  We investigate the statistical properties of fluctuations in active systems
that are governed by non-symmetric responses. Both an underdamped Langevin
system with an odd resistance tensor and an overdamped Langevin system with an
odd elastic tensor are studied. For a system in thermal equilibrium, the
time-correlation functions should satisfy time-reversal symmetry and the
anti-symmetric parts of the correlation functions should vanish. For the odd
Langevin systems, however, we find that the anti-symmetric parts of the
time-correlation functions can exist and that they are proportional to either
the odd resistance coefficient or the odd elastic constant. This means that the
time-reversal invariance of the correlation functions is broken due to the
presence of odd responses in active systems. Using the short-time asymptotic
expressions of the time-correlation functions, one can estimate an odd elastic
constant of an active material such as an enzyme or a motor protein.
