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## Article  in  European Journal of Physics · July 2010

DOI: 10.1088/0143-0807/31/5/012

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Eur. J. Phys. 31 (2010) 1097–1106 doi:10.1088/0143-0807/31/5/012

## Jarzynski’s equality illustrated by

simple examples
Humberto Hı́jar1 and José M Ortiz de Zárate2
1 Facultad de Ciencias, Universidad Nacional Autónoma de México, México DF, Mexico

E-mail: jmortizz@fis.ucm.es

## Received 24 June 2010, in final form 2 July 2010

Published 29 July 2010
Online at stacks.iop.org/EJP/31/1097

Abstract
The Jarzynski theorem is perhaps the most recently discovered simple general
formula in elementary statistical physics. In this paper, written with a
pedagogical aim, we illustrate the physical concepts under the Jarzynski and
related results by a detailed calculation with a representative example. The
physics of the model is sufficiently transparent so that it becomes suitable to
incorporate in a general course of statistical physics.

1. Introduction

The Jarzynski theorem is perhaps the most recently discovered simple general formula in
elementary statistical physics . In its most common expression it is formulated as
e−W/kB T  = e−F /kB T . (1)
Equality (1) refers to two different equilibrium thermodynamic states of an arbitrary system
which we refer to as the initial and the final state. These two states must have the same
temperature T, but differ in some other thermodynamic variable so that there is a difference
in the Helmholtz free energy F between them. For instance, in a regular fluid system,
the variable distinguishing the two states can be the volume V. From a microscopic point of
view, the variable discriminating between the two states enters as an external parameter in
the Hamiltonian of the system. It is known from elementary thermodynamics that when the
system is driven from the initial to the final state by a reversible process, a work W is performed
so that W = −F (isothermal process). If, in contrast, the process connecting initial and
final states is irreversible, the work W differs in general from the free-energy difference. Of
course, there are many irreversible processes (potentially infinity) connecting the initial and
the final states (only one isothermal reversible, though), and in each of these processes the
work performed will be different. Therefore, one can treat the ensemble of possible processes
0143-0807/10/051097+10\$30.00 
c 2010 IOP Publishing Ltd Printed in the UK & the USA 1097
1098 H Hı́jar and J M Ortiz de Zárate

## statistically. As discussed in elementary thermodynamics treatises, by virtue of the second

law, the average work must verify
W   F. (2)
Actually, in thermodynamics books, equation (2) is written without the brackets implying that
it must hold for all processes. We know, however, that the second law is of statistical nature
so that there can be exceptions (very infrequently) and interpreting equation (2) statistically is
more rigorous. Indeed, one of the virtues of the Jarzynski equality is that it gives meaning to
the average in equation (2). As we hope to illustrate in this paper, Jarzynski teaches us how
to consistently perform these kind of averages, opening up the possibility of doing statistical
physics not only of states but also of processes.
Now we are in a position to understand the physical significance of equation (1). In
accordance with the second law of thermodynamics, the work performed in an irreversible
process connecting two isothermal equilibrium states (temperature T) of an arbitrary system
satisfies inequality (2). However, it can be rigorously demonstrated that the average e−W/kB T 
satisfies equation (1), which is an equality. A rigorous formal derivation can be found in the
original papers [1, 2]. Here, our goal is to show Jarzynski’s theorem in action through a simple
example. We note here that due to the well-defined concavity of the exponential function,
equation (1) implies equation (2).
Many theoretical developments based on the Jarzynski equality have appeared during the
Most noted is the so-called Crooks theorem , usually formulated as
PF (W )
= exp [(W − F ) /kB T ] , (3)
PB (−W )
where the left-hand side is the ratio of the probability of work W when going from the initial
to the final state (forward probability) to the probability of work −W when going from the
final to the initial state (backward probability). The equivalence of the Jarzynski and Crooks
formulations can be readily demonstrated . Among other theoretical developments we
mention extensions to cover the Langevin dynamics, when the system is in equilibrium with
a thermal bath . Also equation (1) has been used to formulate nonequilibrium entropies
that could be related to extrema principles . As we discuss later, Jarzynski’s original result
is based on a canonical ensemble; especially interesting is the not so trivial extension to
microcanonical ensembles that might be relevant to adiabatic processes . All this and the
related work has generated in the last 10 years a new sub-field inside nonequilibrium statistical
physics that is usually referred to as nonequilibrium work theorems.
Jarzynski’s and related nonequilibrium work theorems are not only theoretically
interesting, because they have found many practical applications in research. Originally,
Jarzynski proposed his result as a simple and computationally advantageous way of calculating
free energies in molecular simulations, and indeed it was successfully employed with this
aim . More recently, for instance, nonequilibrium work theorems have been used to
computationally estimate free energy differences in a solution of charged particles in water
, or the surface tension of magnetic domains in the Ising model . Another important
field where equation (1) has successfully been used is single-molecule experiments. This
application was originally done by Liphardt et al , who used equation (1) to experimentally
estimate the free-energy difference between the open and the close state of an RNA hairpin.
Subsequently, it has been used to interpret similar experiments .
We should mention that the Jarzynski result has faced some criticism in the scientific
literature . Most criticisms were not about the detailed kind of calculations we will
present here, but about the definition and interpretation of work based on thermodynamics
Jarzynski’s equality illustrated by simple examples 1099

xn 2l0+L-xn
2l0+L

Figure 1. Schematic representation of the system under consideration in section 2. There are
N particles of mass m connected by harmonic springs to two bounding walls. All springs are
identical, with the constant k and equilibrium length l0.

grounds. The objections raised appear to be rebutted in our opinion . We will comment
more on these issues of thermodynamic work and heat at the end of the paper, but a complete
understanding of nonequilibrium work theorems may still require some more scientific debate.
We now stop commenting on Jarzynski’s and related results because our purpose here is not
a systematic review. The goal of this paper is to illustrate the Jarzynski and Crooks fluctuation
theorems by detailed calculations using simple representatives examples. Of course, the
Jarzynski and Crooks theorems are exact and hold for any system. The interest of performing
a detailed calculation is purely pedagogical. In the example that we examine in detail here,
the (irreversible) work probability distribution can be readily calculated analytically. Most
importantly, in our example the central physical insight behind these results is clearly shown,
namely that in an isothermal process starting at an equilibrium state the work performed
depends on the initial phase space position of the system within the corresponding canonical
distribution. Furthermore, our example here is based on one of the typical model systems often
used to teach the introductory courses of statistical physics, being sufficiently transparent to
incorporate in one of these elementary courses, effortlessly exposing the students to cutting-
edge research.

## 2. Two plates joined by springs

Let us consider a system of N identical particles of mass m which are joined to two parallel
plates by a set of 2N springs. The setup under consideration is shown schematically in figure 1.
All the springs are equal, with the constant k and equilibrium length l0, and the corresponding
potentials are assumed to be harmonic.
The Hamiltonian of the mechanical system depicted in figure 1 is readily expressed as
N
pn2 mω2
H (pn , qn ) = + [(xn − l0 )2 + (xn − L − l0 )2 ], (4)
n=1
2m 2
1100 H Hı́jar and J M Ortiz de Zárate

where ω2 = k/m is the frequency corresponding to the set of 2N identical springs and 2l0 + L
is the distance between the bounding plates. The position of the n particle is measured by its
distance xn to the left plane, pn being the corresponding momentum. We note in equation (4)
the dependence of the Hamiltonian on the extensive parameter L; Jarzynski’s equality refers
to the changes in the free energy of the system when L is varied. From Hamiltonian (4) the
equations of motion of the individual particles follow:
d2 xn
m = −2k(xn − l0 ) + kL. (5)
dt 2
If the system is in equilibrium at temperature T, the free energy of the system F(L, T) can
be easily obtained from a canonical distribution, since all integrals are Gaussian. It results in
   
kB T mω2 L2
F (L, T ) = −N kB T ln √ − , (6)
2h̄ω 4kB T
where kB is the Boltzmann constant. In equations (4)–(6) it is implicitly assumed that the
particles move in one dimension and can occupy any position xn ∈ {−∞, +∞}. The model
can be complicated by introducing, for instance, impervious walls or more spatial dimensions.
Any complication introduced into the model does not affect the validity of Jarzynski’s
result, so for the illustration goals intended here it is preferable to keep the model (and the
calculations) as simple as possible. All the thermodynamics of the system can be deduced from
equation (6), which gives the Helmholtz free energy in terms of its natural variables. In
particular, the free-energy difference between a final state with L = L1 and an initial state
with L = L0 , appearing on the right-hand side of Jarzynsk’s equality (1), will be given by
mω2  2 
F = N L1 − L20 . (7)
4
To verify Jarzynski’s theorem for our system we can imagine that at t = 0 the right wall
starts to move with a uniform velocity V, so that L(t) = (L0 + V t)θ (t), where θ (t) is a step
function. By substituting L(t) into the equations of motion, equation (5), it is relatively easy
to obtain the general solution of the corresponding differential equation, namely

xn (t) = An,0 cos( 2ωt + θn,0 ) + l0 + 12 (L0 + V t), for t > 0. (8)
The integration constants An,0 and θn,0 are related to the position and linear momentum of the
n-particle at t = 0. Let us consider next the energy of the n-particle for t > 0, which will be
given by
En (t) = 12 m[ẋn (t)]2 + 12 k[xn (t) − l0 ]2 + 12 k[xn (t) − l0 − L(t)]2 . (9)
The work Wn required to move the n-particle from its position at t = 0 to its position at time
t will be given by the difference between the corresponding energies. Some simple algebra
allows us to express it as
Wn = En (t) − En (0)
 
V √ 1 mωV √
= φ(t) − p0 (1 − cos 2ωt) − x0 − l0 − L0 √ sin 2ωt, (10)
2 2 2
where p0 and x0 are the linear momentum and the position of the n-particle at t = 0, respectively.
The function φ(t) in equation (10) is given by
   
mV 2 2L0 √
φ(t) = 2
1+ω t t + − cos 2ωt ,
4 V
 
mω2 V2 √
= L1 − L0 + 2 (1 − cos 2ωt) ,
2 2
(11)
4 ω
Jarzynski’s equality illustrated by simple examples 1101

where L1 = L0 + V t is the position reached by the right wall after a time t has elapsed. The
total work W required to move the left wall to its location
at time t will be obtained by adding
all the contributions of the individual particles W = n Wn . As is well known, the initial
positions and velocities of the particles at a given temperature are the stochastic variables,
with the probability distribution given by the canonical ensemble. Consequently, the work
required to move the left wall will be a stochastic variable also. The probability distribution
P (W ) of the stochastic variable W can be easily calculated by averaging over initial positions
and velocities of the particles, namely


N ∞
dq
PF (W ) = δ W − Wn = exp[iqW ]exp[−iqWn ]N , (12)
n=1 −∞ 2π
where we represented the delta function as an integral, and used the fact that the system under
consideration is ideal, so that averages are the same for any particle. For later use, we have
introduced the subindex ‘F’ to denote the forward work probability distribution.
The average contained in equation (12) can be exactly calculated for the simple dynamics
under consideration here, since all integrals are Gaussian. Indeed, using equation (10) and the
canonical ensemble with Hamiltonian (4) evaluated at L = L0 , one obtains
 ∞  
βω 1 βp2 V √
exp[−iqWn ] = √ e 4 βmω L0 e−iqφ
2 2
dp0 exp − 0 + iq p0 (1 − cos( 2ωt)
2π −∞ 2m 2
 ∞  2 2
βmω βmω
× dx0 exp − (x0 − l0 )2 − (x0 − l0 − L0 )2
−∞ 2 2
  
mωV L0 √
+ iq √ x0 − l0 − sin( 2ωt)
2 2
 
2 mV
2 √
= exp −iqφ − q (1 − cos 2ωt) , (13)

where, as usual, β = 1/kB T . Next, substituting into equation (12), one obtains a Gaussian
probability distribution for the work performed when the right plate moves, namely
 
1 (W − N φ)2
PF (W ) = √ exp − , (14)
2π σ 2σ 2
where the variance σ is given by
 √ 1/2
√ mV 2 (1 − cos 2ωt)
σ = N . (15)

We√note that the variance of the probability distribution PF (W ) vanishes for times multiples of
π/ 2ω. This strange feature is a consequence of the simplicity of our noninteracting model,
and is also present in other ideal systems . For systems with interaction this oscillatory
behaviour in the moments of the work probability distribution is not present . This feature
will be discussed in more detail in section 3, where a slightly interacting variation of the
current model will be considered.
Now we are in a position to readily verify Jarzynski’s equality, namely
   
β β  
e−βW  = exp (βσ 2 − 2N φ) = exp − N mω2 L21 − L20 = e−βF , (16)
2 4
where the difference between free energies is to be calculated from equation (6) by taking
L = L0 as the initial state and L = L1 as the final state.
1102 H Hı́jar and J M Ortiz de Zárate

To verify the Crooks theorem, we have to study the process inverse to the one considered
so far. This means that the left wall is initially at a position xW = L1 and moves with a uniform
velocity −V , so that after some elapsed time t it reaches the position xW = L0 . The solution
of the equation of motion continues to be represented by (8) with the exchanges L0 by L1 and
V by −V . With the same exchanges, equation (10) continues to represent the work required
to move the n-particle backwards. Furthermore, since the average performed in equation (13)
does not depend on the L0 value, the probability distribution of work performed in the backward
process will also be Gaussian, with the same variance σ as the forward process, but where the
mean value will now be given by φ  , defined as
 
 mω2 V2 √
φ (t) = L0 − L1 + 2 (1 − cos 2ωt) .
2 2
(17)
4 ω
Consequently, the Crooks theorem immediately follows
 2   
PF (W ) exp − (W −Nφ)
2σ 2 mω2  2 
=   )2  = exp β W −N L1 − L0
2
PB (−W ) exp − (W +Nφ2σ 2
4
= exp [β (W − F )] . (18)
From our expression (14) for PF (W ) it readily follows W  = N φ. This result, in view
of equation (11), agrees with the thermodynamic result (2), as expected.
To finalize this section, we discuss the quasi-static reversible limit, that is when the velocity
V is very small. In that case the variance σ of the Gaussian probability work distribution (14)
goes to zero. Consequently, the work distribution approaches a delta function
 
mω2  2 
PF (W ) = δ W − N L1 − L20 , (19)
4
independent of the temperature. Of course this is the result expected on thermodynamic
grounds. For a reversible (infinitely slow) isothermal process the work performed is equal to
the free-energies difference, equation (7).

## 3. A slightly interacting model

We can introduce a little interaction in our model of harmonic oscillators, while maintaining
it analytically tractable, by considering the system depicted in figure 2. The difference with
the system of the previous section is that the two parallel plates are joined now by a chain of
two particles. As before, they are connected by springs to the walls but now there exists a
third spring connecting the two particles of a given pair. To have the same overall number of
particles as in section 2, we assume that there are in this case N/2 pairs of particles joining the
two planes. Furthermore, to facilitate the comparison with the results √of the previous section,
we assume in this section that the mass √of the particles is m = (2/9) 3m, and that the natural
frequency of the springs is ω = (31/4 / 2)ω.
The Hamiltonian of the system is now given by

N/2 2
pn1 2
pn2 mω2
H (pn , qn ) = 
+ 
+ [(xn1 − l0 )2 + (xn2 − xn1 − l0 )2 + (xn2 − L − 2l0 )2 ],
n=1
2m 2m 6
(20)
where xn1 and xn2 are the positions of the first and the second particle of the n-pair measured
with respect to the left wall, while pn1 and pn2 are the respective momenta. From Hamiltonian
(20), the corresponding canonical partition function and free energy F(L, T) can be readily
Jarzynski’s equality illustrated by simple examples 1103

N/2

xn1
xn2 3l0+L- xn2

Figure 2. Schematic representation of the system under consideration in section 3. There are N/2
pairs of particles of mass m connected by harmonic springs between them and to the two bounding
walls. All springs are identical, with the constant k  and equilibrium length l0.

evaluated. Because of our choice for the ratio of masses m /m and ratio of frequencies ω /ω of
the model of section 3 to the model of section 2, it turns out that the free energy corresponding
to Hamiltonian (20) is exactly the same as the free energy corresponding to Hamiltonian
(4), that was given by equation (6). Consequently, the free energy difference F between
two equilibrium states at the same temperature T and different values of L is still given by
equation (7). Note that here, as in section 2, we continue to assume that particles move in
one spatial dimension and can occupy any position xn1 , xn2 ∈ {−∞, +∞}. We note that this
model is slightly more realistic than the simpler one considered before.
The equations of motion corresponding to Hamiltonian (20) are

d2 xn1 3 2
= − ω [(xn1 − l0 ) − (xn2 − xn1 − l0 )] , (21)
dt 2 2

d2 xn2 3 2
2
=− ω [(xn2 − xn1 − l0 ) + (xn2 − 2l0 − L)] . (22)
dt 2

These equations can be readily integrated for a linear wall protocol movement, i.e. when
L(t) = (L0 + V t)θ (t), and positions and momenta at any time can be expressed in terms of
the initial positions and momenta. Similar to equation (10) for the ideal model, the work Wn
required to move a pair of particles from their initial state to the final state can be expressed as
the corresponding energy difference. The explicit expression for Wn is obtained exactly and
analytically from the solution of the equations of motion (21)–(22). This explicit expression
is more complicated now than in the case of the simpler model of section 2, so we will not
display it here. However we mention an important feature: as in equation (10) for the simpler
ideal model, the work Wn depends linearly on the initial positions xn1,0 , xn2,0 and momenta
pn1,0 , pn2,0 .
The total work W required to move the right wall is expressed as a sum over pairs
of Wn’s. Positions and momenta at the initial equilibrium state are the random variables
distributed in accordance to a canonical ensemble. Hence, the total work W will also be an
stochastic variable. The probability distribution of W can be computed in a fashion similar to
1104 H Hı́jar and J M Ortiz de Zárate

equation (12) for the simpler ideal model. Because W linearly depends on the initial position
of momenta, the result turns out to be, again, a Gaussian distribution like in equation (14).
However, for the slightly interacting model of this section, the mean φ  and the variance σ 
differ from the previous results, namely

F m 2 √
φ = + V [10 − 9 cos (ω t) − cos ( 3ω t)] (23)
N 36
  2 1/2

√ mV 
√ 
σ = N [10 − 9 cos (ω t) − cos ( 3ω t)] . (24)
18β

Of course, the different values obtained for the mean and the variance of the Gaussian work
distribution in the case of the slightly interacting model, do not affect the Jarzynski theorem,
since
β
[β(σ  )2 − 2N φ  ] = −βF, (25)
2
see equation (16). A similar argument applies to the Crooks theorem or the limit of quasi-static
process discussed at the end of section 2.
Regarding the time dependence of the variance, we observe that in this case σ 
still features the oscillatory behaviour already encountered for the simpler ideal case in
equation (15). However, when a little interaction is present, the variance is always positive
for t > 0. We conclude, as anticipated after equation (15), that the strange vanishing of the
variance found in section 2 is a consequence of the simplicity of the ideal model, and that
when a slight interaction is added this feature disappears. Furthermore, if more interaction is
added to the model (imagining, for instance, chains of more than two particles between the
two rigid walls) the amplitude of the variance temporal oscillations is diminished. Eventually,
for a realistic interaction, the periodic behaviour is irrelevant, and the variance becomes a
monotonic function of time.

4. Concluding remarks

It is obvious in our examples that when the right wall starts to move, the system is driven
outside of equilibrium. Note that in our developments it was implicitly assumed that while
the wall is moving, the system is disconnected from the thermostat that initially kept the
system in equilibrium at a temperature T. When the wall movement ends, the system is not
in equilibrium, the thermostat has to be reconnected and some energy has to be exchanged
between the system and the thermostat to regain equilibrium at the same initial temperature T.
But the energy exchanged in this re-equilibration step does not modify our results, because from
a thermodynamic point of view it has to be considered as heat, as long as the final position of the
wall is not modified. This way of reasoning (disconnection from the heat reservoir, irreversible
process and re-connection to the reservoir) is not uncommon in elementary thermodynamics.
For instance, similar arguments are implicitly used when calculating the ‘entropy of the
universe’ increase (or entropy production) in the irreversible expansions of a gas against
a constant pressure p0 and in contact with a reservoir. For a recent account of this sort
of thermodynamics calculations see, for instance, section 2.1 of . Of course, in real
processes, actual disconnection from the heat reservoir may not be possible, and coupling with
the reservoir (possibly leading to non-Hamiltonian dynamics) need to be explicitly included
in the calculations. Nonequilibrium work theorems can be established even in the presence
of coupling with a heat bath: the case of a Langevin dynamics was already mentioned in the
Jarzynski’s equality illustrated by simple examples 1105

introduction , and more explicit consideration of some specific thermostats has also been
discussed in the literature .
When working out our examples, we use a simple wall-movement protocol of uniform
velocity. One can imagine more complicated protocols, involving acceleration and/or
deceleration of the wall. Of course, review of the original derivations [1, 2] show that
the Jarzynski identity holds independently of the protocol. However, the actual protocol may
affect the form of the work probability distribution, which in our case of uniform wall velocity
was a simple Gaussian, equation (13).
As a summary, we have illustrated the physical significance of Jarzynski’s and related
nonequilibrium work theorems by working out in detail a representative example. In our
case the probability distribution of work performed when driving the system between two
equilibrium isothermal states can readily be evaluated analytically, and the main physical
insight contained in the Jarzynski equality is immediately apparent, namely that the work
depends on the initial microstate within a canonical distribution of energies.
Before closing we should mention that exact calculations as the one presented in this paper
are of course feasible for almost any ideal system of the sort used as examples in elementary
statistical physics courses. Actually, explicit calculations with pedagogical aims have been
presented for the ideal gas by Lua and Grosberg , or for the Jepsen gas by Bena et al .
Crooks and Jarzynski  considered a simple model of an interacting gas, and Jarzynski
himself  used a system of perturbed harmonic oscillators, somewhat similar to what we
considered here. The examples we have discussed in this paper are different (and simpler
in our opinion) from the others already presented in the literature. In any case, our present
calculations add to the effort of making nonequilibrium work theorems accessible to a wider
audience.

Acknowledgments

During the development of this work we greatly benefitted from discussions with and
suggestions by many people. In particular, we would like to mention Chris Jarzynski,
Jan Sengers, Juan Parrondo, Andrés Santos and the students majoring in the physics of
thermodynamics course that one of us teaches at Complutense University.

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