  The fluctuation relations have received considerable attention since their
emergence and development in the 1990s. We present a summary of the main
results and suggest ways to interpret this material. Starting with a
consideration of the under-determined time evolution of a simple open system,
formulated using continuous Markovian stochastic dy- namics, an expression for
the entropy generated over a time interval is developed in terms of the
probability of observing a trajectory associated with a prescribed driving
protocol, and the probability of its time-reverse. This forms the basis for a
general theoretical description of non-equilibrium thermodynamic pro- cesses.
Having established a connection between entropy production and an inequivalence
in probability for forward and time-reversed events, we proceed in the manner
of Sekimoto and Seifert, in particular, to derive results in stochastic
thermodynamics: a description of the evolution of a system between equilibrium
states that ties in with well-established thermodynamic expectations. We derive
fluctuation relations, state conditions for their validity, and illustrate
their op- eration in some simple cases, thereby providing some introductory
insight into the various celebrated symmetry relations that have emerged in
this field.
