# Bose-Einstein condensate of annihilating paritcles

### modelled as a Zero Range process

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## Abstract

In this thesis the behaviour of a Bose Einstein condensate is explored that consists of bosons that annihilate. In order to do this a system where bose einstein condensation occurs is modeled as a Zero Range process which is a special case of a Markov process.

First we made a single slow site Zero range process with uniform rates and see how this would be influenced by annihilating particles. For this system it is shown that, in the large system limit, the number of particles in the slow site is given by a differential equation. A series of realization of different sizes of this system are done to support that the fraction of the particles in the ground state converges

to the differential equation.

In order to relate this mathematical approach to physical Bose Einstein condensates it is established that the equilibrium distributions of multiple bosons in one system is the same as the detailed balance of a zero range process where the transition rates depend on the occupation of a site, with an indication function, that is equal to 0 when there are no particles at that site and some constant 𝑐 if there is one

or more particles at that site.

In statistical physics we need to take the energy of the state of the system into account. If the particles do not have interaction with each other this is the sum over all particles of the energy of the particle site. As a zero range model only has rates with one particle hop we can see that the difference in energy of the system is equal to the difference in energy of the sites of hop. In order to stay in detailed

balance the rate for the specific hop must be exp(BΔ𝐸) times the opposite rate, where B = 1 / 𝑘_b 𝑇.

Now we can go from any system with a set of spin orbitals and energies of those spin orbitals and design a zero range model where the spin orbitals correspond to sites of the zero range model with rates between those sites, such that the systems detailed balance distribution is the same distribution as the equilibrium of the physical system. In this zero range model an annihilation term is introduced.

This can be any function of occupation for the site, but in this report a simple relation that decreases when Δ𝐸 increases is used.

In order to see the effect on such systems the system is numerically approached for a 100 particles system in a 3 dimensional harmonic potential. This system is too complicated to get an differential equation that can be easily solved. Therefore it is approximated. This numerical approximation appears to have similar behaviour to the numerical approximation done in [3]. However the results are not identical.

The advantage of the model in this thesis is that an annihilation therm can be implemented. This approximation of a 3 dimensional harmonic oscillator with annihilating particles is done. If annihilation is slow enough, the exited sites form a different equilibrium compared to a Bose Einstein condensate of non annihilating particles. This happens as long as there is a condensate in the ground state.