Extended heat-fluctuation theorems for a system with deterministic and stochastic forces.

Abstract

Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.

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arXiv:cond-mat/0311357v2 [cond-mat.stat-mech] 26 Nov 2003
Extended Heat-Fluctuation Theorems for a System with Deterministic and
Stochastic Forces
R. van Zon and E.G.D. Cohen
The Rockefeller University, 1230 York Avenue, New York, New York 10021, USA
(Dated: November 26, 2003)
Heat fluctuations over a time τ in a non-equilibrium stationary state and in a transient
state are studied for a simple system with deterministic and stochastic components: a Brow-
nian particle dragged through a fluid by a harmonic potential which is moved with constant
velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution
of these fluctuations for all τ. By a saddle-point method we obtain analytical results for the
inverse Fourier transform, which, for not too small τ, agree very well with numerical results
from a sampling method as well as from the fast Fourier transform algorithm. Due to the
interaction of the deterministic part of the motion of the particle in the mechanical potential
with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation
theorem is, for infinite and for finite τ, replaced by an extended fluctuation theorem that
differs noticeably and measurably from it. In particular, for large fluctuations, the ratio
of the probability for absorption of heat (by the particle from the fluid) to the probability
to supply heat (by the particle to the fluid) is much larger here than in the conventional
fluctuation theorem.
PACS numbers: 05.40.-a, 05.70.-a, 44.05.+e, 02.50.-r
I.INTRODUCTION
Knowledge of the behavior of heat fluctuations has an intrinsic value, especially for small systems
such as nano-systems and bio-molecules, where the fluctuations are relatively large, but the wide-
spread interest in fluctuation theorems (FTs) stems mostly from the fact that although there are
few general results in non-equilibrium statistical mechanics, these theorems seem to provide some.
The conventional FTs state that the probability distribution function (PDF) π for the average over
a time τ of a physical quantity A to have a value a, satisfies[1, 2, 3, 4, 5, 6]
π(?A?τ= a;τ)
π(?A?τ= −a;τ)≈ exp[aτ].(1)
where ?A?τdenotes the time average of A and the dependence of the PDF on τ has been explicitly
indicated. The asymmetry in Eq. (1) is caused by some external field that is present in these models,
such that a positive value of ?A?τ can correspond to a behavior “with the field” and a negative
value to a behavior “against the field”. Note that for large τ, positive values for a become much
more probable than negative ones. To be more concrete, in dynamical systems theory, the quantity
A pertains to fluctuations in the “phase space contraction rate” [3], while in stochastic systems,
A involves an “action functional”[6]. In specific cases, both have been connected with the heat or
entropy production. Such a connection is based on the expression of the phase space contraction
rate and the action functional, respectively, which typically have the form of a thermodynamic force
times a current, divided by the temperature of the system[7], i.e., the same form as the entropy
production in Irreversible Thermodynamics. It is however not necessary to make the connection
with the entropy production and in this paper, we will simply use the heat.
There are in fact two different kinds of FTs: a transient (TFT) and a stationary state FT
(SSFT). While the conventional SSFT only holds for sufficiently large (strictly infinite) times, the
conventional TFT holds as an identity for all times[8]. Thus, for the SSFT, the ≈ sign in Eq. (1)
indicates the large τ behavior, whereas for the TFT, it can be replaced by an equality sign[9].

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Apart from an (early) laboratory experiment on the SSFT[10], the FTs were restricted to theo-
retical and simulation approaches: it was difficult to make laboratory experiments on macroscopic
systems, since the large number of particles reduces all fluctuations enormously. Recently, Wang
et al.[11] measured a TFT in the laboratory, by studying the motion of a single Brownian particle
dragged through water by a laser-induced moving (confining) potential. But while in Ref. [11] the
entropy production (or heat) fluctuations over a time τ were intended to be studied in a transient
state, in fact the fluctuations in the work done on the system during that time were studied. These
differ from the heat fluctuations due to the joint presence of the confining potential and the water,
as we will explain more clearly later in the paper. While for this system the conventional SSFT
and TFT do hold for the total work done on the system[12], for heat fluctuations, very different
FTs hold, in which the behavior of large heat deviations is notably and measurably different from
the conventional FTs, as shown in a recent letter[13]. Here we present the full theory underlying
the conclusions of Ref. [13].
The paper is organized as follows. Sec. II contains the basic theory. We present the model
(Sec. IIA), explain the difference between work and heat in the model (IIB), and treat the work-
related SSFT and TFT fully (IIC). For the heat fluctuations there is no expression for the PDF
in terms of known functions, but its Fourier transform can be calculated exactly (IID). Since this
has no known exact inverse, we first calculate, in Sec. III, the PDF of the fluctuations using two
numerical methods: a sampling method as well as the algorithm of the fast Fourier transform to
perform the inversion of the Fourier transform. The results of both methods agree well with each
other and point to violations of the conventional SSFT and TFT. Next, this is substantiated by
an analytic method developed Sec. IV based on the saddle-point method, which is shown by a
comparison with the numerical methods, to work well when τ is not too small. In Sec. V we show
how this method can be used to obtain expressions for the extended heat FTs, both in the limit
of τ → ∞, as well as for finite times. We conclude with a discussion of the results in Sec. VI. At
the end, two appendices give some technical details used in the text.
II. BASIC THEORY
A. Model
Theoretical descriptions of the experiment of Wang et al.[11] have been given even before the
experiment by Mazonka and Jarzynski[14], as well as recently by the authors[12, 13]. All were
based on an overdamped Langevin equation for the Brownian particle, which reads:
˙ xt= −τ−1
r (xt− x∗
t) + α−1ζt.(2)
Here xtis the (three-dimensional) position of the Brownian particle at time t, α = 6πηR is the
(Stokes) friction of the particle in water, η is the shear viscosity of water and R is the radius of
the particle. Furthermore, the relaxation time of the position of the particle in Eq. (1), τr, equals
α/k, where k is the strength of the harmonic force. This force is derived from the potential
U(x,t) =k
2|x − x∗
t|2, (3)
where x∗
tis the position of the minimum of the harmonic potential at time t, given in our case by
x∗
t= v∗t, (4)
(for t > 0) with v∗the (constant) velocity of the motion of the potential through the water.
Finally, the random force ζtis taken to be Gaussian and delta function correlated in time: ?ζt? = 0

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and ?ζtζs? = 2kBTαδ(s − t) ?. Here, the brackets denote an average over realizations, T is the
temperature, kBis Boltzmann’s constant and
In the rest of this paper, we will use a reduced set of units. For our time unit we will use τr:
thus wherever t appears one should read t/τr; for our energy unit, we choose kBT and for our
length unit, we use the width of the equilibrium PDF of xtin the potential, i.e.
change of units has been carried through consistently in the paper by simply setting α = 1, k = 1
and kBT = 1. Then the Langevin equation above in Eq. (2) takes the simple form
? denotes the 3 × 3 identity matrix.
?kBT/k. This
˙ xt= −(xt− v∗t) + ζt, (5)
where ζtsatisfies
?ζt? = 0, (6)
?ζtζs? = 2δ(t − s) ?.(7)
In Ref. [12], the distribution of xtand its time autocorrelation function were determined, under
the assumption that the particle has been in the potential from time t = −∞ up to t = 0 and
that during that time the potential did not move, so that the initial PDF at time t = 0 is the
equilibrium one. We found for the PDF of the position x of the Brownian particle at time t:
ρ(x;t) = (2π)−3/2e−|x−v∗(t−1+e−t)|2/2.(8)
The resulting average position ?xt? is therefore
?xt? = v∗(t − 1 + e−t).(9)
Equations (8) and (9) show that the system is not in equilibrium, but that, after a transient time
period of order 1 (after which one may neglect the term e−t) the system becomes stationary in a
co-moving frame — the PDF shifts to the right with a constant velocity v∗. The mean position
of the particle then becomes v∗(t − 1), which is not equal to the position of the minimum of the
potential, x∗
harmonic force −(?xt? − v∗t) = v∗balances the friction force −v∗.
Below, we will need the correlations of the position at different times.
autocorrelation function of xt, Ref. [12] gives in current units:
t= v∗t. The difference −v∗shows we are in a stationary state where the average
For the time-
?xt2xt1? − ?xt2??xt1? = e−|t2−t1|
?.(10)
B.Work versus heat fluctuations
We first consider the fluctuations in the total work done on the system. This work is the total
amount of energy put into the system. By the system, we mean here the particle in the potential
U plus the water. At any time, the harmonic potential exerts a force Ft= −(xt− v∗t) on the
particle. Consequently, the particle exerts a reaction force −Fton the potential. The potential
has to be kept moving at a fixed velocity v∗, for which an external force on it of magnitude Ftis
required to balance the reaction force. Hence, the fluctuating work Wτdone on the system over a
time interval from t to t + τ is given by
Wτ=
?t+τ
t
dt′v∗· Ft′ =
?t+τ
t
dt′v∗· [−(xt′ − v∗t′)].(11)

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We next consider the heat Qτ and its fluctuations, which in contrast to the total work Wτ, is
only that part of the work which goes into the fluid, while the other part of the total work goes
into the potential energy of the particle in the harmonic potential. Therefore
Qτ= Wτ− ∆Uτ, (12)
where
∆Uτ= U(xt+τ,t + τ) − U(xt,t).(13)
The potential energy U was given in Eq. (3)[15]. Note that in Eqs. (11)–(13) we suppressed the t
dependence.
Before we discuss the fluctuations in the work and the heat, we want to say a few words on
their average behavior. Using Eqs. (8) and (11), we find that the average work in a time interval
from t to t + τ is given by
?Wτ? = w?τ − (1 − e−τ)e−t?, (14)
where
w ≡ |v∗|2
(15)
From Eq. (14), one sees that after a transient time t of O(1), the second term within the square
brackets can be neglected. Then, i.e., in the stationary state, the average work is wτ, which means
that the rate of work w done on the system is constant. To calculate the average heat, we use
Eq. (12) and consider first the average potential energy
?U(xt,t)? =
?
dxρ(x,t)1
2|x − v∗t|2=3
2+1
2w(1 − e−t)2,(16)
[by
w?(1 − e−τ)e−t−1
Eqs.(8)and
2(1 − e−2τ)e−2t?.
(15)],fromwhich
Combining this with Eqs. (12) and (14), one obtains
wefind,by Eq.(13),that?∆Uτ?=
?Qτ? = w?τ +1
2(1 − e−2τ)e−2t?. (17)
From this equation, we see that for the heat too, after a transient time of O(1), the rate of heat
production becomes a constant. Furthermore, this constant is the same as that for the work in
Eq. (14). So in the stationary state, on average all work is converted into heat. One might therefore
be tempted to identify heat and work, at least in the stationary state. However, it will become
clear that such identification is not possible for the fluctuations in work and heat.
C.Distribution of work fluctuations
Here we will briefly summarize the results for the work fluctuations from Ref. [12] needed in this
paper. The work in Eq. (11) is defined as an integral over the path the particle. Since Wτis linear
in the path xt, and the xtare Gaussian distributed, it follows that Wτ is Gaussian distributed as
well. Hence, only the first two moments of the PDF need to be considered.
For the work-related TFT, let us first consider the PDF for work done on the system during a
transient state of time τ , i.e., for an ensemble of transient trajectories of duration τ, starting in
equilibrium at t = 0[2, 8]. The first moment can then be found from Eq. (14), with t = 0,
?Wτ?T= w?τ − 1 + e−τ?, (18)

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where subscript T denotes that the average is over transient trajectories. The second moment can
be calculated as in Ref. [12], using the relations in Eqs. (8)–(11), to be
?[Wτ− ?Wτ?]2?
T= 2w?τ − 1 + e−τ?. (19)
This is exactly twice the average in Eq. (18), which turns out to be crucial. Given the mean
m and the variance v of a general Gaussian distributed variable s, its PDF is given by P(s) =
exp[−(s − m)2/2v]/√2πv, hence
P(s)
P(−s)= e2ms/v.
If the variance is twice the mean, v = 2m, the right-hand side (r.h.s.) of this equation becomes es,
which is of a similar form as the FT in Eq. (1). To apply this equation to the work fluctuations in
a convenient way, we define p as the average rate of work in time τ, Wτ/τ, scaled by the average
rate w of work in the stationary state:
(20)
p ≡Wτ
wτ.
(21)
The transient PDF PW
Jacobian:
T(Wτ;τ) of Wτand the transient PDF πW
T(p;τ) of p are related by a constant
πW
T(p;τ) = wτPW
T(Wτ;τ). (22)
Superscript W denotes that these are PDFs related to work. Using Eq. (20) and the fact that for
s = Wτthe variance is twice the mean, one finds the conventional TFT:
πW
πW
T(p;τ)
T(−p;τ)= exp[Wτ] = exp[wτp].(23)
Note that the Jacobian drops out on the left hand side of this equation. Equation (23) coincides
with Eq. (1), with ?A?τ = wp, and shows that the TFT holds for work fluctuations. In fact, it
holds exactly (“as an identity”[8]) for all times τ.
For the work-related SSFT, one should in principle look at a single (half-infinite) trajectory in
the stationary state, and consider the statistics of work done in time intervals of length τ along
that trajectory. However, because of its stochastic nature, our system behaves ergodically and the
distribution of the initial points of these intervals is simply the stationary one. Thus, the desired
statistics can be determined by considering a single interval τ of which the initial point is sampled
from the stationary state. In addition, as shown in the previous sections, the system reaches a
stationary state as t → ∞, in an exponential fashion. Hence we can generate the sampling of the
initial point of the interval by considering the interval [t,t + τ] for t → ∞. In that limit, Eq. (14)
gives
?Wτ?S= wτ.(24)
The subscript S denotes that the average is to be taken over trajectories in the stationary state,
in the t → ∞ limit just explained. The variance can be computed similarly, and the result is[12]
?[Wτ− ?Wτ?]2?
Note that now the variance in Eq. (25) is not equal to twice the mean in Eq. (24), except when
τ → ∞. We therefore expect that only for τ → ∞, we have
πW
πW
S= 2w?τ − 1 + e−τ?. (25)
S(p;τ)
S(−p;τ)≈ exp[Wτ] = exp[wτp],(26)