guest post by Mike Stay
The table in John’s post on quantropy shows that energy and action are analogous:
Statics  Dynamics 
statistical mechanics  quantum mechanics 
probabilities  amplitudes 
Boltzmann distribution  Feynman sum over histories 
energy  action 
temperature  Planck’s constant times i 
entropy  quantropy 
free energy  free action 
However, this seems to be part of a bigger picture that includes at least entropy as analogous to both of those, too. I think that just about any quantity defined by an integral over a path would behave similarly.
I see four broad areas to consider, based on a temperature parameter:
 : statics, or “least quantity”
 Real : statistical mechanics
 Imaginary : a thermal ensemble gets replaced by a quantum superposition
 Complex : ensembles of quantum systems, as in nuclear magnetic resonance
I’m not going to get into the last of these in what follows.
1. “Least quantity”
Lagrangian of a classical particle
is kinetic energy, i.e. the “action density” due to motion.
is potential energy, i.e. minus the “action density” due to position.
The action is then:
where is the particle’s mass. We get the principle of least action by setting
“Static” systems related by a Wick rotation

Substitute q(s = iz) for q(t) to get a “springy” static system.
In John’s homework problem A Spring in Imaginary Time, he guided students through a Wickrotationlike process that transforms the Lagrangian above into the Hamiltonian of a springy system. (I say “springy” because it’s not exactly the Hamiltonian for a hanging spring: here each infinitesimal piece of the spring is at a fixed horizontal position and is free to move only vertically.)
is the potential energy density due to stretching.
is the potential energy density due to position.
We then have
or letting ,
where is the potential energy of the spring. We get the principle of least energy by setting

Substitute q(β = iz) for q(t) to get a thermometer
system.We can repeat the process above, but use inverse temperature, or “coolness”, instead of time. Note that this is still a statics problem at heart! We’ll introduce another temperature below when we allow for multiple possible q‘s.
is the potential energy due to rate of change of with respect to . (This has to do with the thermal expansion coefficient: if we fix length of the thermometer and then cool it, we get “stretching” potential energy.)
is any extra potential energy due to
or letting ,
where is the entropy lost as the thermometer is cooled. We get the principle of “least entropy lost” by setting

Substitute q(T₁ = iz) for q(t).
We can repeat the process above, but use temperature instead of time. We get a system whose heat capacity is governed by a function and its derivative. We’re trying to find the best function , the most efficient way to raise the temperature of the system.
is the heat capacity (= entropy) proportional to .
is the heat capacity due to
or letting
where is the energy required to raise the
temperature. We again get the principle of least energy by setting
2. Statistical mechanics
Here we allow lots of possible q‘s, then maximize entropy subject to constraints using the Lagrange multiplier trick.
Statistical mechanics of a particle
For the statistical mechanics of a particle, we choose a real measure on the set of paths. For simplicity, we assume the set is finite.
Normalize so
Define entropy to be
Our problem is to choose to minimize the “free action” , or, what’s equivalent, to maximize subject to a constraint on
To make units match, λ must have units of action, so it’s some multiple of ℏ. Replace λ by ℏλ so the free action is
The distribution that minimizes the free action is the Gibbs distribution where is the usual partition function.
However, there are other observables of a path, like the position at the halfway point; given another constraint on the average value of over all paths, we get a distribution like
The conjugate variable to that position is a momentum: in order to get from the starting point to the given point in the allotted time, the particle has to have the corresponding momentum.
Other examples from Wick rotation

Introduce a temperature T [Kelvins] that perturbs the spring.
We minimize the free energy i.e. maximize the entropy subject to a constraint on the expected energy
We get the measure
Other observables about the spring’s path give conjugate variables whose product is energy. Given constraint on the average position of the spring at the halfway point, we get a conjugate force: pulling the spring out of equilibrium requires a force.

Statistical ensemble of thermometers with ensemble temperature T₂ [unitless].
We minimize the “free entropy” , i.e. we maximize the entropy subject to a constraint on the expected entropy lost
We get the measure
Given a constraint on the average position at the halfway point, we get a conjugate inverse length that tells how much entropy is lost when the thermometer shrinks by

Statistical ensemble of functions q with ensemble temperature T₂ [Kelvins].
We minimize the free energy i.e. we maximize the entropy subject to a constraint on the expected energy
We get the measure
Again, a constraint on the position would give a conjugate force. It’s a little harder to see how here, but given a nonoptimal function we have an extra energy cost due to inefficiency that’s analogous to the stretching potential energy when pulling a spring out of equilibrium.
3. Thermo to quantum via Wick rotation of Lagrange multiplier
We allow a complexvalued measure as John did in the article on quantropy. We pick a logarithm for each and assume they don’t go through zero as we vary them. We also choose an imaginary Lagrange multiplier.
Normalize so
Define quantropy
Find a stationary point of the free action
We get If we get Feynman’s sum over histories. Surely something like the twoslit experiment considers histories with a constraint on position at a particular time, and we get a conjugate momentum?
A Quantum Version of Entropy
Again allow complexvalued However, this time normalize these by setting
Define a quantum version of entropy

Allow quantum superposition of perturbed springs.
Get If we get the evolution of the quantum state under the given Hamiltonian for a time

Allow quantum superpositions of thermometers.
Get If we get something like a sum over histories, but with a different normalization condition that converges because our set of paths is finite.

Allow quantum superposition of systems.
Get If we get the result of “Measure E, then heat the superposition T₁ degrees in a time much less than t seconds, then wait t seconds.” Different functions q in the superposition change the heat capacity differently and thus the systems end up at different energies.
So to sum up, there’s at least a threeway analogy between action, energy, and entropy depending on what you’re integrating over. You get a kind of “statics” if you extremize the integral by varying the path; by allowing multiple paths and constraints on observables, you get conjugate variables and “free” quantities that you want to minimize; and by taking the temperature to be imaginary, you get quantum systems.
In case you’re wondering, this is a better formatted, somewhat edited version of Mike’s comment on my post about quantropy.
This is too advanced for me. Maybe you could do a new version with more explanation and fewer equations? I want to know things like this: what is new here? Why is this exciting? how could this be used? What new insights does this give into everyday things? What aspects of the analogy exist on one side but are unnamed on the other, so are new discoveries about the other side? Basically a Scientific American style article.
Douglas wrote:
I know you were talking to your brother Mike, but you’ve just nudged me into saying a bit myself…
The first thing to note is that Mike is not talking about an analogy between two sides, but at least four sides, and perhaps more. The main four are:
• In classical dynamics, a system like a thrown rock will trace out the path that minimizes action.
• In classical statics, a system like a hanging spring traces out the path that minimizes energy.
• In thermal statics, a system in thermal equilibrium like a box of gas will occupy different states with different probabilities in a way that maximizes entropy.
• In quantum dynamics, a system like a thrown quantum rock will trace out different paths with different amplitudes in a way that makes the first derivative of quantropy zero.
The analogy between the first two was explained (not in Scientific American fashion) in my problem set A Spring in Imaginary Time. The concept of quantropy is described in my blog post Quantropy.
Infinitely more famous than either of these are other aspects of the analogy between thermal statics and quantum dynamics: people use this analogy to do things like calculate the mass of the proton from first principles. If anyone was really going to write a Scientific American article on this material, that would be the place to start. It’s incredibly important, but I’ve never seen anyone try to explain it to ordinary folks.
Mike is trying to push these analogies further than I’d done. So far I only understand certain portions of what he’s written. The portions I understand seem right, but I can’t vouch for the whole thing yet.
Here are some things I don’t understand:
I don’t understand what’s going on here. Mike seems to be treating the length of a thermometer as a function of temperature, or more precisely the reciprocal of temperature β. But I don’t understand the assumptions in this game—like what laws of physics we’re assuming, or what’s a function of what.
Again, I don’t understand what’s going on here. And now I don’t even have a mental image (like a thermometer) to cling to.
Ditto.
Ditto. I’m afraid I need more help to understand these four items.
In each example, I’m just doing a formal replacement and trying to come up with a plausible story as to why such a system might arise. I thought it was OK to do that, since it’s not particularly clear to me how you’d build your “springy” system, either.
Let’s just talk about the second example I gave, the “thermometer system”. There are composite materials like SiC/Al fibers whose thermal expansion coefficient depends on the coolness . (http://www.waset.org/journals/waset/v78/v78179.pdf) A rod made out of this stuff has an equilibrium length that’s also a function of the coolness, given by integrating If we fix the length of the rod and then cool it, it will have some stretching potential energy We can also toss in some other kind of potential energy that depends on just to make the analogy closer. The entropy lost upon cooling it is the integral of the stretching potential energy plus the extra potential energy. Minimizing this entropy while keeping the initial and final equilibrium lengths fixed becomes a materials science question: which composite produces the best ?
Thanks, that was very helpful.
There exist a deep analogy between statistical mechanics and quantum mechanics. In particular, both statistical theories support the existence of complementary quantities and noncommuting operators, where the entropy appears as a counterpart of classical action. In my opinion, this connection strongly suggests the development of extremal principles for nonequilibrium statistical mechanics, where the notion of complementarity would play a relevant role. More details about this subject can be found in my recent paper
http://dx.doi.org/10.1016/j.aop.2012.03.002
Reading this great post, this paper by Vladimir GarciaMorales came to my mind “Quantum Mechanics and the Principle of Least Radix Economy”
https://arxiv.org/abs/1401.0963