# USING STOCHASTIC THERMODYNAMICS TO ANALYZE NON-THERMODYNAMIC PROPERTIES OF DYNAMIC SYSTEMS

QLS meeting, 12th July 2022

in collaboration with

Tuan Pham

Farita Tasnim

David Wolpert

# OUTLINE

• Stochastic thermodynamics for non-physical systems          (motivation a.k.a. big picture)

• Stochastic thermodynamics of dynamic opinion networks and genetic regulatory networks                                                      (detailed example of ST for non-physical systems)

# Why stochastic thermodynamics in non-physical systems?

### What do non-physical systems have in common with the typical setup of stochastic thermodnyamics?

Linear Markov evolution of probability distribution

$$\dot{p}_m(t) = \sum_n (w_{mn} p_n(t) - w_{nm} p_m(t))$$

This has profound consequences. Regardless of any physical interpretation, the entropy production rate

$$\dot{\Sigma}_t = \sum_{mn} (w_{mn} p_n - w_{nm} p_m ) \log \frac{w_{mn} p_n}{w_{nm} p_m}$$

is a non-negative quantity. Many results of ST including

$$\frac{P(\bar{\Sigma}_t = A)}{\tilde{P}(\bar{\Sigma}_t= -A)} = e^{At}$$

FT's

$$\frac{Var(J_t)}{E(J_t)^2} \geq \frac{2}{\bar{\Sigma}_t}$$

TUR's

$$\frac{L(p(t),p(0))}{2 \Sigma_t \bar{A}_t} \leq t$$

SLT's

# DIFFERENCES

### What ARE DIFFERENCES BETWEEN  NON-PHYSICAL SYSTEMS AN typical setup of stochastic thermodnyamics?

Broken local detailed balance

$$\log \frac{w_{mn}}{w_{nm}} \neq \beta (\epsilon_m - \epsilon_n)$$

That's OK. Many results of ST still remain valid.

No existence of a single potential (energy)

state $$m \nRightarrow$$ energy $$\epsilon_m$$

No global first law of thermodynamics.

While the physical interpretation of the second law is not valid anymore, its applications to ST remain valid.

# NON-PHYSICAL SYSTEMS

subsystem 1

states $$s_i^1$$

potential $$H^1(s_i^1|\textcolor{green}{s_j^2})$$

subsystem LDB

$$\log \frac{w_{ii'}^1}{w_{i'i}^1} = \beta^1(H^1(s_i^1|\textcolor{green}{s_j^2}) - H^1(s^1_{i'}|\textcolor{green}{s_j^2}))$$

subsystem 2

states $$s_i^2$$

potential $$H^2(s_i^2|\textcolor{blue}{s_j^1})$$

subsystem LDB

$$\log \frac{w_{ii'}^2}{w_{i'i}^2} = \beta^2(H^2(s_i^2|\textcolor{blue}{s_j^1}) - H^2(s^2_{i'}|\textcolor{blue}{s_j^2}))$$

No global potential

No global LDB

# NON-PHYSICAL SYSTEMS

Does existence of subsystem potentials (Hamiltonians) lead to existence of global potential?

1

2

3

4

5

Generally not. Subsystem potentials must satisfy certain constraints.

# NON-PHYSICAL SYSTEMS

### 1) lIVING systems

The simplest model of a living system

metabolism

nutrients

energy

ATP

chemical reaction network

energy $$E_n$$

# molecules $$N_n$$

evolution

genotype

phenotype

fitness function $$\Psi_n$$

environment

production of DNA,

proteins

ATP synthase

...

# NON-PHYSICAL SYSTEMS

### 2) Distributed computational systems

by Farita Tasnim

a) parallel bit erasure

b) modularity of computational systems

c) hierarchy of computational systems

d) as a result, computational systems have a similar structure

# NON-PHYSICAL SYSTEMS

### 3) OPINION DYNAMICS OF SOCIAL NETWORKS

influencer

followers

friends

influencer

followers

friends

friends

opinions $$s_i$$, local stress function $$H^i(s_i|s_j)$$

# SIMPLE EXAMPLE of A system with Local potentials driven by CTMC

• Consider a system compound of N subsystems
• Each subsystem has a binary state $$s_i \in \{-1,1\}$$
• The coupling between subsystems is given by (possibly asymmetric) matrix $$J_{ij} \in \{0,1\}$$
• We define the in-neighborhood of a subsystem $$i$$ as $$N_i = \{j|J_{ij} = 1\}$$
• Local Hamiltonian is defined as $$H^i(s_i|s_j) = - \sum_{j \in N_i} s_i s_j$$
• The systems evolves according to CTMC (master equation) $$p(\vec{s}) = \sum_{\vec{s}'} w_{\vec{s}\vec{s}'} p(\vec{s}')$$
• The rate matrix can be decomposed as $$w_{\vec{s}\vec{s}'} = \sum_i w^i_{\vec{s}\vec{s}'}$$ where $$w^i_{\vec{s}\vec{s}'}$$ is non-zero only if $$s_j = s_j'$$ for $$j \neq i$$ (i.e., $$w^i$$ is the rate matrix for i-th subsystem)
• We consider subsystem local detailed balance $$\log \frac{w^i_{\vec{s}\vec{s}'}}{w^i_{\vec{s}'\vec{s}}} = \beta (H^i(\vec{s}')-H^i(\vec{s}))$$
• The whole dynamics and thermodynamics is given by the network topology (and initial conditions)

### local ising spin hamiltonian model

influencer

followers

friends

influencer

followers

friends

friends

opinions $$s_i$$, local stress function $$H^i(s_i|s_j)$$

### local ising spin hamiltonian model

=OPINION DYNAMICS OF A SOCIAL NETWORK

### local ising spin hamiltonian model

=SIMPLE MODEL OF A GENE REGULATORY NETWORK

### What is the dependence of Entropy production on network topology?

In general: the dependence of EP on network topology is complicated.

Question: what is the impact on entropy production?

$$\dot{\Sigma}^{a}_t = \sum_{mn} (w_{mn}p_n - w_{nm} p_m) \log \frac{w_{mn}p^{st}_n}{w_{nm}p^{st}_m}$$

$$\dot{\Sigma}^{na}_t = \sum_{mn} (w_{mn}p_n - w_{nm} p_m) \log \frac{p^{st}_n}{p^{st}_m}$$

## SUMMARY

• Many results of stochastic thermodynamics can be useful in non-physical systems
• These systems are typically composed of many subsystems
• While each subsystem is described by local potential (Hamiltonian) and satisfied subsystem local detailed balance, there is no global Hamiltonian (and no global LDB)
• Dependence of thermodynamic quantities on network topology is quite complicated, however, some preliminary results were obtained

By Jan Korbel

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