Bose-Einstein Condensation: A Network Theory Approach

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Fig.1 Bose-Einstein Condensation at 400, 200, and 50 nano-Kelvins.The peaks show that as the temperature goes down, more and more atoms "condense" to the same energy level.

1 The Concept
2 History and Measurement
3 Connection with Network Theory
4 What all this means
5 References

[edit] The Concept

The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by the Bose-Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons (Such as the photon and helium-4). In Bose-Einstein statistics, the Pauli exclusion principle is not obeyed so that any number of identical bosons can be in the same state. Einstein speculated that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. This transition occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:

$T_c=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{h^2}{2\pi m k_B}$

where:

$T c$	is	the critical temperature,
$n$		the particle density,
$m$		the mass per boson,
$h$		Planck's constant,
$k B$		the Boltzmann constant, and
$ζ$		the Riemann zeta function; $ζ(3 / 2)$ ≈ 2.6124.

[edit] History and Measurement

In the early 1920s Satyendra Nath Bose was studying the new idea that the light came in little discrete packets or quanta. Bose assumed certain rules for deciding when two photons should be counted up as either identical or different. We now call these rules Bose statistics (or sometimes Bose-Einstein statistics). Albert Einstein guessed that these same rules might apply to atoms. He worked out the theory for how atoms would behave in a gas if these new rules applied. What he found was that the equations said that generally there would not be much difference, except at very low temperatures. If the atoms were cold enough, something very unusual was supposed to happen. It was so strange he was not sure it was correct. Not all types of atoms actually follow the rules for Bose statistics. However, some atoms do, and for those Einstein's predictions were right. But even for those kinds of atoms, he did not realize the most important effects that his equations were predicting. The effects come from the fact that, at very low temperatures, most of the atoms are in the same quantum level. If you put an atom in any kind of container, even a mixing bowl, it also can only have certain particular energies. It cannot roll around in there with just any speed it wants. It has to choose from a particular set of allowed energies. They are so close together in energy that you never notice there are tiny steps. What Einstein's equations predicted was that at normal temperatures the atoms would be in many different levels. However, at very low temperatures, a large fraction of the atoms would suddenly go crashing down into the very lowest energy level. It means that all the atoms are absolutely identical (see how this happens). There is no possible measurement that can tell them apart.The first such Bose-Einstein Condensate in alkali gases (Fig.1) was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder and independently by Wolfgang Ketterle of the Massachusettes Institute of Technology. They were awarded the Nobel Prize in Physics in 2001.

[edit] Connection with Network Theory

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^[1].

Fig.2 Schematic illustration of the mapping between the network model and the Bose gas^[1].

Starting from the fitness model, the mapping of a Bose gas to a network can be done by assigning an energy ∈_i to each node, determined by its fitness through the relation^[2]

$\epsilon_i=-\frac{1}{\beta}\ln{\eta_i}$

where β=1/T plays the role of inverse temperature.An edge between two nodes i and j, having energies ∈_i and ∈_j , corresponds to two noninteracting particles, one on each energy level (Fig.2). Adding a new node l to the network corresponds to adding a new energy level ∈_l and 2m new particles to the system. Half of these particles are deposited on the level ∈_l (since all new edges start from the new node), while the other half are distributed between the energy levels of the end points of the new edges, the probability that a particle lands on level i being given by

$\prod_i=\frac{e^{\beta\epsilon_i}k_i}{\sum e^{\beta\epsilon_i}k_i}$

The continuum theory predicts that the rate at which particles accumulate on energy level ∈_i is given by

$\frac{\partial k_i(\epsilon_i,t,t_i)}{\partial t}=m\frac{e^{\beta\epsilon_i}k_i(\epsilon_i,t,t_i)}{Z_t}$

where $k$ _i(∈_i,t,t_i) is the occupation number of the level i and Z_t is the partition function, defined as

$Z_t=\sum_j t e^{\beta\epsilon_i}k_j((\epsilon_i,t,t_i))$

The solution of this differential equation is

$k_i((\epsilon_i,t,t_i)=m\left(\frac{t}{t_i}\right)^{f(\epsilon_i)}$

where the dynamic exponent f(∈) satisfies f(∈)=e^-β(∈-μ), μ plays the role of the chemical potential, satisfying the equation

$\int deg(\epsilon)\frac{1}{e^{\beta(\epsilon-\mu)}-1}=1$

and deg(∈) is the degeneracy of the energy level ∈. In the t → ∞ limit the occupation number, giving the number of particles with energy ∈, follows the familiar Bose statistics

$n(\epsilon)=\frac{1}{e^{\beta(\epsilon -\mu)}-1}$

[edit] What all this means

Fig.3 Numerical evidence for Bose-Einstein condensation in a network model^[1].

The mapping of a Bose gas predicts the exisitance of two distinct phases as a function of the energy distribution. In the fit-get-rich phase, describing the case of uniform fitness, the fitter nodes acquire edges at a higher rate than older but less fit nodes. In the end the fittest node will have the most edges, but the richest node is not the absolute winner, since its share of the edges (i.e the ratio of its edges to the total number of edges in the system) reduces to zero in the limit of large system sizes (Fig.2(b)). The unexpected outcome of this mapping is the possibility of Bose-Einstein Condensation for $T < T$ _BE, when the fittest node acquires a finite fraction of the edges and maintains this share of edges over time (Fig.2(c)).

A representative fitness distribution ρ(η) that leads to a condensationis

ρ(η) = (1 - η) λ

with λ=1 However, the existence of the Bose-Einstein Condensation or the fit-get-rich phase does not depend on the temperature or β of the system but depends only on the functional form of the fitness distribution ρ(η) of the system. In the end, the β falls out of all topologically important quantities. In fact it can be shown that Bose-Einstein Condensation exists in the fitness model even without mapping to a Bose gas^[3]. A similar gelation can be seen in models with superlinear preferential attachment^[4], however, it is not clear whether this is an an accident or a deeper connection lies between this and the fitness model.

[edit] References

^ ^a ^b ^c Bianconi, G., and A.-L. Barabási, 2001, Phys. Rev. Lett. 86, 5632.
^ R. Albert, and Barabási, A.-L., 2002, Rev. Mod. Phys. 74, 47.
^ Dorogovtsev, S. N., and J. F. F. Mendes, 2001, Phys. Rev. E 63, 056125.
^ Krapivsky, P. L., S. Redner, and F. Leyvraz, 2000, Phys. Rev. Lett. 85, 4629.