# Classical treatment of statistical mechanics

• The_Doctor
In summary: I can't find it now, but it said that the efficiency of an E Coli cell was very close to the Carnot Engine.Some researchers have found that the efficiency of an E Coli cell is very close to the Carnot Engine.
The_Doctor
I've been reading, in my own time, a first course in thermodynamics and they present a quantum treatment of statistical mechanics (discrete energy levels), but on the article for the partition function on wikipedia, I find out that there is a classical treatment of statistical mechanics as well, where (I presume) energy levels are continuous and you use integrals.

Is the classical treatment of statistical mechanics taught at university?

If not, are there any important ideas in classical statistical mechanics one should know?

Is it ever still used?

Is it valid?

Is the classical treatment of statistical mechanics taught at university?
Yes - to all questions.

For eg. http://home.comcast.net/~szemengtan/ (scroll to "statistical mechanics")
Classical statistical mechanics is as "valid" as any classical physics.

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Forgive me, but I cannot seem to find classical statistical mechanics from your link. The review of classical thermodynamics doesn't seem to have any statistical mechanics and the fundamentals of statistical mechanics seems to have discrete energy levels.

Also, so at a microscopic level, are molecules' energy levels quantised? And are there well-known experiments that confirm/refute this fact?

When I studied statistical mechanics in graduate school the principle difference was whether you used:

1. Maxwell-Boltzman statistics: classical (each particle can be tagged/distinguished from the others)
2. Bose-Einstein statistics: bosons (they all look alike, and like each other)
3. Fermi-Dirac statistics: fermions (they all look alike, but obey the Pauli "housing rules")

For the similarities and differences see: http://en.wikipedia.org/wiki/Maxwell–Boltzmann_statistics

The principles of thermodynamics are unchanged.

Note: our principle focus today is on the quantum side - because so much of our materials and device engineering depends upon quantum effects. For example: heat engines are classical, but fuel cells depend upon quantum effects/statistics.

Yep - classical statistical mechanics is usually included in thermodynamics courses in some way.
I may have misrecalled the amount that it was included in the SzeTan lectures :(
Try: http://www4.ncsu.edu/~franzen/public_html/CH795N/lecture/XIII/XIII.html
... scroll to "classical partition function". Closer to what you had in mind?

The courses do not tend to go into great depth because of the impact of quantization on the statistics.
Earlier treatments consider stuff like the Rayleigh-Jeans Law - foundational concepts.
You'll also see "semi-classical statistical mechanics".

You realize that classical mechanics is usually thought of as a subset of quantum mechanics?
It follows that classical physics is "valid" up to a point.

Molecular energy levels are quantized - yes.
Look up the operation of CO2 lasers for an example.
http://www.phy.davidson.edu/stuhome/jimn/final/pages/finalmolecular.htm

UltrafastPED said:
Note: our principle focus today is on the quantum side - because so much of our materials and device engineering depends upon quantum effects. For example: heat engines are classical, but fuel cells depend upon quantum effects/statistics.

There is actually a lot of interest in classical Stat Mech still (although, Quantum Stat Mech is probably still more researched). As an answer to the OP, classical Stat Mech is very useful in biological systems where quantum effects are rarely seen. One of the post-doc's that I worked with did all of his research in classical Stat Mech and mostly applied it to proteins.

Simon Bridge said:
Yep - classical statistical mechanics is usually included in thermodynamics courses in some way.
I may have misrecalled the amount that it was included in the SzeTan lectures :(
Try: http://www4.ncsu.edu/~franzen/public_html/CH795N/lecture/XIII/XIII.html
... scroll to "classical partition function". Closer to what you had in mind?
Yeah, but I don't know Hamiltonian mechanics yet :(. Guess I'll have to learn it first.

Simon Bridge said:
The courses do not tend to go into great depth because of the impact of quantization on the statistics.
Earlier treatments consider stuff like the Rayleigh-Jeans Law - foundational concepts.
Yes, we learned the existence of this problem in high school; well we only really learned that Planck resolved it (it was called the 'black-body radiation problem') by quantising energy, E=hf

Simon Bridge said:
You'll also see "semi-classical statistical mechanics".
Semi-classical? Where some energy levels are discrete and some are continuous?

Simon Bridge said:
You realize that classical mechanics is usually thought of as a subset of quantum mechanics?
It follows that classical physics is "valid" up to a point.

Molecular energy levels are quantized - yes.
Look up the operation of CO2 lasers for an example.
http://www.phy.davidson.edu/stuhome/jimn/final/pages/finalmolecular.htm
These links deal with the vibrational energy (I haven't read them yet, although I plan to, just havent' had time yet); but what about the translational kinetic energy? Is this quantised too?

DrewD said:
There is actually a lot of interest in classical Stat Mech still (although, Quantum Stat Mech is probably still more researched). As an answer to the OP, classical Stat Mech is very useful in biological systems where quantum effects are rarely seen. One of the post-doc's that I worked with did all of his research in classical Stat Mech and mostly applied it to proteins.
Now that you mention it, I remember noticing an article on the MIT science page that said some researchers had found out that the effieciency of an E Coli cell was very close to the Carnot Engine! Very efficient indeed.

The paper is here: http://dx.doi.org/10.1063/1.4818538. You can access it without payment, it' s released under creative commons. There are some integral signs, so I presume it does use classical mechanics!

The question is: why are you reading it?

I'm not. It was an example of drew's comment that classical stat mech is used in biological systems, I presume that is an example, though I'm not 100% sure because, as you say, it's advanced for my level.

The_Doctor said:
Yeah, but I don't know Hamiltonian mechanics yet :(. Guess I'll have to learn it first.
Yep - formal statistical mechanics tends to be senior or post-grad college for a reason ;)

Semi-classical? Where some energy levels are discrete and some are continuous?

These links deal with the vibrational energy (I haven't read them yet, although I plan to, just havent' had time yet); but what about the translational kinetic energy? Is this quantised too?
No more than it is quantized for individual particles.

The electrons shared between nuclei in covalent bonds have discrete kinetic energy levels.
If the molecule is confined by a potential, then the system will have discrete kinetic energy levels.
However - the more complicated the system the more complicated the energy levels can be. You can get situations where it is simpler to treat bands of very closely spaced energy levels as if they are continuous. Look up "band theory of solids".

At the level you are at - I'd concentrate on the core concepts first and do the complications later when you've got those down.

I think classical statistical thermodynamics is not of importance any more, even in teaching.
The main problem is that it is not clear how to count states as there aren't any.
E.g. in deriving the entropy of a classical gas, the Sakur Tetrode equation, you have to amend your treatment and introduce phase space cells which becomes only clear once you introduce quantum mechanics.
I would also not consider the Maxwell Boltzmann statistics as classical because you obviously can have distinguishable particles also in quantum mechanics.

## 1. What is the classical treatment of statistical mechanics?

The classical treatment of statistical mechanics is a branch of physics that uses statistical methods to study the behavior of a large number of particles in a system. It is based on classical mechanics, which describes the motion of particles according to Newton's laws of motion.

## 2. How is the classical treatment of statistical mechanics different from quantum statistical mechanics?

The classical treatment of statistical mechanics deals with macroscopic systems where particles are considered to have definite positions and momenta. In contrast, quantum statistical mechanics takes into account the wave-like nature of particles and their probabilistic behavior.

## 3. What is the role of entropy in the classical treatment of statistical mechanics?

Entropy is a measure of the disorder or randomness in a system. In classical statistical mechanics, entropy is used to calculate the probabilities of different states of a system and to describe the tendency of a system to reach equilibrium.

## 4. How does the classical treatment of statistical mechanics explain thermodynamic properties?

The classical treatment of statistical mechanics explains thermodynamic properties, such as temperature, pressure, and energy, in terms of the average behavior of a large number of particles. It uses statistical methods to relate these macroscopic properties to the microscopic behavior of individual particles.

## 5. What are some real-world applications of the classical treatment of statistical mechanics?

The classical treatment of statistical mechanics is used in various fields, including thermodynamics, chemistry, material science, and engineering. It is used to study the behavior of gases, liquids, and solids, as well as complex systems such as polymers and biological systems. It also has applications in understanding phase transitions, chemical reactions, and transport phenomena.

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