Why do physical laws always feature integer indices?

[parsec]

This may be a stupid question or have a pretty obvious answer, but I can't seem to find one so I'll just go ahead and post :)

I was looking at some empirical data for relationships defining (abstracted) values for ionization and recomination coefficients in gases as a function of electric field strength and gas number density. I noticed that none of them had integer indices, rather they featured fractional indices correct to three decimal places.

I had come across similar empirical "laws" as an undergrad studying fluid dynamics, although those seemed to be a bit more acceptable because the variable that was raised to a fractional power was always non dimensional. So for example the equation for the drag force across a flat plate (as determined from experiments) was some function of reynold's number to the power of a fractional index. As reynold's number is non dimensional, the equation would always yield Newtons, not Newtons to the power of some arbitrary (nonsensical) fractional index.

I noticed for these equations however, the variables that are raised to fractional indices indeed have units, and the resulting equation's result is defined to have the nearest integer units.

For example

a/N = 3.4473 x 10³⁴(E/N)^2.985 m²

where E is the electric field (V. m^-1), N is the neutral gas number density (m^-3)

Clearly this equation should yield fractional units but is then redefined to yield m²

How is this possible?

How is it that fundamental physical relationships are always defined in terms of integer indices? Do physical phenomena happen to form perfect functional laws that feature integer indices or are these laws mere approximations that they depart from in reality? This seems absurd so I'm guessing it has something to do with the way that our arbitrary mathematical constructs are formed, or perhaps how we define the dependent variables involved. (I'm thinking of 1/2kT², where T is defined to have a functional relationship to energy involving an integer index, but perhaps there's a better example).

As an unrelated sidenote, what makes e and pi the values that they are? Physical constants are arbitrary, but these constants are ratios of abstract concepts. Changing our number system would change their values superficially, but they would still represent the same quantity.

 

[Jame]

I cannot see how a quantity of dimension could not be raised to an arbitrary fractional power since this would mostly yield nonsense units. One obvious case is taking the square root of an area to get a number of length. I don't know the law you're giving as an example, could you clarify it a bit?

The constants Pi and e are numbers defined to have mathematical properties which are independent of how they are represented numerically. If we were using a different numbering system everything else would have to change the same way, so that number theory still holds true.

Some reading on the subject can be found at: http://en.wikipedia.org/wiki/Dimensional_analysis

 

[Tac-Tics]

The units don't really belong in the laws anyway.

I'm not familiar with the example you selected, but you probably just made a mistake in copying it or in usage. If a value is raised to a fractional power, it was almost certainly unitless to begin with.

Pi and e are defined mathematically. They have no bearing on the real world. Pi is the ratio of a perfect circle's circumference to its diameter. You can calculate its digits by approximating a circle with polygons having a very large number of sides. It's a calculus thing and there's a series for it.

The number e is the base of the natural logarithm. It is weirder than pi. It's not actually a ratio. It's the only exponential function whose derivative is itself. It's also the natural consequence of compound interest. It's actually kind of neat. Think about it like this. Suppose a radioactive pile loses 100% of its mass every year. How long will it take before ALL the mass is gone? Hint: the answer is NOT a year. Why? Because 100% / year is the SAME as 50% / 6 months. So say you start with a 100 kilo pile of radioactive stuff. After 6 months, you lose half of it, so now you only have 50 kilos. Wait 6 MORE months, now you have 25. Wait 6 MORE months, now you have 12.5 kilos. A few years down the line, you'll have 0.0001 kilos, but the damn thing will never fully disappear! The decay of the radioactive stuff obeys a special curve called a logarithm. The number e pops up in there magically when you pick natural units for everything.

 

[vanesch]

This may be a stupid question or have a pretty obvious answer, but I can't seem to find one so I'll just go ahead and post :)

I was looking at some empirical data for relationships defining (abstracted) values for ionization and recomination coefficients in gases as a function of electric field strength and gas number density. I noticed that none of them had integer indices, rather they featured fractional indices correct to three decimal places.

Often, people write down "empirical" laws in which non-dimensionless quantities enter in formulae, with "strange" coefficients who do the unit-correcting work. This is because practical working formulae are easiest to use when you can just plug in the numerical values of quantities in commonly-used units for them.

Take something like your example :

For example

a/N = 3.4473 x 10³⁴(E/N)^2.985 m²

where E is the electric field (V. m^-1), N is the neutral gas number density (m^-3)

Clearly this equation should yield fractional units but is then redefined to yield m²

The "secret" lies in the numerical coefficient 3.4473 10³⁴. It will do the "unit-compensating" work. You can easily see this by telling yourself: imagine that I was living on the planet Zork, empirically deriving the same relationship, but where my "meters" are 5 earth-meters, say. How would I write down this equation, given that the physics is going to be the same ? You will see that the numerical coefficient you will publish in Zork's Anals of Gas Properties will scale with the fractional power as compared to the value on earth. So this numerical coefficient has the "fractional units".

How is it that fundamental physical relationships are always defined in terms of integer indices? Do physical phenomena happen to form perfect functional laws that feature integer indices or are these laws mere approximations that they depart from in reality?

You mean, why is Newton's second law, F = m a and not F = m^(1.032) a^(0.997) or something ? In a way, that's indeed a mystery. Of course, there are reasons why F = m a as a function of certain symmetries of nature, but then this begs the question as of why nature has these symmetries. Yes, it is a big mystery as of why we seem to be able to express fundamental laws of physics by relatively simple mathematical constructs (that said, when looking at modern theoretical physics, those mathematical constructs don't seem to be so simple anymore, which renders the mystery even deeper: why do very good approximate laws turn out to be so simple ?)
However, as to why do the units in natural laws balance (and do they balance ?) ? That's something else: they have to balance perfectly, as long as we consider that our choices of units is arbitrary and could have been different. That is, that our unit of length, the meter, has something arbitrary to it, and that we could have chosen another unit of length. If you consider that you had freedom in fixing your unit of length, then you are entitled to think that the particular relationship between quantities you are studying shouldn't depend on your choice of unit. And that comes down to having a perfect balance of units in your equations. If you consider that you could have set up physics just as well with a different length as "meter", then all your units must have the same "power of meter" on the left hand and the right hand of your equations, because otherwise, switching from one to another wouldn't work anymore.

Let's give a stupid example: suppose that you find a relationship between the mass of a cube of iron and its side (in other words, you are studying the density of iron).

So you find: m_cube = 7874 kg/m^3 L^3
where L is the length of the side of the cube in meter, and m_cube is its mass in kg.

Now, suppose that you do more carefull measurements, and that you think that you now have a better relationship:

m_cube = 7874 kg/m^3 L^3.002

(you could think of some kind of gravitational effect that makes iron more dense when you have more of it - although the effect would be way way smaller than what I have here).

Is this possible ?

Answer: no. Because we take it as a principle that the choice of the unit of "meter" is arbitrary wrt whatever property of iron, and that we could have expressed this property just as well with a different unit of length, say the zork-meter.
If we had been working in zork-meters, but still in kg, then the mass of a given cube would numerically remain the same (m_cube remains the same number).
But numerically, our left-hand side would not be ok anymore.
Let's say,for simplicity, that a zork-meter is 10 earth-meters.
7874 kg/m^3 in earth-meters would become 7874000 kg/(zm)^3 of course.

But then our relationship wouldn't work anymore. If we'd express L in zork-meters, then we would NOT find:

m_cube = 7874000 kg/zm^3 L^3.002

In fact, if we would calculate the mass of a cube of 1 Earth meter (so 0.1 zork meters), we would have found with the "earth" formula: 7874 kg, and with the "zork" formula: 7837.8 kg.
While we are talking about the same cube, with the same dimensions, only in one case, we expressed the size of the cube in "meters" and in the other case, we expressed them in zork-meters.

So IF our property of L^3.002 is a correct physical law (which it COULD be if we think of some effect like self-gravitation and compression), then that means that we have to adapt our coefficient. It means that the numerical coefficient in zork-units is not going to be 7874000, but rather 7874000 x 7874/7837.8 = 7910000. With this number in the second (zork) formula, our same cube will have again its same mass.

But that means that the unit of the coefficient was not kg/ zm^3, but rather kg/zm^(3.002). And the units balance again in the equation.

As we assume the principle that the "meter" has nothing special, and that we were just as entitled to have taken the "zork-meter" as our unit of length than we did take our "meter", we cannot accept the situation that when transforming all quantities in "zork units" our formula doesn't work anymore. It is only if we would have assumed that "meter" has something particular to do with "iron" that we might expect this to be different, but we take it that the meter is just as good a unit to express a property of iron cubes as zork-meters. And then, the units have to balance in order not to have the correctness of the formula to depend on the specific choice of a particular unit of length.

 

[Dr.D]

Parsec, another example where a fractional exponent appears is in the law governing an isentropic expansion,
P*V^(gamma) = constant
where gamma = a property of the gas, typically about 1.4 for air

 

[parsec]

The units don't really belong in the laws anyway.

I'm not familiar with the example you selected, but you probably just made a mistake in copying it or in usage. If a value is raised to a fractional power, it was almost certainly unitless to begin with.

Pi and e are defined mathematically. They have no bearing on the real world. Pi is the ratio of a perfect circle's circumference to its diameter. You can calculate its digits by approximating a circle with polygons having a very large number of sides. It's a calculus thing and there's a series for it.

The number e is the base of the natural logarithm. It is weirder than pi. It's not actually a ratio. It's the only exponential function whose derivative is itself. It's also the natural consequence of compound interest. It's actually kind of neat. Think about it like this. Suppose a radioactive pile loses 100% of its mass every year. How long will it take before ALL the mass is gone? Hint: the answer is NOT a year. Why? Because 100% / year is the SAME as 50% / 6 months. So say you start with a 100 kilo pile of radioactive stuff. After 6 months, you lose half of it, so now you only have 50 kilos. Wait 6 MORE months, now you have 25. Wait 6 MORE months, now you have 12.5 kilos. A few years down the line, you'll have 0.0001 kilos, but the damn thing will never fully disappear! The decay of the radioactive stuff obeys a special curve called a logarithm. The number e pops up in there magically when you pick natural units for everything.

This is a little patronising. I didn't make a mistake copying it (copy-paste rarely makes errors).

I'm well aware of the origins of e and Pi, I was more after a deeper discussion of why they are the values that they are. I have both seen the open form solution of Pi and encountered the natural logarithm...

I find it strange that something abstract (I guess it is spatial) has this well defined ratio between it's circumference and diameter. I guess I can deal with physical constants being arbitrary as they're determined from experiments, but why is Pi its value? Why not 3.1, or 1?

I know the intrinsic value of both e and Pi transcend our number system, but to what extent? Is there a weird mathematical analog to natural units where you can define Pi and e to be 1 and use all of the same mathematical techniques?

 

[parsec]

I cannot see how a quantity of dimension could not be raised to an arbitrary fractional power since this would mostly yield nonsense units. One obvious case is taking the square root of an area to get a number of length. I don't know the law you're giving as an example, could you clarify it a bit?

Some reading on the subject can be found at: http://en.wikipedia.org/wiki/Dimensional_analysis

The law is taken from a paper entitled "A Survey of the Electron and Ion Transport Properties
of SF6" by R. Morrow. It's not the best example, I know.

 

[parsec]

Parsec, another example where a fractional exponent appears is in the law governing an isentropic expansion,
P*V^(gamma) = constant
where gamma = a property of the gas, typically about 1.4 for air

Ah yes, that's a good example.

 

[parsec]

The "secret" lies in the numerical coefficient 3.4473 10³⁴. It will do the "unit-compensating" work. You can easily see this by telling yourself: imagine that I was living on the planet Zork, empirically deriving the same relationship, but where my "meters" are 5 earth-meters, say. How would I write down this equation, given that the physics is going to be the same ? You will see that the numerical coefficient you will publish in Zork's Anals of Gas Properties will scale with the fractional power as compared to the value on earth. So this numerical coefficient has the "fractional units".

Ah yes, this seems pretty obvious now that I think of it. Thanks.

You mean, why is Newton's second law, F = m a and not F = m^(1.032) a^(0.997) or something ? In a way, that's indeed a mystery. Of course, there are reasons why F = m a as a function of certain symmetries of nature, but then this begs the question as of why nature has these symmetries. Yes, it is a big mystery as of why we seem to be able to express fundamental laws of physics by relatively simple mathematical constructs (that said, when looking at modern theoretical physics, those mathematical constructs don't seem to be so simple anymore, which renders the mystery even deeper: why do very good approximate laws turn out to be so simple ?)

Yeah, that's basically what I was getting at. High level abstractions of nature (e.g. in classical mechanics) seem to be able to be described by such simple laws. Fudgy engineering descriptions based on high level descriptions that have no derivation (e.g. a curve fit from an experiment) seem to have fractional indices.

However, as to why do the units in natural laws balance (and do they balance ?) ? That's something else: they have to balance perfectly, as long as we consider that our choices of units is arbitrary and could have been different. That is, that our unit of length, the meter, has something arbitrary to it, and that we could have chosen another unit of length. If you consider that you had freedom in fixing your unit of length, then you are entitled to think that the particular relationship between quantities you are studying shouldn't depend on your choice of unit. And that comes down to having a perfect balance of units in your equations. If you consider that you could have set up physics just as well with a different length as "meter", then all your units must have the same "power of meter" on the left hand and the right hand of your equations, because otherwise, switching from one to another wouldn't work anymore.

Let's give a stupid example: suppose that you find a relationship between the mass of a cube of iron and its side (in other words, you are studying the density of iron).

So you find: m_cube = 7874 kg/m^3 L^3
where L is the length of the side of the cube in meter, and m_cube is its mass in kg.

Now, suppose that you do more carefull measurements, and that you think that you now have a better relationship:

m_cube = 7874 kg/m^3 L^3.002

(you could think of some kind of gravitational effect that makes iron more dense when you have more of it - although the effect would be way way smaller than what I have here).

Is this possible ?

Answer: no. Because we take it as a principle that the choice of the unit of "meter" is arbitrary wrt whatever property of iron, and that we could have expressed this property just as well with a different unit of length, say the zork-meter.
If we had been working in zork-meters, but still in kg, then the mass of a given cube would numerically remain the same (m_cube remains the same number).
But numerically, our left-hand side would not be ok anymore.
Let's say,for simplicity, that a zork-meter is 10 earth-meters.
7874 kg/m^3 in earth-meters would become 7874000 kg/(zm)^3 of course.

But then our relationship wouldn't work anymore. If we'd express L in zork-meters, then we would NOT find:

m_cube = 7874000 kg/zm^3 L^3.002

In fact, if we would calculate the mass of a cube of 1 Earth meter (so 0.1 zork meters), we would have found with the "earth" formula: 7874 kg, and with the "zork" formula: 7837.8 kg.
While we are talking about the same cube, with the same dimensions, only in one case, we expressed the size of the cube in "meters" and in the other case, we expressed them in zork-meters.

So IF our property of L^3.002 is a correct physical law (which it COULD be if we think of some effect like self-gravitation and compression), then that means that we have to adapt our coefficient. It means that the numerical coefficient in zork-units is not going to be 7874000, but rather 7874000 x 7874/7837.8 = 7910000. With this number in the second (zork) formula, our same cube will have again its same mass.

But that means that the unit of the coefficient was not kg/ zm^3, but rather kg/zm^(3.002). And the units balance again in the equation.

As we assume the principle that the "meter" has nothing special, and that we were just as entitled to have taken the "zork-meter" as our unit of length than we did take our "meter", we cannot accept the situation that when transforming all quantities in "zork units" our formula doesn't work anymore. It is only if we would have assumed that "meter" has something particular to do with "iron" that we might expect this to be different, but we take it that the meter is just as good a unit to express a property of iron cubes as zork-meters. And then, the units have to balance in order not to have the correctness of the formula to depend on the specific choice of a particular unit of length.

I'm fine with this, however this relationship should work if we were to define the coefficient (density) with units of kg/m^3.002 right? I guess this could be a linearisation of a density expressed as some function of mass, length and gravitational constants.

Do fractional indices occur in any of the tendrils of modern physics? Could you give me some examples? So far I haven't encountered any, but I haven't really looked very deeply into modern physics.

 

[vanesch]

Yeah, that's basically what I was getting at. High level abstractions of nature (e.g. in classical mechanics) seem to be able to be described by such simple laws. Fudgy engineering descriptions based on high level descriptions that have no derivation (e.g. a curve fit from an experiment) seem to have fractional indices.

This is usually because empirical laws are a kind of approximative summary of a very complicated system ; a kind of "curve fitting". Not only purely empirical laws do so, you can derive, from first principles, also "system laws" which have fractional powers, or are "complicated" functions in other respects. It is because they summarize the behavior of a "complicated" system.

I'm fine with this, however this relationship should work if we were to define the coefficient (density) with units of kg/m^3.002 right? I guess this could be a linearisation of a density expressed as some function of mass, length and gravitational constants.

Yes.

Do fractional indices occur in any of the tendrils of modern physics? Could you give me some examples? So far I haven't encountered any, but I haven't really looked very deeply into modern physics.

I don't know what you call "fundamental". The adiabatic compression law, which can be derived from first principles, has a fractional power, but that's a kind of "system response" of a "complicated system" in a way (though it is not an empirical approximation, in that it is the correct solution of an ideal system).

As "system solutions" there are many examples of fractional powers and otherwise complicated functions. In fact, you are going down the same road as the mathematicians since the 17th century, having to "open up" their set of "acceptable functions" in nature, from "Euclidean constructions (ruler and compass), to algebraic curves, to power series, to fractional power series, to, finally, a general function concept without a specific closed-form prescription, but simply with a certain amount of analytic and topological properties, and even beyond that (in quantum field theory)

However, as to the "fundamental laws" behind this, and not the "solutions" for specific systems no matter how fundamental themselves, usually their formulation is indeed "simpler". And it is indeed a mystery as why this is so.

 

[parsec]

Thanks, I was hoping that it wasn't a mystery, but I guess I sort of have an answer now.

I guess I was thinking of "fundamental" as any sort of low level description of the behavior of individual particles. I wasn't really considering macrostate thermodynamic descriptions involving state variables to be fundamental despite the fact that they're derived from first principles.

Now realize that my categorisation isn't very terse as you could possibly consider a description of the motion and dynamics of atoms to be an abstraction we use to describe the interaction of smaller constituent particles. Sorry, I'm not very educated in modern physics so I might be completely off the ball here.

I guess it would be more surprising to find out that s=klog_2.546\Omega or that E=1/2kT^2.12 than pvⁿ=const. I find the elegance of relationships like this quite remarkable.

 

[ZapperZ]

I've sort of read this thread, but the answer is simpler than what I think I've read here.

There is nothing to prevent anyone from giving such number is "fractional" units. For example, let's look at speed. There's nothing to say that you cannot express speed as something like this

"A travels 30 m in 2.2 seconds".

So you have the speed as 30m/2.2s".

Now, is THIS what you had in mind?

If it is, fine. But take a look at that fraction. The speed is 30/2.2 m/s. I could rescale this and do that fraction, i.e. take 30/2.2. What do I get? I get 13.63. But what's the units? "m/s!"

So in going 30 m in 2.2 seconds, that is equivalent to saying that it goes 13.63 m in one second. The speed has been "renormalized" in units of 1 second. Or if you don't like this, normalize this in units of 1 minute, 1 hour, 1 day, etc... In other words, it is simply carrying out the fraction into more easily read numbers in whole units of the denominator. There nothing profound about such exercises. If you don't like it, you can continue to express it as 30m/2.2s. But you'll soon learn that this is a tedious way to carry such numbers.

Zz.

 

[parsec]

I've sort of read this thread, but the answer is simpler than what I think I've read here.

There is nothing to prevent anyone from giving such number is "fractional" units. For example, let's look at speed. There's nothing to say that you cannot express speed as something like this

"A travels 30 m in 2.2 seconds".

So you have the speed as 30m/2.2s".

Now, is THIS what you had in mind?

If it is, fine. But take a look at that fraction. The speed is 30/2.2 m/s. I could rescale this and do that fraction, i.e. take 30/2.2. What do I get? I get 13.63. But what's the units? "m/s!"

So in going 30 m in 2.2 seconds, that is equivalent to saying that it goes 13.63 m in one second. The speed has been "renormalized" in units of 1 second. Or if you don't like this, normalize this in units of 1 minute, 1 hour, 1 day, etc... In other words, it is simply carrying out the fraction into more easily read numbers in whole units of the denominator. There nothing profound about such exercises. If you don't like it, you can continue to express it as 30m/2.2s. But you'll soon learn that this is a tedious way to carry such numbers.

Zz.

Sorry, I don't think this is what I meant. The discussion is about units being raised to fractional indices. E.g. going 30m in 2.2 s^1.1

 

[ZapperZ]

Sorry, I don't think this is what I meant. The discussion is about units being raised to fractional indices. E.g. going 30m in 2.2 s^1.1

Oh, then I misread the question completely! :(

Zz.

 

[vanesch]

Thanks, I was hoping that it wasn't a mystery, but I guess I sort of have an answer now.

I guess I was thinking of "fundamental" as any sort of low level description of the behavior of individual particles. I wasn't really considering macrostate thermodynamic descriptions involving state variables to be fundamental despite the fact that they're derived from first principles.

The interaction of one single electron with another one is already a problem that doesn't have a "simple solution" (not even a mathematically sound solution at all!).

I guess it would be more surprising to find out that s=klog_2.546\Omega

Nope, that's just a change of value for k. log_2.546 (X) = log_e(X)/log_e(2.546).

or that E=1/2kT^2.12 than pvⁿ=const. I find the elegance of relationships like this quite remarkable.

Well, in relativistic physics, the relationship is already more complicated !

 

[Count Iblis]

A John Cardy explains in one of his books on renormalization group theory (I forgot the exact title), the correct answer has to do with scaling laws. According to Cardy, ordinary dimensional analysis is a special case of using scaling arguments based on the renormalization group in statistical physics or quantum field theory.

The (hidden) assumption in the detailed posting by Vanesh above is the way relations between macroscopic quantities should behave under a rescaling. Now, in principle, this should follow from the laws of physics describing the microscopic degrees of freedom of the system. WHen you eliminate all references to the microscopic degrees of freedom and write down relations involving only macroscopic observables, you can find certain powerlaws. In certain cases you don't get power laws with integer exponents.