# FisMat2017 - Submission - View

**Abstract's title**: Diffusive and evolutionary dynamics from the Master Equation

**Affiliation**: Consorzio RFX

**Affiliation Address**: Corso Stati Uniti 4 35127 Padova

**Country**: Italy

The integro-differential Master Equation (ME) provides a fairly generic framework for describing

the statistical dynamics of complex systems. In this work we address the mathematical analysis of

two different limits of the ME.

1) It well known that diffusive dynamics does arise as long-wavelength limit of the ME, and is

usually described in terms of the Fokker-Planck equation. Fick’s form is another expression for the

diffusion equation. In homogeneous environments the two expressions are trivially identical,

whereas in inhomogeneous environments they are no longer equivalent. Use of the improper form

within a given context leads to appearance of artefacts when modelling experimental data, an

instance being given by the modelling of mass and energy in laboratory plasmas [1,2]. We carry out

a study in terms of microscopic dynamics of the conditions that lead to either the one or the other

form of the diffusive equation.

2) The ME can be interpreted within a Bayesian viewpoint as the mathematical expression of the

rule for a statistical distribution to evolve from a prior one by the gradual assimilation of novel

information (e.g., new data from measurements, etc …); it is, under this regard, an instance of an

evolutionary process and—as such—its dynamics can be modelled using the mathematics of

evolutionary theory, i.e., the replicator equation [3]. We study the asymptotic limit of this equation

pointing out how several common statistical distributions, often encountered in disparate fields of

science, do arise as its limiting solutions.

[1] B. Ph. Van Milligen, B.A. Carreras, R. Sànchez, Plasma Phys. Control. Fusion 47 (2005) B743

[2] F. Sattin, A. Bonato. L. Salasnich, arxiv:1512.07428 (submitted to Springer)

[3] P.D. Taylor, L.B. Jonker, Math. Biosciences 40 (1978) 145