Entropic Dynamics on Gibbs Statistical Manifolds

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.

Entropic Dynamics

Entropic dynamics is a framework in which dynamical laws such as those that arise in physics are derived as an application of entropic methods of inference. No underlying laws of motion such as those of classical mechanics are postulated. Instead, the dynamics is driven by entropy subject to constraints that reflect the information relevant to the problem at hand. In this chapter I review the derivation of three forms of mechanics. The first is a standard diffusion, the second is a form of Hamiltonian mechanics, and the third form is an argument from information geometry used to derive the special case known as quantum mechanics.

The Entropic Dynamics of Quantum Scalar Fields Coupled to Gravity

Entropic dynamics (ED) are a general framework for constructing indeterministic dynamical models based on entropic methods. ED have been used to derive or reconstruct both non-relativistic quantum mechanics and quantum field theory in curved space-time. Here we propose a model for a quantum scalar field propagating in dynamical space-time. The approach rests on a few key ingredients: (1) Rather than modelling the dynamics of the fields, ED models the dynamics of their probabilities. (2) In accordance with the standard entropic methods of inference, the dynamics are dictated by information encoded in constraints. (3) The choice of the physically relevant constraints is dictated by principles of symmetry and invariance. The first of such principle imposes the preservation of a symplectic structure which leads to a Hamiltonian formalism with its attendant Poisson brackets and action principle. The second symmetry principle is foliation invariance, which, following earlier work by Hojman, Kuchař, and Teitelboim, is implemented as a requirement of path independence. The result is a hybrid ED model that approaches quantum field theory in one limit and classical general relativity in another, but is not fully described by either. A particularly significant prediction of this ED model is that the coupling of quantum fields to gravity implies violations of the quantum superposition principle.

Entropic Dynamics in Neural Networks, the Renormalization Group and the Hamilton-Jacobi-Bellman Equation

We study the dynamics of information processing in the continuum depth limit of deep feed-forward Neural Networks (NN) and find that it can be described in language similar to the Renormalization Group (RG). The association of concepts to patterns by a NN is analogous to the identification of the few variables that characterize the thermodynamic state obtained by the RG from microstates. To see this, we encode the information about the weights of a NN in a Maxent family of distributions. The location hyper-parameters represent the weights estimates. Bayesian learning of a new example determine new constraints on the generators of the family, yielding a new probability distribution which can be seen as an entropic dynamics of learning, yielding a learning dynamics where the hyper-parameters change along the gradient of the evidence. For a feed-forward architecture the evidence can be written recursively from the evidence up to the previous layer convoluted with an aggregation kernel. The continuum limit leads to a diffusion-like PDE analogous to Wilson’s RG but with an aggregation kernel that depends on the weights of the NN, different from those that integrate out ultraviolet degrees of freedom. This can be recast in the language of dynamical programming with an associated Hamilton–Jacobi–Bellman equation for the evidence, where the control is the set of weights of the neural network.

Quantum Trajectories in Entropic Dynamics

Entropic Dynamics is a framework for deriving the laws of physics from entropic inference. In an (ED) of particles, the central assumption is that particles have definite yet unknown positions. By appealing to certain symmetries, one can derive a quantum mechanics of scalar particles and particles with spin, in which the trajectories of the particles are given by a stochastic equation. This is much like Nelson’s stochastic mechanics which also assumes a fluctuating particle as the basis of the microstates. The uniqueness of ED as an entropic inference of particles allows one to continuously transition between fluctuating particles and the smooth trajectories assumed in Bohmian mechanics. In this work we explore the consequences of the ED framework by studying the trajectories of particles in the continuum between stochastic and Bohmian limits in the context of a few physical examples, which include the double slit and Stern-Gerlach experiments.

An Entropic Dynamics Approach to Geometrodynamics

In the Entropic Dynamics (ED) framework quantum theory is derived as an application of entropic methods of inference. The physics is introduced through appropriate choices of variables and of constraints that codify the relevant physical information. In previous work, a manifestly covariant ED of quantum scalar fields in a fixed background spacetime was developed. Manifest relativistic covariance was achieved by imposing constraints in the form of Poisson brackets and of intial conditions to be satisfied by a set of local Hamiltonian generators. Our approach succeeded in extending to the quantum domain the classical framework that originated with Dirac and was later developed by Teitelboim and Kuchar. In the present work the ED of quantum fields is extended further by allowing the geometry of spacetime to fully partake in the dynamics. The result is a first-principles ED model that in one limit reproduces quantum mechanics and in another limit reproduces classical general relativity. Our model shares some formal features with the so-called “semi-classical” approach to gravity.