  The Jarzynski identity can be applied to instances when a microscopic system
is pulled repeatedly but quickly along some coordinate, allowing the
calculation of an equilibrium free energy profile along the pulling coordinate
from a set of independent non-equilibrium trajectories. Using the formalism of
Wiener stochastic path integrals in which we assign temperature-dependent
weights to Langevin trajectories, we derive exact formulae for the temperature
derivatives of the free energy profile. This leads naturally to analytical
expressions for decomposing a free energy profile into equilibrium entropy and
internal energy profiles from non-equilibrium pulling. This decomposition can
be done from trajectories evolved at a unique temperature without repeating the
measurement as done in finite-difference decompositions. Three distinct
analytical expressions for the entropy-energy decomposition are derived: using
a time-dependent generalization of the weighted histogram analysis method, a
quasi harmonic spring limit, and a Feynman-Kac formula. The three novel
formulae of reconstructing the pair of entropy-energy profiles are exemplified
by Langevin simulations of a two-dimensional model system prototypical for
force-induced biomolecular conformational changes. Connections to
single-molecule experimental means to probe the functionals needed in the
decomposition are suggested.
