After having read a few different derivations of the Boltzmann distribution I have come to appreciate that some are far more intellectually satisfying than others! I will outline a few derivations below and comment on them, before discussing some of the applications of the Boltzmann distribution that usually go unnoticed.
The first derivation is from Schroeder. This begins by placing a small system (say, one atom) in thermal contact with a much larger reservoir and establishing that the probability of the system being in some (quantum) state s is proportional to the multiplicity of the reservoir (fixing the total energy). Then he uses S = klnw to rewrite this in terms of the entropy, and applies the thermodynamic identity (at fixed volume and particle number). This gives the proportionality stated by Nelson. To normalise the distribution, one simply needs to sum over all of these Boltzmann factors (over all states s, not energy levels!).
This is not very mathetically rigorous, but it does lead to an intuitive understanding of where the distribution comes from. I find this derivation useful in that it highlights the importance of the reservoir in determining the state of the system.
The second is provided by Kittel, and involves a similar process. Rather than rewrite the entropy using the thermodynamic identity, Kittel expands it as a Taylor series and uses the definition of temperature (actually, this is just a slighlty more mathematical way of doing what Schroeder did).
I like the added rigour, but find that it is a little too abstract in that the role of the reservoir is less explicit.
My favourite derivation of the Boltzmann distribution (to date) comes from a quantum mechanics textbook! Sakurai and Neopolitano derive the Boltzmann distribution by maximising the entropy of a system subject to the constraints of constant (expectation value of) energy and particle number. This is achieved by the method of Lagrange multipliers.
This is also one of the most peculiar derivations: note that there is no reference to an external reservoir! The Lagrange multiplier method also doesn't immediately identify the temperature as an important factor controlling the shape of the Boltzmann distribution: this must be added in from further arguments. I think it's beauty derives from the fact that only very basic assumptions have been made (i.e. fixed energy and particle number).
Where does the Boltzmann distribution pop up, even when we might not expect it?
-the concentration of vacancies/impurities in a crystal
-the rate at which a chemical reaction proceeds
-the temperature dependence of the viscosity of a fluid
-the decreasing density of the atmosphere with altitude
-the equilibrium distribution of colloidal particles
-the cooling of an atomic gas cloud by evaporation (to produce BEC!)
-any others?
My only question is why we spend so much time learning the Boltzmann distribution when the majority of life processes occur at constant pressure and temperature (rather than volume and temperature)?
I found a derivation which seems similar to that of Sakurai and Neopolitano in: Molecular driving forces - Statistical thermodynamics in chemistry and biology 2002 - Dill and Bromberg. I find this method – using Lagrange multipliers – rather abstract and difficult to get my head around. The most conceptual derivation I found was given by Feynman in his lecture: The Principles of Statistical Mechanics. Feynman uses “decreasing density of the atmosphere with altitude” as the conceptual foundation of his derivation, very interesting approach. I thought Nelson’s method wasn’t too bad but he didn’t really illustrate its generality.
ReplyDeleteI don’t think the Boltzmann distribution has been taught enough – not in the courses I’ve taken at least. As illustrated by your list of applications, it appears in one form or another in many fields of science and almost every chapter of a Physical Chemistry textbook. To your list I would add: Nernst equation, with application in electrophysiology; phase equilibrium (e.g. vapour pressure); sedimentation.
The paper that is often credited with starting the "information theoretic interpretation" of statistical mechanics is:
ReplyDeleteJaynes, ET "Information Theory and Statistical Mechanics I" Phys. Rev. 106, 620–630 (1957) (http://link.aps.org/doi/10.1103/PhysRev.106.620f).
This is followed up by an extension to quantum statistical mechanics using the density matrix formalism:
Jaynes, ET "Information Theory and Statistical Mechanics II" Phys. Rev. 108, 171–190 (1957) (http://link.aps.org/doi/10.1103/PhysRev.108.171)