#liouville

[dulcamara-words]

Apart from the trivial constant functions then, singularities are a fact of life, and we must learn to live with them.

George Arfken, Mathematical Methods for Physicists

[thetrueoverlordbear]

(Liouville)

[asymptotichigh5]

On dealing with a large number of degrees of freedom (very mathy)

So I’m supposed to be studying for my statistical thermodynamics exam tomorrow, but I can’t be bothered, so I decided to write about it. In part to feel like I’m doing something that will actually help me on tomorrow’s test I guess. 

We know that thermodynamics is all about “equilibrium” and values of quantities at “equilibrium” and how heat can flow from one place to another and so on and so forth. But what is never really defined properly in any thermodynamics class I’ve ever taken is what equilibrium actually is. Furthermore, it is never explored as to how the system reaches equilibrium, or if it even does indeed reach equilibrium. So we shall start this (probably series) of posts by asking us the following questions. 

How can we define equilibrium?

Do all systems naturally evolve towards an equilibrium state as defined by (1)?

How can we describe the time evolution of a system that is not yet in equilibrium?

Without loss of generality (for now at the very least) let us concentrate our efforts into describing a system like for example the ideal gas. Now I know what you’re thinking, obviously you’d pick a gas. 

And yes, to a certain extent it was obvious. You see a gas, in particular an ideal gas, is the easiest system we can consider and for an introduction, it’s the perfect example of exactly what mathematical methods need to be employed in more complex situations.

To examine the behaviour of the gas, let us consider all the information that is technically available to us from a classical mechanics perspective. We could consider for instance, having a super computer and keeping track of the positions \(\vec{q}_i(t)\) and momenta \(\vec{p}_i(t)\) of all \(N \sim 10^{23}\) particles in our typical gas sample. Of course, there is way too much information and this is exactly why this problem is challenging. However, the equations governing the motion of these particles is actually quite simple. 

Due to classical mechanics, we may describe the time evolution of the system by considering a general Hamiltonian of the system given by $$\mathcal{H}= \sum_{n=1}^{N}\left( \frac{\vec{p}_i^2}{2m}+U(\vec{q}_i) \right) + \sum_{i \neq j} V(|\vec{q}_i-\vec{q}_j|)$$ In principle, \(U(\vec{q}_i)\) is free to be any potential you want and \(V(|\vec{q}_i-\vec{q}_j|)\) is the potential due to the interaction of the particles with each other. 

As you can imagine, dealing with this situation would be very difficult, however for now, let’s see for how long we can get away with the full generality of the situation. We conclude also that nature is a hell of a computer.

Of course, because we are treating this classically, the time evolution of this system is simply given by Hamilton’s equations, which are $$\frac{\partial \vec{q}_i}{\partial t} = \frac{\partial \mathcal{H}}{\partial \vec{p}_i} $$ $$ \frac{\partial \vec{p}_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial \vec{q}_i} $$ We define a microstate as being a point \(\mu(t)\) in the \(6N\) dimensional space \(\Gamma = \prod_{n=1}^{N} \{q_i,p_i\}\) whose evolution is governed by the equations above. Notice that as usual, the classical mechanics equations of motion exhibit time-reversal symmetry, so they are invariant under the transformation \(T(\mathbb{p},\mathbb{q}) \to (-\mathbb{p},\mathbb{q})\). That is, if we were to reverse all the momenta of all particles \(\mathbb{p}\) at any point in time, the system would simply come back exactly where it originally came from. This observation of course, is counter intuitive, after all in the case of a gas, once it escapes we know intuitively that the gas does not go back into its original container. And making it go back completely is a very difficult task. So another question we can ask ourselves and that we should ask ourselves in the process of the analysis of this system is: at what point will we lose reversibility?

However, for our purposes, we really couldn’t care less if particle 3 has a certain momentum. In some kind of way, we only care about only what the gas does as a whole. So we care about things like pressure, volume, etc. Defined more precisely, we can say that a macrostate \(M\) of the system is uniquely specified by three so-called state functions such as the energy \(E\), the pressure \(P\), etc. So formally it is a triplet of state functions. 

As you can imagine, given any particular macrostate, it must be that more than one microstate corresponds to it. After all, there so many more microstates than macrostates. The only possibility is thus the previous statement. 

Next, we need to perform somewhat of a weird exercise to come up with a good definition. Imagine that I literally copy paste my system and make \(\mathcal{N}\) copies of it (ain’t nobody got time for that, but suppose you did). We then choose a different representative point \(\mu_n(t)\) in the phase space \(\Gamma\) that corresponds to a microstate that describes indeed our macrostate \(M\). Now consider an infinitesimal volume \(d\Gamma = d^{3N}p_i  d^{3N}q_i\) around the point \(\mu(t)\). If we let \(d\mathcal{N}\) denote the number of representative points within this volume \(d\Gamma\) we can define a probability density function:$$ \rho(\mu) d\Gamma = \lim_{\mathcal{N} \to \infty} \frac{d\mathcal{N}}{\mathcal{N}} $$ So fine, we were meticulously careful, which I usually dislike doing in this blog to arrive at this definition and it’s nice and all but asides just writing definitions and babbling, what have we actually accomplished? Well, at least in theory so far we can take averages of things, which is nice, I guess.

But the thing is, we know really nothing about this mysterious phase space to be able to say anything nice about it so far. So we must dwell into venturing in \(6N\) dimensional space for a while, I guess. If you ever needed some room in your life, you’re about to get it.

I usually don’t make comments about gifs, but concerning the above one, you’re probably shooing away or praying to the lord for the formality to be over. I regret to tell you it’s not, which is nice if you’re a mathematician. But you see, formal systems are not always a bad thing. In fact, in my opinion they’re quite nice. They’re clear and the formal definitions of things doesn’t leave ambiguities. Everything is nice and defined which provided a good mathematical background to deal with a hard problem. That is the main goal of this article.

Next let us try to find out more about this \(\Gamma\)-space we have defined. To do this let us consider a tiny step time \(\delta t\) and imagine we are sitting at point \((\mathbb{p},\mathbb{q})\). Thus after this tiny time step, we will land at the point given by coordinates \((\mathbb{p}’,\mathbb{q}’)\) given by: $$ q_{\alpha}’ = q_{\alpha} + \frac{\partial q_{\alpha}}{\partial t}\delta t + O(\delta t^2) $$ $$ p_{\alpha}’ = p_{\alpha} + \frac{\partial p_{\alpha}}{\partial t}\delta t + O(\delta t^2) $$ But then of course, it follows naturally that because our time step is very small order of \(\delta t^2\) is negligible so we can throw it away (this would be rigorous in non-standard analysis, which I will probably talk about in a future article). Furthermore, because we are looking at a hypercube, the distortions of the sides of the cube will depend on the coordinates of the points of the side of the cube, so we can transform our expressions into: $$ q_{\alpha}’ = q_{\alpha} + \frac{\partial^2 q_{\alpha}}{\partial q_{\alpha} \partial t}dq_{\alpha}\delta t + O(\delta t^2) $$ $$ p_{\alpha}’ = p_{\alpha} + \frac{\partial^2 p_{\alpha}}{\partial p_{\alpha} \partial t}dp_{\alpha}\delta t + O(\delta t^2) $$ Now notice that the new volume of the distorted infinitesimal hypercube is simply the infinitesimals of \(\mathbb{p}’\) and \(\mathbb{q}’\) multiplied so we have $$ dq_{\alpha}’ dp_{\alpha}’ = dq_{\alpha} dp_{\alpha} \left[ 1 + \left( \frac{\partial^2 q_{\alpha}}{\partial q_{\alpha} \partial t}dq_{\alpha} +\frac{\partial^2 p_{\alpha}}{\partial p_{\alpha} \partial t}dp_{\alpha} \right) \delta t + O(\delta t^2) \right] $$ Now we make use of Hamilton’s equations to get rid of the time derivative and swap it for a momentum or position one respectively and we simply obtain: $$dq_{\alpha}’ dp_{\alpha}’ = dq_{\alpha} dp_{\alpha} \left[ 1 + \left( \frac{\partial^2 q_{\alpha}}{\partial q_{\alpha} \partial p_{\alpha}}dq_{\alpha} - \frac{\partial^2 p_{\alpha}}{\partial p_{\alpha} \partial q_{\alpha}}dp_{\alpha} \right) \delta t + O(\delta t^2) \right] =dq_{\alpha} dp_{\alpha} $$

Thus we have that up to this order of \(\delta t\), our phase space remains the same in volume. In some kind of way, we can thus think of it as an incompressible volume. Now notice that this theorem we just proved holds in all generality for any Hamiltonian set of equations. In no way did we invoke the ideal gas specifically. 

But this incompressibility condition now gives us the tools to say something more about our probability density function that we previously defined. However, I think I have tortured you with enough math in one post. The next one will probably be a bit lighter, but nonetheless, the rigour is important, because what we are doing is laying the foundations for our understanding of thermodynamics, thus, we have no choice but to have this level of rigour and consistency.

[panjection]

What's your favourite mathematical theorem you've ever seen/proved?

Ahhhhh!  Hello, greyface!  Thank you!

Okay, focus... god, there are so many, y'know?

My absolute favorite theorem is Liouville's theorem, which states that every bounded entire function must be constant.  That's it.  Like, the whole theorem is just, "Every bounded entire function must be constant."  Just BAM one sentence.  There's something super happy about that for me.  Which is kinda funny because complex analysis is my weakest subject.

[shitphysicspeoplesay]

"it's a theorem you discussed in your classical mechanics course. or you should have. you probably didn't discuss it at all actually."

[spoco-blog]

Harmonic Maps

A geodesic is roughly defined as a path in (not necessarily three-dimensional or Euclidean) space such that every sufficiently small segment of it minimizes the distance between its endpoints. As most know, in Euclidean space all geodesics are straight lines, but this truism does not generalize to curved spaces at all; straight lines don't even typically exist. Note that locality is featured but not globality: a geodesic between A and B is not necessarily the shortest between A and B, it's only the shortest path in all infinitesimally small neighborhoods of each point in the path. 

The Euler-Lagrange equations from variational calculus and basic properties of metric tensors from differential geometry provide the groundwork necessary to derive the geodesic equation - a nonlinear system of partial differential equations which are satisfied by a path if and only if the path is geodesic. In practice, it is impossible (for all we know) to solve it exactly (meaning with an explicit symbolic formula), so we look for theoretical and numerical methods of approximating solutions.

Geodesics are one-dimensional objects imprinted on higher-dimensional objects called manifolds. The reverse is possible; one can "imprint" higher-dimensional spaces onto a one-dimensional continuum. These are called scalar functions.

Scalar functions take points in space and associate to them a single quantity, e.g. altitudes associated to coordinates in the xy plane which map out topography and landforms. Scalar functions have something called a Dirichlet energy associated to them, which is roughly a measure of how "variable" it is. The Euler-Lagrange equations can again be used to minimize Dirichlet energy; the energy E[f] equals zero when a scalar function f(x) satisfies Laplace's equation, which states that the flux divergence vanishes everywhere. Such functions are called harmonic functions - they share the description "harmonic" with musical notes because the latter exemplify harmonic functions on a 1-D space (associating to positions in space between source and receiver varying amounts of air pressure that make up soundwaves). Audio signals are linear combinations of sinusoids just like harmonic functions.

The figure above depicts a harmonic function defined on an annular region (two-dimensional doughnut) with special boundary conditions.

Axler, Bourdon, and Ramey have an excellent book on Harmonic Function Theory freely available in pdf format (link). It is actually one of my favorite digital textbooks so I highly recommend it the curious (and capable). Because of it I found out some gorgeous results on harmonic functions: automatic regularity, mean value property, minimum / maximum principle, Liouville's theorem, and how Poisson kernels resolve Dirichlet's problem on the ball. More generally, harmonic functions also lie in broader contexts of harmonic analysis (looking at signals as compositions of basic waves) and potential theory (physics-related issues, conformality).

So now we have geodesics which are 1-D lines mapped onto n-dimensional spaces, and harmonic functions which are n-dimensional spaces mapped onto 1-D continuums. It turns out these are both special cases of a more general object: harmonic maps. Again, roughly speaking - HMs map points on one (Riemannian) manifold to those on another, both of arbitrary dimension, in such a way that the HM optimizes the Dirichlet energy of the map compared to all maps which differ only infinitesimally from the HM. Wikipedia has a nifty way of visualizing this state of affairs. Call the harmonic map in question φ, mapping the manifold M to the manifold N.

We may imagine N made of marble and M made of rubber. A map from the latter to the former is considered a way to "apply" M to N. The Dirichlet energy E[φ] represents the total amount of elastic potential energy resulting from the tension in the rubber. The generalized analogue of Laplace's equation here is one which describes the tension field once again vanishing, τ(φ)=0. The harmonic map, then, is the application of M onto N which realizes an "energy-minimizing configuration".

A highlight of key issues on this subject is "Harmonic Maps" by Hélein and Wood (link). Way above my grade level and I sorely wish I could follow along beyond the first few pages. Someday.

[isomorphismes]

e, i, pi

Here's a mysterious fact.

What's up with that?

The answer is forthcoming, but first I'll remind you what all the letters mean.

e e e e e e

e = 1 + 1/1 + 1/12 + 1/123 + 1/1234 + 1/12345 + 1/123456 + 1/1234567 + 1/12345678 + ...

I'm writing 1/234 to mean "One over 2×3×4." With the (2×3×4) in parentheses. Get me?

Why is e useful, though?  Because the derivative of e^x is itself.  D[e^x] = e^x.  In other words, the function e^x looks like 1 to the operator D (derivative operation).  Because of that fact, answers to the differential equations that describe the interesting and complicated world we live in, are stated easiest in terms of e^x.  (More specifically, functions like e^x, e^2x, e^ix form a linear basis for the solution space of ODE's.)

π π π π π

2π is the length around a circle.  (Assuming the circle has a radius of 1 in the chosen units.)

i i i i i i i i i

i = √-1.  What does that mean—half-negative?  Answering this question provoked a flourishing of mathematics in the 19th and 20th centuries.  √-1 is used in electrical engineering, quantum mechanics, solving equations, understanding fluid flows, non-Euclidean geometry, and according to some (Roger Penrose & friends), more and more areas of physics will ultimately be explained with complex numbers.

For now, think about this.  When the power company sends electricity to you, it is traveling in the direction "+1".  (With AC I guess it switches between +1 and −1.)  If it traveled in the direction "i", the wire would heat up but no useful energy would be transmitted.

the answer

In fact, e^{ i × any number of ° degrees} behaves like a unit circle -- a circle with radius one -- in the complex plane.

So e^{ i × 90° } = i, e^{ i × 270° } = −i, and e^{ i × 360° } = 1.  In terms of distance around the circle 2π, that's a quarter-turn ↶, a three-quarters turn ⟲, and a full rotation ⥀.  Or, in π terms:

Similarly, going 4π, 6π, 8π, or 173622π times around the circle leaves you just where you started -- whether that was from a quarter-turn anticlockwise ⤿ or at the three o'clock starting point. −1 is 180 degrees ↶ or π arc-length away from the starting point of +1 so voilà,

.

The Punchline

That's all fine, for parlor tricks.  But there's more, much more.  Seeing the gamut of numbers as concentric circles in the complex plane allows you to solve every equation, ever*.  I'll build this up for a paragraph or two.

First, consider that there are two solutions to x²=1, which are ±1.  Meaning, x=1 satisfies the condition and x=−1 satisfies the condition, and those are the only two values of x that satisfy the condition.

Second, there are four solutions to x⁴=1:  1, −1, i, and −i.  Also known as e^{ i × 90°, 180°, 270°, and 360° ⟲}.  And what would 7, −7, 7i, and −7i solve?  x⁴ = 2401.  (Two four oh one is 7⁴.)

Following the same pattern, there are five solutions to x⁵=1 (intervals of 72°), thirteen solutions to x¹³=1 (intervals of 2π/13), and thirty-six solutions to x³⁶=1 (intervals of 10°).  And again, if you doubled the circle's radius, the solutions would be multiplied by 2⁵, 2¹³, or 2³⁶.

So the answer to any problem of powers is shaped like a ship's steering wheel:  evenly spaced points on a circle, possibly a slightly wider circle if the question is x^p = 999.

The answer to polynomial problems is then built up from these sorts of objects.  (But just imagine laying two different-radius wheels on a rectangular grid, and being asked to compute the (x,y) coordinates of sums of pairs of handles.  ouch! gimme a computer)  Anyway, it's nice to know that the answer exists, and it's good to form your imagination around this kind of shape, in case you ever want to explore something practical that involves this lay of the land.

* That's an exaggeration, but at least you can solve every equation you've ever seen.