Stochastic thermodynamics provides a conceptual framework for
describing small systems embedded in a heat bath and mechanically
or chemically driven to non-equilibrium. Both the first law and
entropy production can be consistently defined along single
trajectories. An infinity of integral fluctuation theorems hold,
among which the Jarzynski relation and the one on total entropy
production are prominent ones [1].

After briefly reviewing and illustrating these foundations
using a driven colloidal particle as paradigm, I will present
within this scheme our recent work concerning  (i) optimal
finite-time processes and (ii) extended fluctuation-dissipation
theorems (FDTs) and generalized Einstein relations.

The optimal protocol of an external control parameter minimizes
the mean work required to drive the system from one given
equilibrium state to another in a finite time.  Explicit solutions
both for a moving laser trap and a time-dependent strength of such
a trap show finite jumps of the optimal protocol to be typical both
at the beginning and the end of the process [2].

The Einstein relation connecting diffusion constant and mobility
is violated beyond the linear response regime. Based on our recent
extension of the FDT [3], we have derived and measured an additive
correction term which involves an integral over measurable
correlation functions [4].

[1] U. Seifert, PRL 95: 040602, 2005.
[2] T. Schmiedl and U. Seifert, PRL 98: 108301, 2007.
[3] T. Speck and U. Seifert, EPL 74: 391, 2006.
[4] V. Blickle, T. Speck, U.S., C. Bechinger, PRL 98: 210601, 2007.