The many resent mathematical studies of entropy have lead to some very clear findings. Resent work has shown that reversible dynamics will not give rise to irreversible increases in the entropy of systems since they would be already at their entropic maximum. The world that we observe does present us with a clear case of a system that is not at maximum entropy, thus we are lead to the conclusion that irreversible processes are at the foundation of the behaviors we see in nature. Though providing totally clear criteria for the global evolution of system entropy, at the same time these criteria suggest that all currently formulated physical laws may not be at the foundation of thermodynamic behavior we observe every day of our lives. If we use the tools of mathematics which have given us the amazing degree of control over Nature we enjoy today, we find fractals and Chaos theory as likely candidates to the task.
Fractals are a relatively new concept and unfortunately are unknown to most.
They are derived from ideas in geometry where integers are used to enumerate the degrees of freedom of movement - the quantities of dimension. Fractals represent non-integer dimensions and have been found to describe best the curves that we observe in objects such as coast lines, trees, clouds, etc. Chaos theory stems from the study of systems whose behavior is not able to be simplified into the neat linear package deals as offered by conventional mechanical models. The key difference is that they can not be broken down into separate parts, they must be modeled as wholes. If we take two identical chaotic systems and start them up with initial conditions that vary only slightly, their subsequent behaviors will rapidly diverge such that they end up very different.