Minimum Information Variability in Linear Langevin Systems via Model Predictive Control

Controlling the time evolution of a probability distribution that describes the dynamics of a given complex system is a challenging problem. Achieving success in this endeavour will benefit multiple practical scenarios, e.g., controlling mesoscopic systems. Here, we propose a control approach blending the model predictive control technique with insights from information geometry theory. Focusing on linear Langevin systems, we use model predictive control online optimisation capabilities to determine the system inputs that minimise deviations from the geodesic of the information length over time, ensuring dynamics with minimum "geometric information variability". We validate our methodology through numerical experimentation on the Ornstein-Uhlenbeck process and Kramers equation, demonstrating its feasibility. Furthermore, in the context of the Ornstein-Uhlenbeck process, we analyse the impact on the entropy production and entropy rate, providing a physical understanding of the effects of minimum information variability control.

Speed limit, dissipation bound, and dissipation-time trade-off in thermal relaxation processes

We investigate bounds on speed, nonadiabatic entropy production, and the trade-off relation between them for classical stochastic processes with time-independent transition rates. Our results show that the time required to evolve from an initial to a desired target state is bounded from below by the information-theoretical ∞-Rényi divergence between these states, divided by the total rate. Furthermore, we conjecture and provide extensive numerical evidence for an information-theoretical bound on the nonadiabatic entropy production and a dissipation-time trade-off relation that outperforms previous bounds in some cases..

Global optimization and monotonicity in entropy production of weak drivings

Knowing if an optimal solution is local or global has always been a hard question to answer in more sophisticated situations of optimization problems. In this paper, for finite-time and weak isothermal driving processes, we show the existence of a global optimal protocol for the entropy production. We prove this by showing its convexity as a functional in the derivative of the protocol. This property also proves its monotonicity in such a context, which leads to the satisfaction of the second law of thermodynamics. In the end, we exemplify that the analytical technique of the Euler-Lagrange equation applied to overdamped Brownian motion delivers the global optimal protocol, by comparing it with the results of the global optimization technique of genetic programming.

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics

Stochastic thermodynamics lays down a broad framework to revisit the venerable concepts of heat, work and entropy production for individual stochastic trajectories of mesoscopic systems. Remarkably, this approach, relying on stochastic equations of motion, introduces time into the description of thermodynamic processes-which opens the way to fine control them. As a result, the field of finite-time thermodynamics of mesoscopic systems has blossomed. In this article, after introducing a few concepts of control for isolated mechanical systems evolving according to deterministic equations of motion, we review the different strategies that have been developed to realize finite-time state-to-state transformations in both over and underdamped regimes, by the proper design of time-dependent control parameters/driving. The systems under study are stochastic, epitomized by a Brownian object immersed in a fluid; they are thus strongly coupled to their environment playing the role of a reservoir. Interestingly, a few of those methods (inverse engineering, counterdiabatic driving, fast-forward) are directly inspired by their counterpart in quantum control. The review also analyzes the control through reservoir engineering. Besides the reachability of a given target state from a known initial state, the question of the optimal path is discussed. Optimality is here defined with respect to a cost function, a subject intimately related to the field of information thermodynamics and the question of speed limit. Another natural extension discussed deals with the connection between arbitrary states or non-equilibrium steady states. This field of control in stochastic thermodynamics enjoys a wealth of applications, ranging from optimal mesoscopic heat engines to population control in biological systems.

Interpenetration Networked Polyimide-Epoxy Copolymer under Kinetic and Thermodynamic Control for Anticorrosion Coating

Epoxy (EP) was copolymerized with polyamic acid (PAA, precursor of polyimide (PI)) with termanil monomers of (1) 4,4'-Oxydianiline (ODA) and (2) pyromellitic dianhydride (PMDA) individually to form (PI-O-EP) and (PI-P-EP) copolymers. The FTIR spectrum of PI-O-EP copolymerization intermediates shows that some amide-EP linkages were formed at low temperature and were broken at higher temperature; in additoin, the released amide was available for subsequent imidization to form PI. The curing and imidization of the amide groups on PAA were determined by reaction temperature (kinetic vs. thermodynamic control). In PI-P-EP, the released amide group was very short-lived (fast imidization) and was not observed on FTIR spectra. Formation and breakage of the amide-EP linkages is the key step for EP homopolymerization and formation of the interpenetration network. PI contributed in improving thermal durability and mechanical strength without compromising EP's adhesion strength. Microphase separations were minimal at PI content less than 10 wt%. The copolymerization reaction in this study followed the "kinetic vs. thermodynamic control" principle. The copolymer has high potential for application in the field of higher-temperature anticorrosion.

Geodesic path for the optimal nonequilibrium transition: Momentum-independent protocol

Accelerating controlled thermodynamic processes requires an auxiliary Hamiltonian to steer the system into instantaneous equilibrium states. An extra energy cost is inevitably needed in such finite-time operation. We recently developed a geodesic approach to minimize such energy cost for the shortcut to isothermal process. The auxiliary control typically contains momentum-dependent terms, which are hard to be experimentally implemented due to the requirement of constantly monitoring the speed. In this work, we employ a variational auxiliary control without the momentum-dependent force to approximate the exact control. Following the geometric approach, we obtain the optimal control protocol with variational minimum energy cost. We demonstrate the construction of such protocol via an example of Brownian motion with a controllable harmonic potential.

Minimally Dissipative Information Erasure in a Quantum Dot via Thermodynamic Length

In this Letter, we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimization. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimize dissipation for slow protocols, with a bigger improvement as one approaches perfect erasure. Moreover, the geometric approach also leads to smaller dissipation even when the time of the protocol becomes comparable with the equilibration timescale of the system, i.e., away from the slow driving regime. Our results also illustrate, in a single-electron device, a fundamental principle of thermodynamic geometry: optimal finite-time thermodynamic protocols are those with constant dissipation rate along the process.

Performance of optimal linear-response processes in driven Brownian motion far from equilibrium

Considering the paradigmatic driven Brownian motion, we perform extensive numerical analysis on the performance of optimal linear-response processes far from equilibrium. We focus on the overdamped regime where exact optimal processes are known analytically and most experiments operate. This allows us to compare the optimal processes obtained in linear response and address their relevance to experiments using realistic parameter values from experiments with optical tweezers. Our results help assess the accuracy of perturbative methods in calculating the irreversible work for cases where the exact solution might be difficult to access. For that, we present a performance metric comparing the approximate optimal solution to the exact one. Our main result is that optimal linear-response processes can perform surprisingly well, even far from where they were expected.

Shortcuts to Thermodynamic Quasistaticity

The operation of near-term quantum technologies requires the development of feasible, implementable, and robust strategies of controlling complex many body systems. To this end, a variety of techniques, so-called "shortcuts to adiabaticity," have been developed. Many of these shortcuts have already been demonstrated to be powerful and implementable in distinct scenarios. Yet, it is often also desirable to have additional, approximate strategies available that are applicable to a large class of systems. Hence, in this Letter, we take inspiration from thermodynamics and propose to focus on the macrostate, rather than the microstate. Adiabatic dynamics can then be identified as such processes that preserve the equation of state, and systematic corrections are obtained from adiabatic perturbation theory. We demonstrate this approach by improving upon fast quasiadiabatic driving, and by applying the method to the quantum Ising chain in the transverse field.

Limited-control optimal protocols arbitrarily far from equilibrium

Recent studies have explored finite-time dissipation-minimizing protocols for stochastic thermodynamic systems driven arbitrarily far from equilibrium, when granted full external control to drive the system. However, in both simulation and experimental contexts, systems often may only be controlled with a limited set of degrees of freedom. Here, going beyond slow- and fast-driving approximations employed in previous studies, we obtain exact finite-time optimal protocols for this limited-control setting. By working with deterministic Fokker-Planck probability density time evolution, we can frame the work-minimizing protocol problem in the standard form of an optimal control theory problem. We demonstrate that finding the exact optimal protocol is equivalent to solving a system of Hamiltonian partial differential equations, which in many cases admit efficiently calculable numerical solutions. Within this framework, we reproduce analytical results for the optimal control of harmonic potentials and numerically devise optimal protocols for two anharmonic examples: varying the stiffness of a quartic potential and linearly biasing a double-well potential. We confirm that these optimal protocols outperform other protocols produced through previous methods, in some cases by a substantial amount. We find that for the linearly biased double-well problem, the mean position under the optimal protocol travels at a near-constant velocity. Surprisingly, for a certain timescale and barrier height regime, the optimal protocol is also nonmonotonic in time.

Optimal control with a strong harmonic trap

Quadratic trapping potentials are widely used to experimentally probe biopolymers and molecular machines and drive transitions in steered molecular-dynamics simulations. Approximating energy landscapes as locally quadratic, we design multidimensional trapping protocols that minimize dissipation. The designed protocols are easily solvable and applicable to a wide range of systems. The approximation does not rely on either fast or slow limits and is valid for any duration provided the trapping potential is sufficiently strong. We demonstrate the utility of the designed protocols with a simple model of a periodically driven rotary motor. Our results elucidate principles of effective single-molecule manipulation and efficient nonequilibrium free-energy estimation.

Kibble–Zurek Scaling from Linear Response Theory

While quantum phase transitions share many characteristics with thermodynamic phase transitions, they are also markedly different as they occur at zero temperature. Hence, it is not immediately clear whether tools and frameworks that capture the properties of thermodynamic phase transitions also apply in the quantum case. Concerning the crossing of thermodynamic critical points and describing its non-equilibrium dynamics, the Kibble–Zurek mechanism and linear response theory have been demonstrated to be among the very successful approaches. In the present work, we show that these two approaches are also consistent in the description of quantum phase transitions, and that linear response theory can even inform arguments of the Kibble–Zurek mechanism. In particular, we show that the relaxation time provided by linear response theory gives a rigorous argument for why to identify the “gap” as a relaxation rate, and we verify that the excess work computed from linear response theory exhibits Kibble–Zurek scaling.

Fluctuation theorem for irreversible entropy production in electrical conduction

Linear irreversible thermodynamics predicts that the entropy production rate can become negative. We demonstrate this prediction for metals under AC driving whose conductivity is well described by the Drude-Sommerfeld model. We then show that these negative rates are fully compatible with stochastic thermodynamics, namely, that the entropy production does fulfill a fluctuation theorem. The analysis is concluded with the observation that the stochastic entropy production as defined by the surprisal or ignorance of the Shannon information does not agree with the phenomenological approach.

Environment-Assisted Shortcuts to Adiabaticity

Envariance is a symmetry exhibited by correlated quantum systems. Inspired by this "quantum fact of life," we propose a novel method for shortcuts to adiabaticity, which enables the system to evolve through the adiabatic manifold at all times, solely by controlling the environment. As the main results, we construct the unique form of the driving on the environment that enables such dynamics, for a family of composite states of arbitrary dimension. We compare the cost of this environment-assisted technique with that of counterdiabatic driving, and we illustrate our results for a two-qubit model.

Optimality of nonconservative driving for finite-time processes with discrete states

An optimal finite-time process drives a given initial distribution to a given final one in a given time at the lowest cost as quantified by total entropy production. We prove that for a system with discrete states this optimal process involves nonconservative driving, i.e., a genuine driving affinity, in contrast to the case of a system with continuous states. In a multicyclic network, the optimal driving affinity is bounded by the number of states within each cycle. If the driving affects forward and backwards rates nonsymmetrically, the bound additionally depends on a structural parameter characterizing this asymmetry.

Hierarchical Onsager symmetries in adiabatically driven linear irreversible heat engines

In existing linear response theories for adiabatically driven cyclic heat engines, Onsager symmetry is identified only phenomenologically, and a relation between global and local Onsager coefficients, defined over one cycle and at any instant of a cycle, respectively, is not derived. To address this limitation, we develop a linear response theory for the speed of adiabatically changing parameters and temperature differences in generic Gaussian heat engines obeying Fokker-Planck dynamics. We establish a hierarchical relationship between the global linear response relations, defined over one cycle of the heat engines, and the local ones, defined at any instant of the cycle. This yields a detailed expression for the global Onsager coefficients in terms of the local Onsager coefficients. Moreover, we derive an efficiency bound, which is tighter than the Carnot bound, for adiabatically driven linear irreversible heat engines based on the detailed global Onsager coefficients. Finally, we demonstrate the application of the theory using the simplest stochastic Brownian heat engine model.

Equilibrium free-energy differences from a linear nonequilibrium equality

Extracting equilibrium information from nonequilibrium measurements is a challenge task of great importance in understanding the thermodynamic properties of physical, chemical, and biological systems. The discovery of the Jarzynski equality illumines the way to estimate the equilibrium free-energy difference from the work performed in nonequilibrium driving processes. However, the nonlinear (exponential) relation causes the poor convergence of the Jarzynski equality. Here, we propose a concise method to estimate the free-energy difference through a linear nonequilibrium equality which inherently converges faster than nonlinear nonequilibrium equalities. This linear nonequilibrium equality relies on an accelerated isothermal process which is realized by using a unified variational approach, named variational shortcuts to isothermality. We apply our method to an underdamped Brownian particle moving in a double-well potential. The simulations confirm that the method can be used to accurately estimate the free-energy difference with high efficiency. Especially during fast driving processes with high dissipation, the method can improve the accuracy by more than an order of magnitude compared with the estimator based on the nonlinear nonequilibrium equality.

Negative entropy production rates in Drude-Sommerfeld metals

It is commonly accepted that in typical situations the rate of entropy production is non-negative. We show that this assertion is not entirely correct, not even in the linear regime, if a time-dependent, external perturbation is not compensated by a rapid enough decay of the response function. This is demonstrated for three variants of the Drude model to describe electrical conduction in noble metals, namely the classical free electron gas, the Drude-Sommerfeld model, and the extended Drude-Sommerfeld model. The analysis is concluded with a discussion of potential experimental verifications and ramifications of negative entropy production rates.

Time-Rescaling of Dirac Dynamics: Shortcuts to Adiabaticity in Ion Traps and Weyl Semimetals

Only very recently, rescaling time has been recognized as a way to achieve adiabatic dynamics in fast processes. The advantage of time-rescaling over other shortcuts to adiabaticity is that it does not depend on the eigenspectrum and eigenstates of the Hamiltonian. However, time-rescaling requires that the original dynamics are adiabatic, and in the rescaled time frame, the Hamiltonian exhibits non-trivial time-dependence. In this work, we show how time-rescaling can be applied to Dirac dynamics, and we show that all time-dependence can be absorbed into the effective potentials through a judiciously chosen unitary transformation. This is demonstrated for two experimentally relevant scenarios, namely for ion traps and adiabatic creation of Weyl points.

Geometry of Work Fluctuations versus Efficiency in Microscopic Thermal Machines

When engineering microscopic machines, increasing efficiency can often come at a price of reduced reliability due to the impact of stochastic fluctuations. Here we develop a general method for performing multiobjective optimization of efficiency and work fluctuations in thermal machines operating close to equilibrium in either the classical or quantum regime. Our method utilizes techniques from thermodynamic geometry, whereby we match optimal solutions to protocols parametrized by their thermodynamic length. We characterize the optimal protocols for continuous-variable Gaussian machines, which form a crucial class in the study of thermodynamics for microscopic systems.

Skewed thermodynamic geometry and optimal free energy estimation

Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.

Geometric Optimisation of Quantum Thermodynamic Processes

Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.