Creation and Annihilation Operators For Reaction-diffusion Equations
The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules A diffuse and interact on contact, forming an inert product: A + A → ∅ . To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider particles at a site on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times is proportional to, say to hop left and to hop right. All particles will stay put with a probability .
We can now describe the occupation of particles on the lattice as a `ket' of the form | n1, n2, ... . A slight modification of the annihilation and creation operators is needed so that
and
- .
This modification preserves the commutation relation
- ,
but allows us to write the pure diffusive behaviour of the particles as
The reaction term can be deduced by noting that particles can interact in different ways, so that the probability that a pair annihilates is and the probability that no pair annihilates is leaving us with a term
yielding
Other kinds of interactions can be included in a similar manner.
This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.
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