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Motivated by the notion of perceptual error, as a core concept of the perceptual control theory, we propose an action-amplitude model for controlled entropic self-organization (CESO). We present several aspects of this development that illustrate its explanatory power: (i) a physical view of partition functions and path integrals, as well as entropy and phase transitions; (ii) a global view of functional compositions and commutative diagrams; (iii) a local geometric view of the Kähler–Ricci flow and time-evolution of entropic action; and (iv) a computational view using various path-integral approximations.

We are primarily interested in developing advanced models of military command and control (C2), which we formulate (in its simplest form) as follows. Our approach should also be applicable across a wide range of non-military organizational decision-making settings. Consider a group Γ := {Γ_{i}_{i}_{j}_{i}_{j}

Thus, under this construction, the optimal behavior,
_{i}_{i}

so that the probability Pr_{i}_{i}_{i}_{j}_{j}

The following geometrical interpretations can be given to the optimization Problem (_{i}^{2}-norm:

or to the Euclidean metric:

Its continuous generalization, allowing for a continuum of capability choices, is given by the Banach ^{2}-norm:

and the associated Banach metric between any two real-valued square-integrable functions,

For example, the finite control problem is actually a minimization of the square of the Banach metric (

Upon time discretization using a suitable quadrature scheme, this becomes the following least-squares problem:

Both discretized functions,

The optimization Problem (

with the following particular cases:

From these cases, the set of discrete Euler–Lagrangian equations of motion on the group/graph, Γ, can be derived (see [^{2}-norm (

Next, we assume the probability Pr_{i}_{i}_{i}_{j}_{j}_{i}

The PDF (_{i}^{i}

where
^{i}

provides the Gibbs measure on the system’s state-space, which is a unique statistical distribution that maximizes the entropy for a fixed expectation value of the energy:

The associated system’s order parameter, entropy, is given by:

Entropy describes both “ignorance”, or Heisenberg’s uncertainty, and “randomness”.

A useful particular example of Equation (

where _{i}_{i}^{i}

More generally, we consider Markov random fields/Markov networks, which have a Markov property described by an undirected graph (see [^{i}

where _{i}_{i}^{i}_{i}_{i}f_{i}

We remark here that Markov random fields have been introduced as a Markovian framework for the Ising spin-lattice model, defined by the Hamiltonian energy function (given here in its simplest dot-product form):

where _{i}_{ij}_{i}

In this paper, we present several different views of an action-amplitude model for controlled entropic self-organization (CESO).

We have already seen from example _{i}^{i}^{n}

where

More generally, in quantum field theory, instead of the field Hamiltonian,

and the complex path integral in real time (the so-called Lorentzian path integral):

represent partition functions of the quantum field theory in ℝ^{n}

Finally, we generalize our quantum theory of fields, from ^{n}

Here, _{d}

Recall that Prigogine’s Extended Second Law of Thermodynamics [

considers open (

where _{i}S_{e}S_{e}S

By further extending Prigogine’s open Second Law (

The phase of _{t}S

The phase of _{t}S

The phase of _{t}S

The phase transition from one phase to another, caused by the system’s topology change (see [

where “

The set—or, more appropriately, the category—of generic agents’ behaviors,

such that the following diagram commutes:

The maps,

Here, the Lebesgue integration, in both integrals, is performed over all continuous

The path integral Equations (

(where the overdot denotes the time derivative) and their corresponding action principles:

These correspond to classical Euler–Lagrangian equations of motion on the configuration manifold, _{ij}

with Lagrangians:

The symbolic differentials,

Both adaptive path integral

where ^{w}^{ω}

A complexified extension of the behavioral action _{1}(

Recall that a Kähler manifold (

A set of _{1}_{n}

Hermitian metric tensor:

and the associated Kähler form,

Functional space of Kähler potentials:

Now, consider the normalized Kähler–Ricci flow on a Fano

In a local open chart

We remark here that of central importance in Kähler geometry are the so-called Dolbeault differential operators: _{j}

It was proven in [_{0} has canonical Kähler class, _{1}(

Where

starting from some smooth initial Ricci and scalar curvatures,

From the control-theory perspective, the most important characteristic of the Kähler-Ricci flow is the existence of its solitary solutions (solitons), which are shrinking or decaying in time. This characteristic is associated with the geometrical entropy decrease and gives the global Lyapunov stability to the flow.

Formally, a Riemannian manifold (

In particular, if (_{1}(

or, applying the Bianchi identity, iff the following Schur-identity holds:

For any Kähler metric, _{1}(

A smooth minimizer of the entropy, ε, always exists, though it need not necessarily be unique (see [

To see how the Kähler–Perelman entropic action,

we have:

which implies the Kähler-Perelman monotonicity condition on the geometric entropy [

Roughly speaking, the application of control is trying to reduce the system’s entropy. This is achieved through shrinking Kähler-Ricci solitons.

In this section, we will outline a fast desktop simulation framework for controlled entropic self-organization, based on the preceding idea of the functional composition of path-integrals

In quantum field theory, there is both a theoretical and a numerical approach to solve a similar path integral as a sum-over-fields. A theoretical way consists of its perturbative expansion into a series of Feynman diagrams; although there is a ^{®} package, FeynArts–FeynCalc, devoted to this, this approach does not scale well with respect to increasing numbers of agents and, therefore, is not well suited to our task. A numerical way of handling the problem might be to discretize a path integral on a lattice (of dimension two, three or four) and use the techniques of lattice gauge theory; again, although possible, this approach is not really feasible for the numbers of agents in which we are typically interested. (We remark that military command and control (C2) can involve hundreds of actors, and it is not unreasonable to expect that other non-military decision-making processes could involve similarly large numbers.)

In non-relativistic quantum mechanics, the path integral can be numerically solved, either by a direct implementation of the Feynman formula (see, e.g., [

By analogy, in statistical mechanics, the real path integral in real time is equivalent to the linear Fokker-Planck equation, while its adaptive version is equivalent to the nonlinear Fokker-Planck equation. This approach is developed in the next subsection.

The Fokker–Planck equation, also known as the Kolmogorov forward equation:

is a parabolic partial differential equation (PDE) that describes the time-forward evolution of the probability distribution,

defines the Fokker-Planck differential operator:

Note that the quadratic diffusive term vanishes in the case of zero noise

The PDE (

where

(Note that the backward Fokker–Planck equation or the Kolmogorov backward equation:

describes the time-backward evolution of the PDF,

The Fokker–Planck PDE (_{t}_{t}

where _{t}_{t}_{t}/dt

As a simple demonstration case for the statistical path-integral simulation, we have implemented the following Ito-type SDE with the nonlinear drift: _{t}

where:

In this paper we have presented several different, but complimentary, modelling and computational views of controlled entropic self-organization (CESO). Motivated by the notion of perceptual error from perceptual control theory (PCT) in the setting of group decision making, we have proposed several modelling and simulation frameworks aimed at illuminating controlled self-organization phenomena: (i) a physical view of partition functions and path integrals, as well as entropy and phase transitions; (ii) a global view of functional compositions and commutative diagrams; (iii) a local view of the Kähler–Ricci flow and time-evolution of entropic action; and, finally, (iv) a computational view of various path-integral approximations.

Note that we do not intend these views of CESO to constitute any kind of normative ideal; our underlying formulation of group decision-making according to the PCT hypothesis, that actors make choices, so as to minimize their perceptual error, is merely a modelling construct, not necessarily an accurate representation of human behavior. Yet, we maintain that this construct, embodied in our four formal frameworks, provides powerful new formal tools and the basis for new empirical insights, thereby laying the foundation for other CESO models in our ongoing and future research.

The authors would like to acknowledge the support of Dr Todd Mansell, Chief Joint and Operations Analysis Division, DSTO, for his support of this work.

Jason Scholz conceived of the problem of modelling military C2 in terms of a Perceptual Control Theory construction, concerning the perceptual errors of each agent, and that minimizing the perceptual error would indicate an optimal behavior. His main contribution is the problem motivation (perceptual control and neurological modelling), construction of the perceptual error optimization, with the possibility of generalization to Markov random fields. Darryn Reid developed the research programme, by lifting the construction to more general settings, defining entropy as central to its explanatory power, and outlining the need for a computational view using path-integral approximations. Vladimir Ivancevic is responsible for the largest part of the mathematical development of these ideas, including the adaptive path-integral formalism, its relation to Kähler geometry and Kähler-Ricci flow, and their computer simulations.

The authors declare no conflict of interest.

Illustrative simulation of a (1+1)D Fokker-Planck ^{®} for the simple case of

Illustrative simulation of the Ito stochastic process with a nonlinear drift and a vector Wiener process, including both harmonic and nonlinear waves: 16 paths of this nonlinear random process are depicted (