# Entropic Dynamics in a Theoretical Framework for Biosystems

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Entropic Dynamics

#### 2.2. Kullback Principle of Minimum Information Discrimination

- Uniqueness: the result of the inference should be unique.
- Invariance: the choice of a coordinate system should not matter.
- System independence: it should not matter whether one accounts for independent information about independent systems separately in terms of different densities or together in terms of a joint density.
- Subset independence: it should not matter whether one treats an independent subset of system states in terms of a separate conditional density or in terms of the full system density.

#### 2.3. Biological Continuum (Biocontinuum)

#### 2.4. Information Geometry

#### 2.5. Replicator Dynamics

- ${x}_{i}$ is the proportion of type i in the population with the type being any principal attribute category of determined variation and x is the rate of change.
- ${f}_{i}\left(x\right)$ is the fitness of each type $i$ in the population with fitness being a survival likelihood characteristic in the context of the environment.
- $\mathsf{\phi}\left(x\right)$ is the average population fitness as determined by the weighted average of the fitness of the overall population.

## 3. Results

#### 3.1. Derivation of Equations of Entropic Dynamics for the Biosystem

_{K}

_{L}(Q||P) or I(q,p)) is where P is the prior or current probability distribution of types and Q is the distribution that is the optimal end state. By incorporating this measure into the functioning of the replicator expression, the use of prior information within the living system regarding the state of the biocontinuum is possible using a Bayesian inference approach. The Bayesian updating during the iterative procedure with time also accounts for the Landauer erasure of information required for balancing thermodynamic entropy [33]. The information differential between q and p at any point in the dynamic transition is the remaining information to be learned. The equation for the Kullback–Leibler information divergence is then given by:

**ACTION**measure for all $i$ elements is:

#### 3.2. Information Geometry of the Biological Continuum (Biocontinuum)

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**This graphic below depicts an example of the natural evolution $\frac{d}{dt}I\left(q,p\right)$of the Kullback-Leibler information divergence $I\left(q,p\right)$ as an action gradient driven by the geometry of the biocontinuum created by the system processes and structure, replicator dynamics, fitness function and the entropic procedure of the Kullback principle of minimization of information. This information metric exists as a probability spectrum (i) of uncertainty in the biocontinuum information space of the biologic system process. As new information is assimilated as knowledge into the adapting system, the prior state $p$ emerges as a new target state$q$ along the gradients of the biocontinuum information space.

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Summers, R.L.
Entropic Dynamics in a Theoretical Framework for Biosystems. *Entropy* **2023**, *25*, 528.
https://doi.org/10.3390/e25030528

**AMA Style**

Summers RL.
Entropic Dynamics in a Theoretical Framework for Biosystems. *Entropy*. 2023; 25(3):528.
https://doi.org/10.3390/e25030528

**Chicago/Turabian Style**

Summers, Richard L.
2023. "Entropic Dynamics in a Theoretical Framework for Biosystems" *Entropy* 25, no. 3: 528.
https://doi.org/10.3390/e25030528