Design and optimisation of irreversible heat engines

FACULTAD DE FÍSICA DEPARTAMENTO DE FÍSICA ATÓMICA, MOLECULAR Y NUCLEAR Design and optimisation of irreversible heat engines UNIVERSIDAD DE SEVILLA Irene Prieto Rodríguez 2022-2023 Doble Grado en Física y Matemáticas Supervised by Carlos Alberto Plata Ramos and Antonio Prados Montaño Agradecimientos Quiero dar las gracias a todas las personas que, de una forma u otra, han hecho posible este trabajo. Especialmente a mis tutores, Carlos Plata y Antonio Prados, que me han guiado magistralmente con su sabiduría y cercanía. Ha sido para mí un placer y un honor poder contar con sus excelentes consejos e inestimable ayuda. Siempre con buen humor, me han hecho sentir muy apoyada en todo momento y, desde luego, este trabajo no habría sido posible sin ellos. También quiero agradecer su labor a todos los profesores que, a lo largo de mi vida, me han ayudado a formarme tanto en el aspecto académico como humano. Tal vez muchos de ellos no sean del todo conscientes de la gran huella que dejan en sus alumnos, haciéndonos descubrir nuevas formas de ver el mundo y despertando la curiosidad por responder a preguntas que ni nos habíamos podido plantear. Durante estos años, he tenido la suerte de aprender física y matemáticas junto a maravillosos compañeros, ya amigos, que han hecho que esta etapa, a veces dura, haya sido mucho más amena y feliz. Compartir tiempo, inquietudes y alegrías con ellos ha sido un verdadero regalo. Y, por supuesto, quiero dar también las gracias a mi familia, que con su apoyo y cariño incondicionales me han enseñado las lecciones más importantes de la vida. Resumen Las máquinas térmicas son dispositivos que convierten energía térmica en trabajo útil operando de manera cíclica. Durante siglos, han desempeñado un papel fundamental en el desarrollo industrial y tecnológico. Históricamente, se han empleado únicamente gases y líquidos como sustancia de trabajo, pero los avances técnicos alcanzados en las últimas décadas permiten ampliar las posibilidades experimentales y diseñar también máquinas que operan con una única partícula. Los sistemas de interés en este caso no pueden tratarse a nivel macroscópico y su estudio se enmarca en el campo de la termodinámica estocástica. En este trabajo estudiamos máquinas térmicas mesoscópicas constituidas por una partícula browniana confinada en un potencial armónico e inmersa en un fluido que actúa como baño térmico. Diseñamos un ciclo análogo al clásico motor de Stirling, compuesto por dos ramas isotermas y dos isócoras. Nos centramos en el caso irreversible, no cuasiestático, cuya duración finita permite al dispositivo generar una potencia no nula. Este aspecto es crucial, pues permite plantear la optimización de nuestro ciclo para maximizar la potencia producida, respondiendo así a un interés relevante a nivel práctico. Buscamos los protocolos de control del dispositivo que llevan a la optimalidad, utilizando herramientas del cálculo variacional y de la teoría del control óptimo. Asimismo, exploramos numéricamente la dependencia de la potencia máxima obtenida y la eficiencia correspondiente con los parámetros que caracterizan nuestro sistema. Abstract Heat engines transform thermal energy into useful work, operating in a cyclic manner. For centuries, they have played a key role in industrial and technological development. Historically, only gases and liquids have been used as working substances, but the technical advances achieved over the past decades allow for expanding the experimental possibilities and designing engines operating with a single particle as well. In this case, the systems of interest cannot be addressed at a macroscopic level and their study is framed in the field of stochastic thermodynamics. In the present work, we study mesoscopic heat engines built with a Brownian particle submitted to harmonic confinement and immersed in a fluid acting as a thermal bath. We design a Stirling-like heat engine, composed of two isothermal and two isochoric branches. We focus on the irreversible, non quasi-static, case—whose finite duration enables the device to deliver a non-zero power output. This is a crucial aspect, inasmuch as it permits to put forward the optimisation of our cycle in order to maximise the delivered power, thereby addressing a relevant interest at a practical level. We search for driving protocols yielding optimality by using tools from variational calculus and optimal control theory. Likewise, we explore numerically the dependence of the maximum power output and the corresponding efficiency on the parameters characterising our system. Contents 1 Introduction 1 2 The model system 3 2.1 Harmonically confined Brownian particle . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Fokker-Planck description . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Langevin description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Energy, work and heat . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Dimensionless variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Building blocks for a Stirling cycle 11 3.1 Isothermalprocesses................................. 11 3.1.1 Quasi-static isothermal processes . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Optimal isothermal processes . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Isochoricprocesses.................................. 14 3.2.1 Quasi-static isochoric processes . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Optimal isochoric processes . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Stirling stochastic heat engine 18 4.1 Quasi-static Stirling cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Optimal irreversible Stirling cycle . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.1 Further optimisation of the irreversible Stirling cycle over the operatingpoints.................................. 28 4.2.1.1 Optimisation of the irreversible Stirling cycle in the ideal limit ................................ 29 4.2.1.2 Optimisation of the irreversible Stirling cycle for variable (θmin,θmax)............................ 33 5 Conclusions 38 A Pontryagin’s Maximum Principle 43 A.1 Pontryagin’s Time-Optimal Problem . . . . . . . . . . . . . . . . . . . . . . . 43 A.2 Thermal brachistochrone for an isochoric branch . . . . . . . . . . . . . . . . 48 Addendum: Code example 51 Chapter 1 Introduction Classical thermodynamics is devoted to the study of heat and related properties in macroscopic systems. The effective conversion of thermal energy into mechanical work was a key factor in the course of the industrial revolution, where the development of steam engines played a major role. Blossoming in the 19th century, the field of classical thermodynamics is governed by a concise set of well-known universal laws, which were stated thanks to the work of scientists such as Sadi Carnot or Rudolf Clausius, culminating in an axiomatic formulation schemed by Constantine Carathéodory [1]. Statistical mechanics, primarily thanks to the contributions of Ludwig Boltzmann, unravelled the probabilistic nature lying underneath the principles of equilibrium thermodynamics. Therein, the macroscopic character of the systems of interest allows for neglecting all the fluctuations involved. At the opposite end, pursuing a deterministic microscopic description of a complex system is, besides unattainable, inessential. Finding an adequate separation between the relevant variables in a physical problem and those whose effect can be encoded by a stochastic noise, causing random fluctuations, is one of the cornerstones of statistical mechanics. With this aim, one typically focuses on an intermediate level of description between the microscopic and macroscopic scales: the mesoscopic one. Since its inception, statistical mechanics has gone beyond equilibrium, studying a broad and diverse ensemble of physical systems and leading to numerous interdisciplinary applications ranging from plasma physics to modelling biomolecular dynamics. A colloidal particle immersed in a fluid at equilibrium, known as Brownian particle, embodies a paradigmatic system in this wide framework. Its multiple collisions with the tiny particles composing the surrounding fluid give rise to the random Brownian motion. As the size of a system is reduced, the importance of fluctuations grows, since they may become of the same order of magnitude as the meaningful average values. Present-day miniaturisation of technological devices has brought increasing attention to the extension of thermodynamic results to the mesoscale. Stochastic thermodynamics addresses this 1 CHAPTER 1. INTRODUCTION 2 challenging goal, extending concepts such as heat, work and entropy to individual fluctuating trajectories governed by stochastic equations of motion, thereby incorporating time into the study of thermodynamic processes. Furthermore, this fluctuating nature gives rise to novel phenomena with no analogy in the macroscopic world, such as the occurrence of ‘transient’ violations of the second law of thermodynamics or super Carnot efficiencies in some of the above-mentioned individual trajectories [2,3]. Notwithstanding, the average physical quantities do obey the corresponding universal laws. Moreover, key concepts in thermodynamics, such as that of adiabaticity, require a careful translation to stochastic thermodynamics, with the impossibility of connecting certain states and the emergence of a speed limit for finite-time adiabatic processes [4]. Leveraging the unavoidable fluctuations in a mesoscopic system in order to extract work, as a Maxwell’s demon would do [5], constitutes a revolutionary idea that has been deeply studied —both theoretically and experimentally— in the last decades. Recent advances in novel sensing and manipulation techniques in the micro and nanometre length scales, such as optical trapping, have allowed experimental physicists to transfer originally gedanken experiments, analogous to Feynman’s ratchet and pawl or Szilard’s engine, to actual laboratories [6, 7]. Indeed, we are now equipped to engineer small heat engines performed with a single colloidal particle. Stochastic thermodynamics provides the adequate framework for the design of such mesoscopic heat engines. Finding convenient driving protocols (in terms of delivered power, efficiency, or time duration) for this type of systems has been a major topic in the field [4,8,9]. In addition, interesting connections between information and thermodynamics have arisen as a consequence of the study of feedback protocols [6]. Researchers have explored multiple shortcuts for connecting equilibrium states faster than the corresponding natural relaxation time of the system at hand, often finding inspiration in quantum-control ideas [10,11]. Thus, building mesoscopic finite-time counterparts of classical thermodynamic engines (such as the Carnot, Stirling or Otto cycles) has become a relevant line of research in the field. In reminiscence of its paradigmatic classical counterpart, Carnot-like heat engines have been mainly studied [12–14], encouraging the analysis of other thermal engines as well. The final objective of the present work is to build an irreversible Stirling cycle. The corresponding classical version, invented and patented by engineer Robert Stirling in 1816, encompasses two isothermal and two isochoric branches. Here, we analyse how this motor can be implemented with a Brownian particle in a harmonic trap. Furthermore, we study how the delivered power can be optimised. This purpose is not only theoretically meaningful, but also worthwhile in practice: experimental realisations of Stirling cycles are achievable [15] and their optimisation may be useful in the search of innovative methods for renewable energy production [16]. Chapter 2 The model system 2.1 Harmonically confined Brownian particle We consider an overdamped Brownian particle immersed in a thermal bath at temperature Tand trapped in a one-dimensional harmonic potential of stiffness k. Both of these parameters are externally controlled. The friction coefficient is denoted as λand it is considered as constant. This assumption is consistent with the experimental realisation of the model: a colloidal particle in an optical trap, where the friction coefficient is determined by both the particle geometry and the solvent viscosity.1 Let xdenote the particle’s position with respect to the centre of the trap. Its evolution follows the Langevin equation λdx dt (t)=−k(t)x(t)+ζ(t),(2.1) where ζis a Gaussian white noise, characterised by a null average value and a temperature dependent autocorrelation function, ⟨ζ(t)⟩=0,⟨ζ(t)ζ(t′)⟩=2λkBT(t)δ(t−t′).(2.2) In the above equation, kBis the Boltzmann constant and δis the Dirac delta function. The stochastic dynamics of the system can be equivalently described in the Fokker- Planck or Langevin formalisms. In order to show this equivalence, we study the evolution of the position’s probability distribution in both levels of description. 1Actually, recent experimental realisations of colloidal heat engines operate at constant ‘physical’ temperature and effective kinetic temperatures are implemented by means of external random forces, whose amplitude is controlled. Albeit the friction coefficient λcould depend on the physical temperature, the use of an effective kinetic temperature ensures that considering λas a constant is a rather sensible assumption. 3 CHAPTER 2. THE MODEL SYSTEM 10 In addition, it is useful to adimensionalise time as follows, τ≡t kref λ.(2.40) The state of the system at any time τis thus characterised by the triplet (κ, y, θ). From now on, differentiation with respect to τis denoted as ˙q≡dq/dτ, for any physical quantity q. Hence, the evolution of the non-dimensional variance follows the equation ˙y=−2κy +2θ. (2.41) With the previous definitions, we can write Eq. (2.17) as y(τ)=e−2∫τ 0dτ κ(τ′)[yi+2∫τ 0 dτ′θ(τ′)e2∫τ′ 0dτ′′ κ(τ′′)],(2.42) which gives us the evolution of the variance. Note that, for the special case of constant stiffness κand temperature θ, we have that the variance exponentially decays to its equilibrium value, y(τ)=θ κ+e−2κτ (yi−θ κ).(2.43) Thus, if we consider an infinitely slow process (compared to the characteristic relaxation time of our system), the stiffness of the trap and the temperature behave as constants in our time-scale and the second term on the rhs vanishes. Thence, the normalised variance attains its equilibrium value. This situation corresponds to the quasi-static limit, which will be studied in the upcoming chapters. Chapter 3 Building blocks for a Stirling cycle The aim of this work is to build and investigate a non-equilibrium version of the Stirling cycle, in analogy to the studies of irreversible Carnot-like heat engines [12–14]. Hence, our stochastic cycle should encompass isothermal and isochoric branches. In the following, we define and analyse in detail these branches. 3.1 Isothermal processes In phase space, isothermal processes are represented by curves with a constant value for θ. In this work, we consider two kinds of isothermal processes: quasi-static and optimal. The former is not our main interest; not only are quasi-static processes non-feasible experimentally, but also they lack practical interest, due to their null output power. However, the study of quasi-static transitions is theoretically meaningful, and it is essential to compare both scenarios. Nevertheless, our focus is the power optimisation of an irreversible heat engine, and hence isothermal processes in which the extracted work is maximised play a key role. 3.1.1 Quasi-static isothermal processes A quasi-static process is defined as a succession of equilibrium states [1]. Therefore, an isothermal quasi-static process is represented by an equilibrium curve of the form given in Eq. (2.34), in which the temperature θis fixed. To sweep this equilibrium curve y(τ)=θ/κ(τ), the control parameter κ(τ)must be varied sufficiently slowly. For a quasistatic process, the work required to drive the system from one state to another always equals the Helmholtz free energy difference between the initial and final points ∆Fi→f, which is a function of state. 11 CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 12 The computation of work, heat and energy variation in such a process is straightforward, WQS i→f=1 2∫f i dκ θ κ=θ 2log(κf κi)=∆Fi→f,(3.1) QQS i→f=1 2∫f i dy θ y=θ 2log(yf yi)=−θ 2log(κf κi)=−WQS i→f,(3.2) ∆EQS i→f=0.(3.3) The last equality can be derived as a consequence of the first law, or directly remembering that Ei=Ef=θfor an isothermal process. 3.1.2 Optimal isothermal processes Let us consider an isothermal process lasting a finite time τf. In such a protocol, the system sweeps non-equilibrium states and thus work depends on the selected protocol for the control κ(τ). Therefore, it is possible to minimise the work performed on the system (i.e. maximise the work done by the system) by finding an optimal protocol κ(τ). This optimisation problem has already been solved in Ref. [12]. Hereafter, we revisit the obtaining of such optimal isothermal protocol. Work is a functional of the engineered protocol, Wi→f[κ]=1 2∫f i dκ y = >0 1 2[y κ]f i−1 2∫f i dy κ, (3.4) where integration by parts leads to the second equality. Note that the first term on the rhs vanishes, due to the equilibrium condition at the initial and final states: κiyi=κfyf=θ. Similarly to the quasi-static case, we have that Ei=Ef=θand thus Qi→f[κ]=−Wi→f[κ],∆Ei→f=0.(3.5) The evolution equation of the variance, captured in Eq. (2.41), yields κ=θ y−1 2 ˙y y.(3.6) Introducing this relation in Eq. (3.4), one obtains Wi→f[κ]=−θ 2∫f i d(logy)+1 4∫f i dτ ˙y2 y=∆Fi→f+1 4∫f i dτ ˙y2 y.(3.7) CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 13 The first term in the last expression does not depend on the selected protocol, given that it is the free energy difference, i.e. the quasi-static contribution. Hence, since it is a function of state, its value is determined by the boundary point states, which are fixed. Thereupon, the optimisation problem is solely determined by the second term, which is the ‘irreversible’ or ‘excess’ work, i.e. the difference between the actual work and its value in the quasi-static, reversible, limit. Note that, as expected, the irreversible contribution is positive regardless of the selected protocol. Let us introduce the ‘Lagrangian’ L(y, ˙y)=˙y2 y.(3.8) The Euler-Lagrange equation corresponding to our variational problem, i.e. the minimisation of the irreversible work, is d dτ ∂L ∂˙y−∂L ∂y =0⇐⇒2¨y y −˙y2=0.(3.9) The solution of this second-order differential equation is obtained by standard methods, with the result y(τ)=(aτ +b)2,(3.10) where (a, b)are integration constants. They can be determined by imposing the boundary conditions y(0)=yi,y(τf)=yf, yi=b2⇒b=√yi,(3.11) yf=(aτf+b)2⇒a=1 τf(√yf−√yi).(3.12) Hence, the evolution of the variance in the optimal process is ˜y(τ)=[√yi+(√yf−√yi)τ τf]2 ,∀τ∈[0, τf].(3.13) We observe that ˜y(τ)is continuous in the closed interval [0, τf]and it is a positive-defined quantity, as the variance of any distribution must be. The control ˜κevolution in the optimal process is obtained from Eq. (3.6), in the open time interval τ∈(0, τf), ˜κ(τ)=θ ˜y(τ)−1 2 d dτ log ˜y(τ)=θ [√yi+(√yf−√yi)τ τf]2−⎡⎢⎢⎢⎢⎣τ+τf (√yf/yi−1)⎤⎥⎥⎥⎥⎦ −1 .(3.14) Note that the stiffness of the trap is discontinuous at both the initial and final time CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 14 instants, lim τ→0+˜κ(τ)=κi−1 τf(√yf yi−1)/=κi,(3.15) lim τ→τ− f ˜κ(τ)=κf−1 τf(1−√yi yf)/=κf.(3.16) Similar discontinuities in the control parameter have been repeatedly found in stochastic thermodynamics [9,11,12,14,19]. Note that the continuity at the boundaries is recovered in the quasi-static limit, in which τf→∞. Work associated with the optimal process is obtained from Eq. (3.4),  Wi→f=WQS i→f+1 τf(√yf−√yi)2=WQS i→f+θ τf(1 √κf−1 √κi)2 .(3.17) The work does not only depend on the initial and final equilibrium states connected by the isothermal process, but also on the duration of such transition. The minimum irreversible work scales as τ−1 f, which vanishes in the quasi-static limit τf→∞. We recall that, according to our sign convention, work delivered by the heat engine is negative. Hence, when we refer to ‘optimal work’, we are talking about a minimum value. In Eq. (3.17), we obtain that the optimal work is always greater (i.e. ‘worse’ in energetic exploitation terms) than the corresponding quasi-static value. Albeit the extracted work is maximum in the quasi-static limit, it leads to a vanishing power output. The opposite limit, an instantaneous isothermal transition (τf→0+), leads to the least energetically advantageous case: infinite work is required to perform the process. 3.2 Isochoric processes In this section, our focus of interest are processes in which the harmonic potential is kept constant.1They are represented by curves with a fixed value for the trap stiffness κin the phase space. Similarly to our treatment of isothermal branches, we will study quasi-static and optimal isochoric transitions. In this case, optimality refers to shortest time protocols, given that work vanishes for any isochoric process. Hence, minimal time will imply maximal power output. Therefore, isochoric processes are particularly simple: since the required work is always zero, heat is fully determined by the temperature difference between the final and initial 1In reminiscence of macroscopic heat engines operating with gases, our nomenclature exploits the analogy between (κ, y)and volume and pressure (V, p). Consequently, an isochoric process—associated with constant Vin gases—corresponds here to keeping κconstant. CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 15 states, and so is the energy variation, Wi→f=1 2∫f i dκ y =0,(3.18) Qi→f=1 2∫f i dy κ +1 2(θf−θi)=Õ×× κ(yf−yi)=θf−θi κ 2(yf−yi)+1 2(θf−θi)=θf−θi,(3.19) ∆Ei→f=Wi→f+Qi→f=θf−θi.(3.20) As one may intuitively expect, the system delivers heat to the bath in cooling processes (θf<θi) and absorbs heat from the bath in heating ones (θf>θi). 3.2.1 Quasi-static isochoric processes In order to sweep a quasi-static curve, the control parameter, which in this case is the bath temperature θ(τ), shall be tuned in an infinitely slow manner in order to maintain the equilibrium condition y(τ)=θ(τ)/κ. As outlined above, the energetic characterisation of the isochoric process is identical for any given protocol, independently of the duration of the process or whether the intermediate states are at equilibrium or not. 3.2.2 Optimal isochoric processes We now aim at studying the thermal optimal protocol that minimises the connection time between the equilibrium initial and final states, keeping the potential fixed. This optimal shortcut has been investigated in depth in Ref. [9]. Therein, the problem is solved for arbitrary dimension, which yields a rich phenomenology. Hereupon, we restrict ourselves to the one-dimensional case that we are considering throughout. The external control θ(τ)is submitted to physical constraints, θ(τ)≥0,∀τ. But, moreover, tighter bounds might be brought up by technical limitations in practice. Thus, we consider the general constraints θmin ≤θ(τ)≤θmax for the bath temperature to solve the optimisation problem, particularising later for the ideally relaxed conditions θmin → 0+, θmax →∞. The addressed optimisation problem cannot be solved with the tools of standard variational calculus and less restrictive methods of optimal control theory are needed. The solution of our problem is a bang-bang protocol without ‘switchings’ (i.e. with a single continuity interval), in which θ(τ)is equal to one of its bounds. A detailed derivation of this result can be found in Appendix A. Therein, we obtain that the optimal temperature control in a heating process corresponds to reaching the upper CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 16 bound, ˜ θ(τ)=θmax,∀τ∈(0, τf); whereas, in a cooling procedure, optimality is achieved by setting the temperature control to its lower bound, ˜ θ(τ)=θmin,∀τ∈(0, τf). Thus, the optimal control ˜ θ(τ)can be expressed as ˜ θ(τ)=˜ θ≡⎧⎪⎪⎪⎨⎪⎪⎪⎩ θmax,if θi<θf, θmin,if θi>θf,∀τ∈(0, τf).(3.21) Note that, again, our optimal control presents jumps: ˜ θ(τ)is discontinuous at the initial and final time instants, given that it is submitted to the boundary conditions ˜ θ(0)=θi,˜ θ(τf)=θf.(3.22) The optimal protocol is physically reasonable: it coincides with the intuitive idea of using the largest or smallest possible bath temperatures—depending on whether one intends to heat or cool the system, respectively. Then, one just needs to select the adequate temperature bound and wait until the target state is reached. Since the optimal control is constant in the open interval (0, τf), it is particularly easy to solve the evolution equation for the variance of the confined Brownian particle, obtaining ˜y(τ)=˜ θ κ+(yi−˜ θ κ)e−2κτ ,∀τ∈(0, τf),(3.23) where ˜ θis given by Eq. (3.21). The optimal time connection is obtained by solving the equation yf=lim τ→τ− f ˜y(τ)=˜ θ κ+(yi−˜ θ κ)e−2κτf,(3.24) or, equivalently after dividing by yf, 1=˜ θ θf+(θi θf−˜ θ θf)e−2κτf,(3.25) where we have taken into account that yf=κ θf, since the final state verifies the equilibrium condition. This equation can be readily solved, yielding an optimal time τf=−1 2κlog(θf−˜ θ θi−˜ θ).(3.26) CHAPTER 3. BUILDING BLOCKS FOR A STIRLING CYCLE 17 Therefore, considering the expression of ˜ θgiven in Eq. (3.21), one has τf=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ 1 2κlog(θmax −θi θmax −θf),if θi<θf, 1 2κlog(θi−θmin θf−θmin ),if θi>θf. (3.27) The argument of the logarithmic function is greater than 1 in both cases, thus, the obtained optimal value τfis always positive, as it must be. Note that an opposite selection in Eq. (3.21) (i.e. choosing the largest temperature of the bath for cooling processes and the minimal temperature for heating), besides of being a physically unreasonable choice, would turn into a negative connection time. Thereof, our choosing in Eq. (3.21) has been proven to be correct. It is worth highlighting that, although the control parameter ˜ θ(τ) is not continuous at both ends of the isochoric process, the associated optimal variance ˜y is continuous in the whole time interval, since its expression, given in Eq. (3.23), verifies lim τ→0+˜y(τ)=yi,lim τ→τ− f ˜y(τ)=yf.(3.28) The energetic analysis of this optimal isochoric process is identical to the quasi-static one, given that work is null and heat and energy variation only depend on the boundary temperatures. As previously mentioned, the key difference between the quasi-static and optimal isochoric processes is that the latter lasts a finite time, whereas the former involves an infinite duration. In our discussion, we are particularly interested in the ideal situation θmin →0+, θmax → ∞, for which the minimum connecting time in an isochoric process is τf=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ lim θmax→∞ 1 2κlog(θmax −θi θmax −θf)=0,if θi<θf, lim θmin→0+ 1 2κlog(θi−θmin θf−θmin )=1 2κlog(θi θf),if θi>θf. (3.29) Thus, in the ideal limit of infinite heating power, it is possible to instantaneously heat the system up to any desired temperature; while the physical bound for minimal temperatures leads to a non-zero finite limit of the optimal time for cooling processes. Chapter 4 Stirling stochastic heat engine In this chapter, we study the stochastic version of a Stirling cycle performed by our model system. Experimental versions of such engines have been experimentally built in the last decades [3,15]. Analogously to the classical Stirling heat engine, our cyclic process encompasses four stages, which are illustrated in Fig. 4.1: 1. Isothermal expansion at the hot bath temperature θh≡θA=θB, connecting the phase points Aand B, i.e. (κA, yA, θA)and (κB, yB, θB). The confining harmonic potential is modified via time control of the trap strength κ(τ). Here and onwards, the term expansion (compression) refers to the sign of the variance increment: ∆y> 0(∆y<0), in analogy to the classical thermodynamic cycle. 2. Isochoric compression at ‘loose’ stiffness κl≡κB=κC, starting from the statepoint B≡(κB, yB, θB)up to C≡(κC, yC, θC). The time-dependence of the temperature of the heat bath θ(τ)is now controlled. 3. Isothermal compression at the cold bath temperature θc≡θC=θD<θh, linking states C≡(κC, yC, θC)and D≡(κD, yD, θD). Similarly to process 1, the stiffness of the harmonic trap is externally controlled. 4. Isochoric expansion at ‘tight’ stiffness κt≡κD=κA>κl, departing from state D≡ (κD, yD, θD)and closing the cycle by returning to the initial point A≡(κA, yA, θA). As in process 2, the control variable is the time-dependent temperature of the bath. Since the energy increment in an isothermal process vanishes, from the application of the first law of thermodynamics, we infer that in branches 1 and 3 the absorbed (released) heat equals the delivered (consumed) work, WAB =−QAB <0,WCD =−QCD >0.(4.1) 18 CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 19 Figure 4.1: Scheme of the stochastic Stirling cycle. The harmonic confining potential at the operating points of the cycle is represented by the purple curves. The filled red and blue areas correspond to the probability density functions at those state-points, where red (blue) refers to the hot (cold) equilibrium temperatures. The representation of the heat engine in the (κ, y)plane correspond to the quasi-static version of the described cycle. CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 26 Table 4.3: Energetics of the optimal irreversible Stirling cycle.  Wi→f Qi→f∆ Ei→f (1)A→B1 2logχ−WQS (1+√ν)(1+σ)−1 2logχ+WQS (1+√ν)(1+σ)0 (2)B→C0ν−1ν−1 (3)C→D−ν 2logχ−(√ν 1+√ν)WQS 1+σ ν 2logχ+(√ν 1+√ν)WQS 1+σ0 (4)D→A01−ν1−ν Total σ 1+σWQS −σ 1+σWQS 0 Table 4.3. Note that the total work absorbed by the system is  W=σ 1+σWQS.(4.30) This result exhibits a strong parallelism to the one presented in Ref. [14] for an optimal irreversible Carnot engine. As pointed out therein, the form of Eq. (4.30) yields a physical interpretation for the parameter σ: it is a measurement of the deviation of the total irreversible work from the value corresponding to the quasi-static case. In the limit σ→∞, one has  W→WQS.Note that, from the definition of σin Eq. (4.28), it is clear that the limit σ→∞corresponds, indeed, to infinitely slow isochoric processes: τisoc →∞. Consistently, Eq. (4.29) evinces that the limit σ→∞implies an infinite duration for the optimal isothermal branches as well. Therefore, the optimal power for any given operating points, defined by the temperature ratio between the hot and cold isotherms νand the compression ratio between the isochoric branches χ, is  P=(σ 1+σ)wWQS 1+σ+wτisoc .(4.31) We recall that the optimal time for the isochoric branches τisoc also depends on the bounds of the temperature control. Hence, the optimal time is a function of all the aforementioned parameters, which have been considered as fixed constants in our derivation,  P(ν, χ;θmin, θmax)=[σ(ν, χ;θmin, θmax) 1+σ(ν, χ;θmin, θmax)]w(ν, χ)WQS (ν, χ) 1+σ+w(ν, χ)τisoc (ν, χ;θmin, θmax). (4.32) In the above expression, we explicitly show where the dependence on these four parameters comes from. CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 27 Let us look into the efficiency of the optimised irreversible Stirling cycle, ˜η=−( WAB + WCD)  QAB =1+ WCD  WAB ,(4.33) which, similarly to the optimal power  P, depends on (ν, χ;θmin, θmax), although we omit this dependency in our notation for the sake of conciseness. We can easily check that the Carnot efficiency ηC=1−νis an upper bound for ˜η, ˜η=ηC+ WCD +ν WAB  WAB =ηC+  −ν 2logχ−√νWQS (1+√ν)(1+σ)+  ν 2logχ−νWQS (1+√ν)(1+σ)  WAB =ηC−(√ν+ν)WQS (1+√ν)(1+σ) WAB =ηC−(√ν 1+σ)WQS  WAB ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ >0 ,(4.34) where the factor within parentheses is clearly positive, and the terms WQS, WAB are both negative. We can rewrite the expression of the latter, given in Table 4.3, to make clear that it has a well-defined sign,  WAB =1 2logχ−WQS (1+√ν)(1+σ)=1 2logχ⎡⎢⎢⎢⎢⎣1−1−ν (1+√ν)(1+σ)⎤⎥⎥⎥⎥⎦ =1 2logχ(1−1−√ν 1+σ)=1 2logχ ´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶ <0(σ+√ν 1+σ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ >0<0.(4.35) Nonetheless, a similiar manipulation of Eq. (4.33) shows that the Curzon-Ahlborn efficiency [20] ηCA ≡1−√ν(4.36) is not an upper but a lower bound for our optimal irreversible Stirling cycle. Indeed, ˜η=ηCA + WCD +√ν WAB  WAB =ηCA +−ν 2logχ−√νWQS (1+√ν)(1+σ)+√ν 2logχ−√νWQS (1+√ν)(1+σ)  WAB =ηCA +1  WAB ⎡⎢⎢⎢⎢⎣√ν(1−√ν) 2logχ−2√νWQS (1+√ν)(1+σ)⎤⎥⎥⎥⎥⎦ =ηCA +√νlogχ  WAB ⎡⎢⎢⎢⎢⎣ 1−√ν 2−1−ν (1+√ν)(1+σ)⎤⎥⎥⎥⎥⎦=ηCA +√ν(1−√ν)logχ 2 WAB ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ >0(σ−1 σ+1) ´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶ >0 , (4.37) CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 28 Figure 4.3: Density plots of the optimal power (left) and the corresponding efficiency (right) in the (ν, χ)plane. We have considered that the bounds for the temperature are θmin →0+,θmax →∞. where we have taken into account that ν<1, WAB and logχare both negative and σ>1. Curzon and Ahlborn found ηCA to be the efficiency of a Carnot engine operating at maximum power output when limitations in the rates of heat transfer were considered [20]. They derived this result for a specific model, generating a long-standing debate about its universality and validity as an upper bound. Its generality has been discarded, since efficiencies at maximum power below and above ηCA have been reported in the literature [12,14,21,22]. However, it turns out to be an actual upper bound for the linear response at maximum power, being reached in this regime by systems with strong coupling between work and heat fluxes [23]. We recall that, in linear order, the Curzon-Ahlborn efficiency is exactly half of the Carnot efficiency, ηCA =ηC/2+O(η2 C). Moreover, in the special case of systems with strong coupling between the fluxes, possessing in addition left-right symmetry, the Curzon-Ahlborn result is verified up to quadratic order [21]. Our designed engine is found to operate always above the Curzon-Ahlborn efficiency, as shown by Eq. (4.37). In order to illustrate the results obtained in this section, density plots of the optimal power and the corresponding efficiency as a function of (ν, χ), in the ideal case of θmin →0+,θmax →∞, are presented in Fig. 4.3. 4.2.1 Further optimisation of the irreversible Stirling cycle over the operating points Hitherto, we have studied the optimisation of an irreversible Stirling cycle for which the operating points, defined by the temperature and compression ratios (ν, χ), were CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 29 fixed. Naturally, the question of what choice of ratios renders maximal power arises, and this section is devoted to answering this question. We first analyse the ideal case of nontechnical, but only physical, constraints in the temperature control: θmin →0+,θmax →∞. Secondly, we focus on more realistic limitations for the thermal control. 4.2.1.1 Optimisation of the irreversible Stirling cycle in the ideal limit Let us consider an irreversible Stirling cycle with variable operating points in the limit case of an ideal thermal control, i.e. one in which the temperature has no upper bound, θmax →∞, and the lower bound corresponds to the absolute zero, θmin →0+. The optimal power for any given operating points only depends here on the temperature and compression ratios,  P= P(ν, χ). Herein, we first look into the delivered power optimisation over the compression ratio χ, which will depend on the choice of the temperature ratio νbetween the cold and hot branches of the cycle,  P∗(ν)≡max χ∈(0,1) P(ν, χ)= P(ν, χ∗(ν)),(4.38) where χ∗(ν)denotes the compression ratio yielding optimal power for any fixed temperature ratio ν. Furthermore, we can find the overall optimal Stirling cycle by maximising now  P∗over the temperature ratio ν,  P∗∗ ≡max χ∈(0,1) ν∈(0,1) P(ν, χ)=max ν∈(0,1) P∗(ν)= P(ν∗, χ∗∗)=0.041,(4.39) where ν∗=0.060 is the temperature ratio that gives the overall maximum delivered power and χ∗∗ ≡χ∗(ν∗)=0.507 is the corresponding optimal compression ratio. To illustrate these definitions, we present in Fig. 4.4 the density plot of the optimal power in the (ν, χ)plane with the curve χ=χ∗(ν), as well as two plots showing the behaviour of the optimal power as a function of the compression ratio χfor fixed values of the temperature ratio ν∈{1/2, ν∗}. Interestingly, we find the compression ratio χ∗(ν)yielding optimal power for fixed νto be monotonically increasing with respect the temperature ratio ν. Similarly to the optimal power, the efficiency at maximum power for fixed operating points is a function of the temperature and compression ratios, ˜η=˜η(ν, χ). We denote the efficiency corresponding to the power optimisation over the compression ratio and to the overall maximum power in an analogous manner, ˜η∗(ν)≡˜η(ν, χ∗(ν)),˜η∗∗ ≡˜η(ν∗, χ∗∗)=0.842,(4.40) respectively. CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 30 Figure 4.4: Density plot of the optimal power in the (ν, χ)plane (left) and vertical sections for fixed values of the temperature ratio ν(right). The curve χ=χ∗(ν)(dotted line) gives the compression ratio yielding optimal power for every temperature ratio. On the right, the upper panel corresponds to ν=0.5 and the bottom one to ν=ν∗. The points at which maximum power is reached in each case are also displayed (squares). CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 31 Figure 4.5: Efficiency at maximum power in the present approach (black line), Carnot efficiency (red dashed line), Curzon-Ahlborn efficiency (blue dashed line) and η+(green dashed line), defined in Eq. (4.42), as a function of the temperature ratio ν. In the previous section, we proved that the efficiency of our device is always below the Carnot efficiency, as expected, but above the Curzon-Ahlborn value. Of course, these bounds, which were obtained for arbitrary values of the temperature and compression ratios (ν, χ), also apply when optimisation over χis carried out, ηCA <˜η∗<ηC.(4.41) This behaviour is illustrated in Fig. 4.5. In addition, we observe that the upper bound of the efficiency at low dissipation η+≡ηC 2−ηC=1−ν 1+ν(4.42) presented in Ref. [22] for engines reaching the Carnot efficiency in the reversible limit, as our Stirling cycle does, is verified in our system. We are interested in the asymptotic study of the efficiency at maximum power ˜η∗in the limit ν→1, which corresponds to ηC<<1. Our approach is the following: we find the expansion of χ∗in powers of the Carnot efficiency ηCand introduce this scaling of χ∗ in ˜η∗(ν)=˜η(ν, χ∗(ν))to obtain the power expansion of ˜η∗up to the desired order in ηC. Thus, let us consider the following ansatz for χ∗in the limit as ν→1: χ∗=1+a1ηC+a2η2 C+a3η3 C+O(η4 C),(4.43) where the coefficients (a1, a2, a3)are determined by enforcing the first three terms in the CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 32 expansion of the partial derivative of the optimal power with respect to the compression ratio ∂ P/∂χ to vanish at χ∗, since it corresponds to the optimal power output. Note that we determine the zero-order value of χ∗by using that χ∗→1as ν→1, as the dotted line in Fig. 4.4 shows. The described procedure yields a1=−1 2, a2=−1 48, a3=11 1152.(4.44) In Fig. 4.6, we compare the numerically evaluated optimal values of χ∗with the obtained expansion up to third order in ηC. The agreement of the numerical solutions and the theoretical estimation is quite good, even when considering a range of ηCnot so close to zero. We now introduce the obtained expansion for χ∗in ˜η∗(ν)≡˜η(ν, χ∗(ν)), obtaining the corresponding expansion for the efficiency, ˜η∗=ηC 2+3 16η2 C+41 384η3 C+O(η4 C).(4.45) Note that the linear coefficient 1/2has been proved to be a general upper bound for the linear response at maximum power, as a result of the Onsager reciprocity theorem, which has been considered as the fourth law of thermodynamics [23]. For the Curzon-Ahlborn efficiency, one has ηCA =ηC 2+η2 C 8+η3 C 16 +O(η4 C).(4.46) As expected, we find that the efficiency at maximum power for our engine coincides with the Curzon-Ahlborn efficiency to first order in ηC. Notwithstanding, the non-linear terms deviate from the Curzon-Ahlborn value. This is not surprising either, since we have proved in the previous section that our efficiency is always above the Curzon-Ahlborn bound, even for arbitrary values of the temperature control bounds (θmin, θmax). It has been shown that the quadratic coefficient being equal to 1/8is associated with left-right symmetry in the system [21]. It would be interesting to study the role of the asymmetry between heating and cooling processes, triggered by the upper and lower bounds of the temperature control in the isochoric branches of the cycle, on the observed deviation from the Curzon-Ahlborn efficiency yet in second order. We recall that the expansion of the upper bound η+is η+=ηC 2+η2 C 4+η3 C 8+O(η4 C).(4.47) This efficiency bound, presented in Ref. [22], is reached in a completely asymmetric limit. Note that the quadratic coefficient in the expansion of our efficiency at optimal power, given in Eq. (4.45), lies between the corresponding values for ηCA and η+:1/8<3/16 <1/4. CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 33 0.0 0.1 0.2 0.3 0.4 -3 -2 -1 0 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 4.6: Optimal compression ratio χ∗as a function of the corresponding Carnot efficiency ηC=1−ν. We compare numerical results (black circles) with the perturbative expansion in ηCup to linear order (green dotted line), quadratic order (red dashed line) and cubic order (blue dashed line). The inset shows a comparison beyond first order. 4.2.1.2 Optimisation of the irreversible Stirling cycle for variable (θmin,θmax) In this section, we study again the maximisation of the optimal power with respect to the compression ratio χ, but now considering arbitrary bounds (θmin, θmax)in the temperature control. Restrictions concerning these parameters and the temperatures of the cold and hot branches of the cycle arise, θmin <θc=ν<θh=1<θmax.(4.48) As one may intuitively expect, the most beneficial scenario corresponds to the ideal bounds studied in the previous section: θmin →0+,θmax →∞. To illustrate how more realistic bounds for the thermal control impinge on the optimal power, we present in Fig. 4.7 the behaviour of  Pas a function of χfor multiple finite values in one of the bounds (lower or upper) when the other one (upper or lower) reaches the corresponding ideal limit (∞ or 0). We do so for two meaningful values of the temperature ratio: ν∈{1/2, ν∗ id}, where ν∗ id denotes the temperature ratio yielding the overall maximum power in the ideal limit CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 34 θmin →0+,θmax →∞(note that, in the previous section, this parameter was simply written as ν∗). We observe that, indeed, the optimal power for the ideal bounds θmin →0+,θmax →∞ is always above the corresponding value for more realistic limits in the thermal control. In addition, the optimal power vanishes when the compression ratio reaches its bounds χ→0+and χ→1−in all the studied cases. Hence, there exists an optimal compression ratio χ∗for each temperature ratio νand thermal bounds (θmin, θmax)yielding maximal power. We address this further optimisation of  P(ν, χ;θmin, θmax), given in Eq. (4.32), over the compression ratio χ,  P∗(ν;θmin, θmax)≡max χ∈(0,1) P(ν, χ;θmin, θmax)= P(ν, χ∗;θmin, θmax),(4.49) obtaining the maximal power as a function of the temperature ratio in our cycle and the bounds in the thermal control. The associated compression ratio is defined consistently and it also depends on these three parameters, χ∗=χ∗(ν;θmin, θmax). Similarly to our study of the overall maximal power found in the case of ideal thermal bounds, it is interesting to carry out an even further optimisation over the temperature ratio ν. The interest of such search is transparent: since our physical system is solely submitted to the technical constraints represented by (θmin, θmax), it is natural to wonder what choice of the parameters defining the cycle (ν, χ)gives optimal power,  P∗∗(θmin, θmax)≡max χ∈(0,1) ν∈(θmin,1) P(ν, χ;θmin, θmax)=max ν∈(θmin,1) P∗(ν;θmin, θmax).(4.50) The optimal parameters are denoted consistently, ν∗=ν∗(θmin, θmax), χ∗∗(θmin, θmax)≡χ∗(ν∗;θmin, θmax),(4.51) as well as the corresponding efficiency, ˜η∗∗(θmin, θmax)≡˜η(ν∗, χ∗∗;θmin, θmax),(4.52) which are all function of the temperature bounds in the control. The identification of the optimal parameters (ν∗, χ∗∗)defining the Stirling cycle over the parallelogram defined by the restrictions 0<θmin <0.4and 1.15 <θmax <2.50 has been numerically obtained. It is shown, along with the corresponding evaluation of the maximal power  P∗∗ and efficiency ˜η∗∗, in Fig. 4.8. As expressed by Eq. (4.37), the efficiency of our engine is always above the Curzon- Ahlborn value. This result, particularised for the overall optimal cycle for fixed (θmin, θmax), is shown in Fig. 4.9. Therein, we present a density plot of ˜η∗∗/ηCA on the above men- CHAPTER 4. STIRLING STOCHASTIC HEAT ENGINE 35 0.0 0.2 0.4 0.6 0.8 1.0 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.01 0.02 0.03 0.04 0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.01 0.02 0.03 0.04 0.05 Figure 4.7: Optimal power  Pversus the compression ratio χfor fixed temperature ratios, ν=0.5(top panels) and ν=ν∗ id (bottom panels). In the left (right) panels, the upper (lower) bound in the temperature control reaches its ideal limit, and multiple values of the lower (upper) bound are considered. The optimal power corresponding to the ideal limits of both bounds are also displayed in the four panels (dotted black line). BIBLIOGRAPHY 42 [17] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier, 1992). [18] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, (Springer, 1985). [19] Y. Band, O. Kafri, and P. Salamon, Journal of Applied Physics 53, 8 (1982). [20] F. L. Curzon and B. Ahlborn, American Journal of Physics 43, 22 (1975). [21] M. Esposito, K. Lindenberg, and C. Van den Broeck, Physical Review Letters 102, 130602 (2009). [22] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Physical Review Letters 105, 150603 (2010). [23] C. Van den Broeck, Physical Review Letters 95, 190602 (2005). [24] L. S. Pontryagin, Mathematical Theory of Optimal Processes, (CRC Press, 1987). [25] D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, (Princeton University Press, 2012). Appendix A Pontryagin’s Maximum Principle Pontryagin’s Maximum Principle provides the convenient mathematical tool for the bounded optimisation problem that we have stated in Sec. 3.2.2 [24]. In the following, we enunciate the general time-optimal problem undertaken by a particularisation of Pontryagin’s Principle and later we focus on the one-dimensional case concerning our isochoric branches. A.1 Pontryagin’s Time-Optimal Problem Statement of the Pontryagin’s Time-Optimal Problem Let us consider an object whose law of motion can be written in the form of a system of differential equations dxi dt =fi(x1,...xn, u1,...,ur)≡fi(x,u), i =1,...,n, (A.1) which are equivalent to the differential vector equation dx dt =(f1(x,u),...,fn(x,u))≡f(x,u).(A.2) Here, x≡(x1,...,xn)are the phase coordinates of the system, belonging to the phase space X;u≡(u1,...,ur)are the control parameters, which determine the course of the process and belong to a ‘control region’ U;tdenotes time; and fi∶X×U→Rare continuous functions which are also differentiable with respect to the phase coordinates. Arbitrary piecewise continuous controls u=u(t)are admissible. Thence, for any control function, defined on an interval [t0, tf], there exists, at most, a finite number kof time instants α0≡t0<α1<...<αk<αk+1≡tfat which the control may have finite jump discontinuities (an example is shown in the top panel of Fig. A.1). For any given control u=u(t)and any chosen initial condition x(t0)=x0, in most cases it is possible to uniquely define the evolution of the system x(t)in phase space to be continuous and piecewise differentiable (see Fig. A.1). The procedure that allows 43 APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 44 Figure A.1: Qualitative representation of the control u(top panel) and the corresponding phase coordinate x(bottom panel) time evolutions. For the sake of clarity, we have considered one-dimensional variables in both cases. APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 45 for the construction of such function consists in concatenating the solutions of the initial value problems associated with the system in Eq. (A.1) (or, equivalently, Eq. (A.2)) and appropriate initial conditions for each interval in which the control u(t)is continuous.1 Namely, the first initial value problem that we need to consider is ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ dx dt =f(x,u(t)), α0≤t≤α1, x(α0)=x0. (A.3) Let us denote as x(0)(t)the solution of problem stated in Eq. (A.3), whose existence is assumed. Subsequently, we need to solve the initial value problems ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ dx dt =f(x,u(t)), αj≤t≤αj+1, x(αj)=x(j−1)(αj). j=1,...,k, (A.4) where x(j)(t)is the solution corresponding to the j-th interval (i.e. [αj, αj+1]for j= 0,...,k), which, again, is assumed to exist. The chosen initial condition in Eq. (A.4) guarantees the continuity of the constructed solution, which is said to correspond to the control u(t), and it is given by the following expression, x(t)=x(j)(t), αj≤t≤αj+1;∀j=0,...k. (A.5) If the obtained solution x(t)satisfies the additional boundary condition x(αk+1)=xf, we shall say that the considered control u(t)transfers the phase point from the position x0to the position xf. Now, we are ready to formulate the optimisation problem that we need to assess: Given two points x0,xf∈X, we aim to find, among the admissible controls u(t)which transfer the phase point from position x0to xf(if such controls exist), one for which the functional J[u]=∫tf t0 dt =tf−t0(A.6) attains its minimum possible value. Maximum Principle for Time-Optimal Processes Pontryagin’s Maximum Principle provides a necessary condition for optimality. It can be applied to a wider range of problems in which, instead of the functional given in Eq. (A.6), more general ones with 1We extend the definition intervals for the studied initial value problems to be closed ones, considering the corresponding limit values for the control u(t), which is not strictly defined on the endpoints αj. APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 46 explicit dependence on the control u(t)and the phase-space trajectory x(t)are covered. The proof of the Maximum Principle and its general formulation go beyond the purposes of this text and they are omitted, but can be found in Refs. [24,25]. In the case of the Time-Optimal Problem previously formulated, we shall consider an additional system of equations in the auxiliary variables ψ≡(ψ1,...,ψn), dψi dt =−n ∑ α=1 ∂fα(x,u) ∂xiψα, i =1,...,n. (A.7) For a given control u(t)and the corresponding trajectory x(t), this system becomes linear and homogeneous, and thus it has a uniquely defined solution ψ(t). For clarity, this solution is said to correspond to the chosen control evolution u(t)and phase trajectory x(t). Theorem 1 (Necessary Condition for Time-Optimality).If the control u(t)and the corresponding trajectory x(t)are time-optimal, then there exists a continuous, nonzero vector function ψ(t)corresponding to them which verifies the following conditions: 1. The so-called ‘Pontryagin’s Hamiltonian’, defined as H(ψ(t),x(t),u(t))≡n ∑ i=1 ψnfn(x(t),u(t)),(A.8) reaches its maximum value M(ψ,x)=sup u∈U H(ψ,x,u)(A.9) for all t∈[t0, tf], H(ψ(t),x(t),u(t))=M(ψ(t),x(t)),∀t∈[t0, tf].(A.10) 2. Pontryagin’s Hamiltonian is non-negative at the terminal time: M(ψ(tf),x(tf))≥0.(A.11) Furthermore, if Eq. (A.10) is verified, the function Mis found to be constant in time. Thus, condition in Eq. (A.11) can be equivalently stated ∀t∈[t0, tf], not only at the terminal time tf. In general, existence and sufficiency conditions for optimality are difficult to find and exhibit a strong dependence on the vector function f(x,u)characterising the evolution of the trajectory in the phase space. Fortunately, in our case of interest this function is APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 47 linear in its variables and this yields interesting and useful properties. In particular, the existence of an optimal control is guaranteed. Note that, with the definition of Hgiven in Eq. (A.8), the systems in Eqs. (A.1) and (A.7) are equivalent to ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ dxi dt =∂H ∂ψi , i =1,...,n, dψi dt =−∂H ∂xi, i =1,...,n. (A.12) Linear Time-Optimal Processes Let us now consider the case in which functions describing the trajectory evolution in the phase space are linear: fi(x,u)=n ∑ µ=1 ai µxµ+r ∑ ν=1 bi νuν, i =1,...,n, (A.13) or, in vector form, f(x,u)=Ax+Bu,(A.14) where Aand Bare linear operators defined, respectively, by matrices (ai µ)and (bi ν). Furthermore, in this case we shall consider the control region Uto be a parallelepiped verifying that every vector v∈Uwith the direction of one of its edges satisfies that Bv, ABv,... An−1Bv(A.15) are linearly independent. In this particular case, Pontryagin’s Hamiltonian takes de form H(ψ,x,u)=(ψ, Ax)+(ψ, Bu)=∑ i,µ ψiai µxµ+∑ i,ν ψibi νuν,(A.16) where (⋅,⋅)denotes the scalar product. In this particular case of study, the second equality in Eq. (A.12) can be written as dψi dt =−∂H ∂xi=−n ∑ j=1 aj iψj, i =1,...,n;(A.17) or, equivalently, in vector form, dψ dt =−ATψ.(A.18) Theorem 1 implies that, if u(t)is an optimal control transfering the phase point from x0 to xf, there must exist a solution ψ(t)of Eq. (A.18) verifying Eq. (A.10). In particular, as the first term in Eq. (A.16) does not depend explicitly on the control, the optimal APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 48 solution ˜ usatisfies that P(ψ(t))≡sup u∈U(ψ(t), Bu(t))=(ψ(t), B ˜ u(t)).(A.19) Note that, for any given solution ψ(t)of system in Eq. (A.17), this equation can be used to determine possible optimal controls u(t). Is condition in Eq. (A.19) sufficient for determining the control uniquely? The next theorem (whose proof may be consulted in Ref. [24]) yields an answer for this question. Theorem 2. If all the eigenvalues of matrix Aare real, then, for every non-trivial solution ψ(t)of Eq. (A.18) (or, equivalently, system in Eq. (A.17)), relation in Eq. (A.19) uniquely determines the control function. Moreover, each component uj(t),=1,...r is piecewise constant, taking on only the minimum and maximum values of the interval in which it is bounded, and it has at most n−1jump discontinuities. Hence, the control is a so-called ‘bang-bang’ protocol. A.2 Thermal brachistochrone for an isochoric branch We now return to our particular system of interest: a one-dimensional harmonically confined Brownian particle in an isochoric process. The application of Pontryagin’s Maximum Principle is extremely simple in this special case. We aim to minimise the functional J[θ]=∫τf 0 dτ =τf,(A.20) which corresponds to the duration of the process. We start by recalling that our phase space is three-dimensional, with the state of the system being determined by the triplet (κ, y, θ). Since we are in an isochoric branch, κis fixed and the temperature θis considered as the control. Accordingly, our ‘effective’ phase space in this case is one-dimensional (n=1) and it is characterised by a single coordinate x1=y. Likewise, the control region is also one-dimensional (r=1), since the only control considered is the temperature u1=θ. Furthermore, it is bounded on a closed interval: θmin ≤θ≤θmax, which corresponds to the one-dimensional version of a parallelepiped. Given that n=r=1, instead of the system in Eq. (A.1), we have a single equation dy dτ =f1(y, θ)≡−2κy +2θ. (A.21) It is a linear equation of the form previously studied, with A=−2κand B=2(note that Aand Bare scalars here). In this one-dimensional case, condition in Eq. (A.15) APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 49 is superfluous (a unitary set is not linearly dependent if and only if its only element is zero, i.e. if v=0, but then vwould not determine any direction). Hence, the linearity of Eq. (A.21) and the type of bounds for θimply that our problem corresponds to a linear time-optimal process. Although Theorem 2 proves that the optimal protocol must be constant with either θ(t)=θmax or θ(t)=θmin (this depends on whether we intend to heat or cool our system), we shall derive the result for this simple case. Pontryagin’s Hamiltonian reads H(ψ, y, θ)=(−2κy +2θ)ψ. (A.22) Thus, in this case, Eq. (A.18) is written as follows, dψ dτ =−∂H ∂y =2κψ. (A.23) The general solution for this first-order differential equation is ψ(τ)=ψ0e2κτ ,(A.24) which does not depend on the control parameter θand has a well-defined constant sign, dependent on the initial value ψ0. According to our previous discussion, the Hamiltonian attains a maximum at the optimal value, and thence ∂H ∂θ =2ψ=0(A.25) if the optimal control belonged to the open interval θ(τ)∈(θmin, θmax). Nevertheless, this is not possible, since, as we just mentioned, ψis either positive or negative in the whole time interval. This is an alternative path to prove that the solution of our problem is a bangbang protocol without ‘switchings’ (i.e. with a single continuity interval), in which θ(τ) is equal to one of its bounds. In a heating process, we have that ˜ θ(τ)=θmax,∀τ∈(0, τf), whereas ˜ θ(τ)=θmin,∀τ∈(0, τf)in a cooling procedure. Thus, the optimal control ˜ θ(τ), which is illustrated in Fig. A.2, can be expressed as ˜ θ(τ)=˜ θ≡⎧⎪⎪⎪⎨⎪⎪⎪⎩ θmax,if θi<θf, θmin,if θi>θf,∀τ∈(0, τf).(A.26) Note that ˜ θ(τ)is discontinuous at the initial and final time instants, given that it is submitted to the boundary conditions ˜ θ(0)=θi,˜ θ(τf)=θf.(A.27) APPENDIX A. PONTRYAGIN’S MAXIMUM PRINCIPLE 50 Figure A.2: Sketch of the optimal temperature driving in heating (left panel) and cooling (right panel) processes. Addendum: Code example The numerical optimisation problems presented throughout this work have been solved using the software Mathematica 13.1. In this addendum, we include the notebook used to find the overall optimum in Sec. 4.2.1.1 as an illustrative example. The codes used to obtain the rest of our results are similar and they are omitted for the sake of conciseness. Optimisation of the irreversible Stirling cycle in the limit θmax  ∞,θmin 0 In[]:= ClearAll["Global`*"] In[]:= << MaTeX` Definition of physical quantities and parameters In[]:= Wqs[ν_,χ_] :=1-ν 2Log[χ]; α[χ_] :=1 χ-1 2 ; w[ν_,χ_] :=-Wqs[ν,χ] α[χ] * 1+ν2; tisoc[ν_,χ_] :=-1 2χLog[ν]; σ[ν_,χ_] :=1+w[ν,χ] * tisoc[ν,χ]; tAB[ν_,χ_] :=1+σ[ν,χ] w[ν,χ]*1+ν; tCD[ν_,χ_] :=ν*tAB[ν,χ]; WAB[ν_,χ_] :=Log[χ] 2+α[χ] tAB[ν,χ]; WCD[ν_,χ_] :=-ν*Log[χ] 2+ν*α[χ] tCD[ν,χ]; P[ν_,χ_] := -((WAB[ν,χ] + WCD[ν,χ]) / (tAB[ν,χ] + tCD[ν,χ] + tisoc[ν,χ])); η[ν_,χ_] :=1+WCD[ν,χ] WAB[ν,χ]; ηC[ν_] :=1-ν; ηCA[ν_] :=1-ν; Printed by Wolfram Mathematica Student Edition 51