Snow in April!

I’m beginning to think that Michigan’s weather could be described as a phase transition. For about half the year, it is warm, and humid. The other half of the year, it is bitter cold. And there is no in-between nice and cool. So, is there some sort of order parameter that evolved in time that drastically changes the state of the weather? Would this be a second-order transition?

It seems like this parameter has some variance to it, and we are currently sitting right at the transition point. A few days ago, it was warm (not terribly humid), but now it is snowing. A change of temperature of about 40 degrees.

An analogy for EEs: We’re sitting right at the threshold voltage to turn on a MOSFET—say, for an inverter—and are measuring the logic output. Unfortunately, we have some noise in our input voltage, which causes the output to switch randomly between 0 and 1.

Mr. Wolfram’s crazy Ising model

A few years back, I went to see Steve Wolfram talk at Cornell about his crazy theory of Cellular Autonoma being “A new kind of Science.” The basic gist of his theory is something like this.

Suppose we had an array of cells, being either black or white. Given a certain set of rules, a cell will change it’s color from black to white (or vice versa). The rules are simple; a cell will flip based on the current colors of itself and it’s nearest neighbors. Mr. Wolfram noticed that surprisingly simple rules with a simple initial condition can generate chaotic patterns:

To me, this seems like he’s simply been performing a one-dimensional simulation of the Ising Model. This models magnetic domains in solids. Particles have a property called spin that is related to its magnetic moment. The one-dimensional version of this model is as such:

Suppose we had an array of particles whose spin can either be up or down. The ground state of this array is the configuration that minimizes the energy. The energy is determined by the spins of the nearest neighbors in the array. In a sense, the spin of a particle will flip depending on its nearest neighbors.

This is an interesting thought, but I have no mathematical ‘proof’ of this idea. Thoughts?

Mr. Wolfram’s crazy Ising model

A few years back, I went to see Steve Wolfram talk at Cornell about his crazy theory of Cellular Autonoma being “A new kind of Science.” The basic gist of his theory is something like this.

Suppose we had an array of cells, being either black or white. Given a certain set of rules, a cell will change it’s color from black to white (or vice versa). The rules are simple; a cell will flip based on the current colors of itself and it’s nearest neighbors. Mr. Wolfram noticed that surprisingly simple rules with a simple initial condition can generate chaotic patterns:

To me, this seems like he’s simply been performing a one-dimensional simulation of the Ising Model. This models magnetic domains in solids. Particles have a property called spin that is related to its magnetic moment. The one-dimensional version of this model is as such:

Suppose we had an array of particles whose spin can either be up or down. The ground state of this array is the configuration that minimizes the energy. The energy is determined by the spins of the nearest neighbors in the array. In a sense, the spin of a particle will flip depending on its nearest neighbors.

This is an interesting thought, but I have no mathematical ‘proof’ of this idea. Thoughts?

Thermal Radiation & Photon Gasses

One thing that has always puzzled me is thermal radiation. This is electromagnetic radiation from an object simply because it is warm. I never understood why a warm body emits light (NB: I use “light” and “electromagnetic radiation” interchangeably).

In quantum mechanics, we are taught that light is emitted with a sharply defined color—given by the energy difference between levels. I guess it could make sense that if, naively, hot bodies consisted of atoms in motion and that there were a continuum of motional states, then there could be a continuous energy spectrum.

But this means that the atoms would be losing energy to emit thermal radiation, thus cooling it. Hence it seems like all bodies would then be at zero temperature and all the energy in the universe would be in light.

Here’s the insight I recently found:
Remember that there is a vacuum electromagnetic field, containing an infinite number of modes. Each mode has a certain number of photons. Here’s the kicker: the vacuum field is in thermal contact with the hot body! That is, the electromagnetic field is in thermal equilibrium with the hot body! Therefore, since the body has a nonzero temperature, then there must be a nonzero photon distribution in all the modes. This distribution is exactly the distribution of thermal radiation.

Thermal Radiation & Photon Gasses

One thing that has always puzzled me is thermal radiation. This is electromagnetic radiation from an object simply because it is warm. I never understood why a warm body emits light (NB: I use “light” and “electromagnetic radiation” interchangeably).

In quantum mechanics, we are taught that light is emitted with a sharply defined color—given by the energy difference between levels. I guess it could make sense that if, naively, hot bodies consisted of atoms in motion and that there were a continuum of motional states, then there could be a continuous energy spectrum.

But this means that the atoms would be losing energy to emit thermal radiation, thus cooling it. Hence it seems like all bodies would then be at zero temperature and all the energy in the universe would be in light.

Here’s the insight I recently found:
Remember that there is a vacuum electromagnetic field, containing an infinite number of modes. Each mode has a certain number of photons. Here’s the kicker: the vacuum field is in thermal contact with the hot body! That is, the electromagnetic field is in thermal equilibrium with the hot body! Therefore, since the body has a nonzero temperature, then there must be a nonzero photon distribution in all the modes. This distribution is exactly the distribution of thermal radiation.