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This is a fluctuation of about 10 standard deviations. The probability of a fluctuation
Chapter 3
The Second Law of Thermodynamics this large or larger is found (Prob. 3.26) to be approximately 1/10 200,000,000,000 . The age
10
of the universe is about 10 years. If we measured the density of the gas sample once
every second, it would take (Prob. 3.27) about
0.7 10 200,000,000,000 200,000,000,000
10 (3.58)
3 10 7

years of measurements for the probability of finding a detectable density fluctuation
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of 1 part in 10 to reach 50%. For all practical purposes, such a fluctuation in a macro-
scopic system is “impossible.”
Probability theory shows that we can expect fluctuations about the equilibrium
number density to be on the order of 1N, where N is the number of molecules per
unit volume. These fluctuations correspond to continual fluctuations of the entropy
about its equilibrium value. Such fluctuations are generally unobservable for systems
of macroscopic size but can be detected in special situations (see below). If a system
had 100 molecules, we would get fluctuations of about 10 molecules, which is an
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easily detectable 10% fluctuation. A system of 10 molecules would show fluctuations
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of 0.1%, which is still significant. For 10 molecules ( 10 12 mole), fluctuations are
1 part per million, which is perhaps the borderline of detectability. The validity of the
second law is limited to systems where N is large enough to make fluctuations essen-
tially undetectable.
In certain situations, fluctuations about equilibrium are experimentally observable.
For example, tiny (but still macroscopic) dust particles or colloidal particles suspended
in a fluid and observed through a microscope show a ceaseless random motion
(Fig. 3.14), called Brownian motion (after its discoverer, the botanist Robert Brown).
These motions are due to collisions with the molecules of the fluid. If the fluid pres-
sure on all parts of the colloidal particle were always the same, the particle would re-
main at rest. (More accurately, it would sink to the bottom of the container because of
gravity.) However, tiny fluctuations in fluid pressures on the colloidal particle cause
the random motion. Such motion can be regarded as a small-scale violation of the sec-
ond law.
Similarly, random fluctuations in electron densities in an electrical resistor pro-
duce tiny internal currents, which, when amplified, give the “noise” that is always pre-
sent in an electronic circuit. This noise limits the size of a detectable electronic signal,
since amplification of the signal also amplifies the noise.
In 1993, several workers derived the fluctuation theorem, which gives the proba-
Figure 3.14 bility that the second law is violated in a very small system. The fluctuation theorem
was verified in an experiment that observed the motions of colloidal latex particles of
A particle undergoing Brownian 6300-nm diameter suspended in water and subject to tiny forces due to a laser beam
motion.
[G. M. Wang et al., Phys. Rev. Lett., 89, 050601 (2002)].
Thermal fluctuations impose limitations on the operations of proposed nanoscale
machines and engines. Molecular machines in biological cells operate at the nanoscale
level and so are subject to large thermal fluctuations. Therefore “many central cell
processes, such as protein synthesis, energy generation, and catalysis are inherently
noisy. That the cell somehow manages to coordinate these noisy processes is one of
the remarkable, and still poorly understood, facts of complex biological systems.
How is it that the cell is capable of coordinating all these processes in which the sig-
nals are essentially buried in noise is one of the remarkable facts of complex biolog-
ical systems that are still not well understood” (C. Bustamente et al., arxiv.org/abs/
cond-mat/0511629).
The realization that the second law is not an absolute law but only one for which
observation of macroscopic violations is in general overwhelmingly improbable need