  Nonequilibrium processes of small systems such as molecular machines are
ubiquitous in biology, chemistry and physics, but are often challenging to
comprehend. In the past two decades, several exact thermodynamic relations of
nonequilibrium processes, collectively known as fluctuation theorems, have been
discovered and provided critical insights. These fluctuation theorems are
generalizations of the second law, and can be unified by a differential
fluctuation theorem. Here we perform the first experimental test of the
differential fluctuation theorem, using an optically levitated nanosphere in
both underdamped and overdamped regimes, and in both spatial and velocity
spaces. We also test several theorems that can be obtained from it directly,
including a generalized Jarzynski equality that is valid for arbitrary initial
states, and the Hummer-Szabo relation. Our study experimentally verifies these
fundamental theorems, and initiates the experimental study of stochastic
energetics with the instantaneous velocity measurement.
