An unresolved problem in physics is how the arrow of time due to the Second Law of Thermodynamics arises from dynamics such as Newton's laws that time reversal symmetric. However, there has been much progress over the last decade for a dynamical process that begins in thermal equilibrium. For these processes, one can not only prove the Second Law of Thermodynamics but also more general relations such as Jarzynski's equality and the Crooks fluctuation theorem, two remarkable results which will be explained in this talk. These insights have greatly enhanced our understanding of single molecule experiments which use lasers as chopsticks to manipulate a biological molecule such as RNA.
While these previous results show that an arrow of time exists, we now give a length to this arrow. For a dynamical process beginning in equilibrium, this measure is based on the distinguishability of this forward process from its reverse process, or the time reversal of the forward process. Most importantly, this length of time's arrow can be measured in single molecule RNA pulling experiments. It is also deeply connected with determining equilibrium free energy differences.
We also show the connection between far-from-equilibrium single molecule experiments and an equilibrium quantity called thermodynamic length. Originally, thermodynamic length generalized the notion of curve length in Euclidean space to the surface of equilibrium states in thermodynamics. We define thermodynamic length for a microscopic system such as the RNA molecule in a single molecule experiment. This definition reveals the link between thermodynamic length and equilibrium fluctuations. To measure thermodynamic length in far-from-equilibrium experiments, one must determine the equilibrium properties at an intermediate point of the dynamical process. We show how to re-weight the value of each pulling experiment so that one can determine these equilibrium properties.