Clausius inequality and H-theorems for some models of random wealth exchange.
We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution
equation that describes random wealth exchanges. In such kinetic models economical
agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy.
It appears useful to reformulate the problem and represent the dynamics as a combination
of two processes. The first is a linear transformation of a two-particle distribution
function during the act of exchange while the second one corresponds to new random pairing
of agents and plays a role of some kind of feedback control. This representation leads
to a Clausius-type inequality which suggests a new interpretation of the exchange process
as an irreversible relaxation due to a contact with a reservoir of a special type. Only in
some special cases when equilibrium distribution is exactly a gamma distribution, this inequality
results in the H-theorem with monotonically growing ‘entropy’ functional which
differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule
the evolution has some features of relaxation to a non-equilibrium steady state and it is
still unclear if any general H-theorem could exist.