Date: April 30th, 2011
Subject: Infinity and Infinite Temperatures - from Peter Bailey (Dr. E.P.A. Bailey)
Dear Gary,
The point you make is very pertinent about infinities arising in the mathematics of various equations of theoretical physics .
It touches on a very sensitive and controversial aspect of many equations and phenomena in physics – many physicists have fundamentally opposed views about such infinities.
You are most welcome to post/email/send my reply on this, provided you credit me, Dr. E.P.A. Bailey, for it.
The link you sent is related to the Boltzmann distribution equation for energies of ‘molecules’, and there is a very simple answer about this at absolute zero Kelvin temperature, which I found on the Internet, as explained later..
Other famous equations which throw up infinities in the mathematics include those of Einstein’s Special and General Relativity, e.g. E=m times c squared, what happens at the ‘event horizon/Schwarzchild radius’ with ‘Black Holes’, and time and space measurements as ‘viewed from/by a photon’ travelling at the speed of light. The conventional explanation to avoid an infinite energy of a particle travelling at the speed of light is to say that it has zero mass. This and other explanations related to various infinities are varied and controversial.
It is a vast field of complicated concepts to go through all such equations and infinities, so I cannot go into it all in detail her and now, if ever. Instead it should be recognised that by definition, infinity is larger than any number which anyone can write down or measure, so by definition no infinitely large quantity can be measured. Similarly no infinitely small quantity can be measured.
Nevertheless, we can conceive ( or at least some can, it seems that others cannot or will not) of an infinitely large universe, with infinite variations within it locally, even though we can never measure its size, if it is infinite, by definition. Many physicists seem to be ‘more comfortable’ with the idea of a finite (but unbounded) universe, like the surface of a sphere, and some seem to just have what appears to be a ‘religious’ belief in either an infinite, or a finite, universe, with no proof for or against either concept. Some might say that from the definition of infinity, such proof either way is impossible.
It is probably useful to look at some of the links thrown up by a search on Google for, ‘Definition of Infinity’, for example...
http://www.thefreedictionary.com/infinity
http://encyclopedia2.thefreedictionary.com/Infinity
http://www.brainyquote.com/words/in/infinity178405.html
http://answers.yahoo.com/question/index?qid=20070526151509AAaPEm6
and other links which you can find with a similar search.
It is often useful in the equations of theoretical physics to say that some quantity gets smaller and smaller until it is infinitely small at an infinite distance away – but we cannot measure that this is true an infinite distance away. For example, Newton’s inverse square law of gravity ( which is approximated by Einstein’s General Relativity), has the force between 2 masses decreasing by 1 over the square of the distance apart, which decreases to infinitely small (effectively zero), if the 2 masses are an infinite distance apart – but this can never be measured to be true or not with infinite distances.
Indeed astronomical measurements since around the 1920s suggest that the inverse square law in not correct at large distances involving galaxies, which has led to dark matter, or modified gravitational laws being proposed, controversially. I suggest you do a search on Google for ‘Dark Matter’ and/or the MOND theory of Gravitation if you wish to know more about such concepts.
Even with such ‘patches’ to gravitational theory, such new theories can never be fully verified for their behaviour at infinite distances, by the definition of infinity. All we can do is just go on trying to measure things at larger and larger distances, and also at smaller and smaller distances. For extremely small distances the inverse square laws of gravitation and electrical attraction (coulomb’s law) may also not apply, although no such variations at small distance have yet been found. By definition we can never measure whether such force laws are valid or not at infinitely small distances. I suggest you do a search on Google for ‘ Inverse Square Law Small Distances’, for more on such topics where a number of groups have been trying to look for such variations at small distances quite recently – and have still not found any such variations.
My personal view about the infinities, is that a usually measurable quantity which becomes infinitely large in an equation, implies that the theory behind the equation, or its interpretation, is incomplete, or incorrect.
Exceptions are where some quantity becomes infinitely small in an equation dependent on an infinitely large distance or time after some initial point in time, in which case I think it is acceptable to say that the mathematics of the equation is just saying that to within our current limits of knowledge, such quantities become smaller than we can currently measure at as large distances and time measurements which we can currently measure. For larger distances and times than we can currently make such measurements, however, the equations of the theory may be incorrect or incomplete, so we should always bear this in mind.
Now for the good answer about the zero Kelvin temperature problem which you gave a link about.
I did a search on Google for, ‘Boltzmann’, which gave links which included a good answer about the Boltzmann distribution at zero Kelvin temperature.
The formula for the Boltzmann Distribution is described in...
http://en.wikipedia.org/wiki/Boltzmann_distribution
The above article says that, ‘The Boltzmann distribution applies only to particles at a high enough temperature and low enough density that quantum effects can be ignored, and the particles are obeying Maxwell–Boltzmann statistics.’
Thus his formula is not meant to apply to absolute zero Kelvin, and so the reasoning about ‘negative temperatures’, etc, in the link which you sent is not applicable.
You may be interested in reading a bit about Boltzmann, who was widely recognised as brilliant, but probably had undiagnosed bipolar disorder.
http://en.wikipedia.org/wiki/Ludwig_Boltzmann
You may also be interested in the following link about the solving of some Boltzmann equations which were unsolved for 140 years until around May 2010.
http://www.sciencedaily.com/releases/2010/05/100513162755.htm
It could be said, and it is what I think is probably the case, that any infinitely large quantities arising in an equation of physics indicates that the theory and related equations is most probably incomplete, or incorrect.
A classic example of this is the ‘Ultra Violet Catastrophe’, where a quantity did go to infinity in an old theory (teh Rayleigh-Jeans theory), but it was resolved and pinned down by a later superior theory by Max Planck. The old theory was only a good approximation for a certain range of the variables.
I did a search on Google for ‘Ultraviolet Catastrophe’, and among other links found...
http://en.wikipedia.org/wiki/Ultraviolet_catastrophe
Another infinity problem is that of ‘Renormalisation’ in quantum theory, which you may also like to look up. This quantum theory related problem is, however, very complex, and I have not yet had time myself to look into it, and even understand exactly how it arises or what exactly it is, although I do understand that it is strongly related to how the inverse square laws blow up at a radius of zero, as noted by the Nobel Prize winner Stephen Weinberg in his books on Quantum Field Theory. You may like to do searches on Google on such topics.
I hope the above is of interest and helps. I think that such infinities in physical equations throw up some of the most interesting, important and puzzling problems in theoretical physics, which are usually at the cutting edge of theoretical and, indeed, experimental, physics.
Best Regards,
