Contents - Max-Planck-Institut für Physik komplexer Systeme

p(W)
0.1
0.08
0.06
0.04
0.02
0
0.01
0.005
0.001
9900 ∆F
W
10000
Figure 1: Distributions of the work performed during the direct
process (solid lines) and extracted during the time-reversed process
(dashed lines) for various values of the switching rate δλ.
The parameter δλ, the switching rate, controls how
far from equilibrium the system is driven. With fast
switching (red curve on fig. 1), the system is driven further
away from equilibrium than with slow switching
(blue curve). The mean dissipated work is greater, as
are the fluctuations of the work. The maximum work
theorem implies that in the quasi-static limit δλ → 0,
the distributions of the work for the direct and the timereversed
processes both tend towards a Dirac distribution
centered on ∆F .
〈δW(t)〉/δλ
10000
9960
9920
9880
10 −2
5 · 10 −3
10 −3
9840
0 0.2 0.4 0.6 0.8 1
λ(t)
Figure 2: Evolution of the mean work performed (solid lines) and
extracted (dashed lines) 〈 δW
〉 per time step along the direct and re-
δλ
verse processes for different switching rates δλ.
The evolution of the quantity δW
δλ during switching
helps to understand how the distributions in fig. 1
arise. Fig. 2 shows the evolution of its mean value 〈 δW
δλ 〉
along the process for different values of the switching
rate δλ. Examples of the fluctuations of δW
δλ around its
mean value are shown on fig. 3. The mean value of δW
δλ
during the direct and the reverse process are very close
to one another for a slow process and further apart for
a fast one, implying an increase in the mean dissipated
work as the switching rate increases. The amplitude of
the fluctuations of δW
δλ are not influenced by the switching
rate. However, for slow switching (red curve on
fig. 3), the system has more time to fluctuate and the
fluctuations are partly averaged out when performing
the integration (4).
δW(t)−〈δW(t)〉
δλ
80
40
0
-40
-80
10 −4
10 −3
0 0.2 0.4 0.6 0.8 1
λ(t)
Figure 3: Fluctuations of δW
δλ
along the process for two values of δλ.
For Gaussian distributions of the dissipated work Wd
one can easily verify that Crooks’ relation (1) implies
that the mean value W d and variance σ 2 of the dissipated
work for the direct and the time-reversed processes
coincide. Crooks’ relation then reduces to a generalized
fluctuation-dissipation relation [4]:
σ 2 = 2kBT W d
Fig. 4 shows the distribution of the quantity Wd =
Wd/σ − σ/2kBT for the direct and the time-reversed
process for various values of the switching rate. This
quantity has a Gaussian distribution with zero mean
and unit variance for all values of the switching rate,
implying that the dissipated work Wd has a Gaussian
distribution and satisfies (6) and therefore Crooks’ relation
(1) as well.
p( p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) p( Wd ) Wd )
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 -6 -4 -2 0 W W W W W W W W W W W W W W W d
2 4 6
Figure 4: Distribution of the quantity f Wd = Wd/σ −σ/2kBT for the
direct and reverse process for various values of δλ.
A more careful analysis of the fluctuations suggests
that these are equilibrium heat bath fluctuations in
which only the mean value depends of the protocol and
hence reflects the distance from equilibrium. It is evident
that this cannot hold for arbitrary situations, in
particular not in cases where the driving force creates
dynamical instabilities. The details are still awaiting
further investigation. This issue is relevant for macroscopic
fluctuations in non-equilibrium systems that are
close to the thermodynamic limit, for which pure heat
bath fluctuations should become invisible.
[1] H. B. Callen, THERMODYNAMICS AND AN INTRODUCTION TO
THERMOSTATISTICS, John Wiley & sons (1985).
[2] G. E. Crooks, J. of Stat. Phys. 90 (1998) 1481.
[3] R. Adhikari et al, Europys. Lett. 71 (2005) 473; B. Dünweg,
U. D. Schiller and A. J. C. Ladd, Phys. Rev. E 76 (2007) 036704.
[4] L. Granger, M. Niemann and H. Kantz, J. Stat. Mech.: Th. and
Exp. (2010) P06029
2.14. Work Dissipation Along a Non Quasi-Static Process 69
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