  In this paper, we study a biased version of the nearest-neighbor
transposition Markov chain on the set of permutations where neighboring
elements $i$ and $j$ are placed in order $(i,j)$ with probability $p_{i,j}$.
Our goal is to identify the class of parameter sets ${\bf P} = \{p_{i,j}\}$ for
which this Markov chain is rapidly mixing. Specifically, we consider the open
conjecture of Jim Fill that all monotone, positively biased distributions are
rapidly mixing.
  We resolve Fill's conjecture in the affirmative for distributions arising
from $k$-class particle processes, where the elements are divided into $k$
classes and the probability of exchanging neighboring elements depends on the
particular classes the elements are in. We further require that $k$ is a
constant, and all probabilities between elements in different classes are
bounded away from $1/2$. These particle processes arise in the context of
self-organizing lists and our result also applies beyond permutations to the
setting where all particles in a class are indistinguishable. Additionally we
show that a broader class of distributions based on trees is also rapidly
mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our
work generalizes recent work by Haddadan and Winkler (STACS '17) studying
3-class particle processes.
  Our proof involves analyzing a generalized biased exclusion process, which is
a nearest-neighbor transposition chain applied to a 2-particle system. Biased
exclusion processes are of independent interest, with applications in
self-assembly. We generalize the results of Greenberg et al. (SODA '09) and
Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the
probability of swapping neighboring elements to depend on the entire system, as
long as the minimum bias is bounded away from $1$.
