Overdamped stochastic thermodynamics with multiple reservoirs

After establishing stochastic thermodynamics for underdamped Langevin systems in contact with multiple
reservoirs, we derive its overdamped limit using timescale separation techniques. The overdamped theory is
different from the naive theory that one obtains when starting from overdamped Langevin or Fokker-Planck
dynamics and only coincides with it in the presence of a single reservoir. The reason is that the coarse-grained fast
momentum dynamics reaches a nonequilibrium state, which conducts heat in the presence of multiple reservoirs.
The underdamped and overdamped theory are both shown to satisfy fundamental fluctuation theorems. Their
predictions for the heat statistics are derived analytically for a Brownian particle on a ring in contact with two
reservoirs and subjected to a nonconservative force and are shown to coincide in the long-time limit.