeroth and first inverse temperatures of the reservoirs
are1
␤0 =
冋
册
1
1 1
+
,
2kB T+ T−
␤1 =
冋
册
1
1
1
−
.
k BL T + T −
共1兲
One sees that the zeroth temperature T0 = kB / ␤0 is essentially
the average temperature of the reservoirs and that the first
temperature T1 = kB / ␤1 is essentially the imposed thermal
2
gradient T−1
1 = −共ⵜT兲 / T0.
The zeroth energy moment is the ordinary Hamiltonian
N
E0共⌫兲 = H共⌫兲 = 兺 ⑀i ,
共2兲
i=1
and the first energy moment in the z direction is just
N
E1共⌫兲 = 兺 ⑀izi ,
共3兲
i=1
where ⑀i is the total energy of particle i. The rate of change
of the energy moment, Ė1共⌫兲 = ⌫˙ · ⵜE1共⌫兲, is related to the
heat flux by J = Ė1 / V, where V = AL is the subsystem volume.
Note that previously1,3,4 this was denoted as Ė01; then, as now,
it means the adiabatic evolution. 共Adiabatic means natural or
124, 224103-1
© 2006 American Institute of Physics
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224103-2
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
Hamiltonian motion, which is to say that no heat flows to the
isolated system.兲
In the above ⌫ = 兵qN , pN其 is a point in the phase space of
the subsystem, with q and p denoting position and momentum, respectively. The conjugate point in phase space is the
one with all the velocities reversed
⌫† = 共qN,共− p兲N兲.
共4兲
The conjugation operation is synonymous with time reversal.
Both moments have an even parity, E0共⌫兲 = E0共⌫†兲
and E1共⌫兲 = E1共⌫†兲, since it is assumed that there are no velocity dependent forces in the Hamiltonian. The rate of
change of the energy moment necessarily has an odd parity,
Ė1共⌫†兲 = −Ė1共⌫兲.
Any function f共x兲 can be split into even and odd functions, f ±共x兲 = 关f共x兲 ± f共−x兲兴 / 2, and any non-negative function
can be expressed as the exponential of f共x兲 and consequently
can be factorized as g共x兲 = exp关f +共x兲兴exp关f −共x兲兴 ⬅ ge共x兲go共x兲.
Since the steady state probability distribution is necessarily
non-negative, it can be formally written as
㜷ss共⌫兩␤0, ␤1兲 = 㜷e共⌫兩␤0, ␤1兲㜷o共⌫兩␤1兲.
共5兲
The reservoirs can only enter via the energy moment
derivatives of their entropy, ␤0 and ␤1. The probabilities
must be exponentials of linear functions of ␤0 and ␤1 times
their conjugate energy moment or a linear functional of that
moment. These requirements follow from the recognition
that the exponents represent that part of the reservoir entropy
that depends on the subsystem and that the subsystem is very
much smaller than the reservoir and so a linear Taylor expansion suffices.6 In the limit ␤1 → 0, 㜷ss must reduce to the
equilibrium Boltzmann distribution, which is even, and
hence the odd exponent must be independent of ␤0.
The even term 㜷e is identical with the structural probability given previously,1
㜷e共⌫兩␤0, ␤1兲 =
1
Z̃ss共␤0, ␤1兲
e−␤0E0共⌫兲e−␤1E1共⌫兲 .
共6兲
Here Z̃ss is the normalizing partition function; top and bottom
may be multiplied by the traditional but immaterial factor of
N!h3N to make it dimensionless, and its logarithm may be
taken to form a free energy or total entropy.6 This result was
originally derived by making an explicit physical identification in terms of the change in entropy of the reservoir.1 It is
not possible to add any additional even term without violating that physical interpretation and the requirements of linearity in the inverse temperatures and of extensivity in the
energies. Monte Carlo simulations based on this probability
distribution and a comparison with certain predictions based
on a bulk equation of state provide convincing evidence that
this is indeed correct for the structure.1
In view of the above discussion the odd probability may
be written as
㜷o共⌫兩␤0, ␤1兲 = e
Wmir
1
␤1Wmir
1 共⌫兲
共7兲
,
mir
†
Wmir
1 共⌫ 兲 = −W1 共⌫兲.
with
having an odd parity,
The
label “mir” stands for mirror, which reflects the asymmetric
nature of this term.
The fact that Wmir
1 is multiplied by ␤1 means that it must
be a linear function 共or functional兲 of the conjugate thermodynamic variable E1. However, it also has to have an odd
parity, which suggests that it is a linear function or functional
of Ė1. In Ref. 3 the ansatz Wmir
1 共⌫兲 = ␣Ė1共⌫兲 was explored for
different values of the time constant ␣. The results suggested
that this was a reasonable approximation in certain regimes,
but that this form itself was not exact.
In order to identify Wmir
1 one has to give the change in
the reservoirs’ entropy associated with the point in phase
space ⌫. The difficulty is that E1 is not a conserved variable
and the total change in energy moment arises from its adiabatic evolution, which costs no reservoir entropy, and from
perturbations by the reservoir, which do. The total change in
moment is already accounted for by the even exponent, and
the quantity Wmir
1 is essentially the internal change that must
be subtracted. This particular physical interpretation of Wmir
1
is discussed in more detail at the end of Sec. II E below.
It turns out that ␤1⌬Wmir
1 may also be interpreted as a
type of thermodynamic work, or change in reservoir entropy,
over an interval. Since the system is in a steady state, over
long enough time scales the total change in energy moment
must be zero, which is to say that the internal change in
moment must be canceled by the change in the subsystem
moment induced by the reservoir, ⌬0E1 = −⌬s共r兲E1. But by
energy conservation, the change in the subsystem moment
induced by the reservoir must be equal and opposite to the
change in the reservoir moment, ⌬s共r兲E1 = −⌬E1r. Hence
mir
0
⌬Wmir
1 = ⌬ E1 = ⌬E1r, and one has ␤1⌬W1 = ⌬Sr / kB. Accordmir
ingly, one may indeed interpret ⌬W1 as a type of thermodynamic work performed by the reservoir, which accounts
for the notation used for this term.
These physical interpretations can explain and rationalize Wmir
1 , but they do not replace a precise definition. This
is now given, and one should judge the validity of this forin the light of its mathematical consemulation of Wmir
1
quences. Let ⌫0共t 兩 ⌫兲 be the Hamiltonian 共adiabatic兲 trajectory of the isolated subsystem that passes through ⌫ at t = 0.
Denote the isolated subsystem past and future moments by
E±1 共⌫兲 ⬅ E1共⌫0共±␶ 兩 ⌫兲兲, for some time interval ␶ ⬎ 0. From
the time reversible nature of the equations of motion,
⌫0共t 兩 ⌫兲 = 关⌫0共−t 兩 ⌫†兲兴†, it follows that E+1 共⌫†兲 = E−1 共⌫兲. With
these definitions Wmir
1 may be written in several equivalent
forms,
1 +
−
Wmir
1 共⌫兲 = 关E1 共⌫兲 − E1 共⌫兲兴
2
=
=
1
2
1
2
冕
冕
␶
−␶
dt⬘Ė1共⌫0共t⬘兩⌫兲兲
0
−␶
dt⬘关Ė1共⌫0共t⬘兩⌫兲兲 − Ė1共⌫0共t⬘兩⌫†兲兲兴.
共8兲
The factor of a half outside the integral accounts for double
counting, as the future cost now will also be included as a
past cost in the future; equivalently, the second equality
shows that only the even part of the integrand contributes,
and the past contribution to date is exactly half the total
contribution. The third equality demonstrates that the first
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224103-3
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
two equalities do not really violate causality. Clearly Wmir
1
has the necessary odd parity, and because of this it may be
called the mirror thermodynamic work.
With this result, the steady state probability density is
explicitly
e−␤0E0共⌫兲e−␤1E1共⌫兲 ␤ Wmir共⌫兲
e 1 1
,
㜷ss共⌫兩␤0, ␤1兲 = 3N
h N!Zss共␤0, ␤1兲
共9兲
where h is Planck’s constant and N is the number of atoms in
the subsystem. The first group of factors corresponds to 㜷e,
the final factor corresponds to 㜷o, and the denominator is the
normalizing partition function,
Zss共␤0, ␤1兲 =
1
h3NN!
冕
mir
d⌫e−␤0E0共⌫兲e−␤1E1共⌫兲e␤1W1
共⌫兲
.
共10兲
The nonequilibrium unconstrained total entropy is
Stotal,ss共␤0 , ␤1兲 = kB ln Zss共␤0 , ␤1兲, and its derivatives generate
nonequilibrium statistical mechanical averages, just as in the
equilibrium case.6 The nonequilibrium constrained free energy is −T0 times the constrained total entropy, which is the
subsystem-dependent part of the reservoir entropy 共the exponent of the probability density兲 plus the entropy of the isolated subsystem constrained to be in the macrostate. That is,
F共E0,E1,E±1 兩␤0, ␤1, ␶兲 = E0 + ␤1关E1 − 共E+1 − E−1 兲/2兴/␤0
− T0S共E0,E1,E+1 ,E−1 兩␶兲,
共11兲
where the final term is the entropy of the isolated subsystem
共see also Sec. II F 2 below兲. The constrained free energy is
strictly greater than −kBT0 ln Zss. In the thermodynamic limit
the minimum value of the constrained free energy 共with respect to the constraints, which are written to the left of the
vertical bar兲 is equal to −kBT0 ln Zss with a negligible error,
and this defines the most likely thermodynamic state. Its derivatives in the most likely macrostate generate various relationships between thermodynamic quantities, just as in the
equilibrium case.6
B. Nature of W1mir
The nonzero contributions to this integral come from
about the origin 共from the so-called inertial region兲. This
may be seen from the asymptote of the integrand in the intermediate regime2
Ė1共⌫0共t兩⌫兲兲 ⬃ sign共t兲Ė1共E1共⌫兲兲,
␶short ⱗ 兩t兩 ⱗ ␶long .
discussed below is related to this by ␴ss = −1 / 2␭VT20. Since
the asymptote is odd in time,2,3 it may be added to the integrand without changing the result. Hence Wmir
can be
1
equivalently written as
Wmir
1 共⌫兲 =
Ė1共E1兲 =
− ␭VkBT20
具E21典0
E1 ⬅ − cE1
共future兲,
共13兲
where ␭ is the thermal conductivity. The transport coefficient
冕
␶
−␶
dt⬘关Ė1共⌫0共t⬘兩⌫兲兲 − sign共t⬘兲Ė1共E1共⌫兲兲兴,
共14兲
which shows explicitly that Wmir
1 is dominated by the brief
inertial part of the trajectory. In other words, Wmir
1 is independent of ␶ for ␶ ⲏ ␶short.
In so far as the time integral that is Wmir
1 is dominated by
short times, then the temporally even part of Ė1共t兲 must vanish by 兩t兩 ⱗ ␶short, the inertial time. The simplest approxima2 2
tion is to take Ė1共⌫0共t 兩 ⌫兲兲even ⬇ Ė1共⌫兲e−nt /␶short, for some n
of order unity. In this case
Wmir
1 共⌫兲 ⬇
冑
␲
␶shortĖ1共⌫兲.
4n
共15兲
This result is consistent with the ansatz discussed previously1
and which was shown to be a reasonable approximation in
certain regimes.3
An earlier work gives the inertial time approximately as2
␶short =
=
− 2具Ė1共t兲E1共0兲典0
具Ė21典0
2␭VkBT20
具Ė21典0
,
t Ⰷ ␶short
.
共16兲
This result will be used shortly. The long time limit is given
by2
␶long =
具E21典0
2␭VkBT20
.
共17兲
C. Relationship with Green-Kubo
It is now shown that the above result for the steady state
probability density gives the Green-Kubo expression for the
thermal conductivity. Fourier’s law gives the thermal conductivity as the ratio of the heat flux to the applied temperature gradient,
共12兲
共This is valid if 兩tĖ1兩 Ⰶ 兩E1兩, which in practice is always the
case and will be assumed throughout.兲 It is not explicitly
required here, but the most likely flux going forward in time
is2,3
1
2
␭=
具Ė1共⌫兲典ss
.
VkBT20␤1
共18兲
Hence using the steady state probability distribution in the
linear regime ␤1 Ⰶ 1, the average of Ė1 will yield an expression for the thermal conductivity that can be compared with
the Green-Kubo expression.7
The steady state average of the rate of change of the first
energy moment is
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224103-4
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
mir
具Ė1共⌫兲典ss =
=
兰 d⌫e−␤0E0共⌫兲e−␤1E1共⌫兲e␤1W1
共⌫兲
mir
兰 d⌫e−␤0E0共⌫兲e−␤1E1共⌫兲e␤1W1
兰
E0共⌫兲 = E0共⌫†兲 = E0共⌫*兲,
Ė1共⌫兲
共⌫兲
E1共⌫兲 = E1共⌫†兲 = − E1共⌫*兲,
d⌫e−␤0E0共⌫兲关1 − ␤1E1共⌫兲 + ␤1Wmir
1 共⌫兲兴Ė1共⌫兲
兰 d⌫e−␤0E0共⌫兲关1 − ␤1E1共⌫兲 + ␤1Wmir
1 共⌫兲兴
=
␤1兰 d⌫e−␤0E0共⌫兲Wmir
1 共⌫兲Ė1共⌫兲
兰 d⌫e−␤0E0共⌫兲
=
␤1具Wmir
1 共⌫兲Ė1共⌫兲典0 .
共21兲
Ė1共⌫兲 = − Ė1共⌫ 兲 = − Ė1共⌫ 兲,
†
*
mir †
mir *
Wmir
1 共⌫兲 = − W1 共⌫ 兲 = − W1 共⌫ 兲,
共19兲
Since Ė1 is of an odd parity, the integral of it times the two
even terms from the linearization of the exponents vanishes.
The integral over −␤1E1共⌫兲 in the denominator vanishes because it is odd in z. Inserting the definition of Wmir
1 and using
the Fourier result for the thermal conductivity, one has the
result
and, of course, d⌫ = d⌫† = d⌫*. It is assumed that the equilibrium system is disordered or that it is mirror symmetric
about the midplane. Expanding the exponents to a cubic order in ␤1 and observing that any product of terms that has a
total odd parity under either operation must vanish, one obtains
具Ė1共⌫兲典ss/␤1 = 具Wmir
1 Ė1典0 +
␤21 2 mir
关具E1W1 Ė1典0
2
− 具E21典0具Wmir
1 Ė1典0兴 +
␭共␶兲 =
=
1
冕
冕
2VkBT20
1
VkBT20
␶
−␶
␶
mir
2
− 具共Wmir
1 兲 典0具W1 Ė1典0兴.
dt⬘具Ė1共⌫0共t⬘兩⌫兲兲Ė1共⌫兲典0
dt⬘具Ė1共⌫0共t⬘兩⌫兲兲Ė1共⌫兲典0 .
␤21
3
关具共Wmir
1 兲 Ė1典0/3
2
共20兲
0
The second equality follows because the time correlation
function of two quantities of the same parity is an even function of time. In the intermediate regime, this may be recognized as the Green-Kubo expression for the thermal
conductivity,7 which in turn is equivalent to the Onsager expression for the transport coefficients.2
This result is a very stringent test of the present expression for the steady state probability distribution. There is
one, and only one, exponent that is odd, linear in ␤1, and that
satisfies the Green-Kubo relation. Conversely, the GreenKubo relation alone does not determine the steady state probability distribution, since it is possible to add any even function to the exponent without changing the integral. In other
words, the Green-Kubo relation represents a necessary condition that the steady state probability distribution must satisfy, but it does not provide a sufficient condition to determine the steady state probability distribution. This
observation has implications for the constraints invoked in
the nonequilibrium molecular dynamics technique: demanding that the Green-Kubo relation be satisfied is not sufficient
to determine them uniquely.
The first neglected term is of the order ␤41. Amongst other
things the nonlinear terms account for the coupling of the
induced moment to the heat flux.
E. Adiabatic evolution of W1mir
Let ⌫⬘ = ⌫ + ⌬t⌫˙ , be the adiabatic evolution of ⌫ after an
infinitesimal time step. The adiabatic evolution of Wmir
1 can
be obtained from
Wmir
1 共⌫兲 =
1
2
=
1
2
=
1
2
冕
冕
冕
␶
−␶
dt⬘Ė1共⌫0共t⬘兩⌫兲兲
␶−⌬t
−␶−⌬t
␶−⌬t
−␶−⌬t
dt⬙Ė1共⌫0共t⬙ + ⌬t兩⌫兲兲
dt⬙Ė1共⌫0共t⬙兩⌫⬘兲兲
= Wmir
1 共⌫⬘兲 −
⌬t
关Ė1共⌫0共␶兩⌫⬘兲兲 − Ė1共⌫0共− ␶兩⌫⬘兲兲兴
2
= Wmir
1 共⌫⬘兲 − ⌬tĖ1共E1共⌫⬘兲兲.
The present steady state probability density is not restricted to the linear regime. The first correction to GreenKubo may be obtained as follows. Define the z-mirror operation as ⴱ: qz → −qz, pz → −pz, and under it and the
conjugation one has the useful parities
共23兲
Hence the adiabatic rate of change of Wmir
1 is
Ẇmir
1 共⌫兲 = Ė1共E1共⌫兲兲,
D. Nonlinear conductivity
共22兲
共24兲
which has an even parity, as it ought to have. This result is
exact for ␶ in the intermediate regime.
Despite first appearances, this exact result for Ẇmir
1 is, in
fact, consistent with the approximate result 关Eq. 共15兲兴,
冑
Wmir
1 共⌫兲 ⬇ ␶shortĖ1共⌫兲 ␲ / 4n, even though its derivative,
mir
Ẇ1 共⌫兲 ⬇ ␶shortË1共⌫兲冑␲ / 4n, does not look at all like the exact result.
Consider, however, the quantity
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224103-5
冕
⬁
dt⬘具Ẇmir
1 共⌫0共t⬘兩⌫兲兲Ė1共⌫兲典0 .
共25兲
0
Using the approximate expression, it is given by
−
冑
␲
␶short具Ė21典0 ,
4n
共26兲
since 具Ė1共⌫0共t⬘ 兩 ⌫兲兲Ė1共⌫兲典0 → 0, t⬘ → ⬁. Using the exact expression and the fact that Ė1 = −cE1 关Eq. 共13兲兴, it is given by
冕
⬁
dt⬘具− cE1共⌫0共t⬘兩⌫兲兲Ė1共⌫兲典0
0
=c
冕
⬁
dt⬘具Ė1共⌫0共t⬘兩⌫兲兲E1共⌫兲典0 = − c具E21典0 ,
共27兲
0
since the time correlation of functions of different parities is
odd in time and since 具E1共⌫0共t⬘ 兩 ⌫兲兲E1共⌫兲典0 → 0, t⬘ → ⬁.
Equating these two, one must have
冑
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
c具E21典0 ␭VkBT20
␲
=
.
␶short =
4n
具Ė21典0
具Ė21典0
共28兲
If one takes n = ␲, this is equivalent to the result obtained
previously2 关Eq. 共16兲兴. Such a value is not unreasonable. The
consistency of these two different approaches gives one
some confidence in both the theory and the approximations.
This result for the adiabatic evolution of Wmir
1 provides a
physical interpretation of the steady state probability 关Eq.
共9兲兴. The difficulty with heat flow is the lack of conserved
variables; the change in energy moment ⌬E1 occurs both
internally by the adiabatic evolution of the isolated subsystem ⌬0E1 and externally by the exchange with the reservoir ⌬s共r兲E1 = −⌬E1r. This is in contrast to the canonical Boltzmann distribution where the total energy is conserved, so
that ⌬E0 = ⌬0E0 + ⌬s共r兲E0 = −⌬E0r, which is why the Boltzmann exponent is ⌬Sr / kB = −␤0E0. In the present case, the
change in entropy of the reservoirs is
⌬Sr/kB = ␤0⌬E0r + ␤1⌬E1r
⬇ − ␤0⌬E0 − ␤1关⌬E1 − ⌬tĖ1兴
⬇ − ␤0⌬E0 − ␤1关⌬E1 −
An important aspect of any statistical system is the transitions between states, and nowhere is this more evident than
for nonequilibrium systems. Indeed, Papers II and IV in the
present series focused on this aspect of the problem and established a number of results and theorems that the transition
probability must satisfy.2,4 Relevant to the present analysis is
that the transition probability should 共1兲 yield a stationary
steady state probability, 共2兲 obey the so-called microscopic
transition theorem,4 and 共3兲 reduce to the second entropy
expression for macrostate transitions.2
In the present case of steady heat flow, and indeed in any
realistic statistical problem, it is essential to account for the
stochastic perturbations from the reservoir in the transitions
between states. Consider then the transition between the microstates ⌫ → ⌫⬙ in an infinitesimal time step ⌬t. This can be
considered as comprising a deterministic transition ⌫ → ⌫⬘
due to the internal forces of the subsystem, followed by a
stochastic transition ⌫⬘ → ⌫⬙ due to the perturbations by the
reservoir.
The deterministic transition is just the adiabatic evolution of the isolated subsystem, ⌫⬘ = ⌫ + ⌬t⌫˙ . In terms of the
conditional transition probability this is
⌳d共⌫⬘兩⌫兲 = ␦共⌫⬘ − ⌫ − ⌬t⌫˙ 兲.
共30兲
The stochastic transition probability can be written as the
product of an even and an odd function,
⌳s共⌫⬙兩⌫⬘兲 = ⌳e共⌫⬙兩⌫⬘兲⌳o共⌫⬙兩⌫⬘兲.
共31兲
In the case of transitions, however, parity now refers to the
reversibility or irreversibility of the transition. That is, even
transition probabilities satisfy
⌳e共⌫⬙兩⌫⬘兲 = ⌳e共⌫⬘†兩⌫⬙†兲,
共32兲
and odd transition probabilities satisfy
⌳o共⌫⬙兩⌫⬘兲 = 1/⌳o共⌫⬘†兩⌫⬙†兲.
共33兲
In Ref. 4, the odd transition probability for steady heat
flow was given as
= − ␤0⌬E0 − ␤1关⌬E1 − ⌬0E1兴
⌬Wmir
1 兴.
F. Transition probability
共29兲
From the final approximate expression, one sees that subtracting the change in Wmir
1 from the total change in moment
is equivalent to identifying that part of the change in energy
moment that is due to the reservoir. It is only this external
influence that affects the reservoir entropy and that contributes to the steady state probability.
In Sec. II F 1, which follows shortly, it is shown that the
steady state probability distribution is approximately stationary during the adiabatic evolution of the isolated subsystem.
This confirms the above interpretation that the exponent reflects the entropy of the reservoirs only and that the contribution from internal changes of the subsystem has been correctly removed.
⌳o共⌫⬙兩⌫⬘兲 = e−␤0共E0⬙−E0⬘兲/2e−␤1共E1⬙−E1⬘兲/2 .
共34兲
Notice that the exponent is the difference between two phase
space functions of an even parity, and that this is necessary
for the transition itself to be odd. Hopefully this will not
cause confusion.
The even stochastic transition probability may be taken
as
⌳e共⌫⬙兩⌫⬘兲 = ⍜⌬共兩⌫⬙ − ⌫⬘兩兲.
共35兲
Here ⍜⌬ is a short ranged, even function 共such as a Heaviside step function or a Gaussean兲, with a range ⌬ that represents the strength of the stochastic perturbations from the
reservoir. It is normalized and does not depend on the temperatures to a leading order.
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224103-6
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
evolution, E0共⌫⬘兲 = E0共⌫兲, one can concentrate on the moments. For an infinitesimal time step ⌬t the first moment
changes to
One can give a straightforward physical interpretation to
this result for the stochastic transition probability. The
change in entropy of the reservoirs during the transition
⌫ → ⌫⬙ is
⌬S/kB = − ␤0⌬E0 − ␤1关⌬E1 − ⌬0E1兴
E1共⌫⬘兲 = E1共⌫兲 + ⌬tĖ1共⌫兲,
= − ␤0共E0⬙ − E0兲 − ␤1共E1⬙ − E1 − ⌬tĖ1兲
= − ␤0共E0⬙ − E0⬘兲 − ␤1共E1⬙ − E1⬘兲.
共36兲
共39兲
and the mirror work becomes
Hence the stochastic conditional transition probability is just
the exponential of half the change in the reservoir entropy,
mir
Wmir
1 共⌫⬘兲 = W1 共⌫兲 + ⌬tĖ1共E1共⌫兲兲.
⌳s共⌫⬙兩⌫⬘兲 = ⍜⌬共兩⌫⬙ − ⌫⬘兩兲e−␤0共E0⬙−E0⬘兲/2e−␤1共E1⬙−E1⬘兲/2
= ⍜⌬共兩⌫⬙ − ⌫⬘兩兲e
⌬S/2kB
The overbar signifies the most likely state, and it is in
such states that the steady state probability is significantly
greater than zero. Conversely, states with Ė1 ⫽ Ė1 occur
rarely, with the probability getting smaller as the difference
increases. 共Fluctuations away from the equilibrium or the
steady state are relatively negligible.6兲 The formal result is
共37兲
.
It is possible to multiply the even stochastic transition
mir⬙
共40兲
mir⬘
probability by a factor of e␤1共W1 −W1 兲/2, since the difference between the two exponents of the odd parity gives a
transition of the even parity. All of the results given below
would be unchanged if this factor were to be added. This
includes the microscopic transition theorem established in
Ref. 4, which is independent of the even transition probability.
mir
E1共⌫兲 − Wmir
1 共⌫兲 = E1共⌫⬘兲 − W1 共⌫⬘兲 − ⌬t关Ė1共⌫⬘兲
− Ẇmir
1 共⌫⬘兲兴.
共41兲
1. Stationary steady state probability
In the most likely state the final term vanishes, and for
nearby states it is small. Hence the present steady state probability density is almost stationary during its adiabatic evolution,
For steady heat flow the probability density does not
depend explicitly on time, and so it must be stationary under
the transition probability given above. Consider then the evolution of the probability in time ⌬t, which is given by
㜷̃ss共⌫⬙兩␤0, ␤1兲 =
=
冕
冕
㜷ss共⌫兩␤0, ␤1兲 = 㜷ss共⌫⬘兩␤0, ␤1兲e␤1⌬t关Ė1共⌫⬘兲−Ẇ1
mir
d⌫⬘d⌫⌳s共⌫⬙兩⌫⬘兲⌳d共⌫⬘兩⌫兲㜷ss共⌫兩␤0␤1兲
and it is necessary to show that the left hand side is indeed
㜷ss.
First the deterministic evolution of the probability is explored. Since the energy is conserved during the Hamiltonian
d⌫⬘⌳s共⌫⬙兩⌫⬘兲㜷ss共⌫⬘兩␤0, ␤1兲e␤1⌬t关Ė1共⌫⬘兲−Ẇ1
mir
=
冕
d⌫⬘⍜⌬共兩⌫⬙ − ⌫⬘兩兲e
mir⬙
e−␤0E0⬙e−␤1E1⬙e␤1W1
=
Zss
= 㜷ss共⌫⬙兩␤0, ␤1兲
冕
共42兲
共⌫⬘兲兴
−␤0共E0⬙−E0⬘兲/2 −␤1共E1⬙−E1⬘兲/2 e
冕
.
共To the leading order one can use ⌫ or ⌫⬘ in the final term.兲
The second part of the proof consists in showing that the
present steady state probability density is stationary during
the stochastic step. The proof follows that given previously
in steady state4 and in equilibrium8 contexts, although some
attention has to be paid to the change in probability during
the adiabatic evolution. The stochastic transition integral
becomes
d⌫⬘⌳s共⌫⬙兩⌫⬘兲㜷ss共⌫0共− ⌬t兩⌫⬘兲兩␤0, ␤1兲,
共38兲
冕
共⌫⬘兲兴
⬘
−␤0E0⬘ −␤1E1⬘ ␤1Wmir
1
e
e
e
Zss
mir⬘
d⌫⬘⍜⌬共兩⌫⬙ − ⌫⬘兩兲e␤1共W1
mir⬘
d⌫⬘⌳s共⌫⬘兩⌫⬙兲 ⫻ e␤1共W1
e␤1⌬t关Ė1共⌫⬘兲−Ẇ1
mir
共⌫⬘兲兴
mir
⬙
−Wmir
1 兲 −␤0共E0⬘−E0⬙兲/2 −␤1共E1⬘−E1⬙兲/2 ␤1⌬t关Ė1共⌫⬘兲−Ẇ1 共⌫⬘兲兴
e
e
mir
⬙
−Wmir
1 兲e ␤1⌬t关Ė1共⌫⬘兲−Ẇ1 共⌫⬘兲兴
e
= 㜷ss共⌫⬙兩␤0, ␤1兲.
共43兲
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224103-7
This is the required result which shows the stationarity of the
steady state probability under the present transition probability.
The passage to the final equality requires comment. The
normalization of the conditional transition probability,
兰d⌫⬘⌳s共⌫⬘ 兩 ⌫⬙兲 = 1, is unchanged to the order ⌬2 by the factor of e
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
mir
mir
␤1共W1 ⬘−W1 ⬙兲
. Furthermore, it is assumed that
2
␤1⌬t关Ė1共⌫⬘兲 − Ẇmir
1 共⌫⬘兲兴 ⬃ O⌬ ,
共44兲
2. Unconditional forward and reverse transitions
In Paper IV, the so-called transition theorem and its reverse were given explicitly for the case of the mechanical
work.4 In the present case of steady heat flow, the unconditional microscopic transition probability is
ss共␤0, ␤1兲
e
,
共45兲
for an infinitesimal ⌬t. The Ė1 in the final term can be replaced by 共Ė1⬙ + Ė1兲 / 2 to this order.
Consider the forward transition ⌫ → ⌫⬘ → ⌫⬙, and its reverse, ⌫⬙† → ⌫⵮† → ⌫†. Note that ⌫⵮† ⫽ ⌫⬘†, but that 兩⌫⬙†
− ⌫⬘兩 = 兩⌫† − ⌫⵮†兩. The ratio of the forward to the reverse transition probabilities is
㜷共⌫⬙ ← ⌫兩⌬t兲
mir
mir
= e␤1共W1 共⌫⬙兲+W1 共⌫兲兲e⌬t␤1共Ė1共⌫⬙兲+Ė1共⌫兲兲/2 ,
†
†
㜷共⌫ ← ⌫⬙ 兩⌬t兲
共46兲
the even terms canceling. The exponent of the final term is
part of the change in the reservoir entropy 共the thermodynamic work done兲 on the trajectory.
Consider a trajectory 关⌫兴 = 兵⌫0 , ⌫1 , . . . , ⌫ f 其, over a time
interval t f = f⌬t, and its reverse, 关⌫‡兴 = 兵⌫†f , ⌫†f−1 , . . . , ⌫†0其. The
adiabatic change in E1 over the trajectory is
f−1
⌬0E1关⌫兴 =
⌬t
关Ė1共⌫0兲 + Ė1共⌫ f 兲兴 + ⌬t 兺 Ė1共⌫i兲.
2
i=1
Clearly, ⌬0E1关⌫‡兴 = −⌬0E1关⌫兴. Then
0E /2 −␤ 共Wmir−Wmir兲/2
1
1 1,f
1,0
e
⫻ 关㜷ss共⌫ f 兩␤0, ␤1兲㜷ss共⌫0兩␤0, ␤1兲兴1/2 ,
共48兲
since E1,i−1
⬘ = E1,i−1 + ⌬tĖ1,i−1. The exponent ⌬0E1 should really be replaced by ⌬0E1 + ⌬t关Ė1共⌫0兲 − Ė1共⌫ f 兲兴 / 2, but this
correction is negligible for large f, and in any case it is fixed
up by the ratio that is taken shortly.
Briefly digress to consider a closely related problem,
namely, that instead of the steady state the system is initially
in the so-called static state, which has an even probability
density
㜷st共⌫兩␤0, ␤1兲 =
e−␤0E0共⌫兲e−␤1E1共⌫兲
.
Zst共␤0, ␤1兲
共49兲
In this case the trajectory probability during the evolution to
the steady state is
f
0E /2
1
共50兲
The ratio of the probability of the forward and reverse
trajectories in the steady state is
mir
㜷关⌫兴
0
0
␤1Wmir
1 共⌫ f 兲e ␤1W1 共⌫0兲e ␤1⌬ E1关⌫兴 ⬇ e ␤1⌬ E1关⌫兴
‡ =e
㜷关⌫ 兴
⫻关㜷ss共⌫⬙兩␤0, ␤1兲㜷ss共⌫兩␤0, ␤1兲兴1/2
˙
−Wmir
1 兲/2 ⌬t␤1E1/2
f
= 兿 关⍜⌬共兩⌫i+1 − ⌫i⬘兩兲兴e␤1⌬
⫻ 关㜷st共⌫ f 兩␤0, ␤1兲㜷st共⌫0兩␤0, ␤1兲兴1/2 .
= ⍜⌬共兩⌫⬙ − ⌫⬘兩兲
mir⬙
⬘ 兲/2兴㜷 共⌫ 兩␤ , ␤ 兲
⫻ e−␤1共E1,i−E1,i−1
ss 0 0 1
i=1
−␤0共E0⬙+E0兲/2
mir
˙
e−␤1共E1⬙+E1−⌬tE1兲/2e␤1W1 /Z
⫻ e−␤1共W1
i=1
㜷st关⌫兴 = 兿 关⍜⌬共兩⌫i+1 − ⌫i⬘兩兲兴e␤1⌬
㜷共⌫⬙ ← ⌫兩⌬t兲 = ⌳s共⌫⬙兩⌫⬘兲㜷ss共⌫兩␤0, ␤1兲
⫻
⬘ 兲/2
㜷关⌫兴 = 兿 关⍜⌬共兩⌫i+1 − ⌫i⬘兩兲e−␤0共E0,i−E0,i−1
i=1
which is again second order and neglectable. The rationale
for this result is that the fluctuations away from the most
likely state are relatively negligible in the thermodynamic
limit. Alternatively, one can arrange the second order terms
in the normalization to cancel exactly this term.
= ⍜⌬共兩⌫⬙ − ⌫⬘兩兲e
f
共47兲
⬇ e⌬S关⌫兴/kB ,
共51兲
the even terms canceling. The final two approximations are
valid for large t f , since the retained terms scale with f⌬t. The
final exponent is the change in the entropy of the reservoirs
关Eq. 共36兲兴. This result holds exactly for trajectories evolving
to the steady state from the static state,
㜷st关⌫兴
0
= e␤1⌬ E1关⌫兴 .
㜷st关⌫‡兴
共52兲
The result for the ratio of the probabilities of the forward and
reverse trajectories may be called the reverse transition
theorem.4
The probability of observing the entropy of the reservoir
change by ⌬S over a period t is related to the probability of
observing the opposite change by
㜷共⌬S兩␤0, ␤1,t f 兲 =
冕
d关⌫兴␦共⌬S − ⌬S关⌫兴兲共㜷关⌫兴兲
⬇ kB−1
冕
d关⌫‡兴␦共⌬S/kB − ␤1⌬0E1关⌫兴兲
⫻ 㜷关⌫‡兴e␤1⌬
0E 关⌫兴
1
= e⌬S/kB㜷共− ⌬S兩␤0, ␤1,t f 兲.
共53兲
This result says in essence that the probability of a positive
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224103-8
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow, with the
exponent being the entropy change under consideration. The
change in entropy of the reservoirs may be regarded as a type
of a thermodynamic work performed by the reservoirs during
heat flow. Hence, in essence, this is the temperature gradient
version of the fluctuation theorem that was originally derived
by Evans et al.9 and Evan10 A derivation has also been given
by Crooks,11,12 and the theorem has been verified
experimentally.13 The present derivation based on the microscopic transition probability closely follows that given by
Attard.4
Closely related to the fluctuation theorem is the work
theorem due to Jarzynski,14 which has also been rederived in
different fashions15,16,10 and verified experimentally.17 In this
context one is motivated to consider the average of the exponential of the heat flux,
具e−␤1⌬
0E
1
典ss,t f =
=
冕
冕
d关⌫兴e−␤1⌬
mir
冕
冕
=
冉
= ln
冕
d⌫ exp关− ␤0E0共⌫兲 − ␤1E1共⌫兲
+ ␤1兵E+1 共⌫兲 − E−1 共⌫兲其/2兴␦共E1 − E1共⌫兲兲
⫻␦共E+1 − E+1 共⌫兲兲␦共E−1 − E−1 共⌫兲兲
+
−
+
−
= S共2兲
0 共E1 ,E1,E1 兩␶兲/kB − ␤1E1 + ␤1兵E1 − E1 其/2.
+
−
S共2兲
0 共E1 ,E1,E1 兩␶兲 =
共⌫ f 兲+Wmir
1 共⌫0兲兲
d⌫†f 㜷共⌫†f 兩␤0, ␤1兲e
⫻
共2兲
共E+1 ,E1,E−1 兩␶兲/kB
Stotal
Zst共␤0, ␤1兲
Zss共␤0, ␤1兲
†
−␤1Wmir
1 共⌫ f 兲
mir
冊
共⌫†0兲
2
.
␴ss +
␴ss
共E − E1兲2 +
共E1 − E−1 兲2
2␶ 1
2␶
+
d⌫†0㜷共⌫†0兩␤0, ␤1兲e−␤1W1
共55兲
The second entropy of the isolated system during the adiabatic sequential transition is
㜷共关⌫兴兩␤0, ␤1,t f 兲
d关⌫‡兴㜷共关⌫‡兴兩␤0, ␤1,t f 兲
⫻e␤1共W1
⬇
0E 关⌫兴
1
regarded simply as the difference in two energies over a time
interval of 2␶.兲 The exponent represents the reservoir contribution to the total entropy, since a subsystem microstate, a
point in its phase space, has zero entropy. Hence the total
second entropy for this sequential transition is just the logarithm of the sum over the macrostate of the weights of the
microstates, the latter being the unnormalized steady state
probability,
共54兲
Here it has been assumed that the trajectory is long enough
that the ends are uncorrelated. This result shows that this
particular average is not extensive in time, 共i.e., it does not
scale with t f 兲. In essence the right hand side is the exponential of twice the difference in free energies of the static and
the steady state systems.
3. Second entropy
In Paper II the so-called second entropy was introduced,
and this was said to be the appropriate entropy for transitions
between macrostates, and from it the Onsager reciprocal relations were deduced as well as the Green-Kubo theory for
the linear transport coefficients.2 In Paper IV the second entropy for macrostates was used to formulate a transition
probability for microstates that became the basis for the microscopic transition theorem and the microscopic reverse
transition theorem.4 Now the relationship between the steady
state probability density and the second entropy is derived
for the present case of heat flow.
Three relevant quantities appear in the steady state probability density 关Eq. 共9兲兴: E1, E+1 , and E−1 , since it will be
recalled that the mirror work term can be written as Wmir
1
= 共E+1 − E−1 兲 / 2. Accordingly, the steady state probability density really describes the sequential transition, E−1 → E1 → E−1 ,
with each interval of duration ␶. 共Note that Wmir
1 should always be interpreted in this fashion, and it should never be
S共1兲 2
E,
2 1
␶ ⬎ ␶short .
共56兲
This generalizes slightly the result given previously2 from a
single to a sequential transition. The final term can be replaced by the first or ordinary entropy S共1兲共E1兲, which in turn
to the leading order can be replaced by 关S共1兲共E−1 兲 + 2S共1兲共E1兲
+ S共1兲共E+1 兲兴 / 4 without changing the following results for Ė1
over the entire interval.
The derivatives of the total second entropy are
共2兲
⳵Stotal
共E+1 ,E1,E−1 兩␶兲
⳵E+1
共2兲
⳵Stotal
共E+1 ,E1,E−1 兩␶兲
⳵E−1
= ␴ss
E+1 − E1 ␤1
+ ,
2
␶
= − ␴ss
E1 − E−1 ␤1
− ,
2
␶
共57兲
共2兲
共E+1 ,E1,E−1 兩␶兲
⳵Stotal
E+ − E1
E1 − E−1
= − ␴ss 1
+ ␴ss
− ␤1
⳵E1
␶
␶
+ S共1兲E1 .
Maximizing the second entropy by setting these derivatives
to zero, ones sees that the most likely flux is
ⴰ
ⴰ
E+1 = E−1 = −
−1
␴ss
␤1 ,
2
共58兲
and that the most likely moment is
ⴰ
E1 = 共S共1兲兲−1␤1 .
共59兲
共The circle denotes the coarse-grained velocity,2 Ė±1 ⬅ ± 共E±1
− E1兲 / ␶.兲 These agree with earlier results.1,2 As mentioned
above, the second entropy form of the transport coefficient is
related to the thermal conductivity by ␴ss = −1 / 2␭VT20.
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224103-9
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
One may conclude two things from this analysis. First,
the present steady state probability distribution is entirely
consistent with the second entropy analysis of the heat transport. This demonstrates again the necessity of the mirror
work term Wmir
1 . Second, the steady state probability distribution really describes transitions rather than phase space
microstates per se, since the exponent is so closely related to
the second entropy. This surprising result contrasts with
equilibrium systems where it is the first entropy or state
weight that is relevant to the probability density.
III. NONEQUILIBRIUM MONTE CARLO SIMULATION
A. Algorithm
Similar to earlier papers in the series,1,3,4 a LennardJones fluid was simulated. All quantities were made dimensionless using the welldepth ⑀LJ, the diameter ␴LJ, and the
2
time constant ␶LJ = 冑 共mLJ␴LJ
/ ⑀LJ兲, where mLJ is the mass. In
addition, Boltzmann’s constant was set equal to unity.
The Lennard-Jones potential between atoms was cut and
shifted at Rcut = 2.5, and no tail correction was invoked. A
spatial neighbor table was used with cubic cells of sidelength
⬇0.6, which reduces the number of neighbors required for a
force calculation by almost a factor of 3 compared to the
conventional cells of length Rcut.1 Periodic boundary conditions and the minimum image convention were used.
Both a uniform bulk fluid and an inhomogeneous fluid
were simulated. The latter was in the form of a slit pore,
terminated in the z direction by uniform Lennard-Jones
walls. The distance between the walls for a given number of
atoms was chosen so that the uniform density in the center of
the cell was equal to the nominal bulk density.
Umbrella sampling Monte carlo simulations were performed in 6N-dimensional phase space, where N = 120– 500
atoms. The Metropolis algorithm was used with an umbrella
weight density
˙
␻共⌫兲 = e−␤0E0共⌫兲e−␤1E1共⌫兲e␣␤1E1共⌫兲 .
共60兲
It is emphasized that this is the umbrella weight used in the
Metropolis sampling scheme; the exact steady state probability density 关Eq. 共9兲兴 was used to calculate the averages 共see
below兲. The final term obviously approximates ␤1Wmir
1 , but is
about a factor of 400 faster to evaluate. In the simulations
reported here ␣ was fixed at 0.08. It would be possible to
optimize this choice or to determine ␣ on the fly. 共See, for
example, Ref. 3.兲 A trial move of an atom consisted of a
small displacement in its position and momentum simultaneously. The step lengths were chosen to give an acceptance
rate of about 50%. A cycle consisted of one trial move of
each atom.
Averages were collected after every 50 cycles. Labeling
the current configuration used for an average by i, the Hamiltonian trajectory ⌫0共t 兩 ⌫i兲 was generated forward and backward in time using a second order rule and a time step of
⌬t = 10−2, which gave a satisfactory energy conservation. The
running integral for Wmir
1 共⌫i ; t兲 was calculated along the trajectory using both the trapezoidal rule and Simpson’s rule,
with indistinguishable results. The average flux was calculated as a function of the time interval,
FIG. 1. The dependence of the thermal conductivity on the time interval for
2
the mirror work Wmir
1 共⌫ ; ␶兲. The curves are ␭共␶兲 = 具Ė1共0兲典␶ / VkBT0␤1 for densities of, from bottom to top, 0.3, 0.5, 0.6, and 0.8, and T0 = 2.
˙
具Ė1典␶ =
mir
兺 iĖ1共⌫i兲e−␣␤1E1共⌫i兲e␤1W1
兺 ie
共⌫i;␶兲
−␣␤1E˙1共⌫i兲 ␤1Wmir
1 共⌫i;␶兲
e
.
共61兲
Notice how the umbrella weight used in the Metropolis
scheme is canceled here. The thermal conductivity is reported below as ␭共␶兲 = 具Ė1典␶ / ␤1VkBT20. Not only is the umbrella method orders of magnitude faster in generating configurations, but it also allows results as a function of ␶ to be
collected, and it reduces the correlation between consecutive
costly trajectories by inserting many cheap umbrella steps.
Prior to the generation of each trajectory the velocities of the
particles were scaled and shifted at a constant kinetic energy
to give a zero total z momentum. In the inhomogeneous system, a constraint force was added to keep the total z momentum zero on a trajectory. Of the order of 50 000 trajectories
were generated for each case studied.
B. Results for heat flow
Figure 1 tests the dependence of the thermal conductivity on the time interval used to calculate Wmir
1 共⌫ ; ␶兲. It can be
seen that the thermal conductivity is independent of the integration limit for Wmir
1 for ␶ ⲏ 1. This asymptotic or plateau
value is the thermal conductivity. The value of ␶ required to
reach the respective plateaus here appears comparable to
straight Green-Kubo equilibrium calculations,3 but the
present steady state simulations used about one third of the
number of trajectories for a comparable statistical error.
In the text it was assumed that the change in moment
over the relevant time scales was negligible, ␶兩Ė1兩 Ⰶ 兩E1兩. In
the case of ␳ = 0.8 at the largest value of ␶ in Fig. 1, 具E1典ss
= −432 and 具Ė1典ss = 161, and so this assumption is valid in
this case. Indeed, the reason for making this assumption was
that on long time scales the moment must return to zero and
the rate of change of moment must begin to decrease. There
is no evidence of this occurring in any of the cases over the
full interval shown in Fig. 1.
Table I shows the values of the relaxation time calculated using Eqs. 共16兲 and 共17兲. Both the inertial time and the
long time decrease with the increasing density. This is in
agreement with the trend of the curves in Fig. 1. Indeed, the
actual estimates of the relaxation times in Table I are in
semiquantitative agreement with the respective boundaries of
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224103-10
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
TABLE I. Thermal conductivity and relaxation times for various densities at
T0 = 2. The standard errors of the last few digits are in parentheses.
␳
␭
␶short
Eq. 共16兲
␶long
Eq. 共17兲
0.3
0.5
0.6
0.8
1.63共8兲
2.78共13兲
3.76共16兲
7.34共18兲
0.404共19兲
0.233共11兲
0.197共9兲
0.167共4兲
3.22共16兲
5.31共34兲
3.41共18兲
1.36共3兲
the plateaux in Fig. 1. The estimate of ␶long, the upper limit
on ␶ that may be used in the present theory, is perhaps a little
conservative.
Figure 2 compares the thermal conductivity obtained
from the present nonequilibrium Monte Carlo simulations
with previous NEMD results.18,19 The good agreement between the two approaches validates the present phase space
probability distribution. The number of time steps required
for an error of about 0.1 was about 3 ⫻ 107 共typically 2
⫻ 105 independent trajectories, each of about 75 time steps
forward and backward to get into the intermediate regime兲.
This obviously depends on the size of the applied thermal
gradient, 共the statistical error decreases with the increasing
gradient兲 but appears comparable to that required by NEMD
simulations.19 No attempt was made to optimize the present
algorithm in terms of the number of Monte Carlo cycles
between trajectory evaluations or the value of the umbrella
parameter.
Figure 2 also shows results for the thermal conductivity
obtained for the slit pore, where the simulation cell was terminated by uniform Lennard-Jones walls. The results are
consistent with those obtained for a bulk system using periodic boundary conditions. This indicates that the density inhomogeneity induced by the walls has little effect on the
thermal conductivity. For the bulk system, the minimum image convention was used for all separations that appeared in
the expression for Ė1.3
Figure 3 explores the nonlinear dependence of the heat
flux on the applied temperature gradient. The increase in ␭
with the increasing ␤1 is due primarily to the first nonlinear
term 关Eq. 共22兲兴. This represents a coupling of the induced
density gradient to the heat flux. Nonlinear effects appear
FIG. 3. The nonlinear thermal conductivity ␭共␤1兲 = 具Ė1共0兲典␤1 / VkBT20␤1, for
T0 = 2 共mainly for a cubic cell with N = 120兲. In 共a兲, the density is ␳ = 0.8 and
the fitted quadratic is ␭共␤1兲 = 7.21+ 413␤21. In 共b兲, the density is ␳ = 0.6 and
the fitted quadratic is ␭共␤1兲 = 3.61+ 554␤21.
greater for ␳ = 0.6 than for ␳ = 0.8. The largest gradient shown
corresponds to 1011 K / m in argon, which may be difficult to
achieve in the laboratory. In this case the temperature discontinuity across the periodic z boundaries is 1kB / ⑀LJ, and it is
not clear how this discontinuity and other finite size effects
affect the nonlinear conductivity.
IV. GENERALIZATIONS
A. Hydrodynamic transport
Heat flow is a particular example of a hydrodynamic
transport. The theory for nonequilibrium statistical mechanics and the expression for the steady state probability distribution given above for heat flow may be readily generalized
to other types of flow. It may also be generalized beyond the
single gradient situation to include reservoirs that impose
variations on short spatial wavelengths.
1. Flow in general
Let A0 and A1 be the zeroth and first moments, respectively, of a set of linear additive variables that the subsystem
may exchange with the reservoirs, 共energy, number, volume,
charge, etc.兲. These could represent vectors whose components correspond to two or three spatial dimensions, or the
components could represent different exchangeable variables. Let
FIG. 2. Thermal conductivity at T0 = 2. The circles and squares are the
present steady state results for bulk and inhomogeneous systems, respectively 共horizontally offset by ±0.015 for clarity兲, and the triangles are
NEMD 共Refs. 18 and 19兲 results.
b␣ ⬅
⳵S
,
⳵A␣
␣ = 0,1
共62兲
be the conjugate thermodynamic variables 共one divided by
temperature, minus chemical potential divided by tempera-
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224103-11
ture, etc.兲. Then the steady state probability distribution is
e
㜷ss共⌫兩b0,b1兲 =
−b0·A0共⌫兲/kB −b1·A1共⌫兲/kB b1·Wmir
1 共⌫兲/kB
e
e
h N!Zss共b0,b1兲
3N
, 共63兲
where Zss is the normalizing partition function, and where
the mirror work is
Wmir
1 共⌫兲 =
1
2
冕
␶
−␶
dtȦ1共⌫0共t兩⌫兲兲.
共64兲
By the conservation rules for the linear additive variables,
Ȧ0 = 0 for an isolated system, and so one could formally include Wmir
0 = 0 without changing this result for the probability distribution 共see next兲.
2. Short spatial wavelengths
The preceding subsection and the body of the text are
applicable when the reservoirs impose only long wavelength
variations in the field parameters. As shown in Sec. II of Ref.
1, the first moment is just the leading term in a polynomial
expansion that can accommodate variations on all spatial
wavelengths. In particular, if b共r兲 is the value of the field
variables imposed by the reservoirs at position r, and if A共r兲
is the macrostate of the isolated subsystem consisting of
specified values of the conjugate thermodynamic variables at
that position, then
−
冕
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
drb共r兲 · A共r兲 = − 兺 bn · An
共65兲
n
is the reservoir entropy associated with the subsystem macrostate. Here the summand represents the product of coefficients in an appropriate polynomial expansion of which the
first two correspond in essence to the zeroth and first
moments.1 Accordingly, the steady state probability distribution is
㜷ss共⌫兩关b兴兲 =
兰drb共r兲·Wmir共⌫,r兲/k
e−兰drb共r兲·A共⌫,r兲/kBe
h3NN!Zss关b兴
B
,
1
2
冕
␶
−␶
dtȦ共⌫0共t兩⌫兲,r兲.
˜
㜷̃␮共⌫兩␤,t兲 = Z̃−1e−␤H␮共⌫−,t−␶兲 = Z̃−1e−␤H␮共⌫,t兲e␤W␮共⌫,t兲 .
共68兲
where ⌫− = ⌫␮共t − ␶ 兩 ⌫ , t兲 is the starting point of the Hamiltonian 共adiabatic兲 trajectory, and the work done is W̃␮共⌫ , t兲
= H␮共⌫ , t兲 − H␮共⌫− , t − ␶兲. This is called the YamadaKawasaki distribution.20,21 The problem with it is that it does
not take into account the influence of the heat reservoir while
the work is being performed. A modified thermostatted form
of the Yamada-Kawasaki distribution has been given, but it is
said to be computationally intractable.22–24 Based on calculations performed with the Yamada-Kawasaki distribution,
some have cast doubt on the very existence of a nonequilibrium probability distribution, since, they argue, that such a
distribution is fractal in nature and that it shrinks onto a
low-dimensional attractor.25–27
In view of these difficulties with the Yamada-Kawasaki
distribution and its modifications, one seeks an alternative
nonequilibrium probability distribution that is not restricted
to an isolated system or that does not invoke artificial, deterministic thermostats. The distribution should properly account for the influence of the heat reservoir while the nonequilibrium work is being performed. This influence is
stochastic, and its probability distribution is analogous to that
described above for the case of heat flow down an imposed
thermal gradient. Hence one requires a “mirror work” that
has an odd parity. To obtain this one must continue the work
path into the future by making it even about t,
␮mir
t 共t⬘兲 ⬅
再
␮共t⬘兲, t⬘ 艋 t
␮共2t − t⬘兲, t⬘ ⬎ t.
⌫␮⬘ mir ⬅ ⌫␮mir共t⬘兩⌫,t兲 =
t
共67兲
Note that, in effect, the zeroth mirror work has been included
in the probability distribution by the integration over all
space, but, as mentioned above, due to the conservation laws
this is zero, and no error is introduced by doing this.
t
共69兲
再
⌫␮共t⬘兩⌫,t兲, t⬘ 艋 t
⌫␮共2t − t⬘兩⌫†,t兲† , t⬘ ⬎ t.
冎
共70兲
With these the mirror work is
1
W␮mir共⌫,t兲 = 关H␮mir共⌫+,t + ␶兲 − H␮mir共⌫−,t − ␶兲兴
t
t
2
=
B. Nonequilibrium mechanical work
The nonequilibrium probability distribution is now formulated for the case that a time-dependent work is performed on a subsystem while it is in contact with a thermal
reservoir. Let ␤ be the inverse temperature of the thermal
reservoir, and consider a time-dependent Hamiltonian
H␮共⌫ , t兲, where ␮共t兲 is the work parameter.
If the system were isolated from the thermal reservoir
during its evolution, and if the system were Boltzmann dis-
冎
Denote the corresponding Hamiltonian 共adiabatic兲 trajectory
that is at ⌫ at t by
共66兲
where the mirror work is
Wmir共⌫,r兲 =
tributed at t − ␶, then the probability distribution at time t
would be
=
冕
冕
1
2
1
2
t+␶
t−␶
dt⬘Ḣ␮mir共⌫␮⬘ mir,t⬘兲,
t
t
t
t−␶
dt⬘关Ḣ␮共⌫␮共t⬘兩⌫,t兲,t⬘兲
− Ḣ␮共⌫␮共t⬘兩⌫†,t兲,t⬘兲兴,
共71兲
which clearly has an odd parity W␮mir共⌫ , t兲 = −W␮mir共⌫† , t兲.
With it the nonequilibrium probability distribution for the
subsystem of the thermal reservoir upon which work is being
performed is
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224103-12
J. Chem. Phys. 124, 224103 共2006兲
Phil Attard
mir
e−␤H␮共⌫,t兲e␤W␮
㜷␮共⌫兩␤,t兲 =
Z␮共␤,t兲
共⌫,t兲
共72兲
.
The probability distribution is normalized by Z␮共␤ , t兲, which
is a time-dependent partition function whose logarithm gives
the nonequilibrium total entropy, which may be used as a
generating function.
Asymptotically one expects
¯
Ḣ␮mir共⌫␮mir共t⬘兩⌫,t兲,t⬘兲 ⬃ sign共t⬘兲Ḣ␮共t兲,
t
t
兩t⬘兩 ⲏ ␶short ,
共73兲
which is the most likely rate of doing work at time t. This
assumes that the change in energy is negligible on the rel¯
evant time scales ␶兩Ḣ␮兩 Ⰶ 兩H␮兩. Since this asymptote is odd
in time, one concludes that the mirror work is independent of
␶ for ␶ in the intermediate regime, and that W␮mir is dominated
by the region t⬘ ⬇ t.
In view of this and the fact that d⌫␮共t⬘ 兩 ⌫ , t兲 / dt = 0, the
rate of change of the mirror work along a Hamiltonian trajectory is
¯
Ẇ␮mir共⌫,t兲 = Ḣ␮共t兲.
共74兲
As for the heat flow, the probability distribution is stationary
during the adiabatic evolution on the most likely points of
the phase space.
This result is significant in the context of the time evolution of the probability distribution that includes the stochastic perturbations from the thermal reservoir. As above,
use a single prime to denote the adiabatic development in
time ⌬t, ⌫ → ⌫⬘, and a double prime to denote the final stochastic position due to the influence of the reservoir,
⌫⬘ → ⌫⬙. The conditional transition probability may be taken
to be
⌳␮共⌫⬙兩⌫,t兲 = ⍜⌬共兩⌫⬙ − ⌫⬘兩兲e−␤共H␮⬙ −H␮⬘ 兲/2 .
共75兲
The final, odd term is identical to that given previously,4 and
hence this transition probability obeys the microscopic transition theorem 关see Eqs. 共9兲 and 共11兲 of Ref. 4兴. Hence this
transition probability for the case of nonequilibrium work
yields the fluctuation theorem9 and the work theorem,14 as
was shown in Sec. I C of Ref. 4. This transition probability
preserves the nonequilibrium phase space probability density
关Eq. 共72兲兴 during its time evolution,
㜷␮共⌫⬙兩␤,t + ⌬t兲 =
冕
d⌫⌳␮共⌫⬙兩⌫,t兲㜷␮共⌫兩␤,t兲.
共76兲
This result may be readily confirmed using the fact that
H␮⬘ − W␮mir⬘ = H␮ − W␮mir 共at least for those phase points most
likely to occur兲, together with the usual normalization requirements on the transition probability. 关See the discussion
surrounding Eq. 共44兲 above.兴
C. Nonequilibrium quantum statistical mechanics
Consider a quantum system with a time-dependent
Hamiltonian operator Ĥ共t兲. Nonequilibrium quantum statis-
tical mechanics follows from a development analogous to the
classical case. Define the mirror work operator
Ŵmir共t兲 = 关Ê+共t兲 − Ê−共t兲兴/2,
共77兲
where the past and future energy operators are
ʱ共t兲 = ⍜̂共⫿ ␶ ;t兲Ĥmir
t 共t ± ␶兲⍜̂共± ␶ ;t兲
and where the time-shift operator is
冋
⍜̂共␶ ;t兲 = exp
−i
ប
冕
t+␶
册
dt⬘Ĥmir
t 共t⬘兲 .
t
共78兲
共79兲
The mirror Hamiltonian operator has been continued into the
future,
Ĥmir
t 共t⬘兲 ⬅
再
Ĥ共t⬘兲, t⬘ 艋 t
Ĥ共2t − t⬘兲, t⬘ ⬎ t,
冎
共80兲
and the manipulation of the operators derived from it is famir
cilitated by the symmetry about t, Ĥmir
t 共t⬘兲 = Ĥt 共2t − t⬘兲.
With these definitions, the nonequilibrium density operator for a subsystem of a thermal reservoir of an inverse temperature ␤ is
␳ˆ 共t兲 =
1
exp − ␤关Ĥ共t兲 − Ŵmir共t兲兴,
Z共t兲
共81兲
where Z共t兲 is the normalization factor. Accordingly, the average of an observable at time t is
具Ô典t = tr兵␳ˆ 共t兲Ô共t兲其,
共82兲
and the present density operator can be said to provide a
basis for nonequilibrium quantum statistical mechanics.
V. CONCLUSION
This paper has been concerned with developing a nonequilibrium probability distribution. A specific steady state
system was studied in detail, namely, the heat flow down an
imposed temperature gradient. The key difference from the
Boltzmann distribution for an equilibrium system was a term
of the odd phase space parity, the mirror work, which reflects
the arrow of time inherent in nonequilibrium systems. The
steady state probability distribution that was derived here
was shown to be consistent with the Green-Kubo formula for
the thermal conductivity and with the second entropy formula for macrostate transitions. A Monte Carlo algorithm
was developed based upon the steady state probability distribution. This was shown to be computationally efficient and
to yield known values for the thermal conductivity. Estimates
of various relaxation times based on fluctuation formulas
were shown to be semiquantitative.
The nonequilibrium theory was extended generically beyond the heat flow. The probability distribution for the general hydrodynamic flow due to an imposed thermodynamic
gradient was given, and this was further generalized to imposed fields with arbitrary spatial variations. The theory was
also applied to the case of a time-dependent mechanical
work, and a mirror work term with an odd phase space parity
was identified for the probability density. The associated
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224103-13
J. Chem. Phys. 124, 224103 共2006兲
Statistical mechanical theory for steady state systems. V
transition probability satisfied the microscopic transition
theorem,4 and hence the fluctuation9 and work theorems,14
and conserved the probability density. The analogous density
operator for nonequilibrium quantum statistical mechanics
based on a mirror Hamiltonian operator was also given.
The expressions obtained in this paper appear to be formally exact for the steady state, although they may not be
restricted only to the steady state. At a minimum they are
likely to be good approximations for the quasi-steady-state
共thermodynamic gradients or rates of doing work that change
relatively slowly with time兲, but they may prove even more
general than this. For example, in the case of a nonequilibrium mechanical work, the present probability density can be
explicitly shown to be correct for t ⬍ 0 and for t ⬎ ␶, for a
sudden change in the Hamiltonian at t = 0; it may even prove
correct for all t. That it works in these regimes for this extreme example of a transient nonequilibrium behavior hints
at the broad applicability of the present approach. The
present probability densities possibly represent the optimum
generic formulation for the universal nonequilibrium state
with an arbitrary time dependence when there is an ideal
thermal contact with the heat reservoir, and all other details
of the reservoir can be ignored.
ACKNOWLEDGMENT
This work was financially supported by the Australian
Research Council.
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