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.

In physics, a Bose-Einstein condensate is a state of matter that occurs in certain gases at very low temperatures. Any elementary particle, atom, or molecule, can be classified as one of two types: a Boson or a Fermion. For example, an electron is a Fermion, while a photon or a helium atom is a Boson. In Quantum mechanics, the energy of a (bound) particle is limited to set of discrete values, called energy levels. An important characteristic of a Fermion is that it obeys the Pauli exclusion principle, which states that no two Fermions may occupy the same energy level. Bosons, on the other hand, do not obey the exclusion principle, and any number can exist in the same energy level. As a result, at very low energies (or temperatures), a great majority of the Bosons in a Bose gas can be crowded into the lowest energy state, creating a Bose-Einstein condensate.

In network theory, a network is characterized by a set of nodes or vertices and a set of links between these nodes. This is a very general concept, and may be applied to many situations, such as the world wide web, in which each page is a node in the network, and every link from that page to another page is a link in the network. A random graph is a network which changes in time, with nodes and links appearing and disappearing randomly, but with certain definite probablilites. Again, one possible model is the internet, which also changes in time. In a particular type of random graph, called a fitness model, the nodes and links have simple probabilities of appearing and disappearing. ^[1]

In the late 1990's, Ginestra Bianconi was a graduate student, working with Dr. Albert-László Barabási, a noted network theorist^[1] . At his request, she began investigating the fitness model, and realized that if each node in the network were thought of as an energy level, and each link as a particle, then a perfect analogy could be drawn between the mathematics of the network and the mathematics of a Bose gas. In particular, she found that under certain conditions, a single node could acquire most, if not all of the links in the network, resulting in the network analog of a Bose-Einstein condensate. These results have implications for any real situation involving random graphs, including the world wide web, social networks, and financial markets.

[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 Bose gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:

where:

Tc	is	the critical temperature,
n		the particle density,
m		the mass per boson,
h		Planck's constant,
kB		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 light comes 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^[2].

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^[3]

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

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

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

The solution of this differential equation is

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

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

[edit] What all this means

Fig.3 Numerical evidence for Bose-Einstein condensation in a network model^[2].

The mapping of a Bose gas predicts the existence 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 condensations

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^[4]. A similar gelation can be seen in models with superlinear preferential attachment^[5], however, it is not clear whether this is an accident or a deeper connection lies between this and the fitness model.

[edit] References

1. ^ ^{a b} Barabási, Albert-László (2002). Linked. Cambridge, MA: Perseus Publishing. ISBN 0-7382-0667-9. 
2. ^ ^{a b} Bianconi, G.; Barabási, A.-L. (2001). "Bose-Einstein Condensation in Complex Networks." Phys. Rev. Lett. 86: 5632–35.
3. ^ Albert, R.; Barabási, A.-L. (2002). "Statistical mechanics of complex networks." Rev. Mod. Phys. 74: 47–97.
4. ^ Dorogovtsev, S. N.; Mendes, J. F. F. (2001). "Scaling properties of scale-free evolving networks:  Continuous approach. Phys. Rev. E 63: 056125.
5. ^ Krapivsky, P. L.; Redner, S.; Leyvraz, F. (2000). "Connectivity of Growing Random Networks." Phys. Rev. Lett. 85: 4629–32.