Dear friends,

Here are some facts that shed some light on the problem, although I can't be sure that they provide a full answer.

1. As the temperature decreases, the Maxwell-Boltzmann distribution of velocity moves its peak toward low velocities, but also becomes narrower. The probability of finding two (almost) equal velocity increases. At low temperatures, the Maxwell-Boltzmann distribution will become, for bosons Bose-Einstein, and for fermions Fermi distributions - see below.

2. Assuming also that the considered gas is provided by a reservoir of particles, s.t. with decreasing the temperature the density increases, the atoms become (on average) closer to one another, as said in the question. That has in fact a negative consequence, the bunching is blurred.

3. In consequence of 1 and 2, the average number of particles per phase-space cell increases. (For the usual 3D space, the volume of the unit cell in the phase-space is ħ3.)

4. Bunching of bosons and anti-bunching of fermions are two-particle (or multi-particle) interference phenomena. The anti-symmetry of the wave-function of two fermions identically polarized, has the effect that the closer the two fermions are in linear momentum, the amplitude of probability for them to get close to one another is destroyed. In one cell of the phase-space there can be at most 1 fermion. For identically polarized bosons the situation is opposite due to the symmetry of the wave-function. The closer they are in linear momentum and position, the amplitude of probability is enhanced. Thus in one phase-space cell tend to gather more than 1 boson.

As shown at the points 1 to 3, lowering the temperature increases the density in the phase-space i.e. the above quantum effects become more observable. The Maxwell-Boltzmann distribution splits into a Bose-Einstein one and a Fermi one.

5. The repulsion between clouds of electrons may be a more perturbing or less perturbing effect, depending on the type of atoms and other experimental details.

To discuss this at all, the term ``bunching tendency'' must be defined.  This paper might be a good starting point: http://jilawww.colorado.edu/bec/CornellGroup/theses/lewandowski_thesis.pdf

All this is, of course, just thermodynamics: the existence of different phases at all is the result of the competition between the different interactions, here it's the exchange with the thermal bath that defines the temperature on the one hand and the constraints imposed by Bose-Einstein statistics on the other hand. If the particles in question are subject to additional interactions, these, of course, also come into play.

Stam,

I don't see any relation between what I asked and what you sent me. Did you read the text of the question? I asked about bunching. In a BEC it doesn't happen. This is why the temperature has to be above the temperature of transition to BEC.

Once more, what is, *precisely*, meant by ``bunching''? Using vague terms just leads nowhere. 

Stam, you know what is boson bunching. Which "vague terms"? It is the tendency to appear in pairs (or even triples), two or more particles with the same quantum numbers, or occupying the same quantum state. To the contrary, the fermions won't appear more than one in a given quantum state.

It is said that in one cell of the phase-space one can find 2 or more (identical) bosons, but at most 1 fermion. This is the base of the Bose-Einstein statistic, respectively Fermi statistics.

I saw calculi of 2nd order correlations, but they start from the statistics. I ask about the phenomenology that leads to the statistics. Why the decrease in temperatures leads to increased population of the phase-space? Because of the increase in density? The density also blurs the effect.

(My friend, it's late in my country, we can continue tomorrow. Good night!)

The statistics are the starting point: that was what Bose found for photons, when he found another way of deriving Planck's law, what Einstein generalized for arbitrary integer spin and is called Bose-Einstein statistics. That integer spin is necessary for such statistics  is the spin-statistics theorem, cf. http://ejde.math.txstate.edu/conf-proc/04/w1/wightman.pdf

So it's the other way around: in quantum mechanics one can't label identical particles, particles that carry the same quantum numbers-one can only count them. This is a consequence of the uncertainty theorem. And in three space dimensions it turns out that only the even or odd permutations are relevant, in order to avoid overcounting configurations and, the even permutation is the appropriate one if the spin has integer values. So identical particles, of integer spin, are subject to an attractive interaction, that's long-ranged; identical particles of half-integer spin are subject to a repulsive interaction. 

This leads to the possibility of a phase transition, in the absence of any other interactions- what's called Bose-Einstein condensation and it isn't trivial to show that such a transition is, also, present, when interactions are included-that was what Bogoliubov found for the interactions between electrons and phonons, for example-the attractive interaction, mediated by the phonons, led to an instability, if that interaction could overcome the Coulomb repulsion and then the additional attractive  interaction between the bound states could lead to a condensate. 

By decrease in the temperature probably you meant to decrease the lattice temperature not the spin and/or nuclear spin temperatures. Lattice temperature has two parts translational and rotational, which are all interactions with the electromagnetic field as we have observed in their effects on the Raman Spectral lines.  Bouncing comes out because of the form of the of two body three body interaction potential between the atomic species, which controls the translational motion, quantization of which in the periodic lattice generates simply the boson field. 

The pair wise interaction potential decreases with the bouncing position or the minimum distance between atomic pairs monotonically up the dip then increases in a manner that its second derivative makes a turning point by changing its sign form positive to negative that softening drastically affects on the elastic properties such as thermal expansion coefficient - vibrational amplitude of translational motions, their mean and bouncing positions. The bouncing position deceases with temperature even though the mean position, which shout be  Invariant due the assumed quadratic approximation in the creation of the boson field now increases with temperature. I have presented here completely a classical physics arguments to explain what is going on in reality.

What I am seen every thing  from a different window, where the nature behaves completely nonlinear fashion not the energy of conservation but the dissipation -positive entropy production - controls and dictates the directions of  events. The masses and springs in that world have no meaning what so-ever, unless you go down down where every thing look like almost in the state of frozen. But I like that world as a mathematician since it gives me great opportunity play with. Best Regards to ALL.

Stam,

what I expect is an answer to my question, not a lecture on generalities about bosons. I am busy, very busy. If you don't know the answer, you don't have the obligation to answer. But please, don't waste my time.

The short answer is the spin-statistics theorem, along with standard results from thermodynamics, as explained in a previous message and as is discussed in any textbook on statistical physics. To understand what that means requires a minimum commitment.  On technical subjects there don't exist canned answers. The answer to the question has as prerequisite, knowledge of what the term ``boson'' *means*. 

Stam, if the answer is written in textbooks, I don't need you. Of course it is written in books, in articles. I read these things. My question was not a request for references. What is missing to me is a leading line. With decreasing temperature the population of the phase-space becomes denser. It's not clear why the density in the ordinary space is what leads to this, because as I said already, it also blurs the visibility of the bunching. What I asked for was a phenomenological explanation, not the mathematics in the books.

Books and articles begin from the Bose-Einstein distribution. The question is what leads to this distribution, why the temperature strengthens the bunching.

As to what boson means, all that you explained is known to me. The question is not what is a boson.

People that understand a phenomenon, are able to explain it in a few lines. Look at answers that I give to questions. There are always explanations in my own words, of the major factor standing behind the questioned phenomenon, not sending to references. References are only for details or for the mathematical treatment.

Many, many thanks!

 I don't know any thing about bosons but it is sure any particles no matter what they are if they have mutual attractive interactions they will coalesce if their vibrational energies with respect to their center of mass system show any reduction.  E = (n-3) 1/2kT, where n is the number of particle, and E total energy of the cluster of particles excluding the rotational energy. Where, the complete collapse is avoided by the hard core repulsive interaction.

I gave another explanation from the very nature. When it gets cold hedgehogs come together (bunching) till they feel pain from each other's thorns,  then they try to leave that mess for a while but then they are caught  by freezing  cold again .....That repeats till they got exhausted...

Dear friends,

Here are some facts that shed some light on the problem, although I can't be sure that they provide a full answer.

1. As the temperature decreases, the Maxwell-Boltzmann distribution of velocity moves its peak toward low velocities, but also becomes narrower. The probability of finding two (almost) equal velocity increases. At low temperatures, the Maxwell-Boltzmann distribution will become, for bosons Bose-Einstein, and for fermions Fermi distributions - see below.

2. Assuming also that the considered gas is provided by a reservoir of particles, s.t. with decreasing the temperature the density increases, the atoms become (on average) closer to one another, as said in the question. That has in fact a negative consequence, the bunching is blurred.

3. In consequence of 1 and 2, the average number of particles per phase-space cell increases. (For the usual 3D space, the volume of the unit cell in the phase-space is ħ3.)

4. Bunching of bosons and anti-bunching of fermions are two-particle (or multi-particle) interference phenomena. The anti-symmetry of the wave-function of two fermions identically polarized, has the effect that the closer the two fermions are in linear momentum, the amplitude of probability for them to get close to one another is destroyed. In one cell of the phase-space there can be at most 1 fermion. For identically polarized bosons the situation is opposite due to the symmetry of the wave-function. The closer they are in linear momentum and position, the amplitude of probability is enhanced. Thus in one phase-space cell tend to gather more than 1 boson.

As shown at the points 1 to 3, lowering the temperature increases the density in the phase-space i.e. the above quantum effects become more observable. The Maxwell-Boltzmann distribution splits into a Bose-Einstein one and a Fermi one.

5. The repulsion between clouds of electrons may be a more perturbing or less perturbing effect, depending on the type of atoms and other experimental details.

Dear Sofia,

I do not know the problem of bunching tendency, but it seems not surprising that in a magnetic field a gas appears in a magnetic trap. The magnetic field must increase the energy of condensation. Let me now consider the statistical approach:

The atoms of the gas are supposed in-discernable. For this reason generally people apply them the Bose-Einstein statistic. But as soon as we consider the volume of the atoms the hypothesis of in-discernable particles is wrong.

The statistical problem can be solve like this: How much important the absolute temperature T may be, it is the mean thermal energy U per particle which play a fundamental role in statistics. This leads to determine D(E,U) the density of probability per unit of energy for the energy E of the particles. E is the energy above the energy of condensation.

The problem is to show that the total energy of a given volume of gas shall be calculated by summing, on small segments of energy, the number of atoms with energy on these different segments. For a number N of atoms, this sum should be equal to N times the average U value per atom of thermal energy. To respect the average value of the energy; It should also be that the total probability on the set of the possible energy values must be equal to unity.

To solve the problem it should be noted that the energy exchanges that produce high or low energy particles are less probable than those who leave particles in the vicinity of the average value. There is therefore a maximum of statistical weight determining the probability value around average value.  On the other hand it is their own volume which forbids to two particles to occupy the same place but not the exclusion principle of Pauli.

The study of the variations of UD(E,U) as a function of E/U shows that the highest values are in the vicinity of energy zero. This result is normal because if one atom has a very high energy, it should be that a large number of other atoms have ceded a large part of their energy in order to respect the average U. Taking the parallel with money this is to say that it takes a lot of poor to make one rich.  See “Perturbations and Statistical Distribution of the Thermal Energy” on my site.

Xavier,

The bunching of the bosons comes from the symmetry of the wave-function of undistinguishable bosons. Given two such bosons, one in state a and one in state b, their wave-function is

|ψ> = (1/sqrt(2)) ( |1>_a |1>_b + |1>_b |1>_a )

Now, assume that the difference between the states a and b becomes smaller and smaller. One gets in the limit

|φ> = (1/sqrt(2)) ( |1>_a |1>_a + |1>_a |1>_a ) = sqrt(2) |2>_a 

To feel the difference from the identical fermions, their wave-function is antisymmetrical

|ψ> = (1/sqrt(2)) ( |1>_a |1>_b - |1>_b |1>_a ),

therefore when the difference between the two states diminishes, one gets

|φ> = (1/sqrt(2)) ( |1>_a |1>_a - |1>_a |1>_a ) = 0.

The question here was, why lowering the temperature causes the states a and b to become closer. The state of a particle in a non-interacting gas is given by its energy of movement, or by the absolute value of the linear momentum. Indeed, as I said in my previous post, lowering the temperature the distribution of linear momenta becomes NARROWER. No doubt, the average linear momentum decreases with he temperature, but it is the WIDTH of the distribution that does the job, not its average value. The width decreases with the temperature, you can see this at

https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution .

Dear Sofia;

I think this problem may be handled by using the same approach employed in the nucleation theory advocated by Ya. B. Zel'dovich (1942) where he used Fokker-Planck equation to describe the growth of the nuclei by assuming that it is macroscopic objects. Where the dynamic distribution function insize space may be represented by diffusion equation with source term. The Greens function of the diffusion part is the generalized fct of Dirac type, which is represented by a Gaussian fct  set having amplitude  (Pi Dt)1/3exp- (r-ro)2/4Dt.  which shows absolute condensation of a single particle at ro when T goes to zero since  D=Do exp-Q/kT. All you have to do is generalized that for a pair of particle if you have time?  SEE; Physical Kinetics by Landau&Lifshitz, p. 427

Dear Tarik,

To look for mathematics it is no problem, there are tens of articles. What the question asked for, was a phenomenologic explanation. We are intoxicated with mathematics, but what people yearn for and systematically misses in articles is a phenomenologic description.

About an analogy between boson gases, and growth of nuclei, it is difficult to say, because in a nucleus there are fermions not bosons. For observing the bunching of bosonic atoms the repulsion between the electron clouds of the atoms has to be a weak effect, a weak perturbation of the bunching. To the contrary, in a nucleus the electric repulsion and the nuclear attraction are major effects, at least of the same order of magnitude as the Pauli repulsion between identical fermions. But, again, in a nucleus we have fermions, not bosons.

Dear Sofia,

You have missed the meanings of nuclei or embryo, which are terminologies used in the kinetics and statistics of phase transformation theory. Where the nuclei is a macroscopic entity obeys Maxwell-Boltzmann statistics, and it a embryo having critical size to be allowed  to  leave the dynamical distribution, and to grow. I prefer to deal with the non-linear dynamics version of solitons and solitary particles using analytical as well as numerical methods.

We have performed quite  alot computer experiments on the surface solitons  induced by electromigration employing  a mathematical model deduced from irreversible thermodynamics  of surfaces and interfaces. Best Regards

Note: I am reading your and other colleagues comments with great pleasure but I wander how long this picture of particles, which relies heavily on the conservative forces  even harmonic interactions will continue. 

I am puzzled by your statement that ``in BEC bunching does not occur''. Is not the very formation of the condensate a macroscopic expression of the bunching phenomenon? That, at least, is how I have always understood it. Thus, the particles that do not belong to the condensate may well fail to show bunching, but the particles in the condensate are macroscopically bunched.

If I am not wrong in this, then the increase of microscopic bunching could be understood as a natural precursor to the macroscopic bunching arising at condensation

Dear F. Leyvraz,

I am no specialist in condensates. I deal with gases very close to, though above the temperature of transition to condensate. But I can tell you for sure that in a condensate one doesn't see the bunching. Let me remind you that bunching is a phenomenon, see the attach, by which two or three particles can appear very close to one another comparatively with the distance to other particles. In a condensate you won't see such differences in distances. But, in principle, a condensate is a state with big density, so, it may be that the difference disappears exactly because the particles become anyway close to one another. But, again, I am no specialist in condensates s.t. I am not sure.

Hello,

This is a very interesting question as an underlying problem is the representation that one may make of the system's constituents (structureless bosons or composite bosons) and how this representation relates to results and understanding obtained in idealistic cases to construct a phenomenological model of the boson bunching phenomenon.

Sofia has already raised some elements of answer to her own question. Bunching in momentum space as the bosonic system's temperature is lowered (yet above BEC) is evident if the bosons are structureless; if these are composite bosons made of the same types of "elementary" components, e.g. excitons, exchange effects will play a significant role in the scattering process (see the works of Monique Combescot and co-workers), yet if the system is not too dense bosonic-like behavior is actual as bosonic stimulation may be observed. This bosonic stimulation process originates in the build-up of a coherent matter wave. The width of the velocity distribution mentioned by Sofia will always remain finite because of velocity fluctuations due to various reasons.

Bunching in real space as the system's temperature T is lowered is on the one hand facilitated by the decrease of the speed of the composite bosons' center of mass motion, as on the other hand as the de Broglie wavelength increases as T decreases, the individual matter waves extend more and more, which makes the system effectively denser: this does not mean that all the composite particles gather in a tiny region of space as fundamentally one never knows where the quantum particle is along the spatial extension of its wave function and  in reason of indistinguishability  one absolutely never knows which constituent is which. Note that for a many-body system, the de Broglie wavelength I mention is that of the "fictitious" particle that represents the center of mass. For atoms, I believe that one would usually consider that their internal degrees of freedom are highly constrained and that only the degrees of freedom of the center of mass are relevant for the dynamics. This said the interactions between atoms are usually constituted by a short range hard core followed by an attractive part as the interatomic distance increases, which thus bear some resemblance to a Lennard Jones potential. For cold alkali atoms, one would get the interaction potential from experimental data or ab initio calculations for the short range, plus for the long range a van der Waals dispersion and a contribution  that accounts for the exchange energy (Smirnov and Chibisov). The composite nature of a complex boson constrains highly its interactions with its siblings, but does not completely take away their bosonic character.

Now for a given phase space hypervolume, I would say that composite bosons concentration in the momentum volume would be higher than it is in real space volume (both volumes being projections of the hypervolume in particular sectors of the phase space). Note that this may be related to Heisenberg's uncertainty principle on position and momentum. 

Anyway, thanks for the question! It got me thinking about interesting physics.