  We introduce an analytical model to study the evolution towards equilibrium
in spatial games, with `memory-aware' agents, i.e., agents that accumulate
their payoff over time. In particular, we focus our attention on the spatial
Prisoner's Dilemma, as it constitutes an emblematic example of a game whose
Nash equilibrium is defection. Previous investigations showed that, under
opportune conditions, it is possible to reach, in the evolutionary Prisoner's
Dilemma, an equilibrium of cooperation. Notably, it seems that mechanisms like
motion may lead a population to become cooperative. In the proposed model, we
map agents to particles of a gas so that, on varying the system temperature,
they randomly move. In doing so, we are able to identify a relation between the
temperature and the final equilibrium of the population, explaining how it is
possible to break the classical Nash equilibrium in the spatial Prisoner's
Dilemma when considering agents able to increase their payoff over time.
Moreover, we introduce a formalism to study order-disorder phase transitions in
these dynamics. As result, we highlight that the proposed model allows to
explain analytically how a population, whose interactions are based on the
Prisoner's Dilemma, can reach an equilibrium far from the expected one; opening
also the way to define a direct link between evolutionary game theory and
statistical physics.
