  From bacteria and sperm cells to artificial microrobots, self-propelled
microscopic objects at low Reynolds numbers often perceive fluctuating
mechanical and chemical stimuli and contact exterior wall boundaries both in
nature and the laboratory. In this study, we theoretically investigate the
fundamental features of microswimmers by focusing on their reciprocal
deformation. Although the scallop theorem prohibits the net locomotion of
reciprocal microswimmers, by analyzing a two-sphere swimmer model, we show that
in a fluctuating and geometrically confined environment, reciprocal
deformations can afford a statistically average displacement. After designing
the shape gait, a reciprocal swimmer can migrate in any direction, even in the
statistical sense, while the statistical average of passive rigid particles
statistically diffuses in a particular direction in the presence of external
boundaries. To elucidate this symmetry breakdown, by introducing an impulse
response function, we derive a general formula for predicting the nonzero net
displacement of a reciprocal swimmer. Using this theory, we determine the
relation between the shape gait and net locomotion as well as the net diffusion
constant increase and decrease owing to a reciprocal deformation. Based on
these findings and a theoretical formulation, we provide a fundamental basis
for environment-coupled statistical locomotion. Thus, this study is valuable
for understanding biophysical phenomena in fluctuating environments, designing
artificial microrobots, and conducting laboratory experiments.
