  Local Shannon entropy lies at the heart of modern thermodynamics, with much
discussion of trajectory-dependent entropy production. When taken at both
boundaries of a process in phase space, it reproduces the second law of
thermodynamics over a finite time interval for small scale systems. However,
given that entropy is an ensemble property, it has never been clear how one can
assign such a quantity locally. Given such a fundamental omission in our
knowledge, we construct a new ensemble composed of trajectories reaching an
individual microstate, and show that locally defined entropy, information, and
free energy are properties of the ensemble, or trajectory-independent true
thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and
Landauer's principle can be generalized naturally as properties of the
ensemble, and that trajectory-free state functions of the ensemble govern the
exact mechanism of non-equilibrium relaxation.
