  We consider the correlations and the hydrodynamic description of random
walkers with a general finite memory moving on a $d$ dimensional hypercubic
lattice. We derive a drift-diffusion equation and identify a memory-dependent
critical density. Above the critical density, the effective diffusion
coefficient decreases with the particles' propensity to move forward and below
the critical density it increases with their propensity to move forward. If the
correlations are neglected the critical density is exactly $1/2$. We also
derive a low-density approximation for the same time correlations between
different sites. We perform simulations on a one-dimensional system with
one-step memory and find good agreement between our analytical derivation and
the numerical results. We also consider the previously unexplored special case
of totally anti-persistent particles. Generally, the correlation length
converges to a finite value. However in the special case of totally
anti-persistent particles and density $1/2$, the correlation length diverges
with time. Furthermore, connecting a system of totally anti-persistent
particles to external particle reservoirs creates a new phenomenon: In almost
all systems, regardless of the precise details of the microscopic dynamics,
when a system is connected to a reservoir, the mean density of particle at the
edge is the same as the reservoir following the zeroth law of thermodynamics.
In a totally anti-persistent system, however, the density at the edge is always
higher than in the reservoir. We find a qualitative description of this
phenomenon which agrees reasonably well with the numerics.
