# STOCHASTIC THERMODYNAMICS OF COMPLEX SYSTEMS

Jan Korbel

CSH Friday Seminar "Analysis of complex systems" - 5.3.2021

Slides available at: https://slides.com/jankorbel

# Introduction

Microscopic systems

Classical mechanics (QM,...)

Mesoscopic systems

Stochastic thermodynamics

Macroscopic systems

Thermodynamics

Statistical mechanics

Trajectory TD

Ensemble TD

Stochastic Thermodynamics is a thermodynamic theory
for mesoscopic, non-equilibrium physical systems
interacting with equilibrium thermal (and/or chemical)
reservoirs

# History

### Equilibrium thermodynamics(19 th century)

- Maxwell, Boltzman, Planck, Claussius, Gibbs...

- Macroscopic systems ($$N \rightarrow \infty$$) in equilibrium (no time dependence of measurable quantities - thermoSTATICS)

-  General structure of thermodynamics

• Laws of thermodynamics (general)
• Response coefficients (system-specific)

- Applications: engines, refridgerators, air-condition,...

efficiency $$\leq 1-\frac{T_2}{T_1}$$

# History

### Laws of thermodynamics

Zeroth law:

Temperature can be measured. $$T_A = T_B \quad \mathrm{if} \quad A \ \mathrm{and} \ B \ \mathrm{are} \ \mathrm{in} \ \mathrm{equilibrium}.$$

First law (Claussius 1850, Helmholtz 1847):

Energy is conserved.
$${\color{aqua} d}U = {\color{orange} \delta} Q - {\color{orange} \delta} W$$ Second law (Carnot 1824, Claussius 1854, Kelvin):

Heat cannot be fully transformed into work. $${ \color{aqua} d} S \geq \frac{{\color{orange} \delta} Q}{T}$$ Third law: We cannot bring the system into the absolute zero
temperature in a finite number of steps. $$\lim_{T \rightarrow 0} S(T) = 0$$

# History

### Local equilibrium thermodynamics(1st half of 20th cent.)

- Onsager, Rayleigh...

- Systems close to equilibrium - linear response theory

-  Local equilibrium: subsystems a,b,c are each in equilibrium

Total entropy $$S \approx S^a + S^b + S^c + \dots$$

Entropy production $$\sigma^a = \frac{d S^a}{d t} = \sum_i Y_i^a J_i^a$$

$$Y_i^a$$ - thermodynamic forces; $$J_i^a$$ - thermodynamic currents

4th Law of thermodynamics (Onsager 1931): $$\sigma = \sum_{ij} L_{ij} \Gamma_i \Gamma_j$$

$$\Gamma_i = Y_i^a - Y_i^b$$ - afinity, $$L_{ij}$$ - symmetric

# History and now

### Stochastic thermodynamics(90s of 20th century - present)

- Evans, Jarzynski, Crooks, Seifert, van den Broek,....

- Mesoscopic systems far from equilibrium

- Combines stochastic calculus and non-equilibrium thermodynamics

- Main results: Trajectory thermodynamics, Fluctuation theorems, Thermodynamic uncertainty relations, Speed limit theorems,...

- Applications: colloidal particles and soft matter, biochemistry, molecular motors

### Molecular motor: kinesine walking on microtobulus

efficiency $$\leq 1$$

# Stochastic thermodynamics

1.) Consider linear Markov (= memoryless) with distribution $$p_i(t)$$.

Its evolution is described by master equation

$$\dot{p}_i(t) = \sum_{j} [w_{ij} p_{j}(t) - w_{ji} p_i(t) ]$$

$$w_{ij}$$ is transition rate.

2.) Entropy of the system - Shannon entropy  $$S(P) = - \sum_i p_i \log p_i$$. Equilibrium distribution is obtained by maximization of $$S(P)$$ under the constraint of average energy $$U(P) = \sum_i p_i \epsilon_i$$

$$p_i^{eq} = \frac{1}{Z} \exp(- \beta \epsilon_i) \quad \mathrm{where} \ \beta=\frac{1}{k_B T}, Z = \sum_j \exp(-\beta \epsilon_j)$$

# Stochastic thermodynamics

3.) Detailed balance - stationary state ($$\dot{p}_i = 0$$ ) coincides with the equilibrium state ($$p_i^{eq}$$). We obtain

$$\frac{w_{ij}}{w_{ji}} = \frac{p_i^{eq}}{p_j^{eq}} = e^{\beta(\epsilon_j - \epsilon_i)}$$

4.) Second law of thermodynamics:

$$\dot{S} = - \sum_i \dot{p}_i \log p_i = \frac{1}{2} \sum_{ij} (w_{ij} p_j - w_{ji} p_i) \log \frac{p_j}{p_i}$$

$$=\underbrace{\frac{1}{2} \sum_{ij} (w_{ij} p_j - w_{ji} p_i) \log \frac{w_{ij} p_j}{w_{ji} p_i}}_{\dot{S}_i} + \underbrace{\frac{1}{2} \sum_{ij} (w_{ij} p_j - w_{ji} p_i) \log \frac{w_{ji}}{w_{ij}}}_{\dot{S}_e}$$

$$\dot{S}_i \geq 0$$ - entropy production rate (2nd law of TD)

$$\dot{S}_e = \beta \dot{Q}$$ entropy flow rate

# Stochastic thermodynamics

5.) Trajectory thermodynamics - consider stochastic trajectory

$$x(t)= (x_0,t_0;x_1,t_1;\dots)$$. Energy $$E_x = E_x(\lambda(t))$$, $$\lambda(t)$$ - control protocol

Probability of observing $$x(t)$$: $$\mathcal{P}(x(t)$$)

Time reversal $$\tilde{x}(t) = x(T-t)$$

Reversed protocol $$\tilde{\lambda}(t) = \lambda(T-t)$$

Probability of observing reversed trajectory under reversed protocol $$\tilde{\mathcal{P}}(\tilde{x}(t))$$

# Stochastic thermodynamics

6.) Fluctuation theorems

Trajectory entropy: $$s(t) = - \log p_x(t)$$

Trajectory 2nd law $$\Delta s = \Delta s_i + \Delta s_e$$

Relation to the trajectory probabilities

$$\log \frac{\mathcal{P}(x(t))}{\tilde{\mathcal{P}}(\tilde{x}(t))} = \Delta s_i$$

Detailed fluctuation theorem $$\frac{P(\Delta s_i)}{\tilde{P}(-\Delta s_i)} = e^{\Delta s_i}$$

Integrated fluctuation theorem $$\langle e^{- \Delta s_i} \rangle = 1 \quad \Rightarrow \langle \Delta s_i \rangle = \Delta S_i \geq 0$$

# Complex systems

• Complex systems are composed of many interacting subsystems
• Standard stochastic thermodynamics works with weakly-interacting subsystems that can be treated separately
• Shannon entropy and/or linear Markov dynamics do not have to be adequate description

We will discuss two examples of complex systems

## Thermodynamics of structure-forming systems

(with S. Lindner, R. Hanel, S. Thurner - Nat. Comm. 12 (2021) 1127)

• Many systems form structures: molecules of atoms, clusters of colloidal particles, (bio)polymers or micelles
• Toy model: magnetic coin model

$$W(N) \sim e^{N \ln N}$$

"Simple systems"  $$W(N) \sim a^N$$

## Multiplicity of structure-forming systems

Boltzmann entropy formula: $$S = k_B \log W$$

$$W$$ - is multiplicity, i.e., number of microstates corresponding to a mesostate

Microstate: state of each particle

if more particles are bound to a molecule, then state each molecule

Mesostate: how many particles and/or molecules are in given state

Example: magnetic coin model: 3 coins, magnetic

microstates                          mesostate                 multiplicity

3

3

## Multiplicity

### General formula: $$W = \frac{n!}{\prod_{ij} n_i^{(j)}! (j!)^{n_i^{(j)}}}$$

Boltzmann's 1884 paper

## Entropy of structure-forming systems

$$S = \log W \approx n \log n - \sum_{ij} \left(n_i^{(j)} \log n_i^{(j)} - n_i^{(j)} + {\color{aqua} n_i^{(j)} \log j!}\right)$$

Introduce "probabilities" $$\wp_i^{(j)} = n_i^{(j)}/n$$

$$\mathcal{S} = S/n = - \sum_{ij} \wp_i^{(j)} (\log \wp_i^{(j)} {\color{aqua}- 1}) {\color{aqua}- \sum_{ij} \wp_i^{(j)}\log \frac{j!}{n^{j-1}}}$$

Finite interaction range: concentration $$c = n/b$$

$$\mathcal{S} = S/n = - \sum_{ij} \wp_i^{(j)} (\log \wp_i^{(j)} {\color{aqua}- 1}) {\color{aqua}- \sum_{ij} \wp_i^{(j)}\log \frac{j!}{{\color{orange}c^{j-1}}}}$$

Equilibrium distribution:

$$\hat{\wp}_i^{(j)} = \frac{c^{j-1}}{j!} \exp(-\alpha j - \beta \epsilon_i^{(j)})$$

# Examples

## Currie-Weiss model with molecules

$$H(\sigma_i) = - \frac{J}{n-1} \sum_{i \neq j, \ free} \sigma_i \sigma_j - h \sum_{j, \ free} \sigma_j$$

## Stochastic thermodynamics of structure-forming systems

Detailed fluctuation theorem

$$\frac{P(\Delta \sigma)}{\tilde{P}(-\Delta \sigma)} = e^{\Delta \sigma}$$

where

$$\Delta \sigma = \Delta s_i + \log j_0 - \log j_f$$

## Stochastic thermodynamics of       non-linear systems

(with D. Wolpert - New J. Phys.  doi:10.1088/1367-2630/abea46)

• Many complex, long-range systems are non-linear
• Question: can we use stochastic thermodynamics?

Requirements:

1. Non-linear Markov dynamics $$\dot{p}_i(t) = \frac{1}{C(p)} \sum_{j} [w_{ij} \Omega(p_{j}(t)) - w_{ji} \Omega(p_i(t)) ]$$
2. Detailed balance $$\frac{w_{ij}}{w_{ji}} = \frac{\Omega(p_i^{eq})}{\Omega(p_j^{eq})}$$
3. Second law of thermodynamics:
• ​​For some (generalized) entropy $$S = f(\sum_i g(p_i))$$: $$\dot{S}_i \geq 0$$.

## Stochastic thermodynamics of       non-linear systems

Theorem: Requirements 1-3 imply

1) $$\Omega(p_m) = \exp(-g'(p_m))$$

2) $$C(p) = f'(\sum_m g(p_m))$$

Stochastic entropy: $$s(t) = \log\left(\frac{1}{\Omega(p_x(t))}\right) = g'(p_x(t))$$

Detailed fluctuation theorem holds:

$$\frac{P(\Delta s_i)}{\tilde{P}(-\Delta s_i)} = e^{\Delta s_i}$$

## Here's more!

1) CSH Talk: Friday 12th March 3pm - Gülce Kardes

Thermodynamic uncertainty relations for multipartite processes

2) WOST II: Virtual workshop on stochastic thermodynamics

Tutorials 13th May, Conference 17th - 21st May

Web: https://wiki.santafe.edu/index.php/Stochastic_Thermodynamics_II

By Jan Korbel

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