Extremal Principles in Classical, Statistical and Quantum Mechanics

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:

1. : statics, or “least quantity”
2. Real : statistical mechanics
3. Imaginary : a thermal ensemble gets replaced by a quantum superposition
4. 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

1. 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 Wick-rotation-like 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

2. 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

3. 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

1. 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.

   

2. 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

   

3. 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 non-optimal 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 complex-valued 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 two-slit 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 complex-valued However, this time normalize these by setting

Define a quantum version of entropy

1. Allow quantum superposition of perturbed springs.

   Get If we get the evolution of the quantum state under the given Hamiltonian for a time

2. 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.

3. 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 three-way 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.