# Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales

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## Abstract

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## 1. Introduction

## 2. Phase-Transition for Selection in Two-State Systems

## 3. Simplest Case for Selection: A Three-State System

## 4. Time-Scale Separation and Equilibrium Hints

#### 4.1. Fast-Dissipation Chemical Sub-Networks in Two-Box Models

#### 4.2. Fast-Dissipation Ensemble Distribution in Two-Box Models

#### 4.3. Numerical Results and Energy Landscapes

#### 4.4. Fast-Dissipation Chemical Sub-Networks for Continuous Systems

## 5. Diffusion-Controlled Switch of Selection

## 6. Equilibrium Hints for Entropy Production

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Maximum Possible Ratio of Two-State System

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**Figure 1.**(

**a**) ${R}_{BA}=\overline{{P}_{B}}/\overline{{P}_{A}}$ as a function of the energetic barrier $\epsilon $ for both finite and infinite diffusion cases. Small insets are sketches of the reaction network in different conditions, where the thickness of the arrows reflects the speed of the corresponding reaction. The solutions in these settings are also reported next to the insets. (

**b**) Sketch of a two-state, two-box reaction network in a temperature gradient, where the inner circular arrow represents the direction of non-equilibrium stationary flux J. (

**c**) The theoretical critical point ${\epsilon}_{d}$ scales linearly with $d/{k}_{0}$, showing a clear agreement with numerical estimations, with small deviations only for large values of d.

**Figure 2.**(

**a**) Three-state, two-box reaction network. The same color corresponds to the same value of the energy. The energy difference between A and B (or C) is $\u2206E$; (

**b**) $\u2206E=1,{T}_{c}=0.1,{\epsilon}_{C}=1,{\epsilon}_{B}=2$. We report the logarithm of the selection parameter, ${R}_{CB}=\overline{{P}_{C}}/\overline{{P}_{B}}$, as a function of d for different values of the thermal gradient. ${R}_{CB}$ is maximised at infinite diffusion. (

**c**) In this case, $\u2206E=0.1$, while all other parameters and the color code are the same as in panel (

**b**). Here, we show a peak in the selection strength for finite diffusion.

**Figure 3.**(

**a**) Two-box model for a complex reaction network. Solid lines represent fast-dissipation reactions, while slow-dissipation reactions are indicated by dashed lines. Gray dashed circles indicate the fast-dissipative sub-networks in the cold box. The distribution inside each sub-network follows the Boltzmann equilibrium. Overall, the probability is redistributed by diffusion so that half of the total particles populate each box; (

**b**) Kullback–Leibler divergence between exact solutions and theoretical predictions for the stationary probability distribution of different reaction networks composed of 40 states each. Networks are generated by randomly assigning fast-dissipation ($\epsilon =1$) and slow-dissipation ($\epsilon =5$) reactions between states that uniformly populate a given energy range, ${R}_{E}$. The Kullback–Leibler divergence is shown for ${10}^{3}$ reaction networks with the same energy range (identified by the same color), and for an increasing energy range (here estimated through the variance of the energies). As the roughness of the landscape increases, the proposed framework starts failing; (

**c**) theoretical and exact probabilities in comparison for ${R}_{E}=1,2$, and $2.5$ from left to right (here different colors represent different networks).

**Figure 4.**Total probability of states B, $\overline{{P}_{B}}=P\left({B}_{1}\right)+P\left({B}_{2}\right)$, and C, $\overline{{P}_{C}}=P\left({C}_{1}\right)+P\left({C}_{2}\right)$, as a function of d. The diffusion coefficient can trigger a switch of the selected state at stationarity, which is due to the competition between dissipation-driven and energetic selection. The diffusion coefficients at which each transition occurs can be estimated by comparison with the chemical reaction rate in the cold box. The upper inset sketch the chemical network here investigated. Parameters are ${T}_{c}=0.1$, ${T}_{w}=0.2$, ${\epsilon}_{C}=1$, and ${\epsilon}_{B}=2$.

**Figure 5.**(

**a**) For a simple three-state system, we compare the exact entropy production (solid red line) with the theoretical approximated one obtained using our framework (blue dashed line), and the formula obtained by a small gradient expansion, Equation (25) (yellow dot-dashed line). In the exact ${\dot{S}}_{\mathrm{tot}}$ the probabilities stem from direct solution of the master equation, while in the theoretical ${\dot{S}}_{\mathrm{tot}}$, and in Equation (25), the probabilities are obtained employing the equilibration of fast-dissipative sub-networks. To consistently apply our approach, we choose an intermediate value of the diffusion with respect to the chemical rates (see also Figure 4), showing an excellent agreement among the curves presented. (

**b**) It is evident that the main contribution to the entropy production comes from the reaction $A\leftrightharpoons C$, ${\dot{S}}_{A\leftrightarrow C}$, which supports a much faster dissipation with respect to the slow-dissipation branch $A\leftrightharpoons B$. Indeed, we also see that ${\dot{S}}_{A\leftrightarrow B}\ll {\dot{S}}_{A\leftrightarrow C}$. Parameters are reported in panel (

**a**).

**Figure 6.**The logarithm of the selection strength as a function of the critical barrier ${\epsilon}_{d}$ for a three-state, two-box model (solid line), as sketched in the inset, and for a time-periodic driven three-state system (dashed line). These two paradigms are qualitatively and quantitatively equivalent to determine non-isothermal selection of states.

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**MDPI and ACS Style**

Liang, S.; De Los Rios, P.; Busiello, D.M.
Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales. *Entropy* **2021**, *23*, 1068.
https://doi.org/10.3390/e23081068

**AMA Style**

Liang S, De Los Rios P, Busiello DM.
Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales. *Entropy*. 2021; 23(8):1068.
https://doi.org/10.3390/e23081068

**Chicago/Turabian Style**

Liang, Shiling, Paolo De Los Rios, and Daniel Maria Busiello.
2021. "Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales" *Entropy* 23, no. 8: 1068.
https://doi.org/10.3390/e23081068