Turbulence & Information Field Dynamics

Turbulence & Information field dynamics Torsten Enßlin MPI für Astrophysik

Information field dynamics

Computer simulation of fields are essential in astrophysics and elsewhere: hydrodynamics, MHD, cosmic structure formation, plasma processes, cosmic ray transport, …

Computer simulations need to discretize the space. This erases information on small-scale structures/processes.

How can knowledge on sub-grid processes be used to improve the simulation accuracy?

Idea: use Information Field Theory (IFT) to bridge between discretized field (data in computer memory) and

continuous field configurations (possible physical field realization). data in computer memory

data in computer memory

configuration space

signal inference

data in computer memory

configuration space configuration space

time evolution

signal inference

data in computer memory

configuration space configuration space

time evolution

signal inference

data in computer memory data in computer memory

configuration space configuration space

time evolution

signal entropic inference matching

data in computer memory data in computer memory

Recipe

1. Field dynamics: Specify the field dynamics equations. 2. Prior knowledge: Specify the ignorance knowledge for absent data. 3. Data constraints: Establish the relation of data and the ensemble of field configurations being consistent with data and background knowledge. Assimilation of external measurement data into the simulation scheme is naturally done during this step. 4. Field evolution: Describe the evolution of the field ensemble over a short time interval. 5. Prior update: Specify the background knowledge for the later time. 6. Data update: Invoke again the relation of data and field ensemble to construct the data of the later time. Use for this entropic matching based on the Maximum Entropy Principle (MEP). Thereby a transformation rule is constructed that describes how the initial data determines the later data. This transformation forms the desired numerical simulation scheme. It has incorporated the physics of the sub-grid degrees of freedom into operations solely in data space. 7. Implementation: Implement and test the resulting algorithm.

Information Field Theory Free Theory Gaussian signal & noise, linear response WienerFree filterTheory theory Gaussian signalknown & for noise, 60 yearslinear response

Interacting Theory non-Gaussian signal, noise, or non-linear response use field theory toobox: Feynman diagrams, renormalization, resummation, effective action (=MaxEnt), mean field theory (variational Bayes), ... Numerical Information Field Theory Selig et al. (arXIv:1301.4499) Code & Docu @ http://www.mpa-garching.mpg.de/ift/nifty/ Galactic Faraday rotation Oppermann et al. (2012, 2014)

Galactic Faraday rotation Oppermann et al. (2012, 2014)

Angular Power Spectrum

p re vi t ou hi s s wo wo rk rk

Turbulence prior

Statistical homogeneity:

Minimal informative PDF (MaxEnt):

Time evolution

Maximum entropy principle

Entropy ranks PDFs according how well they represents a knowledge state. Its functional form is determined by three requirements (Jaynes, 1957):

● Locality: Local information has local effects; information that affects only some part of the phase space should not modify the entropy and the implied MEP PDF in case this area is discarded. ● Coordinate invariance: The system of coordinates of the phase space does not carry information. Entropy should be invariant under coordinate transformation as well as the determined MEP PDF. ● Independence: Independent systems can be treated jointly or separately, yielding the same entropy in both cases. The joint MEP PDF must therefore be separable into a product of PDFs for the individual systems.

Entropic matching

Thermally exited Klein-Gordon field

Discretized differential operator

discretized differential operator in Fourier space (dispersion relation)

IFD scheme (t=0) t = 0 spectral scheme t = 10⁻⁴ finite differences t = π/2

Accuracy

evolved field (t = 10) accuracy with time

finite differences spectral scheme IFD scheme (t=0)

Ensemble dynamics of stochastic systems Ramalho et al. (2013)

Gaussian ansatz:

Linear noise approximation:

Ensemble dynamics of stochastic systems Ramalho et al. (2013)

Gaussian ansatz:

Entropic matching:

Stochastic van der Pol oscillator Ramalho et al. (2013)

weak non-linearity

mean

linear noise approx.

variance entropic matching

strong non-linearity

linear noise entropic approx. matching mean

linear noise approx.

variance

entropic matching genetic circuit model

entropic linear noise matching approx. mean

linear noise approx.

variance entropic matching

Summary

Information field dynamic (IFD) permits the construction of simulation schemes for field dynamics that incorporate knowledge on the field's sub- grid statistics in an information theoretical optimal way.

IFD uses information field theory (IFT) to establish a relation between data in computer memory and an ensemble of plausible configurations, which are consistent with all information (data & sub-grid knowledge).

Each plausible configuration is time evolved with the exact dynamics. The time developed ensemble is recast into a representation in computer memory via entropic matching.

The resulting scheme to update the computer data form then the IFD simulation scheme, incorporating the PDE & the sub grid information in

an (nearly) optimal way.