  We study the Langevin dynamics of the adaptive immune system, modelled by a
lymphocyte network in which the B cells are interacting with the T cells and
antigen. We assume that B clones and T clones are evolving in different thermal
noise environments and on different timescales. We derive stationary
distributions and use statistical mechanics to study clonal expansion of B
clones in this model when the B and T clone sizes are assumed to be the slow
and fast variables respectively and vice versa. We derive distributions of B
clone sizes and use general properties of ferromagnetic systems to predict
characteristics of these distributions, such as the average B cell
concentration, in some regimes where T cells can be modelled as binary
variables. This analysis is independent of network topologies and its results
are qualitatively consistent with experimental observations. In order to obtain
full distributions we assume that the network topologies are random and locally
equivalent to trees. The latter allows us to employ the Bethe-Peierls approach
and to develop a theoretical framework which can be used to predict the
distributions of B clone sizes. As an example we use this theory to compute
distributions for the models of immune system defined on random regular
networks.
