[Paper Review] Maximizing free energy gain

Artemy Kolchinsky, Iman Marvian+7 more|arXiv (Cornell University)|Apr 28, 2017

Advanced Thermodynamics and Statistical Mechanics2 references3 citations

TL;DR

This paper derives closed-form solutions for maximizing free energy gain and extractable work in classical and quantum stochastic processes by optimizing the initial state distribution. It shows the problem is convex and solvable via gradient descent, with the optimal free energy increase quantified by the difference in Kullback-Leibler divergences between initial and final states, enabling direct comparison between optimal and sub-optimal protocols.

ABSTRACT

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesisto energy storage systems such as fuels and batteries. Here we consider the maximization of free energy -- and by extension, the maximum extractable work -- that can be gained by a classical or quantum system that undergoes driving by its environment. We consider how the free energy gain depends on the initial state of the system, while also accounting for the cost of preparing the system. We provide simple necessary and sufficient conditions for increasing the gain of free energy by varying the initial state. We also derive simple formulae that relate the free energy gained using the optimal initial state rather than another suboptimal initial state. Finally, we demonstrate that the problem of finding the optimal initial state may have two distinct regimes, one easy and one difficult, depending on the temperatures used for preparation and work extraction. We illustrate our results on a simple model of an information engine.

Motivation & Objective

• To identify the initial probability distribution that maximizes free energy gain in classical and quantum stochastic processes.
• To compare the maximum extractable work from optimal vs. sub-optimal initial states.
• To establish conditions under which the maximum free energy increase is guaranteed, especially when temperature and potentials vary during the process.
• To show that the optimization problem for free energy harvesting is convex, enabling global optimization via gradient descent.
• To extend non-equilibrium thermodynamics and quantum resource theory to derive general formulae for maximum work extraction and free energy increase under arbitrary constraints.

Proposed method

• Derives the non-equilibrium free energy as $ F(p) = \langle E \rangle - \tau S $, where $ \tau $ is temperature and $ S $ is entropy.
• Expresses extractable work as $ W^{ex} = \tau D(p \| p_{th}) $, using the Kullback-Leibler divergence between the system's state and its thermal equilibrium state.
• Applies variational calculus with Lagrange multipliers to maximize free energy gain under constraints on energy, volume, particle number, etc.
• Derives Theorem 4: $ \Delta G(q_0) - \Delta G(r_0) = \tau_0 D(r_0 \| q_0) - \tau_1 D(r_1 \| q_1) $, quantifying the difference in free energy gain between optimal and sub-optimal initial states.
• Proves convexity of the free energy gain function, ensuring global optimality via gradient descent.
• Extends results to quantum systems evolving under completely positive maps, showing the same formalism applies in the quantum regime.

Experimental results

Research questions

• RQ1 What initial probability distribution maximizes the free energy gain in a classical stochastic process with time-varying temperature and external potentials?
• RQ2 How does the extractable work differ between an optimal initial state and a sub-optimal one, and what determines this difference?
• RQ3 Under what conditions is the maximum free energy gain guaranteed to be achieved, particularly when the final temperature is lower than the initial temperature?
• RQ4 Can the problem of optimal free energy harvesting be formulated as a convex optimization problem, and if so, how can it be solved efficiently?
• RQ5 To what extent do the results generalize to quantum stochastic processes governed by completely positive maps?

Key findings

• The maximum free energy gain is achieved when the initial state minimizes the Kullback-Leibler divergence with respect to the final state, under the constraints of the process.
• The difference in free energy gain between optimal and sub-optimal initial states is given by $ \Delta G(q_0) - \Delta G(r_0) = \tau_0 D(r_0 \| q_0) - \tau_1 D(r_1 \| q_1) $, which is always non-negative when $ \tau_1 \leq \tau_0 $, ensuring optimality.
• The optimization problem for free energy harvesting is convex, meaning there are no local optima and global convergence is guaranteed via gradient descent.
• The results apply to both classical and quantum systems evolving under completely positive maps, extending the framework beyond equilibrium thermodynamics.
• The framework supports varying temperature, pressure, chemical potential, and other generalized conjugate forces throughout the process, enabling realistic modeling of energy harvesting in dynamic environments.
• For biological systems like photosynthetic organisms, the model explains evolutionary pressure to optimize initial state preparation to maximize free energy gain, especially when harvesting occurs under changing environmental conditions.