There are very few things I find unsatisfactory in L.C. Evans’ wonderful textbook on Partial Differential Equations; one of them is the illustration (on p.53 of the second edition) of the “heat ball”.

The heat ball is a region with respect to which an analogue of the mean value property of solutions to Laplace’s equation can be expressed, now for solutions of the heat equation. In the case of the Laplace’s equation, the regions are round balls. In the case of the heat equation, the regions are somewhat more complicated. They are defined by the expression

where is the fundamental solution of the heat equation

In the expressions above, the constant is the number of spatial dimensions; is the analogue of the radius of the ball, and in , the point is the center. Below is a better visualization of the heat balls: the curves shown are the boundaries in dimension , for radii between 0.75 and 4 in steps of 0.25 (in particular all the red curves have integer radii). In higher dimensions the shape is generally the same, though they appear more “squashed” in the direction.

1-dimensional heat balls centered at (0,5) for various radii. (Made using Desmos)

Joseph O’Rourke’s question at MathOverflow touched on an interesting characterization of geodesics in pseudo-Riemannian geometry, which was apparently originally due to Einstein, Infeld, and Hoffmann in their analysis of the geodesic hypothesis in general relativity. (One of my two undergraduate junior theses is on this topic, but I certainly did not appreciate this result as much when I was younger.) Sternberg’s book has a very good presentation on the theorem, but I want to try to give a slightly different interpretation in this post.

Geodesics and variation
One of the classical formulation of the criterion for a curve to be geodesic is that it is a stationary point of the length functional. Let be a Riemannian manifold, and let $latex: \gamma:[0,1]\to M$ be a mapping. Define the length functional to be

A geodesic then is a curve that is a critical point of under perturbations that fix the endpoints $\latex \gamma(0)$ and .

One minor annoyance about the length functional is that it is invariant under reparametrization of , and so it does not admit unique solutions. One way to work around this is to instead consider the energy functional (which also has the advantage of also being easily generalizable to pseudo-Riemannian manifolds)

It turns out that critical points of the energy functional are always critical points of the length functional. Furthermore, the energy functional has some added convexity: a curve is a critical point of the energy functional if it is a geodesic and that it has constant speed (in the sense that is independent of the parameter ).

The standard way to analyze the variation of is by first fixing a coordinate system . Writing the infinitesimal perturbation as , we can compute the first variation of :

Integrating the second term by parts we recover the familiar geodesic equation in local coordinates.

There is a second way to analyze the variation. Using the diffeomorphism invariance, we can imagine instead of varying while fixing the manifold, we can deform the manifold while fixing the curve . From the point of view of the energy functional the two should be indistinguishable. Consider the variation , which can be regarded as a vector field along which vanishes at the two end points. Let be a vector field on that extends . Then the infinitesimal variation of moving the curve in the direction should be reproducible by flowing the manifold by and pulling back the metric. To be more precise, let be the one parameter family of diffeomorphisms generated by the vector field , the first variation can be analogously represented as

By the definition of the Lie derivative we get the following characterizing condition for a geodesic:

Theorem
A curve is an affinely parametrized geodesic if and only if for every vector field vanishing near and , the integral

Noticing that , where is the Levi-Civita connection, we have that the above integral condition is equivalent to requiring

Using the boundary conditions and integrating by parts we see this also gives us, without passing through the local coordinate formulation, the geodesic equation

The Einstein-Infeld-Hoffmann theorem
The EIH theorem reads:

Theorem (EIH)
A curve is geodesic if and only if there exists a non-vanishing contravariant symmetric two tensor along such that for every vector field vanishing near and , the integral

(where is the induced length measure on ).

The EIH theorem follows immediately from the discussion in the previous section and the following lemma.

Lemma
A contravariant symmetric two tensor that satisfies the assumptions in the previous theorem must be proportional to .

Proof: Choose an orthonormal frame along for such that is tangent to . Write . Suppose . Then there exists a vector field such that and the symmetric part of is equal to . (We can construct by choosing a local coordinate system in a tubular neighborhood of such that . Then can be prescribed by its first order Taylor expansion in the normal direction to .) Let be a non-negative cut-off function and setting we note that since vanishes along . Therefore we have that the desired integral condition cannot hold. q.e.d.

References:

Shlomo Sternberg, Curvature in Mathematics and Physics

Einstein, Infeld, Hoffmann, “Gravitational Equations and the Problem of Motion”

but with all the terms, whose decimal expansion includes the digit ‘9’, removed, in fact converges to some number below 80. The original proof is given in the Wikipedia article linked above, so I will not repeat it. But to make it easier to see the idea: let us first think about the case where the number is expressed in base 2. In base 2, all the positive integers has the leading binary bit being 1 (since it cannot be zero). Therefore there are no binary positive numbers without the bit ‘1’ in its expansion. So the corresponding series converges trivially to zero. How about the case of the bit ‘0’? The only binary numbers without any ‘0’ bits are

.

So the corresponding series actually becomes

So somewhere from the heavily divergent harmonic series, we pick up a rapidly converging geometric series. So what’s at work here? Among all the n-bit binary numbers, exactly 1 has all bits not being 0. So the density of these kinds of numbers decays rather quickly: in base 2, there are numbers that are exactly n-bit long. So if a number has a binary representation that is exactly n bits long (which means that ), the chances that it is one of the special type of numbers is . This probability we can treat then as a density: replacing the discrete sum by the integral (calculus students may recognize this as the germ of the “integral test”) and replacing the by the density , we get the estimate

.

Doing the same thing with the original Kempner series gives that the chances a n-digit number does not contain the digit nine to be

The length of the decimal expansion of a natural number is basically . So the density we are interested in becomes

From this we can do an integral estimate

The integral can be computed using that

to get

Notice that this estimate is much closer to the currently known value of roughly 22.92 than to the original upper bound of 80 computed by Kempner.

Kempner’s estimate is a heavy overestimate because he performed a summation replacing every n-digit long number that does not contain the digit 9 by ; this number can be many times (up to 9) times smaller than the original number. Our estimate is low because among the n-digit long numbers, the numbers that do not contain the digit 9 are not evenly distributed: they tend to crowd in the front rather than in the back (in fact, we do not allow them to crowd in the back because none of the numbers that start with the digit 9 is admissible). So if in the original question we had asked for numbers that do not contain the digit 1, then our computation will give an overestimate instead since these numbers tend to crowd to the back.

I use JabRef as my reference manager. In this post, however, I will discuss how we can abuse it to do some other things.

The problem

Let’s start with a concrete example: I keep a “lab notebook”. It is where I document all my miscellaneous thoughts and computations that come up during my research. Some of those are immediately useful and are collected into papers for publication. Some of those are not, and I prefer to keep them for future reference. These computations range over many different subjects. Now and then, I want to share with a collaborator or a student some subset of these notes. So I want a way to quickly search (by keywords/abstract) for relevant notes, and that compile them into one large LaTeX document.

Another concrete example: I am starting to collect a bunch of examples and exercises in analysis for use in my various classes. Again, I want to have them organized for easy search and retrieval, especially to make into exercise sheets.

The JabRef solution

The “correct” way to do this is probably with a database (or a document store), with each document tagged with a list of keywords. But that requires a bit more programming than I want to worry about at the moment.

JabRef, as it turns out, is sort of a metadata database: by defining a customized entry type you can use the BibTeX syntax as a proxy for JSON-style data. So for my lab notebook example, I define a custom type lnbentry in JabRef with

Required fields: year, month, day, title, file

Optional fields: keywords, abstract

I store each lab notebook entry as an individual TeX file, whose file system address is stored in the file field. The remaining metadata fields’ contents are self-evident.

(Technical note: in my case I actually store the metadata in the TeX file and have a script to parse the TeX files and update the bib database accordingly.)

For generating output, we can use JabRef’s convenient export filter support. In the simplest case we can create a custom export layout with the main layout file containing the single line

\\input{\file}

with appropriate begin and end incantations to make the output a fully-formed TeX file. Then one can simply select the entries to be exported, click on “Export”, and generate the appropriate TeX file on the fly.

(Technical note: JabRef can also be run without a GUI. So one can use this to perform searches through the database on the command line.)

One of my current research interests is in the geometry of constant-mean-curvature time-like submanifolds of Minkowski space. A special case of this is the study of cosmic strings under the Nambu-Goto action. Mathematically the classical (as opposed to quantum) behavior of such strings are quite well understood, by combining the works of theoretical physicists since the 80s (especially those of Jens Hoppe and collaborators) together with recent mathematical developments (such as the work of Nguyen and Tian and that of Jerrard, Novaga, and Orlandi). To get a better handle on what is going on with these results, and especially to provide some visual aids, I’ve tried to produce some simulations of the dynamics of closed cosmic strings. (This was also an opportunity for me to practice code-writing in Julia and learn a bit about the best practices for performance and optimization in that language.)

The code

After some false starts, here are some reasonably stable code.

function MC_corr3!(position::Array{Float64,2}, prev_vel::Array{Float64,2}, next_vel::Array{Float64,2}, result::Array{Float64,2}, dt::Float64)
# We will assume that data is stored in the format point(coord,number), so as a 3x1000 array or something.
num_points = size(position,2)
num_dims = size(position,1)
curr_vel = zeros(num_dims)
curr_vs = zeros(num_dims)
curr_ps = zeros(num_dims)
curr_pss = zeros(num_dims)
pred_vel = zeros(num_dims)
agreement = true
for col = 1:num_points #Outer loop is column
if col == 1
prev_col = num_points
next_col = 2
elseif col == num_points
prev_col = num_points - 1
next_col = 1
else
prev_col = col -1
next_col = col + 1
end
for row = 1:num_dims
curr_vel[row] = (next_vel[row,col] + prev_vel[row,col])/2
curr_vs[row] = (next_vel[row,next_col] + prev_vel[row,next_col] - next_vel[row,prev_col] - prev_vel[row,prev_col])/4
curr_ps[row] = (position[row,next_col] - position[row,prev_col])/2
curr_pss[row] = position[row,next_col] + position[row,prev_col] - 2*position[row,col]
end
beta = (1 + dot(curr_vel,curr_vel))^(1/2)
sigma = dot(curr_ps,curr_ps)
psvs = dot(curr_ps,curr_vs)
bvvs = dot(curr_vs,curr_vel) / (beta^2)
pssps = dot(curr_pss,curr_ps)
for row in 1:num_dims
result[row,col] = curr_pss[row] / (sigma * beta) - curr_ps[row] * pssps / (sigma^2 * beta) - curr_vel[row] * psvs / (sigma * beta) - curr_ps[row] * bvvs / (sigma * beta)
pred_vel[row] = prev_vel[row,col] + dt * result[row,col]
end
agreement = agreement && isapprox(next_vel[:,col], pred_vel, rtol=sqrt(eps(Float64)))
end
return agreement
end
function find_next_vel!(position::Array{Float64,2}, prev_vel::Array{Float64,2}, next_vel::Array{Float64,2}, dt::Float64; max_tries::Int64=50)
tries = 1
result = zeros(next_vel)
agreement = MC_corr3!(position,prev_vel,next_vel,result,dt)
for j in 1:size(next_vel,2), i in 1:size(next_vel,1)
next_vel[i,j] = prev_vel[i,j] + result[i,j]*dt
end
while !agreement && tries < max_tries
agreement = MC_corr3!(position,prev_vel,next_vel,result,dt)
for j in 1:size(next_vel,2), i in 1:size(next_vel,1)
next_vel[i,j] = prev_vel[i,j] + result[i,j]*dt
end
tries +=1
end
return tries, agreement
end

This first file does the heavy lifting of solving the evolution equation. The scheme is a semi-implicit finite difference scheme. The function MC_Corr3 takes as input the current position, the previous velocity, and the next velocity, and computes the correct current acceleration. The function find_next_vel iterates MC_Corr3 until the computed acceleration agrees (up to numerical errors) with the input previous and next velocities.

Or, in notations: MC_Corr3: ( x[t], v[t-1], v[t+1] ) --> Delta-v[t]
and find_next_vel iterates MC_Corr3 until Delta-v[t] == (v[t+1] - v[t-1]) / 2

The code in this file is also where the performance matters the most, and I spent quite some time experimenting with different algorithms to find one with most reasonable speed.

function make_ellipse(a::Float64,b::Float64, n::Int64, extra_dims::Int64=1) # a,b are relative lengths of x and y axes
s = linspace(0,2π * (n-1)/n, n)
if extra_dims == 0
return vcat(transpose(a*cos(s)), transpose(b*sin(s)))
elseif extra_dims > 0
return vcat(transpose(a*cos(s)), transpose(b*sin(s)), zeros(extra_dims,n))
else
error("extra_dims must be non-negative")
end
end
function perturb_data!(data::Array{Float64,2}, coeff::Vector{Float64}, num_modes::Int64)
# num_modes is the number of modes
# coeff are the relative sizes of the perturbations
numpts = size(data,2)
for j in 2:num_modes
rcoeff = rand(length(coeff),2)
for pt in 1:numpts
theta = 2j * π * pt / numpts
for d in 1:length(coeff)
data[d,pt] += ( (rcoeff[d,1] - 0.5) * cos(theta) + (rcoeff[d,2] - 0.5) * sin(theta)) * coeff[d] / j^2
end
end
end
nothing
end

This file just sets up the initial data. Note that in principle the number of ambient spatial dimensions is arbitrary.

using Plots
pyplot(size=(1920,1080), reuse=true)
function plot_data2D(filename_prefix::ASCIIString, filename_offset::Int64, titlestring::ASCIIString, data::Array{Float64,2}, additional_data...)
x_max = 1.5
y_max = 1.5
plot(transpose(data)[:,1], transpose(data)[:,2] , xlims=(-x_max,x_max), ylims=(-y_max,y_max), title=titlestring)
if length(additional_data) > 0
for i in 1:length(additional_data)
plot!(transpose(additional_data[i][1,:]), transpose(additional_data[i][2,:]))
end
end
png(filename_prefix*dec(filename_offset,5)*".png")
nothing
end
function plot_data3D(filename_prefix::ASCIIString, filename_offset::Int64, titlestring::ASCIIString, data::Array{Float64,2}, additional_data...)
x_max = 1.5
y_max = 1.5
z_max = 0.9
tdata = transpose(data)
plot(tdata[:,1], tdata[:,2],tdata[:,3], xlims=(-x_max,x_max), ylims=(-y_max,y_max),zlims=(-z_max,z_max), title=titlestring)
if length(additional_data) > 0
for i in 1:length(additional_data)
tdata = transpose(additional_data[i])
plot!(tdata[:,1], tdata[:,2], tdata[:,3])
end
end
png(filename_prefix*dec(filename_offset,5)*".png")
nothing
end

This file provides some wrapper commands for generating the plots.

include("InitialData3.jl")
include("MeanCurvature3.jl")
include("GraphCode3.jl")
num_pts = 3000
default_timestep = 0.01 / num_pts
max_time = 3
plot_every_ts = 1500
my_data = make_ellipse(1.0,1.0,num_pts,0)
perturb_data!(my_data, [1.0,1.0], 15)
this_vel = zeros(my_data)
next_vel = zeros(my_data)
for t = 0:floor(Int64,max_time / default_timestep)
num_tries, agreement = find_next_vel!(my_data, this_vel,next_vel,default_timestep)
if !agreement
warn("Time $(t*default_timestep): failed to converge when finding next_vel.")
warn("Dumping information:")
max_beta = 1.0
max_col = 1
for col in 1:size(my_data,2)
beta = (1 + dot(next_vel[:,col], next_vel[:,col]))^(1/2)
if beta > max_beta
max_beta = beta
max_col = col
end
end
warn(" Beta attains maximum at position $max_col")
warn(" Beta = $max_beta")
warn(" Position = ", my_data[:,max_col])
prevcol = max_col - 1
nextcol = max_col + 1
if max_col == 1
prevcol = size(my_data,2)
elseif max_col == size(my_data,2)
nextcol = 1
end
warn(" Deltas")
warn(" Left: ", my_data[:,max_col] - my_data[:,prevcol])
warn(" Right: ", my_data[:,nextcol] - my_data[:,max_col])
warn(" Previous velocity: ", this_vel[:,max_col])
warn(" Putative next velocity: ", next_vel[:,max_col])
warn("Quitting...")
break
end
for col in 1:size(my_data,2)
beta = (1 + dot(next_vel[:,col], next_vel[:,col]))^(1/2)
for row in 1:size(my_data,1)
my_data[row,col] += next_vel[row,col] * default_timestep / beta
this_vel[row,col] = next_vel[row,col]
end
if beta > 1e7
warn("time: ", t * default_timestep)
warn("Almost null... beta = ", beta)
warn("current position = ", my_data[:,col])
warn("current Deltas")
prevcol = col - 1
nextcol = col + 1
if col == 1
prevcol = size(my_data,2)
elseif col == size(my_data,2)
nextcol = 1
end
warn(" Left: ", my_data[:,col] - my_data[:,prevcol])
warn(" Right: ", my_data[:,nextcol] - my_data[:,col])
end
end
if t % plot_every_ts ==0
plot_data2D("3Dtest", div(t,plot_every_ts), @sprintf("elapsed: %0.4f",t*default_timestep), my_data, make_ellipse(cos(t*default_timestep), cos(t*default_timestep),100,0))
info("Frame $(t/plot_every_ts): used $num_tries tries.")
end
end

And finally the main file. Mostly it just ties the other files together to produce the plots using the simulation code; there are some diagnostics included for me to keep an eye on the output.

The results

First thing to do is to run a sanity check against explicit solutions. In rotational symmetry, the solution to the cosmic string equations can be found analytically. As you can see below the simulation closely replicates the explicit solution in this case.

The video ends when the simulation stopped. The simulation stopped because a singularity has formed; in this video the singularity can be seen as the collapse of the string to a single point.

Next we can play around with a more complicated initial configuration.

In this video the blue curve is the closed cosmic string, which starts out as a random perturbation of the circle with zero initial speed. The string contracts with acceleration determined by the Nambu-Goto action. The simulation ends when a singularity has formed. It is perhaps a bit hard to see directly where the singularity happened. The diagnostic messages, however, help in this regard. From it we know that the onset of singularity can be seen in the final frame:

The highlighted region is getting quite pointy. In fact, that is accompanied with the “corner” picking up infinite acceleration (in other words, experiencing an infinite force). The mathematical singularity corresponds to something unreasonable happening in the physics.

To make it easier to see the “speed” at which the curve is moving, the following videos show the string along with its “trail”. This first one again shows how a singularity can happen as the curve gets gradually more bent, eventually forming a corner.

This next one does a good job emphasizing the “wave” nature of the motion.

The closed cosmic strings behave like a elastic band. The string, overall, wants to contract to a point. Small undulations along the string however are propagated like traveling waves. Both of these tendencies can be seen quite clearly in the above video. That the numerical solver can solve “past” the singular point is a happy accident; while theoretically the solutions can in fact be analytically continued past the singular points, the renormalization process involved in this continuation is numerically unstable and we shouldn’t be able to see it on the computer most of the time.

The next video also emphasizes the wave nature of the motion. In addition to the traveling waves, pay attention to the bottom left of the video. Initially the string is almost straight there. This total lack of curvature is a stationary configuration for the string, and so initially there is absolutely no acceleration of that segment of the string. The curvature from the left and right of that segment slowly intrudes on the quiescent piece until the whole thing starts moving.

The last video for this post is a simulation when the ambient space is 3 dimensional. The motion of the string, as you can see, becomes somewhat more complicated. When the ambient space is 2 dimensional a point either accelerates or decelerates based on the local (signed) curvature of the string. But when the ambient space is 3 dimensional, the curvature is now a vector and this additional degree of freedom introduces complications into the behavior. For example, when the ambient space is 2 dimensional it is known that all closed cosmic strings become singular in finite time. But in 3 dimensions there are many closed cosmic strings that vibrate in place without every becoming singular. The video below is one that does however become singular. In addition to a fading trail to help visualize the speed of the curve, this plot also includes the shadows: projections of the curve onto the three coordinate planes.

Recently I have been playing around with the Julia programming language as a way to run some cheap simulations for some geometric analysis stuff that I am working on. So far the experience has been awesome.

A few random things …

Juno

Julia has a decent IDE in JunoLab, which is built on top of Atom. In terms of functionality it captures most of the sort of things I used to use with Spyder for python, so is very convenient.

Jupyter

Julia interfaces with Jupyter notebooks through the IJulia kernel. I am a fan of Jupyter (I will be using it with the MATLAB kernel for a class I am teaching this fall).

Plots.jl

For plotting, right now one of the most convenience ways is through Plots.jl, which is a plotting front-end the bridges between your code and various different backends that can be almost swapped in and out on the fly. The actual plotting is powered by things like matplotlib or plotlyJS, but for the most part you can ignore the backend. This drastically simplifies the production of visualizations. (At least compared to what I remembered for my previous simulations in python.)

Automatic Differentiation

I just learned very recently about automatic differentiation. At a cost in running time for my scripts, it can very much simplify the coding of the scripts. For example, we can have a black-box root finder using Newton iteration that does not require pre-computing the Jacobian by hand:

module NewtonIteration
using ForwardDiff
export RootFind
function RootFind(f, init_guess::Vector, accuracy::Float64, cache::ForwardDiffCache; method="Newton", max_iters=100, chunk_size=0)
### Takes input function f(x::Vector) → y::Vector of the same dimension and an initial guess init_guess. Apply Newton iteration to find solution of f(x) = 0. Stop when accuracy is better than prescribed, or when max_iters is reached, at which point a warning is raised.
### Setting chunk_size=0 deactivates chunking. But for large dimensional functions, chunk_size=5 or 10 improves performance drastically. Note that chunk_size must evenly divide the dimension of the input vector.
### Available methods are Newton or Chord
# First check if we are already within the accuracy bounds
error_term = f(init_guess)
if norm(error_term) < accuracy
info("Initial guess accurate.")
return init_guess
end
# Different solution methods
i = 1
current_guess = init_guess
if method=="Chord"
df = jacobian(f,current_guess,chunk_size=chunk_size)
while norm(error_term) >= accuracy && i <= max_iters
current_guess -= df \ error_term
error_term = f(current_guess)
i += 1
end
elseif method=="Newton"
jake = jacobian(f, ForwardDiff.AllResults, chunk_size=chunk_size, cache=cache)
df, lower_order = jake(init_guess)
while norm(value(lower_order)) >= accuracy && i <= max_iters
current_guess -= df \ value(lower_order)
df, lower_order = jake(current_guess)
i += 1
end
error_term = value(lower_order)
else
warn("Unknown method: ", method, ", returning initial guess.")
return init_guess
end
# Check if converged
if norm(error_term) >= accuracy
warn("Did not converge, check initial guess or try increasing max_iters (currently: ", max_iters, ").")
end
info("Used ", i, " iterations; remaining error=", norm(error_term))
return current_guess
end
end

This can then be wrapped in finite difference code for solving nonlinear PDEs!

(The following is somewhat rough and may have typos.)

Let us begin by setting the notations and recalling what happens without the Stieltjes part.

Defn (Partition)
Let be a closed interval. A partition is a finite collection of closed subintervals such that

is finite;

covers , i.e. ;

is pairwise almost disjoint, i.e. for distinct elements of , their intersection contains at most one point.

We write for the set of all partitions of .

Defn (Refinement)
Fix a closed interval, and two partitions. We say that refines or that if for every there exists such that .

Defn (Selection)
Given a closed interval and a partition, a selection is a mapping that satisfies .

Defn (Size)
Given a closed interval and a partition, the size of is defined as , where is the length of the closed interval .

Remark In the above we have defined two different preorders on the set of all partitions. One is induced by the size: we say that if . The other is given by the refinement . Note that neither are partial orders. (But that the preorder given by refinement can be made into a partial order if we disallow zero-length degenerate closed intervals.) Note also that if we must have .

Now we can define the notions of integrability.

Defn (Integrability)
Let be a closed, bounded interval and be a bounded function. We say that is integrable with integral in the sense of

Riemann if for every there exists such that for every and every selection we have

Generalised-Riemann if for every there exists such that for every and every selection we have

Darboux if

From the definition it is clear that “Riemann integrable” implies “Generalised-Riemann integrable”. Furthermore, we have clearly that for a fixed

and that if we have

so “Darboux integrable” also implies “Generalised-Riemann integrable”. A little bit more work shows that “Generalised-Riemann integrable” also implies “Darboux integrable” (if the suprema and infima are obtained on the intervals , this would follow immediately; using the boundedness of the intervals we can find such that the Riemann sum approximates the upper or lower Darboux sums arbitrarily well.

The interesting part is the following Theorem
Darboux integrable functions are Riemann integrable. Thus all three notions are equivalent.

Proof. Let be partitions. Let , and let be the number of non-degenerate subintervals in . We have the following estimate

The estimate follows by noting that “most” of the will be proper subsets of , and there can be at most of the that straddles between two different non-degenerate sub-intervals of . To prove the theorem it suffices to choose first a such that the upper and lower Darboux sums well-approximates the integral. Then we can conclude for all with sufficiently small the Riemann sum is almost controlled by the -Darboux sums. Q.E.D.

Now that we have recalled the case of the usual integrability. Let us consider the case of the Stieltjes integrals: instead of integrating against , we integrate against , where is roughly speaking a “cumulative distribution function”: we assume that is a bounded monotonically increasing function.

The definition of the integrals are largely the same, except that at every step we replace the width of the interval by the diameter of , i.e. . The arguments above immediately also imply that

However, Darboux-Stieltjes integrable functions need not be Riemann-Stieltjes integrable. The possibility of failure can be seen in the proof of the theorem above, where we used the fact that is allow to be made arbitrarily small. The same estimate, in the case of the Stieltjes version of the integrals, has replaced by , which for arbitrary partitions need to shrink to zero. To have a concrete illustration, we give the following:

Example
Let . Let if and otherwise. Let if and otherwise. Let be the partition . We have that

while

so we have that in particular the pair is Darboux-Stieltjes integrable with integral 0. However, let be any odd integer, consider the partition of into equal portions. Depending on the choice of the selection , we see that the sum can take the values

which shows that the Riemann-Stieltjes condition can never be satisfied.

The example above where both and are discontinuous at the same point is essentially sharp. A easy modification of the previous theorem shows that Prop
If at least one of is continuous, then Darboux-Stieltjes integrability is equivalent to Riemann-Stieltjes integrability.

Remark The nonexistence of Riemann-Stieltjes integral when and has shared discontinuity points is similar in spirit to the idea in distribution theory where whether the product of two distributions is well-defined (as a distribution) depends on their wave-front sets.

In the two previous posts, I shot particles (okay, simulated the shooting on a computer) at a single potential barrier and looked at what happens. What happens when we have more than one barrier? In the classical case the picture is easy to understand: a particle with insufficient energy to escape will be trapped in the local potential well for ever, while a particle with sufficiently high energy will gain freedom and never come back. But what happens in the quantum case?

If the intuition we developed from scattering a quantum particle against a potential barrier, where we see that depending on the frequency (energy) of the particle, some portion gets transmitted and some portion gets reflected, is indeed correct, what we may expect to see is that the quantum particle bounces between the two barriers, each time losing some amplitude due to tunneling.

But we also saw that the higher frequency components of the quantum particle have higher transmission amplitudes. So we may expect that the high frequency components to decay more rapidly than the low frequency ones, so the frequency of the “left over” parts will continue to decay in time. This however, would be wrong, because we would be overlooking one simple fact: by the uncertainty principle again, very low frequency waves cannot be confined to a small physical region. So when we are faced with two potential barriers, the distance between them gives a characteristic frequency. Below this frequency (energy) it is actually not possible to fit a (half) wave between the barriers, and so the low frequency waves must have significant physical extent beyond the barriers, which means that large portions of these low frequency waves will just radiate away freely. Much above the characteristic frequency, however, the waves have large transmission coefficients and will not be confined.

So the net result is that we should expect for each double barrier a characteristic frequency at which the wave can remain “mostly” stuck between the two barriers, losing a little bit of amplitude at each bounce. This will look like a slowly, but exponentially, decaying standing wave. And I have some videos to show for that!

In the video we start with the same random initial data and evolve it under the linear wave equation with different potentials: the equations look like

where is a non-negative potential taken in the form

which is a difference of two Gaussians. For the five waves shown the values of are the same throughout. The coefficients (taken to be ) and (taken to be ) increases from top to bottom, resulting in more and more-widely separated double barriers. Qualitatively we see, as we expected,

The shallower and narrower the dip the faster the solution decays.

The shallower and narrower the dip the higher the “characteristic frequency”.

As an aside: the video shown above is generated using Python, in particular NumPy and MatPlotLib; the code took significantly longer to run (20+hours) than to write (not counting the HPDE solver I wrote before for a different project, coding and debugging this simulation took about 3 hours or less). On the other hand, this only uses one core of my quad-core machine, and leaves the computer responsive in the mean time for other things. Compare that to Auto-QCM: the last time I ran it to grade a stack of 400+ multiple choice exams it locked up all four cores of my desktop computer for almost an entire day.

As a further aside, this post is related somewhat to my MathOverflow question to which I have not received a satisfactory answer.

In the previous post we shot a classical particle at a potential barrier. In this post we shoot a quantum particle.

Whereas the behaviour of the classical particle is governed by Newton’s laws (where the external force providing the acceleration is given as minus the gradient of the potential), we allow our quantum particle to be governed by the Klein-Gordon equations.

Mathematically, the Klein-Gordon equation is a partial differential equation, whereas Newton’s laws form ordinary differential equations. A typical physical interpretation is that the state space of quantum particles are infinite dimensional, whereas the phase space of physics has finite dimensions.

Note that physically the Klein-Gordon equation was designed to model a relativistic particle, while in the previous post we used the non-relativistic Newton’s laws. In some ways it would’ve been better to model the quantum particle using Schroedinger’s equation. I plead here however that (a) qualitatively there is not a big difference in terms of the simulated outcomes and (b) it is more convenient for me to use the Klein-Gordon model as I already have a finite-difference solver for hyperbolic PDEs coded in Python on my computer.

To model a particle, we set the initial data to be a moving wave packet, such that at the initial time the solution is strongly localized and satisfies . Absent the mass and potential energy terms in the Klein-Gordon equation (so under the evolution of the free wave equation), this wave packet will stay coherent and just translate to the right as time goes along. The addition of the mass term causes some small dispersion, but the mass is chosen small so that this is not a large effect. The main change to the evolution is the potential barrier, which you can see illustrated in the simulation.

The video shows 8 runs of the simulation with different initial data. Whereas in the classical picture the initial kinetic energy is captured by the initial speed at which the particle is moving, in the quantum/wave picture the kinetic energy is related to the central frequency of your wave packet. So each of the 8 runs have increasing frequency offset that represents increasing kinetic energy. The simulation has two plots, the top shows the square of the solution itself, which gives a good indication of where physically the wave packet is located. The bottom shows a normalized kinetic energy density (I have to include a normalization since the kinetic energy of the first and last particles differ roughly 10 fold).

One notices that in the first two runs, the kinetic energy is sufficiently small that the particle mostly bounces back to the left after hitting the potential.

For the third and fourth runs (frequency shift and respectively) we see that while a significant portion of the particle bounces back, a noticeable portion “tunnels through” the barrier: this caused by a combination of the quantum tunneling phenomenon and the wave packet form of the initial data.

The phenomenon of quantum tunneling manifests in that all non-zero energy waves will penetrate a finite potential barrier a little bit. But the amount of penetration decays to zero as the energy of the wave goes to zero: this is known as the semiclassical regime. In the semiclassical limit it is known that quantum mechanics converge toward classical mechanics, and so in the low-energy limit we expect our particle to behave like a classical particle and bounce off. So we see that naturally increasing the energy (frequency) of our wave packet we expect more of the tunneling to happen.

Further, observe that by shaping our data into a wave packet it necessarily contains some high frequency components (due to Heisenberg uncertainty principle); high frequency, and hence high energy components do not “see” the potential barrier. Even in the classical picture high energy particles would fly over the potential barrier. So for wave packets there will always be some (perhaps not noticeable due to the resolution of our computation) leakage of energy through the potential barrier. The quantum effect on these high energy waves is that they back-scatter. Whereas the classical high energy particles just fly directly over the barrier, a high energy quantum particle will leave some parts of itself behind the barrier always. We see this in the sixth and seventh runs of the simulation, where the particle mostly passes through the barrier, but a noticeable amount bounces off in the opposite direction.

In between during the fifth run, where the frequency shift is 2, we see that the barrier basically split the particle in two and send one half flying to the right and the other half flying to the left. Classically this is the turning point between particles that go over the bump and particles that bounces back, and would be the case (hard to show numerically!) where a classical particle comes in from afar with just enough energy that it comes to a half at the top of the potential barrier!

And further increasing the energy after the seventh run, we see in the final run a situation where only a negligible amount of the particle scatters backward with almost all of it passing through the barrier unchanged. One interesting thing to note however is that just like the case of the classical particle, the wave packet appears to “slow down” a tiny bit as it goes over the potential barrier.

Here’s a small animation of what happens when you try to shoot a classical particle when there’s a potential barrier. For small initial kinetic energies, the particle bounces back. For large initial kinetic energies, the particle goes over the hump, first decelerating and then accelerating in the process.

(It may be best to watch this full screen with HD if the network supports it.)

(The NumPy code is pretty simple to write for this; and it runs relatively fast. The one for my next post is a bit more complicated and takes rather much longer to run. Stay tuned!)