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## 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.