A FREE ENERGY PRINCIPLE FOR A PARTICULAR PHYSICS Karl Friston The Wellcome Centre for Human Neuroimaging, UCL Queen Square Institute of Neurology, London, UK WC1N 3AR. Email: [email protected] (This work is under consideration for publication by The MIT Press) Abstract This monograph attempts a theory of every ‘thing’ that can be distinguished from other ‘things’ in a statistical sense. The ensuing statistical independencies, mediated by Markov blankets, speak to a recursive composition of ensembles (of things) at increasingly higher spatiotemporal scales. This decomposition provides a description of small things; e.g., quantum mechanics – via the Schrödinger equation, ensembles of small things – via statistical mechanics and related fluctuation theorems, through to big things – via classical mechanics. These descriptions are complemented with a Bayesian mechanics for autonomous or active things. Although this work provides a formulation of every ‘thing’, its main contribution is to examine the implications of Markov blankets for self- organisation to nonequilibrium steady-state. In brief, we recover an information geometry and accompanying free energy principle that allows one to interpret the internal states of something as representing or making inferences about its external states. The ensuing Bayesian mechanics is compatible with quantum, statistical and classical mechanics and may offer a formal description of lifelike particles. Key words: self-organisation; nonequilibrium steady-state; active inference; active particles; free energy; entropy; random dynamical attractor; autopoiesis; Markov blanket; Bayesian; variational. Contents Abstract................................................................................................................................................................... 1 Introduction ............................................................................................................................................................ 4 Part One: the setup .................................................................................................................................................. 7 Something or nothing ......................................................................................................................................... 7 Some preliminaries........................................................................................................................................................... 8 Nonequilibrium steady states ......................................................................................................................................... 10 The free energy principle Fluctuations and information length ............................................................................................................................... 13 Random dynamical systems and Markov blankets ......................................................................................................... 16 Markov blankets and marginal flows ............................................................................................................................. 17 Summary ........................................................................................................................................................................ 18 Symmetry breaking and self-organisation ....................................................................................................... 19 Self-organization and self-evidencing ............................................................................................................................ 24 Self-organisation, frustration and supersymmetry .......................................................................................................... 24 Self-organisation and information length ....................................................................................................................... 28 Summary ........................................................................................................................................................................ 31 Synthetic soups and active matter .................................................................................................................... 32 An active soup ................................................................................................................................................................ 33 A random dynamical attractor and its Markov blankets ................................................................................................. 35 The Markov blanket ....................................................................................................................................................... 35 The emergence of order.................................................................................................................................................. 36 Summary ........................................................................................................................................................................ 37 States, particles and fluctuations ...................................................................................................................... 38 Starting at the end .......................................................................................................................................................... 39 The Markovian partition................................................................................................................................................. 41 The adiabatic reduction .................................................................................................................................................. 44 Elimination and renormalisation .................................................................................................................................... 47 Summary ........................................................................................................................................................................ 50 Part Two: some special cases ............................................................................................................................... 53 A theory of small things – quantum mechanics ............................................................................................... 53 The Schrödinger equation from first principles .............................................................................................................. 55 Wave particle duality and the de Broglie hypothesis ..................................................................................................... 56 Heisenberg uncertainty principle .................................................................................................................................... 58 Inference, measurement and wave function collapse? .................................................................................................... 58 Summary ........................................................................................................................................................................ 60 A theory of lots of little things – statistical mechanics .................................................................................... 64 Stochastic thermodynamics ............................................................................................................................................ 65 Stochastic energetics ...................................................................................................................................................... 68 Fluctuation theorems ...................................................................................................................................................... 72 Summary ........................................................................................................................................................................ 75 2 The free energy principle A theory of big things – classical mechanics ................................................................................................... 76 Conservative systems ..................................................................................................................................................... 77 Random fluctuations and generalised motion................................................................................................................. 79 Summary ........................................................................................................................................................................ 80 Part Three: a particular case ................................................................................................................................. 84 A theory of autonomous things – Bayesian mechanics.................................................................................... 84 Risk and ambiguity ........................................................................................................................................................ 88 Inference and measurement ............................................................................................................................................ 92 Information geometry ....................................................................................................................................................