Thursday, January 19, 2017

Degrees of freedom

The number of degrees of freedom in a system is the number of numerical parameters that need to be set to fully determine the system. Scientists have an epistemic preference for theories that posit systems with fewer degrees of freedom.

But any system with n real-valued degrees of freedom can be redescribed as a system with only one real-valued degree of freedom, where n is finite or countable. For instance, consider a three-dimensional system which is fully described at any given time by a position (x, y, z) in three-dimensional space. We can redescribe x, y and z by real-valued variable X, Y and Z in the interval from 0 and 1, for instance by letting X = 1/2 + π−1arctan x and so on. Now write out these new variables in decimal:

  • X = 0.X1X2X3...
  • Y = 0.Y1Y2Y3...
  • Z = 0.Z1Z2Z3...

Finally, let:

  • W = 0.X1Y1Z1X2Y2Z2X3Y3Z3....

Then W encodes all the information about X, Y and Z, which in turn encode all the information about (x, y, z) and hence about our system at a given time. (This obviously generalizes to any finite number of degrees of freedom. For a countably infinite one, things are slightly more complicated, but can still be done.)

There is a lesson here, even if not a particularly deep one. The epistemic preference for theories that have fewer degrees of freedom cannot be separated from the the epistemic preference for simpler theories. For of course rewriting a theory that made use of (x, y, z) in terms of W is in practice going to make for a significantly messier theory. So we cannot replace a simplicity preference by a preference for a low number of degrees of freedom.

Objection: Instead of a simplicity preference, we may a priori specify that laws of nature be given by differential equations in terms of the variables involved. But when, say, x, y and z vary smoothly over time, it is very unlikely that W will do so as well.

Response: But one can find a replacement for W that is smoothly related to x, y and z up to any desired degree of precision, and hence we can give a differential-equation based theory that fits the experimental data pretty much equally well but has only one degree of freedom.

5 comments:

entryleveljobscam said...
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William said...

I'm not so sure about this. We can encode both the numeral 1 or the complete text of one of your books in a computer memory as a single binary number on disk, but are they really at the same level of complexity? There seems to be a genuine difference between the complexity of the internal properties of a system and the complexity of our representation of the system.

I think that for example if we consider a system composed of systolic blood pressure (121), diastolic blood pressure (070), and mean blood pressure (087), we might encode this as 0.100278107, or maybe 3.100278107 with the 3 to indicate how many numbers to decode (leaving aside positioning decimal places).

But this seems to leave 3 independent variables to consider in the dataset. We can however remove one independent variable by noting that mean blood pressure MBP = (SBP + 2 * DBP)/3, so we actually need only two numbers to get the third. which means we can simplify our measurement of the system from 3 to 2 degrees of freedom. This kind of reduction of symmetries is much more significant than how we encode the data itself.

Maybe the degrees of freedom of a such a system is more like the order of a group in group theory, which is rather independent of the simplicity of the language we use to define that group. Such a group's size is not changed to 1 just because I can set up a way to name it with a single number.

Alexander R Pruss said...

In continuous systems, we can expect every single numerical parameter to have infinite complexity at any given time. It is extremely rare for an object to have a speed that, say, is a rational number (in some pre-specified unit system) or anything else that can be specified in a finite way. So it's more like different countably infinite groups--they all have the same order.

Note, too, that a single particle quantum system and a 10^100 particle quantum system are both described by a Hilbert space of the same dimension (i.e., countably infinite).

William said...

If two theories both have models that require an infinite number of measurements to be done for the theory to be used, I agree there is little reason to consider one more complicated for our use than another. They both have a parameter group order of infinite complexity, countable or not.

On the other hand, if we take a parameter of a theory to be measurable as something like 2 * pi * 1.2345 meters, that calculates out as an irrational number, but a theory that only requires 2 such irrational numbers ( a "group size" of 2) seems to me to be less complex than one which requires three.

bethyada said...

x y and z are numbers, as are X Y and Z albeit transformed. They have a meaning as numbers. But W is merely a code that is represented as a numeral. You have changed the language but are using the same glyphs (numerals) to represent fundamentally different things. I would say this is similar to binary as a number and binary as code.