USING STOCHASTIC THERMODYNAMICS TO ANALYZE NONTHERMODYNAMIC PROPERTIES OF DYNAMIC SYSTEMS
QLS meeting, 12th July 2022
in collaboration with
Tuan Pham
Farita Tasnim
David Wolpert
USING STOCHASTIC THERMODYNAMICS TO ANALYZE NONTHERMODYNAMIC PROPERTIES OF DYNAMIC SYSTEMS
OUTLINE

Stochastic thermodynamics for nonphysical systems (motivation a.k.a. big picture)

Stochastic thermodynamics of dynamic opinion networks and genetic regulatory networks (detailed example of ST for nonphysical systems)
Why stochastic thermodynamics in nonphysical systems?
What do nonphysical 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 nonnegative 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 NONPHYSICAL 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
NONPHYSICAL 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
NONPHYSICAL 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
NONPHYSICAL 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
NONPHYSICAL 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
EXamples OF
NONPHYSICAL SYSTEMS
3) OPINION DYNAMICS OF SOCIAL NETWORKS
influencer
followers
friends
influencer
followers
friends
friends
opinions \(s_i\), local stress function \(H^i(s_is_j)\)
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 inneighborhood of a subsystem \(i\) as \(N_i = \{jJ_{ij} = 1\}\)
 Local Hamiltonian is defined as \(H^i(s_is_j) =  \sum_{j \in N_i} s_i s_j\)
 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 nonzero only if \(s_j = s_j'\) for \(j \neq i\) (i.e., \(w^i\) is the rate matrix for ith 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 \(H^i(s_is_j)\)
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.
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 nonadiabatic EP?
$$ \dot{\Sigma}^{a}_t = \sum_{mn} (w_{mn}p_n  w_{nm} p_m) \log \frac{w_{mn}p^{st}_n}{w_{nm}p^{st}_m}$$
$$ \dot{\Sigma}^{na}_t = \sum_{mn} (w_{mn}p_n  w_{nm} p_m) \log \frac{p^{st}_n}{p^{st}_m}$$
What is the dependence of Entropy production on network topology?
MORE RESULTS
Speed limit theorems
MORE RESULTS
THERMODYNAMIC UNCERTAINTY RELATIONS
SUMMARY
 Many results of stochastic thermodynamics can be useful in nonphysical 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
 53