# Statistical Physics of Evolving Systems

## Abstract

**:**

## 1. Introduction

_{B}). Yet, in the quest for the microscopic formulation, Clausius, Helmholtz, and Boltzmann equated entropy, S = k

_{B}lnW, with disorder enumerating the so-called microstates, W (i.e., energetically indistinguishable constituent configurations) [5]. Moreover, Schrödinger stating that life feeds on negative entropy led to a worsened confusion [6].

## 2. Materials and Methods

#### 2.1. The State Equation

_{1}P

_{j}= ϕ

_{1}ϕ

_{2}ϕ

_{3}… = ∏

_{k}ϕ

_{k}, as a product, ∏

_{k}, of substrates, labeled with k, guarantees that if a k-substrate is absent entirely, i.e., ϕ

_{k}= 0, also

_{1}P

_{j}= 0. For example, a given protein in a cellular system could not be present if any of its constituents, such as an essential amino acid, was absent altogether.

_{k}, to count them all since all entities ultimately comprise the same basic constituents.

_{j}, say, a cellular system holds many copies of a protein, the population probability, P

_{j}= [

_{1}P

_{j}][

_{1}P

_{j}][

_{1}P

_{j}]…/N

_{j}! = [

_{1}P

_{j}]

^{Nj}/N

_{j}! is

_{1}P

_{j}in power N

_{j}. Once more, the product guarantees that if any one entity is absent entirely, i.e.,

_{1}P

_{j}= 0, also P

_{j}= 0. Since the order among identical entities makes no difference, indistinguishable configurations (i.e., microstates) should not be counted as different states unless truly independent [36]. Hence, P

_{j}is scaled by the permutations, N

_{j}!

_{j}, over P

_{j}

_{k}= N

_{k}exp[(−ΔG

_{jk}+ iΔQ

_{jk})/k

_{B}T], encompasses the substrates, N

_{k}, each of free energy, −ΔG

_{jk}+ iΔQ

_{jk}, per, k

_{B}T, the average energy. When fluxes of quanta perturb k

_{B}T only a little, the system changes gradually, as if in a continuous manner. Then, an energy difference can be approximated with the selfsimilar exponential function de

^{x}/dt = e

^{x}[37]. The energy difference, ΔG

_{jk}, between the substrate, labeled with k, and the product, labeled with j, is spanned by ΔQ

_{jk}, such as insolation, from the surroundings. The quotient, i, distinguishing vector potential from the scalar one, indicates that the system is open for the flows of quanta that couple to a jk-transformation. For example, insolation makes photosynthesis happen and dissipation enables metabolism.

_{B}

_{k}= k

_{B}Tln[N

_{k}exp(G

_{k}/k

_{B}T)] is the potential bound in the k substrate, μ

_{j}is the potential bound in the j product, and their difference is Δμ

_{jk}= μ

_{j}− μ

_{k}. The form of chemical potential qualifies for any bound potential, postulating everything comprising the same fundamental entities, the quanta. The surrounding quanta, carrying energy ΔQ

_{jk}, that couple to the jk-transformation contribute to the system’s energetic status. The approximation sign stands for Stirling’s approximation for factorial, lnN

_{j}! ≈ N

_{j}lnN

_{j}− N

_{j}, in Equation (2). It is excellent for a many-body system.

_{j}k

_{B}T, and free, ΣN

_{j}(−Δμ

_{jk}+ iΔQ

_{jk}) energy. Thus, the system keeps changing until free energy is consumed altogether. Only then does the textbook form of entropy, S = k

_{B}ΣN

_{j}, apply.

#### 2.2. The Evolutionary Equation

_{j}can be given to a good approximation by continuous differentials, dN

_{j}. Thus, the total energy of the system, TS, evolves with time, t,

_{jk}+ iΔQ

_{jk}, is consumed in transformations from N

_{k}to N

_{j}or vice versa. The average energy, k

_{B}T, is not explicitly differentiated using the chain rule since its variation contains variations in dS.

_{j},

_{jk}> 0 [12]. A factor, σ

_{jk}, such as a catalyst, facilitates the jk-transformation from N

_{k}to N

_{j}, and the other way around. The natural process naturally selects the most efficient mechanisms of free energy consumption to attain balance in the least time. In a sense, the various processes by Equation (4) compete for the flows of quanta. Thus, the least-time principle can be recognized as the long-yearned teleological imperative of evolution, the final cause [38]. The quest for attaining balance in the least time explains, for example, the proliferation of insolation-absorbing compounds during abiogenesis [39,40].

#### 2.3. The Continuous Equation of Motion

_{j}= (∂U/∂N

_{j}) and radiation Q

_{j}= (∂Q/∂N

_{j}) in terms of continuous potentials U and Q serves to cast Equation (3) to

_{j}depends on ∂/∂N

_{j}while being independent of ∂/∂N

_{k}. Since fluxes of photons of Q transform U and K, TdS corresponds to the change in kinetic energy d2K. In other words, energy and period, as well as momentum and wavelength, change when the quanta settle on new trajectories. For example, when an electron settles on a higher orbit, the Coulomb field in an atom changes.

**p**·d

**x**= ∫2Kdt that totals quanta with momenta,

**p**, on paths,

**x**, or equivalently with kinetic energy, 2K, on periods, t [45]. The free-energy variational principle describes complex processes [46], such as morphogenesis [47]. Moreover, the revived interest in the Maupertuis action has led to recognizing causality as its partial derivative of along the spatial coordinate with the smallest momentum change along the trajectory [48] and connection between time and thermodynamics for equilibrium phenomena and its extension to irreversible processes assuming local equilibrium [49].

**p**/dt, in momentum,

**p**= m

**v**, with velocity,

**v**, and decomposing kinetic energy, 2K =

**v ∙ p**= ∑v

_{j}mv

_{k}, that vanishes for j ≠ k, and taking into account that d

**v**/dt

**∙ p**= 0 since acceleration

**a**= d

**v**/dt ⊥

**v**. The change in mass, dm/dt = dE/c

^{2}dt = dQ/v

^{2}dt, denotes, in geometric terms, changes in the quantized trajectories as they open up and dissipate quanta into the surroundings. Concurrently to any state transformation (e.g., nuclear, chemical reaction, also mass changes, dm). Accordingly, masses change until the system develops into a steady state. Mass-energy equivalence, E = mc

^{2}, the relativistic formula, is understood here to follow from the action, Et = mc

^{2}t = px. Thus, the 2nd law of thermodynamics, the original (Maupertuis’) principle of least action, and the complete form of Newton’s second law of motion are the same law.

## 3. Results

#### 3.1. The Ubiquitous Patterns

_{k}σ

_{jk}, limit free energy, −Δμ

_{jk}+ iΔQ

_{jk}, consumption, and define

_{j}/dN

_{j}= d(G

_{j}+ k

_{B}T lnN

_{j})/dN

_{j}= k

_{B}T/N

_{j}, since μ

_{k}, Q

_{j}and Q

_{k}have no explicit but only a stoichiometric dependence on N

_{j}. The initial growth by Equation (7) is approximately exponential since initially, the amount of free energy seems as if infinite. In turn, when the free energy dwindles down, the final growth decreases slowly, almost exponentially.

_{j}as the product of the basic elements, N

_{1}, and using the atomistic axiom, N

_{j}= ∏

_{k}ϕ

_{k}= α

_{j}N

_{1}

^{j}, where α

_{j}= ∏

_{mn}exp[ΣN

_{j}(−Δμ

_{mn}+ iΔQ

_{mn})/k

_{B}T] contains the free energy terms that force the assembly of N

_{j}from N

_{1}through various mn-transformations, the change,

_{j}= j lnN

_{1}+ constant. The form is familiar from various self-organizing processes, including technological ones such as the one leading to Moore’s law [52]. The power law in the continuous form, d ln

**p**= d ln

**v**+ d lnm, follows from dividing Newton’s second law of motion by momentum,

**p**.

^{−}βN

_{j}to Equation (7)

_{j}(t

_{o}), at an instant, t

_{o}, determines the population, N

_{j}(t), at a later instant, t. By this model [53], evolution is almost predictable when the change in free energy compared with average energy, |(−Δμ

_{jk}+ iΔQ

_{jk})/k

_{B}T| << 1, is small [51]. And when not, oscillations, bifurcations, and chaos occur, exemplified by soaring and sinking populations.

_{jk}+ iΔQ

_{jk})/k

_{B}T| << 1. As the variation, n, is small, n << j, around a representative, mean, or an average factor, ϕ

_{j}, the distribution of factors, given in logarithmic terms, lnϕ

_{j}= j lnϕ

_{1}, of the elemental factor, ϕ

_{1},

#### 3.2. Non-Deterministic Motion

_{j}, cannot be separated from its driving force since Δμ

_{jk}is a function of N

_{j}. As Equation (3) cannot be solved, the course of events is, so to say, non-deterministic, non-computable, and intractable [57]. This conclusion contradicts the common misconception that evolution would be a random (i.e., an indeterministic process). A mutation in a gene may appear such as an arbitrary event. However, it also has its cause, such as natural background radiation or exposure to a chemical agent. The significance of a mutation depends on the free energy it may unleash when the gene is expressed. Thus, it is the consumption of free energy, not the facilitating mutations, that drives evolution. In short, evolution follows forces.

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Any system can be depicted by the same general energy level diagram, assuming everything comprising quanta. Entities, in numbers, N

_{k}, with energy, G

_{k}, are on the same level. Their mutual exchange (bow arrows) brings about no change, so the average energy, k

_{B}T, does not change either. However, when quanta from surroundings with energy, ΔQ

_{jk}, (wavy arrows) couple to transformations from starting materials, N

_{k}, into products, N

_{j}, the entities move (horizontal arrows) from one level to another. Through transformations, the system and its surroundings move toward mutual thermodynamic balance. The system evolves toward higher average energy when the surroundings are higher in energy and vice versa. The logarithm of the sigmoid cumulative probability, P, distribution curve (dashed line) is entropy, S = k

_{B}lnP. On the logarithm-logarithm scale (inset), S vs. (chemical) potential energy, μ, closely follows a power law (i.e., a straight line).

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Annila, A.
Statistical Physics of Evolving Systems. *Entropy* **2021**, *23*, 1590.
https://doi.org/10.3390/e23121590

**AMA Style**

Annila A.
Statistical Physics of Evolving Systems. *Entropy*. 2021; 23(12):1590.
https://doi.org/10.3390/e23121590

**Chicago/Turabian Style**

Annila, Arto.
2021. "Statistical Physics of Evolving Systems" *Entropy* 23, no. 12: 1590.
https://doi.org/10.3390/e23121590