  In this paper, we consider a diffusive mobile molecular communication (MC)
system consisting of a pair of mobile transmitter and receiver nano-machines
suspended in a fluid medium, where we model the mobility of the nano-machines
by Brownian motion. The transmitter and receiver nano-machines exchange
information via diffusive signaling molecules. Due to the random movements of
the transmitter and receiver nano-machines, the statistics of the channel
impulse response (CIR) change over time. We introduce a statistical framework
for characterization of the impulse response of time-variant MC channels. In
particular, we derive closed-form analytical expressions for the mean and the
autocorrelation function of the impulse response of the channel. Given the
autocorrelation function, we define the coherence time of the time-variant MC
channel as a metric that characterizes the variations of the impulse response.
Furthermore, we derive an analytical expression for evaluation of the expected
error probability of a simple detector for the considered system. In order to
investigate the impact of CIR decorrelation over time, we compare the
performances of a detector with perfect channel state information (CSI)
knowledge and a detector with outdated CSI knowledge. The accuracy of the
proposed analytical expression is verified via particle-based simulation of the
Brownian motion.
