Ariel Caticha From Information Geometry to Newtonian Dynamics July 8, 2007 (62)

E. T. Jaynes “Information Theory and Statistical Mechanics” Physical Review, 1957

From Information Geometry to Newtonian Dynamics Ariel Caticha and Carlo Cafaro Department of Physics University at Albany - SUNY MaxEnt 2007

The laws of Physics are not laws of Nature; they are rules for processing information about nature. Outrageous!! But the evidence accumulates... Statistical Mechanics Quantum Mechanics This is where we specify the interpretation. Geometry Our objective: Classical Mechanics

Objective: To derive from principles of inference. Challenge: To codify the relevant prior information into an appropriate statistical model. To derive the dynamics without appealing to additional postulates from physics.

Configuration Space: a single particle The particle lives in a 3-dimensional space. Positions are uncertain: space is fuzzy. (Small uncertainties.)

To each “point” we associate a probability distribution. A “point” is not just a dot. These “points” have structure. Configuration Space is a statistical manifold. The degree to which a point can be distinguished from a neighbor is a measure of distance. The induced “information” geometry is unique.

Information geometry:

To each “point” we associate a probability distribution: Given we can find , and vice versa! In fact, caution!! Information distance is measured in units of the local uncertainty .

Entropic Dynamics: ?? The entropic dynamics trajectory is a geodesic.

The trajectory minimizes the length We are done, but ... does it look like classical mechanics? Yes ... it is identical to Jacobi’s action principle: where

The parameter is quite arbitrary. Define time t so that motion looks simple, then, and Quite impressive, but...

Objections: The energy is a fixed constant: the particle is isolated. Is t really a universal time? Or just a parameter for one specific particle? Solution: Apply the theory to the whole universe.

The whole Universe: N = 2 particles To each “point” in configuration space, we associate a probability distribution, ... a product: are independent, but...

For particle 1, a spherically symmetric Gaussian Each particle has its own mass. A single conformal factor affects all particles equally.

For the whole Universe, where mass matrix Information metric: Entropic Dynamics:

Define time t so that motion looks simple, then, Newton’s equation for interacting particles!! and

Conclusions and remarks: On mass and interactions: On time: There is no external time. Internal time: Ephemerides time The universe is the ultimate clock. It measures universal time. This is just a model... ...but it is a statistical model!

Statistical manifolds or statistical fiber bundles? 9-dim space of Gaussians: Standard information metric: But we want the metric induced on a 3-dim submanifold.

On the 9-dim manifold On the 3-dim submanifold: Substitute ? No: not covariant!! better: Therefore Given find . Solution:

Entropic Dynamics: ?? The entropic dynamics trajectory is a geodesic.

We only need to consider very short steps.

Entropic Dynamics: and form a straight line. The entropic dynamics trajectory is a geodesic.

Information Theory and Classical Mechanics Ariel Caticha and Carlo Cafaro Department of Physics University at Albany - SUNY MaxEnt 2007