Monday, December 19, 2011

Bose-Einstein statistics

Increasing disorder

In statistical mechanics, Bose–Einstein statistics ... determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Fermi–Dirac and Bose–Einstein statistics apply when quantum effects are important and the particles are "indistinguishable"... Fermi–Dirac statistics apply to fermions... and Bose–Einstein statistics apply to bosons...

As the quantum concentration depends on temperature; most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit unless they have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperature or at low concentration.

Bosons, unlike fermions, are not subject to the Pauli exclusion principle: an unlimited number of particles may occupy the same state at the same time. This explains why, at low temperatures, bosons can behave very differently from fermions; all the particles will tend to congregate at the same lowest-energy state, forming what is known as a Bose–Einstein condensate.

B–E statistics was introduced for photons in 1924 by Bose and generalized to atoms by Einstein in 1924-25.

Errare humanum est
After Satyendra Nath Bose surprised himself by getting unexpected results during that famed lecture at the University of Dhaka he carefully went over his "error". Repeated calculations proved to his own great amazement that, in fact, it was not an error but rather a discovery of something quite important - a truth that had not been previously known by the entire human race.

Erring is human, as the Romans said.

But not recognizing the truth is also a very human characteristic.

When there is a commonly accepted scientific truth and some Bengali fellow comes with a short article to the publisher claiming it is not true - the editor in chief has a dilemma. Should he or she allow the publication of a possibly embarrassing article by some nobody in the distinguished scientific journal?

The answer to that question depends on the ability of the editors to distinguish truth from error - not an easy task when someone nobody presents a paper that challenges the common knowledge and accepted scientific traditions and theory.

Satyendra Bose was not the first to have his paper on a new discovery, observation, theory turned down by almost all respected scientific journals. Charles Darwin had his problems in the 19th century. Just now, in 2011, we have been told the story of the Israeli Nobel Prize Winner in Physics, Dan Shechtman, how his discovery of chemical structures called quasicrystals was shunned by the scientific community and his papers left unpublished.

Scientific research fixes itself as knowledge increases. But the process is painful and possibly even fatal to those thrown into the dynamics where human vanity and glory produce dislike of anything challenging the peace of mind of the best and brightest.

Einstein to the help
Physics journals refused to publish Bose's paper. Various editors ignored his findings, contending that he had presented them with a simple mistake. Discouraged, he wrote to Albert Einstein, who immediately agreed with him. His theory finally achieved respect when Einstein sent his own paper in support of Bose's to Zeitschrift für Physik, asking that they be published together. This was done in 1924.

There you have it.

Respected editors of Physics journals refuse to publish a controversial paper talking about some "error". These people know a lot - but not enough.

That former patent office worker in Bern with sometimes funny hairstyle immediately understood that this paper is true. Science does not like authority faith - for good reasons! - but the authority of Albert Einstein was already in 1924 strong enough to give Bose's paper a chance. He not only understood the Bengali guy's math but applied the brilliant idea from photons to the more general level of atoms.

Not that Einstein had not experienced himself how it is to get letters of rejection from respected scientific publishers.

Simple way to look at it
A much simpler way to think of Bose–Einstein distribution function is to consider that n particles are denoted by identical balls and g shells are marked by g-1 line partitions. It is clear that the permutations of these n balls and g-1 partitions will give different ways of arranging bosons in different energy levels.

Say, for 3(=n) particles and 3(=g) shells, therefore (g-1)=2, the arrangement may be like |..|. or ||... or |.|.. etc.

Hence the number of distinct permutations of n + (g-1) objects which have n identical items and (g-1) identical items will be:


How simple can you get!

So what?
Bose-Einstein statistics have a truly significant dimension in interdisciplinary applications.

Here is where theoretical mathematical thinking meats not only Physics but also other branches of science involving various types of networks. For example in Economics, in understanding growth patterns of computer networks or - and here is for me the most surprising point - even in Evolutionary Biology.

The evolution of many complex systems, including the World Wide Web, business, and citation networks, is encoded in the dynamic web describing the interactions between the system’s constituents. Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can undergo Bose–Einstein condensation. Addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts that the “first-mover-advantage,” “fit-get-rich(FGR),” and “winner-takes-all” phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks.

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