S´eminaire Poincar´e XV Le Temps (2010) 77 – 102 S´eminaire Poincar´e
Equalities and Inequalities : Irreversibility and the Second Law of
Thermodynamics at the Nanoscale
Christopher Jarzynski
Department of Chemistry and Biochemistry
and Institute for Physical Science and Technology
University of Maryland
College Park MD 20742, USA
1 Introduction
On anyone’s list of the supreme achievements of the nineteenth-century science,
both Maxwell’s equations and the second law of thermodynamics surely rank high.
Yet while Maxwell’s equations are widely viewed as done, dusted, and uncontro-
versial, the second law still provokes lively arguments, long after Carnot published
his Reflections on the Motive Power of Fire (1824) and Clausius articulated the
increase of entropy (1865). The puzzle at the core of the second law is this : how can
microscopic equations of motion that are symmetric with respect to time-reversal
give rise to macroscopic behavior that clearly does not share this symmetry ? Of
course, quite apart from questions related to the origin of “time’s arrow”, there is
a nuts-and-bolts aspect to the second law. Together with the first law, it provides a
set of tools that are indispensable in practical applications ranging from the design
of power plants and refrigeration systems to the analysis of chemical reactions.
The past few decades have seen growing interest in applying these laws and
tools to individual microscopic systems, down to nanometer length scales. Much
of this interest arises at the intersection of biology, chemistry and physics, where
there has been tremendous progress in uncovering the mechanochemical details of
biomolecular processes. [1] For example, it is natural to think of the molecular com-
plex φ29 – a motor protein that crams DNA into the empty shell of a virus – as a
nanoscale machine that generates torque by consuming free energy. [2] The deve-
lopment of ever more sophisticated experimental tools to grab, pull, and otherwise
bother individual molecules, and the widespread use of all-atom simulations to study
the dynamics and the thermodynamics of molecular systems, have also contributed
to the growing interest in the “thermodynamics of small systems”, as the field is
sometimes called. [3]
Since the rigid, prohibitive character of the second law emerges from the sta-
tistics of huge numbers, we might expect it to be enforced somewhat more leniently
in systems with relatively few degrees of freedom. To illustrate this point, consider
the familiar gas-and-piston setup, in which the gas of N ∼ 10
23
molecules begins
in a state of thermal equilibrium, inside a container enclosed by adiabatic walls. If
the piston is rapidly pushed into the gas and then pulled back to its initial location,
there will be a net increase in the internal energy of the gas. That is,
W > 0, (1)
78 C. Jarzynski S´eminaire Poincar´e
where W denotes the work performed by the agent that manipulates the piston. This
inequality is not mandated by the underlying dynamics : there certainly exist micro-
scopically viable N-particle trajectories for which W < 0. However, the probability
to observe such trajectories becomes fantastically small for large N. By contrast,
for a “gas” of only a few particles, we would not be surprised to observe – once in a
rare while, perhaps – a negative value of work, though we still expect Eq. 1 to hold
on average :
hW i > 0. (2)
The angular brackets here and below denote an average over many repetitions of
this hypothetical process, with the tiny sample of gas re-equilibrated prior to each
repetition.
This example suggests the following perspective : as we apply the tools of ther-
modynamics to ever smaller systems, the second law becomes increasingly blurred.
Inequalities such as Eq. 1 remain true on average, but statistical fluctuations around
the average become ever more important as fewer degrees of freedom come into play.
This picture, while not wrong, is incomplete. It encourages us to dismiss the
fluctuations in W as uninteresting noise that merely reflects poor statistics (small
N). As it turns out, these fluctuations themselves satisfy rather strong, interesting
and useful laws. For example, Eq. 2 can be replaced by the equality,
he
−W/k
B
T
i = 1, (3)
where T is the temperature at which the gas is initially equilibrated, and k
B
is
Boltzmann’s constant. If we additionally assume that the piston is manipulated in
a time-symmetric manner, e.g. pushed in at a constant speed and then pulled out
at the same speed, then the statistical distribution of work values ρ(W ) satisfies the
symmetry relation
ρ(+W )
ρ(−W )
= e
W/k
B
T
. (4)
The validity of these results depends neither on the number of molecules in the gas,
nor (surprisingly !) on the rate at which the process is performed.
I have used the gas and piston out of convenience and familiarity, but the
predictions illustrated here by Eqs. 3 and 4 – and expressed more generally by Eqs. 15
and 30 below – are not specific to this particular example. They apply to any system
that is driven away from equilibrium by the variation of mechanical parameters,
under relatively standard assumptions regarding the initial equilibrium state and
the microscopic dynamics. Moreover, they belong to a larger collection of recently
derived theoretical predictions, which pertain to fluctuations of work, [4, 5, 6, 7, 8, 9]
entropy production, [10, 11, 12, 13, 14, 15, 16, 17, 18] and other quantities [19, 20] in
systems far from thermal equilibrium. While these predictions go by various names,
both descriptive and eponymous, the term fluctuation theorems has come to serve
as a useful label encompassing the entire collection of results. There is by now a
large body of literature on fluctuation theorems, including reviews and pedagogical
treatments. [21, 22, 23, 24, 25, 26, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]
In my view these are not results that one might naturally have obtained, by star-
ting with a solid understanding of macroscopic thermodynamics and extrapolating
down to small system size. Rather, they reveal genuinely new, nanoscale features of
the second law. My aim in this review is to elaborate on this assertion. Focusing
Vol. XV Le Temps, 2010 Irreversibility and the Second Law of Thermodynamics at the Nanoscale 79
on those fluctuation theorems that describe the relationship between work and free
energy – these are sometimes called nonequilibrium work relations – I will argue that
they have refined our understanding of dissipation, hysteresis, and other hallmarks
of thermodynamic irreversibility. Most notably, when fluctuations are taken into ac-
count, inequalities that are related to the second law (e.g. Eqs. 5, 24, 28, 35) can be
rewritten as equalities (Eqs. 15, 25, 30, 31). Among the “take-home messages” that
emerge from these developments are the following :
– Equilibrium information is subtly encoded in the microscopic response of a
system driven far from equilibrium.
– Surprising symmetries lurk beneath the strong hysteresis that characterizes
irreversible processes.
– Physical measures of dissipation are related to information-theoretic measures
of irreversibility.
– The ability of thermodynamics to set the direction of time’s arrow can be
quantified.
Moreover, these results have practical applications in computational thermodyna-
mics and in the analysis of single-molecule manipulation experiments, as discussed
briefly in Section 8.
Section 2 of this review introduces definitions and notation, and specifies the
framework that will serve as a paradigm of a thermodynamic process. Sections 3 -
6 address the four points listed above, respectively. Section 7 discusses how these
results relate to fluctuation theorems for entropy production. Finally, I conclude in
Section 8.
2 Background and Setup
This section establishes the basic framework that will be considered, and intro-
duces the definitions and assumptions to be used in later sections.
2.1 Macroscopic thermodynamics and the Clausius inequality
Throughout this review, the following will serve as a paradigm of a nonequili-
brium thermodynamic process.
Consider a finite, classical system of interest in contact with a thermal reservoir
at temperature T (e.g. a rubber band surrounded by air), and let λ denote some
externally controlled parameter of the system (the length of the rubber band). I
will refer to λ as a work parameter, since by varying it we perform work on the
system. The notation [λ, T ] will specify an equilibrium state of the system. Now
imagine that the system of interest is prepared in equilibrium with the reservoir, at
fixed λ = A, that is in state [A, T ]. Then from time t = 0 to t = τ the system is
perturbed, perhaps violently, by varying the parameter with time, ending at a value
λ = B. (The rubber band is rapidly stretched.) Finally, from t = τ to t = τ
∗
the
work parameter is held fixed at λ = B, allowing the system to re-equilibrate with
the thermal reservoir and thus relax to the state [B, T ].
In this manner the system is made to evolve from one equilibrium state to
another, but in the interim it is generally driven away from equilibrium. The Clausius
inequality of classical thermodynamics [39] then predicts that the external work
performed on the system will be no less than the free energy difference between the
80 C. Jarzynski S´eminaire Poincar´e
terminal states :
W ≥ ∆F ≡ F
B,T
− F
A,T
(5)
Here F
λ,T
denotes the Helmholtz free energy of the state [λ, T ]. When the parameter
is varied slowly enough that the system remains in equilibrium with the reservoir at
all times, then the process is reversible and isothermal, and W = ∆F .
Eq. 5 is the essential statement of the second law of thermodynamics that will
apply in Sections 3 - 6 of this review. Of course, not all thermodynamic processes
fall within this paradigm, nor is Eq. 5 the broadest formulation of the Clausius in-
equality. However, since complete generality can impede clarity, I will focus on the
class of processes described above. Most of the results presented in the following
sections apply also to more general thermodynamic processes – such as those invol-
ving multiple thermal reservoirs or nonequilibrium initial states – as I will briefly
mention in Section 7.
Three comments are now in order, before moving down to the nanoscale.
(1) As the system is driven away from equilibrium, its temperature may change
or become ill-defined. The variable T , however, will always denote the initial tem-
perature of the system and thermal reservoir.
(2) No external work is performed on the system during the re-equilibration
stage, τ < t < τ
∗
, as λ is held fixed. In this sense the re-equilibration stage is
somewhat superfluous : Eq. 5 remains valid if the process is considered to end at
t = τ – even if the system has not yet re-equilibrated with the reservoir ! – provided
we always define ∆F to be a free energy difference between the equilibrium states
[A, T ] and [B, T ].
(3) While in general it is presumed that the system remains in thermal contact
with the reservoir for 0 < t < τ, the results discussed in this review are also valid if
the system is isolated from the reservoir during this interval.
2.2 Microscopic definitions of work and free energy
Now let us “scale down” this paradigm to small systems, with an eye toward
incorporating statistical fluctuations. Consider a framework in which the system of
interest and the thermal reservoir are represented as a large collection of microsco-
pic, classical degrees of freedom. The work parameter λ is an additional coordinate
describing the position or orientation of a body – or some other mechanical variable
such as the location of a laser trap in a single-molecule manipulation experiment [27]
– that interacts with the system of interest, but is controlled by an external agent.
This framework is illustrated with a toy model in Fig. 1. Here the system of interest
consists of the three particles represented as open circles, whose coordinates z
i
give
distances from the fixed wall. The work parameter is the fourth particle, depicted
as a shaded circle at a distance λ from the wall.
Let the vector x denote a microscopic state of the system of interest, that is
the configurations and momenta of its microscopic degrees of freedom ; and let y
similarly denote a microstate of the thermal reservoir. The Hamiltonian for this
collection of classical variables is assumed to take the form
H(x, y; λ) = H(x; λ) + H
env
(y) + H
int
(x, y) (6)
where H(x; λ) represents the energy of the system of interest – including its interac-
tion with the work parameter – H
env
(y) is the energy of the thermal environment,
Vol. XV Le Temps, 2010 Irreversibility and the Second Law of Thermodynamics at the Nanoscale 81
Figure 1 – Illustrative model. The numbered circles constitute a three-particle system of interest,
with coordinates (z
1
, z
2
, z
3
) giving the distance of each particle from the fixed wall, as shown for
z
1
. The shaded particle is the work parameter, whose position λ is manipulated externally. The
springs represent particle-particle (or particle-wall) interactions. The system of interest interacts
with a thermal reservoir whose degrees of freedom are not shown.
and H
int
(x, y) is the energy of interaction between system and environment. For the
toy model in Fig. 1, x = (z
1
, z
2
, z
3
, p
1
, p
2
, p
3
) and we assume
H(x; λ) =
3
X
i=1
p
2
i
2m
+
3
X
k=0
u(z
k+1
− z
k
) (7)
where u(·) is a pairwise interaction potential, z
0
≡ 0 is the position of the wall, and
z
4
≡ λ is the work parameter.
Now imagine a process during which the external agent manipulates the work
parameter according to a protocol λ(t). As the parameter is displaced by an amount
dλ, the change in the value of H due to this displacement is
¯dW ≡ dλ
∂H
∂λ
(x; λ) (8)
Since dλ · ∂H/∂λ is the work required to displace the coordinate λ against a force
−∂H/∂λ, we interpret Eq. 8 to be the work performed by the external agent in
effecting this small displacement. [40] Over the entire process, the work performed
by the external agent is :
W =
Z
¯dW =
Z
τ
0
dt
˙
λ
∂H
∂λ
(x(t); λ(t)) (9)
where the trajectory x(t) describes the evolution of the system of interest. This will
be the microscopic definition of work that will be used throughout this review. (For
discussions and debates related to this definition, see Refs. [40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 37].)
