Physics of Life Reviews 7 (2010) 477–498 www.elsevier.com/locate/plrev Review Natural patterns of energy dispersal Teemu Mäkelä a , Arto Annila a,b,c,∗ a Department of Physics, POB 64, University of Helsinki, Finland b Institute of Biotechnology, POB 56, University of Helsinki, Finland c Department of Biosciences, POB 56, University of Helsinki, Finland Received 17 August 2010; accepted 8 October 2010 Available online 13 October 2010 Communicated by J. Fontanari Abstract Universal patterns such as power-law dependences, skewed distributions, tree-like structures, networks and spirals are associated with energy dispersal processes using the principle of least action. Also ubiquitous temporal courses such as sigmoid growth, bifurcations and chaos are ascribed to the decrease of free energy in the least time. Moreover, emergence of natural standards such as the common genetic code and chirality consensus of amino acids are understood to follow from the quest to maximize the dispersal of energy. Many mathematical functions that model natural patterns and processes are found as approximations of the evolutionary equation of motion that has been derived from statistical physics of open systems. The evolutionary processes can be described as flows of energy that run from high energy sources to low energy sinks in the least time. However, the equation of evolution cannot be solved in general because the flows of energy and their driving forces are inseparable. Since the energy of the system keeps changing, the paths of evolution cannot be integrated from a given initial state to a final state. Although evolutionary courses of these non-Hamiltonian systems with two or more alternative ways of dissipation cannot be predicted, the flows of energy will search and naturally select paths of least action, known as geodesics, to consume free energy in the least time. The scale-invariant natural patterns follow from this natural law that impinges on processes at all scales of space and time.  2010 Elsevier B.V. All rights reserved. Keywords: Entropy; Evolution; Free energy; Least action; Natural selection; Power law; Scale-free; Self-similarity; Thermodynamics Contents 1. Introduction ........................................................................ 478 2. Dispersal of energy .................................................................... 479 2.1. Energy transduction systems ........................................................ 479 2.2. Entropy as a statistical measure of energy dispersal ......................................... 481 2.3. Equations of evolution ............................................................. 482 2.4. The steady-state system ............................................................ 484 2.5. Continuous energy density in evolution ................................................. 485 * Corresponding author at: Department of Physics, POB 64, University of Helsinki, Finland. E-mail address: arto.annila@helsinki.fi (A. Annila). 1571-0645/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.plrev.2010.10.001