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Nonequilibrium thermodynamics of uncertain stochastic processes

Jan Korbel & David Wolpert

WOST IV

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

Reference: https://arxiv.org/abs/2210.05249

Idealized experiment

$T$

$X$

$P(T)$

$X$

3rd trial

$T_3$

$X_3$

...

In reality

Measure a quantity $X$ and the assume that the temperature $T$ can be measured with infinite precision

Temperature is measured with limited precision, can change between experiments

$T=?$

In many experiments

We do not know the exact value of
- number of heat baths
- temperatures
- chemical potentials
- energy spectrum
- control protocol
- transition rates
- initial distribution

In real experiments, there is always some uncertainty about the system and its environment

Example: 3-state system

Consider a simple example of a 3-state system coupled to one of three apparatuses.
We want to measure the distribution $p_{t_f}(E)$ at final time $t_f$
There are two possible scenarios:

Example: 3-state system

Effective scenario: we do not know which apparatus is coupled to our system but we can repeat the experiment many times with the same apparatus

Phenomenological scenario: the apparatus is randomly rechosen each time we run the experiment

Consequences for the experiment

Whether the experimenter can adjust the experiment to the actual value of the apparatus has significant consequences for the experiment.

Let's illustrate the difference on a simple example of a moving optical tweezer with uncertain stiffness.
We consider a particle in a laser trap potential described by Langevin equation $\dot{x} = -\frac{\partial V}{\partial x} + \xi$ with the quadratic potential $V_k(x,t) = \frac{k}{2} (x - \lambda(t))^2$ where $k$ is the stiffness.
We want to move the trap from $\lambda_i=0$ at $t_i = 0$ to $\lambda_f$ at time $t_f$
It is possible to calculate the optimal protocol that minimizes the work

Consequences for the experiment

Let us suppose that the stiffness can change each time due to imprecisions in the experimental setup. We assume the stiffness is sampled from a distribution $p(k)$
We compare two scenarios:

Adapted: The experimenter can measure $k$ for each run and can adapt the protocol accordingly
Unadapted: The experimenter can only measure average stiffness $\bar{k}$ and adopts the protocol for that single stiffness $\bar{k}$ for all runs

The average work as a function of standard deviation of the distribution $p(k)$

Thermodynamics of systems coupled to uncertain environment

Consider a set of apparatuses $\mathcal{A}$.
For each apparatus $\alpha \in \mathcal{A}$, we have a system with a precise number of baths, temperatures, chemical potentials, etc. satisfying local detailed balance
We consider a probability distribution $P^\alpha$ over the apparatuses

Effective value over the apparatuses can be defined as

$$ \overline{X}:= \int \mathrm{d} P^\alpha X^\alpha$$

Effective distribution $\bar{p}_x(t)$ fulfills the equation

$ \dot{\bar{p}}_x(t) = \sum_{x'} \int \mathrm{d} P^\alpha K^\alpha_{xx'} p^\alpha_{x'}(t) $ which is generally non-Markovian

Effective ensemble stochastic thermodynamics

Expected internal energy is $\bar{U} = \int \mathrm{d} P^\alpha \sum_x p^\alpha_t(x) u^\alpha(x)$
Expected first law of thermodynamics

$$\dot{\bar{U}} = \dot{\bar{Q}} + \dot{\bar{W}}$$

Expected ensemble entropy $\bar{S} = - \sum_x \int \mathrm{d} P^\alpha p_t^\alpha(x) \ln p_t^\alpha(x)$
Expected second law of thermodynamics

$$\dot{\bar{S}} = \dot{\bar{\Sigma}} + \dot{\bar{\mathcal{E}}}$$

where $\dot{\bar{\Sigma}} \geq 0$
and $\dot{\bar{\mathcal{E}}} = \overline{\beta \dot{Q}}$ - no explicit relation between $\dot{\bar{\mathcal{E}}}$ and $\dot{\bar{Q}}$

Two types of optimal work

For each scenario, we can define the minimum dissipated work

Adapted: For each apparatus $\alpha$, we can choose the optimal protocol $\lambda_\alpha(t)$ minimizing the work. The effective optimal work is $$W^{ad}_{\min} = \int \mathrm{d} P^\alpha \min_{\lambda_\alpha(t)}(W^\alpha[\lambda_\alpha(t)]) $$
Unadapted: Since we cannot adopt the protocol for each apparatus separately, we find a single protocol that minimizes the effective work $$W^{unad}_{\min} = \min_{\lambda(t)} \left( \int \mathrm{d} P^\alpha W^{\alpha}[\lambda(t)] \right) $$

The effective dissipated work is in both cases the difference between the average work obtained from the given set of protocols (adapted scenario) or a single protocol (unadapted scenario, from the optimal work

Uncertainty in initial distribution

We consider that the apparatus is fixed, except for $p_{t_i}$. We denote the set of possible initial distributions $p_{t_i}^\alpha$ where each distribution has probability $ p(\alpha) $ to appear.
We consider a distribution $q_i$ that minimizes the expected work.
For any other distribution, it is possible to express the dissipated work as $$W_{diss}(p_{t_i}^\alpha) = D_{KL}(p_{t_i}^\alpha || q_{t_i}) - D_{KL}(p_{t_f}^\alpha || q_{t_f}) + W_{diss}(q_{t_i})$$
If $p_{t_f}^\alpha = q_{t_f}$ for all $\alpha$, i.e., the process has the same ending distribution regardless of the initial distribution, we obtain $$ \overline{W}_{diss} =\int \mathrm{d} P^\alpha W_{diss}(p_{t_i}^\alpha) \geq \int \mathrm{d} P^\alpha D_{KL}(p_{t_i}^\alpha || \bar{p}_{t_i}) = D_{JS}(\{p_{t_i}^\alpha\}^\alpha) $$

Phenomenological EP and FTs

Let us now focus on the case where the apparatus changes after each stochastic trajectory $\pmb{x}$ is generated. We are not able to measure $\pmb{P}(\pmb{x}|\alpha)$ but only $\overline{\pmb{P}}(\pmb{x})$
Denote $\pmb{P}^\alpha(\pmb{x}) \equiv \pmb{P} (\pmb{x}|\alpha)$ and $\pmb{P}(\pmb{x},\alpha) = \pmb{P}(\pmb{x}|\alpha) P^\alpha$. $^{\dag}$ denotes time-reversal
Effective ensemble EP $\bar{\Sigma}= D_{KL}(\pmb{P}(\pmb{x},\alpha)||\pmb{P}^\dag(\pmb{x}^{\dag},\alpha)) $
By using the chain rule for KL-divergence $$\underbrace{D_{KL}(\pmb{P}(\pmb{x},\alpha)||\pmb{P}^\dag (\pmb{x}^{\dag},\alpha))}_{\bar{\Sigma}} = \underbrace{D_{KL}(\overline{\pmb{P}}(\pmb{x})||\overline{\pmb{P}}^{\dag}(\pmb{x}^{\dag}))}_{\Phi} + \underbrace{D_{KL}(\pmb{P}(\alpha|\pmb{x})||\tilde{\pmb{P}}(\alpha|\tilde{\pmb{x}}))}_{\Lambda}$$
Here $\pmb{P}(\alpha|\pmb{x}) = \frac{\pmb{P}(\pmb{x}|\alpha) P^\alpha}{\pmb{P}(\pmb{x})}$
We have three types of EP

Effective EP $\bar{\Sigma}$
Phenomenological EP $\Phi$
Likelihood EP $\Lambda$

Phenomenological EP

Phenomenological EP $\Phi = D_{KL}(\overline{\pmb{P}}(\pmb{x})||\overline{\pmb{P}}^{\dag}(\pmb{x}^{\dag})) = \int \mathcal{D} \pmb{x} \pmb{P}(\pmb{x}) \color{pink}{\ln \frac{\pmb{P}(\pmb{x})}{{\pmb{P}^\dag}({\pmb{x}}^\dag)}}$

Phenomenological trajectory EP is $\color{pink}{\phi(\pmb{x}) = \ln \frac{\overline{\pmb{P}}(\pmb{x})}{\overline{\pmb{P}}^\dag(\pmb{x}^{\dag})}}$

It is straightforward to show that $\phi$ fullfills detailed fluctuation theorem

$$ \frac{P(\phi)}{P^\dag(-\phi)} = e^{\phi}$$

Phenomenological EP describes thermodynamics for the case of expected probability

It is a lower bound for the effective EP: $\overline{\Sigma} \geq \Phi$

Likelihood EP

Likelihood EP $\Lambda = D_{KL}(\pmb{P}(\alpha|\pmb{x})||{\pmb{P}^{\dag}}(\alpha|\pmb{x}^\dag)) = \int \mathrm{d} P^\alpha \pmb{P}(\alpha|\pmb{x}) {\color{pink} \ln \frac{\pmb{P}(\alpha|\pmb{x})}{\pmb{P}^\dag(\alpha|\pmb{x}^\dag)}}$

Likelood trajectory EP as ${\color{pink}\lambda(\alpha|\pmb{x})}:= \sigma(\pmb{x}|\alpha) - \phi(\pmb{x}) = {\color{pink}\ln \frac{\pmb{P}(\alpha|\pmb{x})}{\pmb{P}^\dag(\alpha|\pmb{x}^\dag)}}$

We can also show that $\omega$ fulfills Detailed FT:

$$\frac{P(\lambda_{\pmb{x}})}{\tilde{P}(-\lambda_{\pmb{x}^\dag})} = e^{\lambda_{\pmb{x}}} $$

From Integrated FT, we obtain that $\Lambda_{\pmb{x}} = \langle \lambda_{\pmb{x}} \rangle_{P(\lambda_{\pmb{x}})} \geq 0$

By averaging over all trajectories $\Lambda = \langle \Lambda_{\pmb{x}} \rangle_{\pmb{P}(\pmb{x})} \geq 0$

Likelihood EP tells us how much more one can learn about the apparatus by observing a forward trajectory versus a backward trajectory

Nonequilibrium thermodynamics of uncertain stochastic processes

Idealized experiment

In reality

In many experiments

In real experiments, there is always some uncertainty about the system and its environment

Example: 3-state system

Example: 3-state system

Consequences for the experiment

Consequences for the experiment

Thermodynamics of systems coupled to uncertain environment

Effective ensemble stochastic thermodynamics

Two types of optimal work

Uncertainty in initial distribution

Phenomenological EP and FTs

Phenomenological EP

Likelihood EP

Example: 2-state system with uncertain temperature

Other topics