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
Example: 2-state system with uncertain temperature
Other topics
Further topics included in the paper
- Maximal work extractions with uncertain temperatures
- Dynamics of the thermodynamic value of information when there are uncertain thermodynamic parameters
Future steps:
- Systems with uncertain energy spectrums
- Experiments with uncertain control protocols
- Complete analysis of maximal work extraction
- Extension to time-dependent apparatuses
- Extension to Hidden Markov models
Thanks!