Recent advances in technology have enabled studying the dynamics of single molecules. What new
insights can be revealed by these experiments which cannot be seen in bulk measurements? In this talk
I will focus on two examples where we showed that studying single molecules reveals new phenomena.
The first is the case of anomalous diffusion in an equilibrium environment. The continuous time
random walk with a power-law distribution of sojourn times is a common model to describe subdiffusion
processes. A particle which undergoes such a process will visit all sites of a finite system, yet
the process is non-ergodic. We clarify the concept of weak ergodicity breaking and calculate the
distribution of occupation times from which the time averages of physical observables can be derived.
Unlike in ergodic systems even at the infinite long time limit, the occupation times (and therefore the
physical observables) are random quantities. I will also discuss another model for anomalous diffusion
due to coupling between stochastic processes which can lead to ergodic or non-ergodic behavior for
different coupling functions. In both models the single molecule properties provide an insight into the
dynamical mechanism, which could not be obtained from ensemble measurements. The second
example I will discuss is a non-equilibrium system, a single molecule excited by a monochromatic laser
field. I will introduce an extension of the generating function technique for the calculation of photon
emission statistics for systems governed by multi-level quantum dynamics. This extension enables
studying the statistics of photons that are emitted from specific transitions and subject to quantum
coherence. Several model calculations illustrate the generality of the technique and highlight
quantitative and qualitative differences between quantum mechanical models and related stochastic
approximations. I will also introduce the moment-generating function for photon emissions in which
the frequencies of the fluoresced photons are explicitly considered. Calculations were performed for the
case of a two-level dye molecule, showing that measured photon statistics will display a strong and
nonintuitive dependence on detector bandwidth. It will also be demonstrated that the anti-bunching
phenomenon, associated with negative values of Mandel's Q parameter, results from correlations
between photons with well separated frequencies. These two examples show that single molecule
measurements can provide new information about the observed system both in equilibrium and non equilibrium
conditions. Moreover, new phenomena which are observed only at the single molecule
level can favor certain microscopic models over others.
