  Continuous-time Markov chains associated to finite-volume discretization
schemes of Fokker-Planck equations are constructed. Sufficient conditions under
which quantitative exponential decay in the $\phi$-entropy and Wasserstein
distance are established, implying modified logarithmic Sobolev, Poincar\'e,
and discrete Beckner inequalities. The results are not restricted to additive
potentials and do not make use of discrete Bochner-type identities. The proof
for the $\phi$-decay relies on a coupling technique due to Conforti, while the
proof for the Wasserstein distance uses the path coupling method. Furthermore,
exponential equilibration for discrete-time Markov chains is proved, based on
an abstract discrete Bakry-Emery method and a path coupling.
