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Abstract |
At the mesoscopic scale, the equilibrium dynamics of a system are typically modeled as reversible and
Markovian, preserving the Boltzmann measure. The macroscopic properties of the system are fully described
by ensemble averages following this measure and they can be estimated by time expectations along a carefully
chosen Markov processes. Designing efficient Markovian dynamics leaving invariant some target distribution
comes down to finding how to design a global structure of probability flows from local information. In the
general case, this is made possible by breaking the time-reversal symmetry of the usual detailed-balance
dynamics.
This presentation will discuss a collection of works pertaining to non-reversible Markov processes and explore
the interplay between the invariance of the stationary probability distribution and certain key symmetries.
These key symmetries now serve as the basis for necessary probability flow conservation, having replaced the
time-reversal symmetry that is now broken. Notably, these investigations span both equilibrium and non-
equilibrium systems. It is indeed possible to define the universality classes of some active particle systems
based on the satisfaction or not of some skew-detailed symmetry. Interestingly, this reflection extends to
reversible cluster algorithms as well, as it revolves around the same core concept of phase space extension. In
such algorithms, the flip involutive symmetry plays a crucial role in ensuring correctness, and can be mapped
to the conservation of random currents in the flow or loop graphical representation.
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