Exponential distribution / Energy states / Scaling denominator kT
In this section,
Feynman explains the Boltzmann factor, the distribution of molecules across
energy states, and the role of the scaling denominator kT. A more
fitting title might simply be The Boltzmann Distribution, which is not a
strict deterministic law but rather a statistical model of molecular behavior. Some
historians and physicists also use the term Boltzmann–Gibbs distribution,
acknowledging that while the Boltzmann factor itself originates with Boltzmann,
the broader framework of statistical ensembles was developed by Gibbs.
1. Exponential distribution
Equation (40.3) n = (constant)e−P.E./kT,
known as Boltzmann’s law, is another of the principles of statistical
mechanics: that the probability of finding molecules in a given spatial
arrangement varies exponentially with the negative of the potential energy of
that arrangement, divided by kT (Feynman et al., 1963, p. 40-3).
It is misleading to
describe the equation n=(constant)e−P.E./kT as Boltzmann’s law, which is used
for the probability of finding molecules. To avoid confusion, it is useful to
distinguish between the Boltzmann factor and Boltzmann distribution.
The Boltzmann factor, e−E/kT, gives the relative weight or likelihood of an energy state E,
but by itself it is not a probability distribution because it lacks
normalization. On the other hand, the Boltzmann distribution is obtained by
normalizing the Boltzmann factor over all possible states to ensure a proper
probability distribution. By calling the formula “Boltzmann’s law” complicates the
matter, since it is not really an absolute law but a derived statistical
mechanical distribution that holds under equilibrium conditions. In short,
keeping the distinctions clear among the Boltzmann factor, the Boltzmann
distribution, and the resulting number density helps prevent the misconception
that number density is itself a probability or that the formula represents a
fundamental law.
Boltzmann’s
distribution of molecular number density is a statistical tendency, but not a
deterministic law. It does not describe the trajectory of any single molecule,
but rather the collective behavior of a large ensemble. The distribution expresses
the most probable arrangement of molecules consistent with energy
conservation and thermal equilibrium, rather than a rule that every molecule
follows at every instant. This distinction lies at the heart of statistical
mechanics: predictable regularities emerge from the random motion of
individuals, yet they do not have the absoluteness of Newtonian laws. The
deeper lesson is that approximate order can arise from apparent chaos — a
conceptual shift that anticipates the probabilistic foundations of quantum
theory. Pedagogically, the Boltzmann distribution offers a clear first step
into the probabilistic way of thinking that lies at the heart of statistical
mechanics.
2. Energy states
“Here we note the
interesting fact that the numerator in the exponent of Eq. (40.1) is
the potential energy of an atom… (Feynman et al., 1963, p. 40-2).
In general, the
numerator in the exponent of the Boltzmann factor is not limited to potential
energy; it can represent any form of energy relevant to the system. In the case
of an isothermal atmosphere, it is specifically the gravitational potential
energy that sets the relative probabilities. When the potential energy of a
state is low, the exponential factor stays close to one, so the state is
relatively common. As potential energy increases, the exponent becomes more
negative, and the probability decreases exponentially, so higher-energy states become
progressively less likely. More broadly, the Boltzmann distribution applies to
all kinds of energy states— whether due to position, motion, or quantum levels.
Pigeonhole analogy:
We can picture a
vast wall of pigeonholes (energy states) and a great many pigeons (molecules). Each
pigeon gets bumped around by random hits (thermal collisions), which can push
it up into a higher hole (more energy) or let it drop into a lower one (less
energy). Although each collision is random, the overall pattern is highly
regular: the lower holes are much more crowded, and progressively fewer pigeons
occupy the higher ones. The guiding rule is that the likelihood of a pigeon
settling in a specific hole decreases exponentially with the hole’s height. The
key lesson is that we cannot track any particular pigeon at a given moment —
what matters is the collective distribution. This statistical pattern—where the
chance of finding a pigeon in a hole is predictable—is precisely what the
Boltzmann distribution describes.
“Therefore what we noticed
in a special case turns out to be true in general. (What if F does
not come from a potential? Then (40.2) has
no solution at all. Energy can be generated, or lost by the atoms running
around in cyclic paths for which the work done is not zero, and no equilibrium
can be maintained at all (Feynman et al., 1963, p. 40-3).”
According to
Feynman, the equation F=kT d(ln n)/dx has
no solution when the force F is non-conservative (i.e.,
it cannot be expressed as the gradient of a potential F≠−∇U). The equation is derived from balancing forces in equilibrium: F dx=kTd(ln
n), which implies: F/kT = d(ln n)/dx.
For this equation
to have a solution, F must be the derivative of some function.
If F is conservative (e.g., gravity):
F=−dU/dx Þ (−1/kT)dU/dx=d(ln
n)/dx Þ ò(−1/kT)dU=òd(ln n) Þ ln n = (−1/kT)U
+ C Þ n=n0e−U(x)/kT.
This yields the Boltzmann distribution.
If F is non-conservative (e.g.,
friction): F cannot be written as −dU/dx. In other
words, the equation F=kTd(ln n)/dx cannot
be integrated to find n(x) because
1. Mathematically: The path-dependence
of its work integral makes it impossible to define a single-valued potential
energy function U.
2. Physically: It transforms
mechanical energy into thermal energy, violating the conservation of energy
that the concept of a potential function is built upon.
In short, a
non-conservative force prevents the system from reaching a true thermodynamic
equilibrium.
3. Scaling denominator kT
“Thermal equilibrium cannot
exist if the external forces on the atoms are not conservative (Feynman et al., 1963, p. 40-3).”
From a practical
perspective, the Boltzmann distribution is a powerful tool for approximating
the behavior of systems near thermal equilibrium, even when the temperature is
only roughly constant. Its shape is governed by the Boltzmann factor, where the
product kT has the dimensions of energy, allowing E/kT to be a
dimensionless ratio and the exponent is unitless. At high temperatures (large kT),
the distribution flattens: higher-energy states gain significant probability,
and populations spread across a wider range of energies. At low temperatures
(small kT), the distribution “sharpens”: low-energy states dominate, and
the population is concentrated within a narrower range of energies (See fig
below). Conceptually, kT acts as nature’s probability scale, setting how
energy spreads across states in terms of their relative likelihood. Even when
temperature is not strictly constant, this scaling relation enables the
Boltzmann factor to provide reliable probabilistic predictions about complex
systems.
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| Source: Boltzmann distribution – Wikipedia |
What if k were larger or smaller?
The value of the
Boltzmann constant is crucial because it sets the scale at which thermal energy
translates into probabilities; if it were different, the entire statistical
structure of the universe would change. A larger k would make the ratio E/kT
smaller at a given temperature, increasing the likelihood of higher-energy
states and broadening the distribution. A smaller k, by contrast, would
have a narrower range of lower-energy states, making the universe more rigid,
with molecules rarely accessing higher energies. In a sense, the magnitude of k
governs the balance between stability and variability—the very balance on which
chemistry, biology, and cosmic structure depend.
The Boltzmann
constant k can be viewed as a unit-conversion factor: it connects the
macroscopic scale of temperature (Kelvin) to the microscopic scale of energy
(joules). If k were numerically larger or smaller, we would simply
measure temperature in different units, while the product kT—the true
physical quantity that sets the energy scale in the Boltzmann factor—would
remain unchanged. The underlying physics of distributions, fluctuations, and
equilibrium would be identical; only the human-assigned numerical values of
temperature would shift. For example, if k were ten times bigger, the
same gas with the same kinetic energy per particle would have a temperature
reading ten times smaller. Thus, k does not control the universe’s
behavior; rather, it calibrates the conversion between microscopic kinetic
energy and macroscopic temperature—much like how c links space and time
units in relativity.
Review questions:
1.
Is the Boltzmann distribution of molecular number density a strict physical
law, or is it better understood as a statistical tendency?
2.
In what way does potential energy influence the form of the Boltzmann
distribution?
3.
Why is thermal equilibrium an essential condition for applying the Boltzmann
distribution?
A Brief History of
the Boltzmann Constant
Phase 1 — Radiation
Constant (1900):
In the famous
December 14, 1900 paper, Max Planck introduced two new constants, h and k,
as fitting parameters (Hilfsgrößen) in his derivation of the blackbody
spectrum. At this stage, k was called die Strahlungskonstante
(radiation constant), because Planck regarded it as specific to thermal
radiation and the resonators in the cavity walls (Planck, 1900; Kuhn, 1978). In
the same paper, h was framed by Planck as a first constant of nature, k
was described as a second constant of nature, though still within the
restricted context of radiation theory.
Phase 2 — Universal
Constant (1901 onward):
In his 1901 paper On
the Law of Distribution of Energy in the Normal Spectrum (Annalen der
Physik, 4, 553–563), Planck introduced k as one of the universal constants.
Similarly, H. A. Lorentz emphasized the universal role of k in
connecting the gas constant R with Avogadro’s number N, reinforcing its status
as a fundamental link between microscopic and macroscopic physics (Lorentz,
1905).
Phase 3 — Boltzmann
Constant (1906 onward):
Paul Ehrenfest was
among the first to refer to k as the Boltzmannsche Konstante
(Boltzmann constant) (Ehrenfest, 1906). By 1920, the name had become common enough
that Planck, in his Nobel lecture, noted with some irony: “This constant is
often referred to as Boltzmann’s constant, although, to my knowledge, Boltzmann
himself never introduced it – a peculiar state of affairs, which can be
explained by the fact that Boltzmann, as appears from his occasional
utterances, never gave thought to the possibility of carrying out an exact
measurement of the constant…” Since Planck had originally introduced k
alongside h in his radiation theory, it is understandable that some of
his contemporaries referred to k as the “Planck coefficient,” “Planck constant,” or “Boltzmann–Planck
constant.” However, Planck mentioned Boltzmann-Drude constant (a = 3k/2)
that is related to k in his 1900 paper, but extending the term to
“Boltzmann–Drude–Planck constant” would have been too cumbersome for use.
In 1900, Planck
was the first to determine the constant k, assigning it the value 1.346
× 10⁻¹⁶ erg/deg. Over the following century, successive refinements improved
its precision, resulting in the 2019 redefinition of the International System
of Units (SI), where the Boltzmann constant was fixed at the exact value k
= 1.380 649 × 10⁻²³ J K⁻¹. This redefinition established k as a
fundamental constant of measurement, marking its transformation from a
provisional fitting parameter in Planck’s blackbody theory into a cornerstone
of modern physics and metrology (BIPM, 2019).
The Moral of the
Lesson:
The Boltzmann
distribution shows that order can emerge from apparent chaos. Imagine a vast
wall of pigeonholes (energy states) filled with countless pigeons (molecules),
each jostled randomly by collisions. While we cannot track which pigeon is in
which hole at any moment, a predictable pattern emerges: lower-energy holes are
more crowded, higher-energy holes less so. This reminds us that structure and
success often arise not from controlling every detail, but from understanding
the overall tendencies of a complex system.
Key Takeaway (In
Feynman’s Spirit):
Don’t waste energy
trying to follow every pigeon (molecule). Focus on the probabilities instead. The universe
doesn’t obey our deterministic expectations, but by appreciating the
statistical patterns, we can understand and even predict the collective
behavior of the system.
References
Bureau
International des Poids et Mesures (BIPM). (2019). SI Brochure, 9th edition.
Ehrenfest, P.
(1906). Zur Planckschen Strahlungstheorie. Physikalische Zeitschrift, 7(2),
528-532.
Feynman,
R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on
Physics, Vol I: Mainly
mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
Kuhn, T. S. (1978).
Black-Body Theory and the Quantum Discontinuity, 1894–1912. Oxford:
Clarendon Press.
Lorentz, H. A.
(1905). Einige Bemerkungen über die Molekulartheorie. Verslagen en
Mededeelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 14,
1273–1280.
Planck, M. (1900). Zur
Theorie des Gesetzes der Energieverteilung im Normalspectrum. Verhandlungen
der Deutschen Physikalischen Gesellschaft, 2, 237–245.
Planck, M. (1901). Über
das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 4,
553–563.
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