USING STOCHASTIC THERMODYNAMICS TO ANALYZE NON-THERMODYNAMIC PROPERTIES OF DYNAMIC SYSTEMS
QLS meeting, 12th July 2022
in collaboration with
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9585853/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/8965659/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9701874/pasted-from-clipboard.png)
Tuan Pham
Farita Tasnim
David Wolpert
USING STOCHASTIC THERMODYNAMICS TO ANALYZE NON-THERMODYNAMIC PROPERTIES OF DYNAMIC SYSTEMS
OUTLINE
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Stochastic thermodynamics for non-physical systems (motivation a.k.a. big picture)
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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
are still valid.
Similarities
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.
TYPICAL SETUP OF
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
TYPICAL SETUP OF
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.
EXamples OF
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
...
EXamples OF
NON-PHYSICAL SYSTEMS
2) Distributed computational systems
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9702306/fig2_important_issues_v3.jpg)
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
EXamples OF
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
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9702546/pasted-from-clipboard.png)
What is the dependence of Entropy production on network topology?
In general: the dependence of EP on network topology is complicated.
Experiment: start with a directed acyclic graph and try to
a) add more links
b) make some links reciprocal
Question: what is the impact on entropy production?
what is the impact on adiabatic and non-adiabatic EP?
$$ \dot{\Sigma}^{a}_t = \sum_{mn} (w_{mn}p_n - w_{nm} p_m) \log \frac{w_{mn}p^{st}_n}{w_{nm}p^{st}_m}$$
$$ \dot{\Sigma}^{na}_t = \sum_{mn} (w_{mn}p_n - w_{nm} p_m) \log \frac{p^{st}_n}{p^{st}_m}$$
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9702670/newfig1.jpg)
What is the dependence of Entropy production on network topology?
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9702675/compare_adding_n20.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9702681/compare_adding_dag_n20.jpg)
MORE RESULTS
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9704393/SPL_MPP.jpg)
Speed limit theorems
MORE RESULTS
THERMODYNAMIC UNCERTAINTY RELATIONS
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9704402/TUR1.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9704403/TUR2.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9704404/TUR3.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/9704405/TUR4.jpg)
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
QLS presentation
By Jan Korbel
QLS presentation
- 54