There’s a paradox of white noise: in its standard idealization it has equal energy at all frequencies, hence infinite energy. Curiously this paradox is linked to two of Einstein’s great papers from his annus mirabilis.
The ultraviolet catastrophe. The first paradox was a big problem implicit in 19th century physics. Statistical mechanics predicted that a cavity should support electromagnetic waves of all frequencies, each with an equal amount of energy. So at each instant a cavity should radiate infinite energy! Planck’s quantization, it was later realized, resolved this difficulty by positing that in fact, higher frequencies have less energy in them. Einstein investigated more deeply the implications of quantization for the behavior of light photons, for which he won the Nobel Prize.
Brownian motion. Meanwhile another work of Einstein’s that year eventually led to today’s standard idealization of Brownian motion, which has become a central subject in mathematics. The idealization is this: Brownian motion is the indefinite integral of white noise. But white noise has infinite variance! This time the physicists are not concerned, since bouncing molecules don’t go down to the infinitesimal scale, but the paradox still needs a mathematical resolution if we are to work with these ideas rigorously. In effect mathematicians take the view that white noise itself is never measured, only integrals of it, and these are finite because of sign cancellations. The higher the frequency, the greater the cancellation. But the details make stochastic analysis technical and forbidding.
Did Einstein notice that blackbody radiation and Brownian motion are linked in this way?
[6 May 2017]
I’m sitting in a coffee shop in Boise, Idaho. I spend a lot of time working in coffee shops. But here the music is too loud, and it has been hard to work.
Yet just now they’ve switched from current songs I don’t know to an oldie that I know very well, “All along the watchtower” by Jimi Hendrix. How amazing — it’s still loud, but somehow less intrusive. I can work! This is an effect I’ve noticed before. My theory is that my brain is familiar with this track and knows there is no new information to be processed, so it tunes it out.
[14 March 2017]
The other night at a dinner for graduate students at Balliol, I had a German on my left and a Frenchman on my right. The question came up, what happened around 1870 involving Germany and France?
The German student was unaware of any events from this period.
The French student was aware of one event from 1870: the founding of the French Third Republic. He did not know of any connection of this with Germany.
(In fact, in 1870-71 Germany conquered France. Napoleon III surrendered in September, and Paris fell in January.)
[1 February 2017]
The love of parents for their children, it is well known, is the purest love of all. There is nothing we would not do for them, and we keep giving to the end of our lives without thought of what we get in return.
The irony is that from an evolutionary point of view, love of one’s children is obviously selfish, since they carry forward one’s genes. It is so perfectly in our self-interest to take care of our children that natural selection would not dream of letting us deliberate about the matter! No, our love is hardwired. We deeply want to give to them, and who cares what it costs?
And because it’s hardwired, that’s why it feels selfless. This is the love that bypasses our brains and inhabits our bones.
[19 January 2017]
Of course it’s trivial in principle, but one of my notes from 1990 records that I found it surprisingly tricky in practice to calculate how far Perth is from Boston, based on their latitudes and longitudes. In fact I remember working out the formula while on the train crossing the Nullarbor Plain.
It’s not tricky any more. On a whim I just asked my phone, “How far is Perth from Boston?”, and it responded instantly in a friendly man’s voice. “Perth, Australia is about 11,621 miles from Boston, Massachusetts as the crow flies.”
However, it is still only 2017. I followed up with “How far is that in kilometers?” and the friendly man responded, “Sorry, I don’t know where that is”.
[13 January 2017]
About 19% of SIAM’s non-student members are women, but at the next SIAM Annual Meeting, 9 out of 16 invited speakers will be women. These figures imply that the fraction invited to speak will be around 5.5 times higher for women than for men.
That’s the math, which is easy. The politics? Not so easy.
[6 January 2017]
“Did you see the tombs of Mary of Burgundy and Charles of Burgundy in the chancel?”
(Kate asked me this just now in the Church of Our Lady in Bruges.)
I couldn’t help thinking of when Emma and Jacob were little and they knew every detail of Pikachu and Charmander and the other Pokémon characters, each with its particular abilities and weaknesses and path of evolution. Pokémon for the kids, Harry Potter for the teenagers, saints for the Catholics, celebrities for the masses. Our appetite for particulars about people is infinite, and it hardly matters how small the facts are or whether the people are even real.
[31 December 2015]