  For driven open systems in contact with multiple heat reservoirs, we find the
marginal distributions of work or heat do not satisfy any fluctuation theorem,
but only the joint distribution of work and heat satisfies a family of
fluctuation theorems. A hierarchical structure of these fluctuation theorems is
discovered from microreversibility of the dynamics by adopting a step-by-step
coarse-graining procedure in both classical and quantum regimes. Thus, we put
all fluctuation theorems concerning work and heat into a unified framework. We
also propose a general method to calculate the joint statistics of work and
heat in the situation of multiple heat reservoirs via the Feynman-Kac equation.
For a classical Brownian particle in contact with multiple heat reservoirs, we
verify the validity of the fluctuation theorems for the joint distribution of
work and heat.
