# Stochastic Thermodynamics of Multiple Co-Evolving Systems—Beyond Multipartite Processes

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## Abstract

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## 1. Introduction

**subsystems**.

**composite systems**, and their overall dynamics is called a

**composite process**. There has been some preliminary work extending the stochastic thermodynamics of MPPs to consider composite processes [4]. (It should also be noted that [5] considered scenarios where different subsystems share mechanisms, restricted to the specific issue of how to extend the concept of the “learning rate” to biochemical diffusion processes that have this characteristic.) Here, we extend the preliminary work in [4] and obtain new results on the stochastic thermodynamics of composite processes.

## 2. Stochastic Thermodynamics of Composite Processes

#### 2.1. Background on Composite Processes

**puppet set**and write it as $\mathcal{P}\left(v\right)\subseteq \mathcal{N}$.

**dependency network**, each edge $j\to i$ means that the state of subsystem j affects the rate of state transitions in subsystem i. (We do not assign the self-dependency of a subsystem’s dynamics its own edge.) We refer to the set of subsystems whose states affect the dynamics of i as the

**leaders**of i. Thus, $j\to i$ means that j is a leader of i, in addition to i itself. In any dependency network, the leaders of each subsystem i are i itself together with its parents in the dependency graph, $\mathrm{pa}\left(i\right)$.

**leader set**for a mechanism v is defined to be the union of the leaders of each subsystem in the puppet set of v: $\mathcal{L}\left(v\right)={\bigcup}_{i\in \mathcal{P}\left(v\right)}\mathrm{pa}\left(i\right)$, where $\mathrm{pa}\left(i\right)$ represents the parents of i in the dependency network. As an example, although the puppet set of mechanism ${v}_{2}$ in Figure 2 is $\{A,C,D\}$, the leader set of ${v}_{2}$ is $\{A,B,C,D\}$.

#### 2.2. Background on Units

**unit**$\omega \subseteq \mathcal{N}$ is a collection of subsystems such that, as the full system’s state evolves via a master equation according to $K\left(t\right)$, the marginal distribution over the states of the unit also evolves according to its own CTMC,

**unit structure**if it obeys the following properties [4].

- The union of the units in the set equals $\mathcal{N}$, i.e., they cover $\mathcal{N}$:$${\mathcal{N}}^{*}=\{{\omega}_{1},{\omega}_{2},\dots \}:\bigcup _{\omega \in {\mathcal{N}}^{\u2020}}\omega =\mathcal{N}$$
- The set is closed under intersections of its units:

**inclusion–exclusion sum**(or simply “in–ex sum” for short) of f for the unit structure ${\mathcal{N}}^{*}$. (See [31] for background on the inclusion–exclusion principle.)

**in–ex information**as

## 3. Strictly Positive Lower Bounds on EP from Its In–Ex Decomposition

#### 3.1. Mismatch Cost

**prior**, due to its Bayes optimality when running a full thermodynamic cycle [33], and the drop in KL divergence in Equation (12) is called the

**mismatch cost**. (It is important to note that Equation (12) applies for an extremely wide range of dynamic processes implementing P, for many types of state space, and for many other thermodynamic costs generated during the process, in addition to EP; see [15,34].)

#### 3.2. Periodic Processes

#### 3.3. Example Where In–Ex Decomposition Is Necessary for Lower-Bound EP

## 4. Thermodynamics Due to Multiplicity of Mechanisms

#### 4.1. Additional Decompositions of Thermodynamic and Dynamical Quantities in Composite Processes

#### 4.2. Thermodynamic Uncertainty Relations for Composite Processes

#### 4.3. Information Flow TURs

## 5. Strengthened Thermodynamic Speed Limits for Composite Processes

## 6. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Examples of systems whose dynamics can be modeled as composite processes. Each system consists of multiple subsystems (blue circles). Mechanisms are denoted as r, and their puppet sets $\mathcal{P}\left(r\right)$ are indicated by translucent white bubbles. (

**a**) An example stochastic CRN consists of four co-evolving species $\{{X}_{1},{X}_{2},{X}_{3},{X}_{4}\}$ that change state according to three chemical reactions $\{A,B,C\}$. (

**b**) An example toy circuit consists of four conductors $\{1,2,3,4\}$ that change state via interactions with three devices $\{A,B,C\}$.

**Figure 2.**The dependency network specifies how the dynamics of each subsystem is governed by the states of other subsystems. This network defines the leader sets in a composite process.

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**MDPI and ACS Style**

Tasnim, F.; Wolpert, D.H.
Stochastic Thermodynamics of Multiple Co-Evolving Systems—Beyond Multipartite Processes. *Entropy* **2023**, *25*, 1078.
https://doi.org/10.3390/e25071078

**AMA Style**

Tasnim F, Wolpert DH.
Stochastic Thermodynamics of Multiple Co-Evolving Systems—Beyond Multipartite Processes. *Entropy*. 2023; 25(7):1078.
https://doi.org/10.3390/e25071078

**Chicago/Turabian Style**

Tasnim, Farita, and David H. Wolpert.
2023. "Stochastic Thermodynamics of Multiple Co-Evolving Systems—Beyond Multipartite Processes" *Entropy* 25, no. 7: 1078.
https://doi.org/10.3390/e25071078