Untangling Complex Systems, page 11
of energy dissipation can be the actions played by artificial intelligent agents that, like the Maxwell’s
demon, detect and manipulate microscopic bodies.
2.3.6 The surPrising behavior of small sysTems
As we have ascertained in the previous paragraph, the recent refinement of the micromanipulation
technology has opened new opportunities to shed light on the unique properties of microscopic
systems (Grier 2003). It is now possible to perturb single small systems, driving them away from
the equilibrium and observe their subsequent responses. Whereas the thermodynamic state of a
macroscopic system is defined by specifying parameters such as temperature, pressure, and concen-
trations of the chemical species, the state of a microscopic system is determined by measuring the
so-called “controlled parameters” (Bustamante et al. 2005). Examples of controlled parameters are
the elongation of a macromolecule, the force acting on a colloidal particle (having linear dimensions
in the range between a few units and hundreds of nanometers) and its position in space with respect
to a reference’s system. As a small system is pushed out of equilibrium, its dynamics are effectively
random, because it is soaked into a thermal bath where unpredictable fluctuations become relevant.
The behavior of the small system is described by the Fluctuation Theorem, formulated in the mid-
1990s within the theory of non-equilibrium statistical mechanics (Evans and Searles 2002). The
state of a small system is represented by a point in its phase space. The phase space is a multidimen-
sional reference system whose coordinates are the controlled parameters needed to define the states
of the small system. When a small system evolves in time, it describes a trajectory in its phase space.
Every trajectory is characterized by a value of the “entropy production” P*:
*
dq
P =
, [2.19]
Tdt
which is the ratio of the rate at which the system exchanges heat with the bath over its temperature.
The Fluctuation Theorem defines the ratio between the probability that the small system traces
a trajectory generating a positive entropy production P* (i.e., Pr( *
P )) and the probability that the
same small system follows the respective anti-trajectory generating a negative entropy production
−P* (whose probability is Pr(
*
− P )). Its formulation for arbitrary averaging times is the following:
Pr ( P*
P*( t
)
∆ )
e kB . [2.20]15
Pr (− P*) =
15 The Fluctuation theorem presents different formulations. A formulation refers to non-equilibrium steady-state fluctuations ([2.21]) whereas the other [2.20] refers to transient fluctuations.
*
Pr( P )
k
lim B
*
ln
= P [2.21]
*
t →∞
t
Pr(− P )
36
Untangling Complex Systems
The Fluctuation Theorem gives a precise mathematical expression for the probability of violating
the Second Law of Thermodynamics. The probability of observing an event reducing entropy is
inversely proportional to the exponential of the product P*(Δ t). In simple words, it means that the
probability of consuming entropy is negligible when we deal with events occurring over long-time
scales (Δ t large) and involving macroscopic systems producing large entropy production values.16
We may rewrite equation [2.20] in the following form:
*
P ( t
∆ )
Pr
*
*
P
Pr ( P ) e kB
( ) = −
[2.22]
Reminding that Pr ( *) + Pr(
*
P
− P ) =1, it derives that:
1
Pr
*
(− P ) =
[2.23]
*
1+
(∆ )
eP t kB
1
Pr
*
( P ) =
. [2.24]
*
1+ e− P ( t∆) kB
When we are at equilibrium, Δ t is 0, and Pr(
*
P ) Pr( *
−
=
P ) = .
0 5, in agreement with the postulate of
microscopic reversibility. In fact, at equilibrium, a transformation of a microscopic system tracing a
trajectory in its phase space is counterbalanced by the corresponding anti-trajectory. The anti-trajectory
is overlapped to the trajectory, but it is crossed in the opposite direction and with equal probability.
From equations [2.23] and [2.24], we see that outside of equilibrium Pr(
*
− P ) decays from 0.5 to
0, and the larger the system, the faster the decay. On the other hand, Pr( P*) grows from 0.5 to 1 (see
Figure 2.9).
1.0
0.9
Pr (+P∗)
0.8
0.7
0.6
Pr 0.5
0.4
0.3
0.2
0.1
Pr (−P∗)
0.0 0
1
2
3
4
5
Time (Δ t)
FIGURE 2.9 Trends of the probabilities that the entropy production is positive (Pr(+ P*)) and negative (Pr(− P*)) expressed by equations [2.24] and [2.23], respectively. The three curves in the graph have been
traced for three values of P* that grows going from the black to the gray up to the light gray one.
16 Note that the entropy production is an extensive variable: its values depend on the extent of the system.
Reversibility or Irreversibility? That Is the Question!
37
Laser beam
Focal
point
FIGURE 2.10 Sketch representing the latex particle trapped by the laser beam. The laser beam draws the
particle towards its focal point by a harmonic force.
Fluctuation Theorem has been proved experimentally. An important empirical demonstration
has been offered by the team of Denis Evans at the Australian National University in Canberra
(Wang et al. 2002). Evans and his colleagues followed the trajectory of a 6.3 μm diameter latex
particle contained in a glass sample cell filled with water and put on the stage of an inverted micro-
scope.17 The latex particle was trapped within a laser beam having 980 nm as the wavelength (see
Figure 2.10).
The infrared (IR) rays of the laser beam are refracted at different intensity over the volume of the
sphere and draw the particle towards the focus of the beam, i.e., the region of highest intensity. The
force exerted by the laser beam ( F ) was of the order of pico-Newton (10−12 N) and was assumed to
opt
be harmonic near the focal point. It was defined by the following equation:
F
( ∆
0 )
opt = − k x t − x
, [2.25]
where in k is the trapping constant which was tuned to about 1 × 10−5 pN/Å by adjusting the laser
power. The team first recorded x which represents the position of the particle in the absence of any
0
stage translation (averaging for a minimum of 2 sec). Then, the stage was translated at a constant
velocity ( v = 1.25 μm/s), and the position of the particle was recorded at different time intervals, xΔ .
t
The entropy produced along a trajectory of duration Δ t was estimated by the following equation:
∆ t
∆ S = T −1 vFoptdt
∫
, [2.26]
0
where T is the temperature of the heat sink surrounding the system. The determination of Δ S
has been repeated over 500 trajectories and for different Δ t. They found that when Δ t was
10−2 sec, the trajectories were distributed nearly symmetrically around Δ S = 0, that is with
entropy-consuming and entropy-producing trajectories equally probable. At longer times, the
entropy-consuming trajectories occurred less often, and the mean value of Δ S shifted towards
positive numbers. For Δ t greater than a few seconds, the entropy-consuming trajectories could
not be detected at all.
The Fluctuation Theorem has received further deft empirical proofs (see for instance the work
by Dillenschneider and Lutz (2009), who proved Landauer’s Principle in the limit of long erasure
17 An inverted microscope is an optical microscope “upside down”. In fact, its light source and condenser (a lens concentrat-ing light from the illumination source) are on the top, above the stage, pointing down, while the objectives are below the stage, pointing up.
38
Untangling Complex Systems
cycles for a system of a single colloidal bead 2 μm in diameter), and it is now accepted as a valid
generalization of the laws of thermodynamics to small systems (Bustamante et al. 2005). Its content
is revolutionary, because small systems, such as the molecular machines we encounter in biology
and nanotechnology, manifest striking behaviors if compared with the workings of the macroscopic
world. Biological and nanotech machines can transform heat in work over short time scales when
the work performed during a cycle is comparable to the thermal energy available for each degree
of freedom. In other words, the nano-machines spend some time working in “reverse mode.” It is
as if a car could transform its products of combustion and the released heat into fuel and oxygen to
go ahead.
TRY EXERCISE 2.2
2.3.7 There is sTill an oPen QuesTion
In the last paragraph, we have discovered that small systems (having dimensions not larger than
a few μm) can break the second law of thermodynamics over short time scales (less than tens of
milli-sec). It means that heat can be transformed entirely into work only in the microscopic world
and over a limited lapse of time.
Now a further question arises. How can we conciliate the Second Law of Thermodynamics with
the theory of evolution? The theory of evolution asserts that life on earth evolved from simple forms
(unicellular organisms) towards more and more organized and complex species (multi-cellular liv-
ing beings). Biological evolution requires very long-time scales, being of the order of billions of
years. Moreover, how is it possible that after fertilizing an egg, a multi-cellular organism, which
is highly organized in both space and time, emerges? It seems that amazing phenomena, such as
biological evolution and the birth of any living being, are in sharp contrast with the Second Law of
Thermodynamics. This antinomy cannot be solved by invoking the Fluctuation Theorem, because
it predicts that only microscopic systems can decrease entropy, and just for short time scales. To
address this apparent conundrum, we must go ahead on our journey and learn the principle of non-
equilibrium thermodynamics. So, let us plunge into Chapter 3.
2.4 KEY QUESTIONS
• What are the fundamental Conservation Laws?
• What kind of system can we encounter in Thermodynamics?
• How many definitions of entropy do you know?
• What happens to entropy in an irreversible transformation?
• What is a reversible transformation?
• What happens to entropy in a reversible transformation?
• What are the two contributions to statistical entropy?
• What is the connection between information and entropy?
• Is it possible to violate the Second Law of Thermodynamics?
• What is the role of Maxwell’s demon?
• Why did the mechanical attempts at violating the Second Principle fail?
• What is the significant concept proposed by Landauer in the analysis of the Szilard’s
machine?
• What does the Fluctuation Theorem tell us?
2.5 KEY WORDS
Conservation Laws; Irreversibility and Reversibility; Configurational and Thermal Entropy;
Degrees of freedom; Uncertainty and Information; Fluctuations.
Reversibility or Irreversibility? That Is the Question!
39
2.6 HINTS FOR FURTHER READING
To deepen the relationship between thermodynamic entropy and information, read Brillouin (1956),
Leff and Rex (1990), and Maruyama et al. (2009). To meditate more on the Maxwell’s demon, read
Maddox (2002).
To collect more information about optical trapping, read Ashkin (1997).
A pleasant book to meditate on the rise of complexity in nature is that written by Chaisson (2002).
2.7 EXERCISES
2.1. If we erase 8 kilobytes in the memory of our computer working at 300 K, how much heat
do we generate according to Landauer’s Principle? How much entropy do we generate?
Remember that one byte is a unit of information that consists of eight bits.
2.2. In the experiment of Evans and his team, the latex particle trapped by the laser beam
feels a harmonic potential; it is like a ball bound to a spring following Hooke’s Law (see
Figure 2.11).
After
Δ t = 10−2 sec, the minimum value of Δ S obtained was −6 k . Knowing that the
B
force constant k of the spring was 0.1 pN/μm and the speed of the stage v was −1.25 μm/s,
which was the value of ∆ x = x∆ t − x 0? Assume that the temperature of water’s bath wherein the latex particle was immersed was 300 K.
2.8 SOLUTIONS TO THE EXERCISES
2.1. According to Landauer’s Principle, any logically irreversible transformation of classical
information is necessarily accompanied by the dissipation of at least k T
B ln 2 of heat per lost
bit. Since we erase 64000 bits, the heat produced will be 64000
−
B
2 = 1 8
. ×10 16
k Tln
J. The
entropy is S = q T =
× −
6 1 10 19
.
J/K. Note that in current silicon-based digital circuits, the
energy dissipation per logic operation is about a factor of 1000 greater than the ultimate
Landauer’s limit.
2.2. After applying the laser beam and maintaining constant its power, the internal energy of
the particle is constant because it is surrounded by a thermal bath.
dU = dq + dw = dq − Foptdx = 0
The entropy production P becomes:
P*
dq
Foptdx
Foptvdt
=
=
=
kB
k T
B dt
k T
B dt
k T
B dt
*
Foptv
P =
T
X′
0
X 0
0
X″
0
(b)
(a)
(c)
FIGURE 2.11 The latex particle trapped by the laser beam behaves like a ball bound to a spring. In (a), (b),
and (c), the spring is at equilibrium, stretched and compressed, respectively.
40
Untangling Complex Systems
Foptv
dS =
dt
T
∆ t
∆
1
S =
Foptvdt
∫
T
0
F
(−
)
opt v∆ t
k∆ x v∆ t
∆ S =
=
T
T
Introducing the values of variables, we obtain ∆ x = −19 9
. µm. Note that within the same
time interval ∆ t , the displacement due to the movement of the stage is − 0.0125 μm. This
result means that the latex particle exploits the thermal energy to do work and go further
away from the focal point of the laser.
Out-of-Equilibrium
3 Thermodynamics
Nature, to be commanded, must be obeyed.
Francis Bacon (1561–1626 AD)
Science is built up with facts, as a house is with stones. But a collection of facts is no more a
science than a heap of stones is a house.
Jules Henri Poincaré (1854–1912 AD)
3.1 INTRODUCTION
Out-of-equilibrium systems are widespread. The Universe, although it is an isolated system, is very
far-from-equilibrium because it is expanding and its cosmic background thermal radiation, of about
3 Kelvin (K), is not in thermodynamic equilibrium with the matter concentrated in the galaxies.
Each star is out-of-equilibrium; in fact, nuclear reactions of fusion occur in their cores, which have
higher temperatures and different compositions than their outer shells. The same is true for each
planet, particularly for Earth. The core of our planet is melted with a higher temperature and pres-
sure than its surface. Moreover, the influx of electromagnetic radiation coming from the sun main-
tains our planet far-from-equilibrium. Also, humans, like every other living being on Earth, are
out-of-equilibrium. Every living creature is an open system exchanging matter and energy with its
