Finite-time adiabatic processes: derivation and speed limit

TL;DR

This paper develops a method to construct finite-time adiabatic processes for Brownian particles by controlling potential and temperature, and establishes a fundamental speed limit dictated by the second law of thermodynamics.

Contribution

It provides a systematic way to engineer finite-time adiabatic processes and derives an explicit speed limit for such transformations.

Findings

Finite-time adiabatic processes can be explicitly constructed for Brownian particles.

A minimum time for adiabatic transformations is derived from the second law.

The speed limit depends on the system's initial and final states.

Abstract

Obtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant both at the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression/decompression situation.

Equations

$$\lambda dX/dt=-{\partial }_{X}U(X,t)+\sqrt{2\lambda k_{B}T(t)}\xi (t),$$

$$\lambda {\partial }_{t}P(X,t)={\partial }_X\left[{\partial }_{X}U(X,t)P(X,t)\right]+k_{B}T(t){\partial }_X^{2}P(X,t).$$

$$\dot{p}(x,\tau )=$$

$$\frac{j(x,\tau )}{t_{\text{f}}^{\ast }}=$$

$$\dot{S}={\dot{S}}_{\text{irr}}=\frac{1}{t_{\text{f}}}\frac{1}{\theta (\tau )}\int dx\frac{j^2(x,\tau )}{p(x,\tau )}.$$

$$p_{\text{i}}(x)=Z_{\text{i}}^{-1}{e}^{-u_{\text{i}}(x)/{\theta }_{\text{i}}},p_{\text{f}}(x)=Z_{\text{f}}^{-1}{e}^{-u_{\text{f}}(x)/{\theta }_{\text{f}}},$$

$$\Xi (\tau )=\theta (\tau )e^{2S_x(\tau )}=1+\frac{2}{t_{\text{f}}}{\int }_0^{\tau }d\zeta e^{2S_x(\zeta )}\int dx\frac{j^2(x,\zeta )}{p(x,\zeta )}.$$

$$t_{\text{f}}=\frac{2}{\Delta \Xi }{\int }_0^{1}d\tau e^{2S_x(\tau )}\int dx\frac{j^2(x,\tau )}{p(x,\tau )},$$

$$\theta (\tau ){\sigma }_x^2(\tau )-1=\frac{1}{2t_{\text{f}}}{\int }_0^{\tau }d\zeta {\left[\frac{d}{d\zeta }{\sigma }_x^2(\zeta )\right]}^2,$$

$$t_{\text{f}}=\frac{J[{\sigma }_x]}{2\Delta (\theta {\sigma }_x^2)}withJ[{\sigma }_x]\equiv {\int }_0^{1}d\tau {\left[\frac{d}{d\tau }{\sigma }_x^2(\tau )\right]}^2$$

$$u(x,\tau )=-\frac{1}{2t_{\text{f}}}\frac{{\dot{\sigma }}_x(\tau )}{{\sigma }_x(\tau )}{x}^2+\theta (\tau )u_{\text{i}}\left(\frac{x}{{\sigma }_x(\tau )}\right).$$

$${\tilde{\sigma }}_x(\tau )=\sqrt{1+\tau \Delta ({\sigma }_x^2)},and\widetilde{t_{\text{f}}}=\frac{{\left(\Delta ({\sigma }_x^2)\right)}^2}{2\Delta (\theta {\sigma }_x^2)}.$$

$$\tilde{\theta }(\tau )=\frac{1+\tau \Delta \left(\theta {\sigma }_x^2\right)}{1+\tau \Delta \left({\sigma }_x^2\right)},$$

$$\kappa (\tau )=-\frac{1}{t_{\text{f}}}\frac{d\ln {\sigma }_x(\tau )}{d\tau }+\frac{\theta (\tau )}{{\sigma }_x^2(\tau )}.$$

$$\tilde{\kappa }(\tau )=\frac{C}{{\tilde{\sigma }}_x^4(\tau )},C=\frac{\Delta \theta }{\Delta ({\sigma }_x^{-2})}=\frac{\Delta \theta }{\Delta (\kappa /\theta )}.$$

$${\sigma }_W^2\sim \frac{1}{2t_{\text{f}}}{\int }_0^{1}d\tau {\dot{\kappa }}^2(\tau )\frac{{\theta }^2(\tau )}{{\kappa }^3(\tau )}.$$

$${\sigma }_Q^2-({\sigma }_Q^{\text{qs}}{)}^2\sim {\sigma }_W^2-\frac{1}{2t_{\text{f}}}\Delta \left(\dot{\kappa }\frac{{\theta }^2}{{\kappa }^2}\right).$$

$$m\frac{d^{2}X}{d{t}^2}=-\lambda \frac{dX}{dt}-U^{\prime }(X,t)+\lambda \sqrt{2D}\xi (t)$$

$$\lambda =6\pi \eta r,$$

$$\lambda \frac{dX}{dt}=-{\partial }_{X}U(X,t)+\lambda \sqrt{2D}\xi (t).$$

$$\lambda \frac{dX(t)}{dt}=-kX(t)+\sqrt{2\lambda k_{B}T}\xi (t).$$

$$\lambda {\partial }_{t}P(X,t)=k{\partial }_X\left[XP(X,t)\right]+k_{B}T{\partial }_X^{2}P(X,t).$$

$$G(s,t)\equiv ⟨e^{isX}⟩={\int }_{-\infty }^{+\infty }dX{e}^{isX}P(X,t).$$

$$\lambda {\partial }_{t}G(s,t)=-ks{\partial }_{s}G(s,t)-k_{B}T{s}^{2}G(s,t).$$

$$\ln G(s,t)=\sum _{n=1}^{\infty }\frac{(is{)}^n}{n!}{\chi }_n(t).$$

$$\lambda {\partial }_t\ln G(s,t)=-ks{\partial }_s\ln G(s,t)-k_{B}T{s}^2.$$

$$\lambda \frac{d{\chi }_n(t)}{dt}=-nk{\chi }_n(t)+2k_{B}T{\delta }_{n,2},n\ge 1.$$

$${\mu }_1(t)\equiv ⟨X⟩(t)=⟨X⟩(0)\exp \left[-\frac{1}{\lambda }{\int }_0^{t}d{t}^{\prime }k(t^{\prime })\right].$$

$$\lambda \frac{d}{dt}{\sigma }_X^2=-2k{\sigma }_X^2+2k_{B}T.$$

$${\sigma }_X^2(t)=$$

$${\sigma }_{X,\text{eq}}^2(t)\equiv$$

$$\kappa =\frac{k}{k_{\text{i}}},\theta =\frac{T}{T_{\text{i}}},x=\frac{X}{({\sigma }_{X,\text{eq}}{)}_{\text{i}}},$$

Full text

I Introduction

Adiabatic processes are a cornerstone in the thermodynamics of macroscopic systems. Therein, energy is solely exchanged as work–there is no heat–and the large system size makes fluctuations mostly irrelevant. If, in addition, the system always sweeps equilibrium states, i.e., the process is reversible, there is no entropy change. These processes played a central conceptual role in laying the foundations of thermodynamics, culminating with the works of Carathéodory and Planck Callen (1985). Besides, such processes are essential to build thermal engines. The Carnot cycle indeed consists of two reversible isothermal and two reversible adiabatic branches Callen (1985) 111We understand adiabatic in the thermodynamical sense, not in the often found “slow enough” quantum mechanical meaning, to which we refer as quasi-static..

The relevance of mesoscopic systems spreads out across a wide range of fields in physics and technology, such as nanodevices Jin et al. (2010); Roche et al. (2015), biomolecules Collin et al. (2005); Prados et al. (2013); Lin et al. (2018) or active matter Bertin et al. (2013); Battle et al. (2016). Statistical methods are typically applicable to mesoscopic systems, but their smallness entails that fluctuations play an important role Sekimoto (2010); Seifert (2012). It is thus challenging but also compelling to extend macroscopic concepts to the mesoscale because new physics often emerges, like the fluctuation theorems or transient violations of the second law Gallavotti and Cohen (1995); Jarzynski (1997); Crooks (1998); Ritort (2004); Marconi et al. (2008).

At the mesoscale, defining and characterising adiabatic processes is crucial, e.g. to build a mesoscopic version of the Carnot engine. But this is far from trivial: it is meaningless to imagine an inherently fluctuating Brownian particle thermally isolated from its environment for each of its trajectories: over them, both work and heat contribute to the energy change Crooks and Jarzynski (2007); Martínez et al. (2015). However, one can think of processes in which the average heat vanishes, not only between the initial and final states, but along the whole dynamics; thus the average work yields the average energy increment. This is the concept of adiabatic process that we employ throughout.

Finite-time adiabatic processes have not been devised so far. In fact, adiabatic processes have been only analysed in simple limiting cases: vanishing or infinite time operation. In the overdamped regime, instantaneous processes in which the position distribution does not change have been termed adiabatic Schmiedl and Seifert (2008); Holubec (2014); Rana et al. (2014); Singh and Johal (2018) because the configurational contribution to the heat vanishes. However, these processes are not actually adiabatic, since there is a contribution to the heat–and to the entropy change–coming from the velocity degree of freedom: the temperature varies in such instantaneous processes Hondou and Sekimoto (2000); Schmiedl and Seifert (2008). For underdamped dynamics, only quasi-static reversible adiabatic processes have been analysed, mainly for the harmonic case. Therein, this has led to the condition $T^{2}/k=\text{const}$, where $T$ is the bath temperature and $k$ is the stiffness of the trap Bo and Celani (2013); Martínez et al. (2015, 2015).

Engineering adiabatic processes requires the joint time control of both the bath temperature and the confining potential, which can be implemented in experiments with micron-size colloids manipulated by laser tweezers in a suspending fluid Martínez et al. (2013); Ciliberto (2017). Optical confinement makes it possible to control the time dependence of the effective temperature seen by the Brownian particle Martínez et al. (2013). The dynamics is neatly overdamped for the broad class of systems consisting of mesoscopic objects suspended in a solvent 222The time variation of the physical properties controlling the dynamics–trap stiffness and bath temperature–must be much slower than the relaxation of velocity to equilibrium, governed by the fluid viscosity.. We shall thus carry our analysis in this framework, see Appendix A.

Hereafter, we answer two physically relevant questions. First, we show how finite-time adiabatic processes can be built for a colloidal particle driven by an arbitrary potential. Not only does this have theoretical importance but also practical consequences. For example, shortening the duration of the adiabatic branches of a Brownian Carnot engine–like the one investigated in Ref.  Martínez et al. (2015)–increases the delivered power. Second, we show that there appears a fundamental speed limit for such adiabatic processes. This is at variance with the isothermal case, where equilibration can be arbitrarily accelerated Martínez et al. (2016). The emergence of such speed limits is also of fundamental interest, with relevance in control theory and the foundations of non-equilibrium statistical mechanics Sekimoto and Sasa (1997); Aurell et al. (2012); Ito (2018); Okuyama and Ohzeki (2018); Shiraishi et al. (2018); Ito (2018); Rosales-Cabara et al. (2020).

The paper is organised as follows. In Sec.  II, we rigorously show the feasibility of finite-time adiabatic processes in the context of stochastic thermodynamics. Section III is devoted to the optimisation of such processes, either minimising the connecting time or optimising the target temperature. Finally, we summarise the conclusions of this work in Sec. IV, along with a discussion of future perspectives. The appendices deal with some technical aspects and complementary discussions that are not essential for the understanding of our results, and thus are omitted in the main text.

II Engineering finite-time adiabatic processes and speed limit

We consider a Brownian particle immersed in a heat bath at temperature $T(t)$ and trapped in a generic potential $U(X,t)$. The particle position obeys the Langevin equation

[TABLE]

with $\lambda$ the friction coefficient and $\xi(t)$ a unit-variance Gaussian white noise. The Fokker-Planck (FP) equation for the probability density function (PDF) $P(X,t)$ of finding the particle at position $X$ at time $t$ thus reads

[TABLE]

We are interested in processes that connect two given equilibrium states in a running time $t_{\text{f}}$. Dimensionless variables are introduced with the definitions $\tau=t/t_{\text{f}}$ ($0\leq\tau\leq 1$), $x=X/\sigma_{X,\text{i}}$, $\theta=T/T_{\text{i}}$, $u=U/(k_{B}T_{\text{i}})$, $p(x,\tau)=\sigma_{X,\text{i}}\,P(\sigma_{X,\text{i}}\,x,t_{\text{f}}\,\tau)$. For any physical quantity $Y$, we denote throughout the paper derivatives by $\dot{Y}\equiv\partial_{\tau}Y$ and $Y^{\prime}\equiv\partial_{x}Y$, the initial (final) value by $Y_{\text{i}}$ ($Y_{\text{f}}$), the difference between final and initial values by $\Delta Y\equiv Y_{\text{f}}-Y_{\text{i}}$, and the variance by $\sigma_{Y}^{2}$. The FP equation is then

[TABLE]

where $t_{\text{f}}^{*}=k_{B}T_{\text{i}}t_{\text{f}}/(\lambda\sigma_{X,\text{i}}^{2})$ is the dimensionless connecting time 333We drop the asterisk not to clutter our formulas..

Energy has two contributions: a kinetic one, and a configurational one coming from the potential $u(x,\tau)$. Within the overdamped description, the average kinetic energy always has the equilibrium value $\theta/2$, which is time-dependent. Thus, the average energy is $\overline{E}=\theta/2+\overline{u}(x,\tau)$, where $\overline{u}(x,\tau)=\int dx\,u(x,\tau)p(x,\tau)$. Work and heat exchange rates are $\dot{\overline{W}}=\int dx\,\dot{u}\,p$ and $\dot{\overline{Q}}=\dot{\theta}/2+\dot{\overline{Q}}_{x}$, with $\dot{\overline{Q}}_{x}\equiv\int dx\,u\,\dot{p}=\int dx\,u^{\prime}\,j$ the configurational heat rate. The first principle then holds: $\dot{\overline{E}}=\dot{\overline{Q}}+\dot{\overline{W}}$ Sekimoto (2010).

Entropy is introduced as Sekimoto (2010): $S=S_{\text{kin}}+S_{x}$, where $S_{\text{kin}}=\frac{1}{2}\ln\theta$, $S_{x}(\tau)=-\int dx\,p(x,\tau)\ln\!p(x,\tau)+K$. We choose the constant $K$ to make $S_{x,\text{i}}=0$ and simplify some formulas. From the FP equation, extended forms of the second principle have been derived, $\dot{S}=\dot{S}_{\text{irr}}+\dot{\overline{Q}}/\theta$, where $\dot{S}_{\text{irr}}\geq 0$ is the entropy production rate Ge (2009); Sekimoto (2010); Seifert (2012). For adiabatic processes, $\dot{S}_{\text{irr}}$ only contributes to the entropy change and one gets in dimensionless variables

[TABLE]

Let us consider given equilibrium initial and final states, corresponding to temperature and potential pairs $(\theta_{\text{i}},u_{\text{i}}(x))$ and $(\theta_{\text{f}},u_{\text{f}}(x))$, respectively. Our first aim is to show the feasibility of connecting these states adiabatically, i.e. find solutions of Eq. (3) that (i) have the canonical form at both the initial and final times,

[TABLE]

with $Z_{\text{i},\text{f}}$ ensuring the normalisation of the distributions, and (ii) make the total heat exchange rate $\dot{\overline{Q}}=0$ for all times. We show below how this can be done by tuning the temperature $\theta(\tau)$ and the potential $u(x,\tau)$. Note that $\overline{W}=\Delta\overline{E}$, regardless of the process duration.

We build explicitly these adiabatic processes by an inverse-engineering procedure. Starting from any $p(x,\tau)$ connecting these two fixed states, Eq. (3a) gives $j(x,\tau)=\int_{x}^{+\infty}d\xi\,\dot{p}(\xi,\tau)$. If we knew $\theta(\tau)$ (we do not yet), integration of Eq. (4) from $\tau=0$ to $1$ would yield the value of $t_{\text{f}}$ and Eq. (3b) would finally give the force field $u^{\prime}(x,\tau)$. This remark suggests to get rid of $\theta(\tau)$ by introducing the change of variable $\Xi(\tau)=e^{2S(\tau)}$, with $\Xi_{\text{i}}=1$. Then, $\Xi$ evolves according to

[TABLE]

Starting again from a PDF $p(x,\tau)$ verifying Eq. (5), $j(x,\tau)$ and also $S_{x}(\tau)$ follow. Thus, we know $\Xi(\tau)$ for all times and we can complete the inverse-engineering procedure: (i) particularising Eq. (6) for $\tau=\tau_{\text{f}}=1$, we obtain the value of the connecting time

[TABLE]

(ii) turning to Eq. (6), we get the temperature program $\theta(\tau)$–$\theta(\tau)>0$ for all times–and (iii) Eq. (3b) provides us with the force $u^{\prime}(x,\tau)$ that does the job.

Equation 6 shows that two arbitrary states cannot be connected with an adiabatic transformation. The positiveness of the right hand side ensures that $\Delta\Xi\geq 0$ or $\Xi_{\text{f}}\geq\Xi^{\text{qs}}=1$. The equality only holds for the quasi-static case: if $\Delta\Xi=0$, we have that $t_{\text{f}}$ diverges 444The current associated to the quasi-static solution $p^{\text{qs}}(x,\tau)\propto\exp\left[-u(x,\tau)/\theta(\tau)\right]$ does not vanish in general, $j^{\text{qs}}(x,\tau)=\int_{x}^{\infty}d\xi\dot{p}^{\text{qs}}(\xi,\tau)\neq 0$. Then, Eq. (3b) poses no problem because it becomes an identity. and $\Xi(\tau)=1$. Note that, with the exception of the quasi-static case, the adiabatic process cannot be reversed in time because that would violate the second principle.

Moreover, the second principle imposes a speed limit for finite-time adiabatic processes: there appears a minimum non-vanishing value for the connecting time $t_{\text{f}}$, except for a trivial “configurationally-static” case. Starting from Eq. (6), this can be proved by a reductio ad absurdum argument. Let us assume that there is no lower bound for $t_{\text{f}}$ and thus an instantaneous adiabatic process with $t_{\text{f}}=0$ is possible. The rhs of Eq. (7) then vanishes and $j(x,\tau)=0$ everywhere. This entails that the FP equation must have a time-independent solution $p(x,\cancel{\tau})$, but $p_{\text{i}}(x)\neq p_{\text{f}}(x)$ in general. This contradiction completes the argument. If $p_{\text{i}}(x)=p_{\text{f}}(x)$, i.e. $u^{\prime}_{\text{f}}(x)/\theta_{\text{f}}=u^{\prime}_{\text{i}}(x)/\theta_{\text{i}}$ as follows from Eq. (5), we deal with a “configurationally-static” situation and $t_{\text{f}}$ may vanish–see below. In that case, the system cannot cool since $\Xi(\tau)$ is non-decreasing, and thus so is $\theta(\tau)$.

III Compression/decompression processes: Optimal connection

Let us analyse a generic and physically relevant case: the compression or decompression of a system around a fixed average value $\overline{x}$ (the axis origin for convenience). We take $p(x,\tau)=(Z\sigma_{x}(\tau))^{-1}\exp[-u_{\text{i}}(x/\sigma_{x}(\tau))]$, thus of shape-preserved form, where $\sigma_{x}$ is the variance of the distribution and the normalisation constant $Z=\int dy\exp[-u_{\text{i}}(y)]$ does not depend on $\sigma_{x}$. The system is being decompressed (compressed) for $\dot{\sigma}_{x}>0$ ($\dot{\sigma}_{x}<0$) 555All the central moments are $\overline{x^{n}(\tau)}=(\overline{x^{n}})_{\text{i}}[\sigma_{x}(\tau)]^{n}$; $\sigma_{x,\text{i}}=1$ and $\theta_{\text{i}}=1$ with our choice of units.. The corresponding current and entropy follow immediately as $j=(\dot{\sigma}_{x}/\sigma_{x})xp$ and $\Xi=\theta\sigma_{x}^{2}$. The adiabatic inequality simplifies to $\theta_{\text{f}}\sigma_{x,\text{f}}^{2}\geq 1$.

For each choice of the function $\sigma_{x}(\tau)$ obeying $\sigma_{x}(0)=1$, $\sigma_{x}(1)=\sigma_{x,\text{f}}$, the initial and final states are connected. Herein, explicit expressions for the connecting time $t_{\text{f}}$, the temperature program $\theta(\tau)$ and the binding potential $u(x,\tau)$ can be given. Indeed, Eqs. 6 and 7 reduce to

[TABLE]

and the potential stems from Eq. (3b),

[TABLE]

The speed limit for the adiabatic process can be explicitly worked out as well. Since the denominator of $t_{\text{f}}$ in Eq. (8b) is fixed, the minimum time $\tilde{t}_{\text{f}}$ corresponds to the variance profile $\tilde{\sigma}_{x}$ that minimises $J[\sigma_{x}]$. We get

[TABLE]

Note that $\tilde{t}_{\text{f}}>0$ unless $\Delta\sigma_{x}=0$: consistently, the connection time cannot vanish except for the “configurationally-static” case. The optimal temperature evolution follows from Eq. (8a),

[TABLE]

Eqs. 10 and 11 are valid in the whole time interval $t\in[0,\tilde{t}_{\text{f}}]$ or $0\leq\tau\leq 1$. Both $\tilde{\sigma}_{x}$ and $\tilde{\theta}$ are monotonic functions of time, the sign of their derivatives being those of $\Delta\sigma_{x}$ and $\Delta\theta$, respectively. The optimal potential $\tilde{u}(x,\tau)$ is obtained after inserting $\tilde{\sigma}_{x}$ and $\tilde{\theta}$ into Eq. (9). This expression holds only for $t\in(0,\tilde{t}_{\text{f}})$ because $\dot{\tilde{\sigma}}_{x}\neq 0$ for $t=0,t_{\text{f}}$ 666As found in other problems, the optimal control has finite jumps at $t=0$ and $t=t_{\text{f}}$ Band et al. (1982); Schmiedl and Seifert (2007); Aurell et al. (2011); Solon and Horowitz (2018); Plata et al. (2019). Adiabaticity is not broken: there is no instantaneous heat transfer at $t=0$ and/or $t=t_{\text{f}}$.

We complement the study above with the analysis of the harmonic case having time-dependent stiffness $u(x,\tau)=\frac{1}{2}\kappa(\tau)x^{2}$, a standard experimental situation. Therein, $p(x,\tau)$ remains Gaussian for all times, which guarantees shape preservation as shown in Appendix B. With our choice of units, $\kappa_{\text{i}}=1$ and $\kappa_{\text{f}}=\theta_{\text{f}}/\sigma_{x,\text{f}}^{2}$. The adiabatic inequality is $\theta_{\text{f}}^{2}/\kappa_{\text{f}}\geq 1$. Eq. (9) gives the relation between $\kappa(\tau)$ and $\sigma_{x}(\tau)$, which reduces to 777This expression ensures adiabaticity $d\overline{Q}=0$, which simplifies to $d\theta+\kappa d\sigma_{x}^{2}=0$ for harmonic confinement as said in Appendix C.

[TABLE]

After some simple algebra, the optimal stiffness results

[TABLE]

Time evolutions of the state point $(\sigma_{x}^{2}(\tau),\theta(\tau),\kappa(\tau))$, in both optimal (solid lines) and non-optimal adiabatic processes, are illustrated on panels (a)-(c) of Figure 1. Optimal evolutions are obtained by particularising Eqs. 10, 11 and 13 for each case. Non-optimal evolutions are obtained starting from a fourth-order polynomial for the variance $\sigma_{x}^{2}(\tau)=1+b\tau+c\tau^{2}+d\tau^{3}+e\tau^{4}$, similarly to the approach in Ref. Martínez et al. (2016) for isothermal processes. The values $(b,c,d,e)$ are chosen to fulfil the boundary conditions for $(\sigma_{x}^{2},\theta,\kappa)$ and the desired connecting time $t_{\text{f}}=r\tilde{t}_{\text{f}}$. For each $r$ value, there are two paths that connect the initial and final states, see Appendix C.

Figure 1(d) shows a phase diagram in the plane of final states $(\kappa_{\text{f}},\theta_{\text{f}})$–recall that $\sigma_{x,\text{f}}^{2}=\theta_{\text{f}}/\kappa_{\text{f}}$. Over the reversible line $\theta_{\text{f}}=\theta_{\text{f}}^{\text{qs}}=\sqrt{\kappa_{\text{f}}}$, the denominator $\tilde{t}_{\text{f}}$ in Eq. (10) vanishes and the minimum time $\tilde{t_{\text{f}}}$ diverges. The bath always must be heated to get compression (region I), whereas the trap must be loosened to allow for cooling (IV). However, at odds with the isothermal case, the signs of $\Delta\kappa$ and $\Delta\sigma_{x}^{2}$ are not univocally related in an adiabatic process: stiffening the trap may lead to expansion (II). Loosening entails expansion but the bath may need to be heated (III).

We turn to the characterisation of the minimum time. For both loosening and stiffening, $\tilde{t}_{\text{f}}$ is a non-monotonic function of $\theta_{\text{f}}$ for fixed $\kappa_{\text{f}}$; $\tilde{t}_{\text{f}}$ decreases from infinity for the quasi-static value $\theta_{\text{f}}=\theta_{\text{f}}^{\text{qs}}$ to its minimum $\tilde{t}_{\min}^{d\text{ or }c}$ at $\theta_{\text{f}}=\theta_{\text{f}}^{d\text{ or }c}$ and increases therefrom to $t_{\text{f}}^{(1)}=(2\kappa_{\text{f}})^{-1}$ at large $\theta_{\text{f}}$, see Fig. 2. For loosening, $\theta_{\text{f}}^{d}=1$ ($\Delta\theta=0$) and $\tilde{t}_{\min}^{d}=(2\kappa_{\text{f}})^{-1}-1/2>0$. For stiffening, $\theta_{\text{f}}^{c}=\kappa_{\text{f}}$ and $\tilde{t}_{\min}^{c}=0$. The horizontal dashed red line marks the minimum time $\tilde{t}_{\min}^{d\text{ or }c}$, the horizontal blue dashed line the asymptotic value $t_{\text{f}}^{(1)}$, and the dotted vertical asymptote the minimum temperature $\theta_{\text{f}}^{\text{qs}}$.

Instead of fixing the final temperature $\theta_{\text{f}}$, we can fix the connecting time $t_{\text{f}}$ and investigate the range of reachable final temperatures. For instance, a question of experimental relevance for stiffening is: what is the minimum final fluid temperature for a given $t_{\text{f}}$? Interestingly, Fig. 2 yields the answer, if read “horizontally” rather than “vertically” as before. A fresh look at either panel shows that for “long” connecting times $t_{\text{f}}\geq t_{\text{f}}^{(1)}$, temperatures below the only one verifying $\tilde{t}_{\text{f}}(\tilde{\theta}_{\text{f}},\kappa_{\text{f}})=t_{\text{f}}$ are inaccessible, because they demand a longer $t_{\text{f}}$. This is illustrated with the horizontal dot-dashed line above $t_{\text{f}}^{(1)}$, where $\tilde{\theta}_{\text{f}}$ is marked with a purple circle. For “short” connecting times, $\tilde{t}_{\min}^{d\;{\rm or}\;c}\leq t_{\text{f}}<t_{\text{f}}^{(1)}$, there are two temperatures verifying $\tilde{t}_{\text{f}}(\tilde{\theta}_{\text{f}},\kappa_{\text{f}})=t_{\text{f}}$, as exemplified by the horizontal dot-dashed line below $t_{\text{f}}^{(1)}$: only the temperatures between the two purple circles can be reached. Both the minimum time for fixed final state and the extremal temperature(s) for fixed connection time can be obtained by means of a variational approach, as shown in Appendix D.

For a quasi-static–not necessarily adiabatic–process, the PDF of the work is delta-peaked around its average value. The heat distribution is more complex and has been explicitly obtained for the harmonic potential, being asymmetric around its mean Martínez et al. (2015). For finite-time operation, calculating these PDFs is far more challenging because position values at different times are correlated. Yet, the dominant (up to order of $1/t_{\text{f}}$) contributions to the variance of work and heat can be obtained for slow driving. Work is Gaussian-distributed with variance

[TABLE]

The change in the heat PDF is also more complex: shape is not conserved and the variance shift is

[TABLE]

The last term in $\sigma_{Q}^{2}$ stems from the cross-correlation between work and heat. For a detailed derivation of work and heat fluctuations, see Appendix E.

IV Conclusion

The reported results are very general, being applicable to an overdamped Brownian particle bound by an arbitrary non-linear potential. We have shown how two equilibrium states can be connected with an adiabatic–zero heat–process in a finite time, by explicitly building such a transformation. The second principle entails the existence of (i) a forbidden region, i.e. the impossibility of reaching certain final states from a given initial one, and (ii) a speed limit for the adiabatic connection, when it is indeed possible: in general, an instantaneous adiabatic process does not exist.

For compression/decompression of the Brownian particle, further characterisation of these adiabatic transformations can be done. Both the physical discussion and the conclusions stemming from Figs. 1 and 2 remain valid for any non-linear potential, by defining a final “effective” stiffness $\kappa_{\text{f}}=\theta_{\text{f}}/\sigma_{x,\text{f}}^{2}$ in the non-linear case. Specifically, the emergence of a speed limit–for fixed final state–or a range of reachable temperatures–for fixed connecting time–in adiabatic transformations are robust features of our theory. Also, the phase diagram in Fig. 1(d) applies to the general non-linear case 888Even the time dependent profiles of the variance and the temperature, panels (a) and (b) of Fig. 1, are still valid in the non-linear case. The only result that is specific to the harmonic potential is the time evolution of the stiffness, as given by Eq. 12, in panel (c) of Fig. 1..

In the underdamped case, building finite-time adiabatic processes remains an open problem. We may surmise though that they cannot be instantaneous, which would point to the robustness of finite speed limits. An instantaneous process requires again that the initial and target PDFs be coincident, but now for the joint position-velocity PDF. Thus $\theta_{\text{f}}=\theta_{\text{i}}$ and $u_{\text{f}}(x)=u_{\text{i}}(x)$: there would be no room for the entropy to increase, a scenario we can dismiss.

Perspectives concern the stability of the optimal solutions found here with respect to small perturbations in the trap stiffness, bath temperature or other constraints. Besides, our work paves the way for a theory of control in statistical physics, based on stochastic thermodynamics. Such optimal solutions clarify the role of fluctuations and identify fundamental bottleneck with, for instance, the existence of a speed limit. The extension of such ideas to quantum thermodynamics is a promising perspective Caldeira and Leggett (1983); Weiss (2008); Rivas and Huelga (2011).

Appendix A The overdamped formalism: why?

We are interested in the dynamics of a mesoscopic object in a suspending fluid (solvent), driven by a time-dependent force field stemming from the potential $U(X,t)$. The fluid is at equilibrium at temperature $T$. For simplicity, we investigate a one dimensional situation, without loss of generality. The position $X$ of a ‘particle’ of mass $m$ (a colloid such as a macromolecule, or, at a much smaller scale, a large molecule) obeys the Langevin equation

[TABLE]

where $D=kT/\lambda$ is the diffusion coefficient, and $\xi(t)$ stands for Gaussian white noise of zero mean and unit variance. The drag coefficient $\lambda$ originates from viscous friction and reads

[TABLE]

where $\eta$ is the fluid dynamic viscosity and $r$ the particle radius. The associated time scale $m/\lambda$ governs the equilibration of velocity degrees of freedom; it depends on particle size: explicitly through $r$, and also through $m\propto r^{3}$. For a micron-size particle in water at room temperature, we find $m/\lambda$ in the range of $10^{-7}\,$s. This largely exceeds the microscopic solvent correlation time, which justifies the Langevin description with white noise. A third important scale in the problem is the time $t_{r}$ needed to diffuse over a particle diameter (hence, $r^{2}\sim D\,t_{r}$). It sets the scale of position evolution; on the other hand, $m/\lambda$ is the time scale ruling velocity relaxation. While $m/\lambda\propto r^{2}$, $t_{r}\propto r^{3}$, and we have $t_{r}\gg m/\lambda$. For instance, $t_{r}\simeq 1\,$s for the above micron-size colloid. The scale separation $t_{r}\gg m/\lambda$ holds down to to small dimensions, being still true for $r$ in the nanometer range. This gap makes it possible to simplify Eq. (16): as far as positional degrees of freedom are concerned, inertial terms are irrelevant and we have

[TABLE]

This yields the overdamped framework, of much relevance for practical applications, and the starting point of our treatment. Protocols that would require consideration of the inertial term in (16) would need to involve time scales below a tenth of microsecond for micron-size particles.

Appendix B Evolution equation for the variance of the position

B.1 From Langevin to Fokker-Planck

We now address external driving through a harmonic potential with stiffness $k$. Both the temperature and the stiffness, which are externally controlled, may be time-dependent. The Langevin equation (18) for the particle position reads

[TABLE]

The dynamics of the system can be studied using the probability density function $P(X,t)$ for finding the Brownian particle at position $X$ at time $t$. Its time evolution is governed by the Fokker-Planck equation

[TABLE]

The Langevin equation (19) and the Fokker-Planck equation (20) are equivalent; both completely characterise the time evolution of the Brownian particle position–mathematically, the stochastic process Van Kampen (1992).

In light of the above, we can obtain the time evolution of all the moments or, alternatively, all the cumulants of the position from either Eq. (19) or Eq. (20). If the initial condition $P(X,0)$ is Gaussian, as is the case if the system starts from the corresponding equilibrium state, $P(X,t)$ remains Gaussian for all times: the two first cumulants, i.e. position’s average $\langle X\rangle$ and variance $\sigma_{X}^{2}=\langle X^{2}\rangle-\langle X\rangle^{2}$, completely characterise the evolution of the Brownian particle. This can be readily understood from the Fokker-Planck equation by going to Fourier space. This is the route we take in the following.

B.2 Evolution of moments

First, we define the Fourier transform of $P(X,t)$ as

[TABLE]

Therefore, taking the Fourier transform in Eq. (20) leads to

[TABLE]

On the one hand, the expansion of $G(s,t)$ generates the moments $\mu_{n}(t)\equiv\langle X^{n}\rangle(t)$, since $G(s,t)=\sum_{n=0}^{\infty}(is)^{n}\mu_{n}(t)/n!$. On the other hand, the expansion of $\ln G(s,t)$ generates the cumulants $\chi_{n}(t)$,

[TABLE]

We have $\chi_{1}=\mu_{1}$ (the mean) and $\chi_{2}=\mu_{2}-\mu_{1}^{2}$ (the variance).

Equation (22) can be rewritten as

[TABLE]

Introducing the expansion (23) into (24) and equating the coefficients sharing the same power of $s$, the equations for the cumulants are obtained as

[TABLE]

First, we consider the equation for $n=1$. Its solution is

[TABLE]

Then, the average position remains zero for all times if it is so initially. Second, the equation for $n=2$ gives the time evolution of the variance $\chi_{2}\equiv\sigma_{X}^{2}$,

[TABLE]

Third, the equations for $n\geq 2$ entail that an initially Gaussian distribution remains Gaussian for all times: if $\chi_{n}(0)=0$ for all $n\geq 2$, we have that $\chi_{n}(t)=0$ for all $n\geq 2$. Eq. (27) can be solved, with the result

[TABLE]

If the stiffness of the trap $k$ and the temperature of the fluid $T$ are time-independent, the third term on the rhs is not present and $\sigma_{X}^{2}(t)$ decays exponentially towards its equilibrium value $\sigma_{X,\text{eq}}^{2}$.

For the discussion that follows, we introduce dimensionless variables as in the main text,

[TABLE]

except for dimensionless time that is defined as

[TABLE]

Note that, consistently with our notation in the paper, $t_{\text{f}}^{*}$ is the connection time in the $t^{*}$ variable. Therefore, $\tau=t^{*}/t_{\text{f}}^{*}$ is the dimensionless time scale that we have employed throughout the main text. In these appendices, we will make use of both time scales, $t^{*}$ and $\tau$, depending on which is most useful for each situation. In agreement with the notation followed in the paper, we drop the asterisk for simplicity. Also, for the sake of consistency, $\dot{}\equiv d/d\tau$ and thus we explicitly write $d/dt$ for derivatives in the time scale $t$.

In dimensionless variables, the evolution equation of the variance is given by

[TABLE]

The equilibrium variance of the position is

[TABLE]

and the time evolution of the mean and variance of the position are

[TABLE]

where we have defined

[TABLE]

Appendix C Non-optimal adiabatic processes with fourth-order polynomial in the variance

Herein, we describe the non-optimal adiabatic protocols considered in Fig. 1(a)-(c) of the paper. For a general compression/decompression, the adiabaticity condition $\dot{\overline{Q}}=0$ reduces to $\sigma_{x}^{2}d\theta+\langle xu^{\prime}\rangle d\sigma_{x}^{2}=0$, where $u$ is the binding potential. For the harmonic case, $u^{\prime}=\kappa x$ and $\langle xu^{\prime}\rangle=\kappa\sigma_{x}^{2}$, so that adiabaticity is further simplified to $d\theta+\kappa d\sigma_{x}^{2}=0$ or $\kappa=-\dot{\theta}/(2\sigma_{x}\dot{\sigma}_{x})$, which provides us with the stiffness.

The construction of these protocols follows the recipe described in the main text: starting from a given time-dependence for the variance, first we compute the time duration for the process, second the temperature protocol, and finally the stiffness protocol. Specifically, our starting point here is a fourth-order polynomial for the time evolution of the variance,

[TABLE]

which satisfies the initial condition $\sigma_{x}^{2}(0)=1$. The set of parameters $(b,c,d,e)$ are chosen by imposing:

• •

Given final value for the variance, $\sigma_{x}^{2}(\tau=1)=\sigma_{x,\text{f}}^{2}$, which leads to

[TABLE]

This constraint (i) reduces the degrees of freedom of our polynomial from $4$ to $3$ and (ii) is necessary for the consistency of the proposed protocol, i.e. that which connects the initial and final states.

• •

Fixed value of the connecting time $t_{\text{f}}$, which we give in terms of the minimum time $\tilde{t}_{\text{f}}$ as $t_{\text{f}}=r\tilde{t}_{\text{f}}$. For our specific shape of $\sigma_{x}^{2}(\tau)$, the functional $J[\sigma_{x}]$ in Eq. (8b) reduces to a function $J(b,c,d,e)$ of the polynomial parameters. Thus, Eqs. (8b) with the condition $t_{\text{f}}=r\tilde{t}_{\text{f}}$ entails that

[TABLE]

Note that this condition ensures that the temperature protocol $\theta(\tau)$ obtained from Eq. (8a) verifies the boundary conditions for both the initial and final times, $\theta(\tau=0)=1$ and $\theta(\tau=1)=\theta_{\text{f}}$.

• •

Continuity in the stiffness protocol at both the initial and final times, i.e. $\kappa(0)=1$ and $\kappa(\tau=1)=\kappa_{\text{f}}$. Following our discussion above,

[TABLE]

The system of equations Eqs. (38)-(40) can be exactly solved and provides us with two sets of parameters

[TABLE]

where the up and down signs correspond to the first and second solutions, respectively, and we have introduced

[TABLE]

Thus, these non-optimal protocols are limited to $r\geq 6/5$ and allow us to obtain connection times that are, at least, $20\%$ longer than the minimum time $\tilde{t}_{\text{f}}$. This restriction stems from our imposing of continuous stiffness at the boundaries, as given by Eq. (40). Had we relaxed this condition, we would have obtained a larger set of solutions for the parameters $(b,c,d,e)$ including the optimal solution $(\Delta y,0,0,0)$ for $r=1$, whose associated optimal stiffness has finite jumps at the boundaries, as discussed in the main text.

Appendix D Optimisation problems

D.1 Optimal (extremal) temperature for fixed running time

D.1.1 Statement of the variational problem

We would like to minimise the final temperature in an adiabatic process for the trapped Brownian particle. Therefore, consider the temperature increment

[TABLE]

This is a “constrained” minimisation problem, since we seek the minimisation of $\Delta\theta$ that is compatible with (i) the time evolution of the variance of the Brownian particle, Eq. (27), and (ii) the adiabaticity condition, $d\theta+\kappa\,d\sigma_{x}^{2}=0$, i.e.

[TABLE]

Therefore, we have to introduce Lagrange multiplier functions $\lambda(t)$ and $\mu(t)$ ensuring that the above conditions hold for all times, as explained in Ref. Lanczos (1970) for minimisation problems with “auxiliary conditions”–or in Ref. Gelfand and Fomin (2000) for minimisation problems with “subsidiary conditions”.

Throughout this section, we use the abbreviation $y\equiv\sigma_{x}^{2}$ to simplify the notation. Then, we look for functions that make

[TABLE]

stationary. We have to minimise the “action”

[TABLE]

in which we have the “Lagrangian”

[TABLE]

Note that the “Lagrangian” does not depend on $d\kappa/dt$. This means that the corresponding Euler-Lagrange for $\kappa$ can be used to eliminate $\kappa$ in favour of the remainder of the variables 999The same is true for the Lagrange multipliers $\lambda$ and $\mu$, but this is not a peculiarity of the problem with which we deal here, but a general property of the Lagrange multiplier method: by construction, the Lagrangian never depends on the time derivatives of the Lagrangian multipliers..

The boundary conditions for the minimisation problem are the following: (i) given initial equilibrium state, i.e. given values of $\kappa_{\text{i}}$, $y_{\text{i}}$ and $\theta_{\text{i}}$.

[TABLE]

and (ii) given value of the final stiffness $\kappa_{\text{f}}$ and equilibrium condition at the final time, $\kappa_{\text{f}}y_{\text{f}}=\theta_{\text{f}}$.

[TABLE]

By taking an infinitesimal variation of $\mathcal{S}$ and equating it to zero, not only do we get the Euler-Lagrange equations for the minimisation problem, but also the adequate boundary conditions–as discussed in Ref. Lanczos (1970), section II.15. The boundary term in $\delta\mathcal{S}$ must vanish,

[TABLE]

We have here introduced the canonical momenta, which for our problem read

[TABLE]

Since $\kappa_{\text{i}}$, $y_{\text{i}}$ and $\theta_{\text{i}}$ are fixed, there is no boundary contribution coming from $t=0$. For $t=t_{\text{f}}$, however, we have a different situation, $\delta\kappa_{\text{f}}=0$ but $y_{\text{f}}$ and $\theta_{\text{f}}$ are simply linked by the equilibrium condition, which entails that $\kappa_{\text{f}}\delta y_{\text{f}}=\delta\theta_{\text{f}}$. Therefore, we have that

[TABLE]

so that

[TABLE]

since $\delta y_{\text{f}}$ is arbitrary. By employing the expressions for $p_{y}$ and $p_{\theta}$ found above, we get

[TABLE]

for the lacking boundary condition, i.e.

[TABLE]

D.1.2 Euler-Lagrange equations

Now, we write the Euler-Lagrange equations for the minimisation problem. First, taking into account Eq. (51a) and $\partial_{\kappa}\mathcal{L}=2\lambda y+\mu\,{dy}/{dt}$,

[TABLE]

Second, we bring to bear Eq. (51b) and $\partial_{y}\mathcal{L}=2\kappa\lambda$,

[TABLE]

Third, we make use of Eq. (51c) and $\partial_{\theta}\mathcal{L}=-2\lambda$ to write

[TABLE]

In addition, since by construction the Lagrangian does not depend on the time derivatives of the Lagrange multipliers ${\lambda}$ and ${\mu}$, the Euler-Lagrange equations for $\lambda$ and $\mu$ reduce to the constraints or auxiliary conditions (44).

It is straightforward to get rid of the Lagrange multipliers by first inserting Eq. (58) into (56), which gives

[TABLE]

where $c_{1}$ is an arbitrary constant, to be determined later by imposing the boundary conditions. Moreover, Eq. (58) yields

[TABLE]

These expressions for the multipliers in terms of $y$ and $dy/dt$ allow us to work out the solution, as detailed below. The constant $c_{1}$ should be non-zero because its vanishing leads to $\lambda(t)=\mu(t)=0$, i.e. the situation without constraints.

Inserting Eqs. (59) and (60) into (57), we get

[TABLE]

after taking into account that $c_{1}\neq 0$. By employing Eq. (44) to take the time derivative of the evolution equation for $y$ and make use of the adiabatic condition, it is also shown that

[TABLE]

Combining Eqs. (61) and (62), we obtain

[TABLE]

where $c_{2}$ is an arbitrary constant.

Finally, taking into account Eq. (63), we find the expressions for the variance and the temperature. The adiabatic condition is now simplified to

[TABLE]

in which $c_{3}$ is another arbitrary constant–the factor $1/2$ on the rhs is convenient later. Substituting Eqs. (63) and (64) into the evolution equation for the variance $y$ gives

[TABLE]

Once more, $c_{4}$ is an arbitrary constant.

D.1.3 Solution of the problem

Equations (63), (64) and (65) provide the solution to the minimisation problem. The constants $(c_{1},c_{2},c_{3},c_{4})$ have to be written in terms of physical quantities by imposing the boundary conditions. It may seem odd at first sight that there are $4$ constants but $6$ boundary conditions. The reason is the same as in other problems in stochastic thermodynamics: $\kappa$ may have jumps at the boundaries. In the present context, this peculiar behaviour is readily understood: the conjugate moment $p_{k}=\partial_{\kappa}\mathcal{L}$ identically vanishes and therefore $\delta\kappa(t=0)$ and $\delta\kappa(t=t_{\text{f}})$ are in fact arbitrary when imposing the extremality condition $\delta\mathcal{S}=0$. This means that $\kappa$ can indeed have finite-jump discontinuities at the initial and final times: $\kappa(t=0^{+})$ and $\kappa(t=t_{\text{f}}^{-})$ do not coincide in general with $\kappa_{\text{i}}$ and $\kappa_{\text{f}}$.

Following the discussion above, we now impose the four relevant boundary conditions

[TABLE]

The constants $c_{3}$ and $c_{4}$ are directly obtained as

[TABLE]

Note that $y_{\text{f}}$ does not have a definite value but is related to $\theta_{\text{f}}$ by the equilibrium condition; this will be brought to bear later. The optimal time evolution for the variance is then

[TABLE]

Now, particularising Eq. (64) for $t=0$ makes it possible to obtain $c_{2}$,

[TABLE]

Using again Eq. (64) but for an arbitrary time $t$, one gets after some simple algebra

[TABLE]

Substituting $t=t_{f}$ into this equation, we obtain

[TABLE]

so that

[TABLE]

with the equality holding in the limit as $t_{\text{f}}\to\infty$.

We have yet to impose the boundary condition $y_{\text{f}}=\theta_{\text{f}}/\kappa_{\text{f}}$. We do so in Eq. (71),

[TABLE]

This is a quadratic equation for $\theta_{\text{f}}$ in terms of the fixed parameters $\kappa_{\text{f}}$, $y_{\text{i}}$, $\theta_{\text{i}}$, and $t_{\text{f}}$. Solving it for $t_{\text{f}}$, we find

[TABLE]

which is equivalent to Eq. (10) in the main text, particularised for the harmonic potential.

It is worth noting that the constant $c_{1}$ has not been necessary to obtain the solution for the physical quantities, the stiffness $\kappa(t)$, the variance $y(t)$, and the temperature $\theta(t)$. It is only needed to derive the final expressions for the Lagrange multipliers $\lambda(t)$ and $\mu(t)$. For the sake of completeness, we give the expression for $c_{1}$ that follows from Eq. (55),

[TABLE]

D.2 Minimum time for fixed initial and final states

We turn our attention to another optimisation problem: that of obtaining the minimum time to connect two given equilibrium states with an adiabatic process. This problem has been solved in the paper by an ad-hoc procedure, but it can be addressed in a way similar to the one employed in the previous section. In this case, we would like to minimise

[TABLE]

submitted again to the constraints given by Eq. (44). Therefore, we have to minimise a new “action”

[TABLE]

in which we have the new “Lagrangian”

[TABLE]

Since $\hat{\mathcal{L}}$ and $\mathcal{L}$ differ by the total derivative of a function that depend only on the “coordinates”–and not on the velocities, we know that the Euler-Lagrange equations for both minimisation problems will be the same. Anyhow, we cannot yet conclude that the solution to both problems is the same, since the boundary conditions for them are not 101010In this particular case, the fact that the Euler-Lagrange equations do not change can be easily seen, without “invoking” the equivalence of Lagrangians differing by the time derivative of a function. The equality $\hat{\mathcal{L}}=\mathcal{L}+1-\dot{\theta}$ implies that all the partial derivatives of $\hat{\mathcal{L}}$ and $\mathcal{L}$ are equal, except for the momentum $\hat{p}_{\theta}\equiv\partial\hat{\mathcal{L}}/\partial\dot{\theta}=p_{\theta}-1$. However, since it is the derivative of the momenta that enter into the Euler-Lagrange equations, they obviously remain unchanged..

In this case, the boundary conditions are simpler than those addressed in section D.1, because $(\kappa,y,\theta)$ have prescribed values at the initial and final times, although the latter is not fixed; it is the quantity that we want to minimise. Specifically, Eq. (48) and Eq. (49) remain valid but Eq. (55) must be substituted with

[TABLE]

Therefore, we deal with a “standard” variational problem, for which $\delta\kappa$, $\delta y$ and $\delta\theta$ vanish at the boundaries, similarly to the situation found in Classical Mechanics. Notwithstanding, once more we have that $\kappa$ may have finite jump discontinuities at the boundaries: recall that its corresponding canonical momentum verifies $\hat{p}_{\kappa}\equiv 0$.

Since the Euler-Lagrange equations are unchanged, Eq. (59), Eq. (60), Eq. (63), Eq. (64) and Eq. (65) still hold. In principle, we should have to reobtain the constants $(c_{2},c_{3},c_{4})$ with the new boundary conditions. However, it is readily realised that the substitution of Eq. (55) with Eq. (79) leaves their expressions unchanged, because Eq. (55) was not employed in their derivation for the optimal temperature problem. The only difference is that $\theta_{\text{f}}$ is now fixed and $t_{\text{f}}$ is the variable being minimised, instead of the other way round. In light of the previous discussion, it appears that the same function relates the optimal values $\theta_{\text{f}}$ and $t_{\text{f}}$ for both physical situations, as argued in the main text on physical grounds.

Appendix E Fluctuations of the energy increment, work and

heat

In the quasi-static limit, the PDFs for the increment of potential energy $\Delta u$, work $W$ and heat $Q$ have been obtained for the harmonic potential Martínez et al. (2015). The calculations rely on the values of $x(t)$ and $x(t^{\prime})$ being uncorrelated for all times, which is strictly true only for an infinite connecting time. Here, we consider how these results are changed by a finite-time but slow driving, i.e. the situation when the dimensionless connecting time $t_{\text{f}}\gg 1$ and both the stiffness and the temperature are slowly varied, i.e. their time derivatives in the “fast” scale $t$ is of the order of $t_{\text{f}}^{-1}$.

E.1 Time correlations

For slow drivings, it is convenient to go to the time scale $\tau=t/t_{\text{f}}$, over which the evolution equation of the variance of the position is given by

[TABLE]

Therefore, the lowest order approximation for the variance is

[TABLE]

This expression is uniformly valid in time: it verifies both boundary conditions in Eq. (5) of the main text. Therefore, the one-time PDF for the position at time $t$ is Gaussian with zero mean and this variance, i.e.

[TABLE]

The situation is more subtle for two-times objects. Indeed, let us consider the same equation but for a given initial condition, i.e. when we are interested in the transition probability $p(x,\tau|x_{\text{i}},0)$, such that $p(x,\tau|x_{\text{i}},0)|_{\tau=0}=\delta(x-x_{\text{i}})$ and $x(t=0)=x_{\text{i}}$. Over the time scale $\tau$, Eq. (81) again holds but it does not verify the initial condition $\sigma_{x,\text{i}}=0$. Note that over the time scale $\tau$,

[TABLE]

Therefore, the “external” approximation Eq. (81)–using the terminology in Ref. Bender and Orszag (1999)–holds for $\tau=O(1)$, such that $\varphi=O(t_{\text{f}})\gg 1$, but not for short times, such that $\varphi=O(1)$, where a boundary layer emerges. In the boundary layer, we obtain the “internal” solution

[TABLE]

This can be justified either by a dominant balance argument in the differential equation (32) or by showing that the last term on the rhs of Eq. (35) is subdominant against the first one. Consistently with the notation employed in the main text, $\sigma_{x,\text{i}}^{2}$ stands for the initial value of the variance $\sigma_{x.\text{eq}}^{2}(t=0)$. A uniform solution in time is obtained by adding Eq. (81) and (84) and subtracting their common limit for $\varphi(t,0)\gg 1$ and $\tau\ll 1$, which is $\sigma_{x,\text{eq}}^{2}(t)$. Therefore, Eq. (84) gives the uniform solution and $p(x,t|x_{\text{i}},0)$ is the Gaussian distribution with that variance and mean $x_{\text{i}}\exp[-\varphi(t,0)]$, as predicted by Eq. (34),

[TABLE]

This equation can be readily generalised to a given initial condition $x^{\prime}$ at time any $t^{\prime}$ as

[TABLE]

The lowest order approximation given by Eqs. (82) and (86) is consistent, these PDFs obey the Chapman-Kolmogorov conditions $\int dx^{\prime}p(x,t|x^{\prime},t^{\prime})p(x^{\prime},t^{\prime}|x^{\prime\prime},t^{\prime\prime})=p(x,t|x^{\prime\prime},t^{\prime\prime})$ and $\int dx^{\prime}p(x,t|x^{\prime},t^{\prime})p(x^{\prime},t^{\prime})=p(x,t)$.

For calculating the probability distributions of energy increment, work and heat, we will need to calculate correlation functions of the form

[TABLE]

In our lowest order approximation, we have that $\overline{x(t)}=0$ and

[TABLE]

for all times. Then, the two-time correlations reduce to

[TABLE]

Therefore, we first evaluate the conditioned average

[TABLE]

which inserted into Eq. (89) leads to

[TABLE]

Correlations are relevant over the “fast” time scale $t$, as long as $\varphi(t,t^{\prime})=O(1)$, but become exponentially small over the “slow” time scale $\tau$ because, consistently with Eq. (83),

[TABLE]

and $\varphi(\tau,\tau^{\prime})=O(t_{\text{f}})\gg 1$. In the quasi-static limit $t_{\text{f}}\to\infty$, $C(t,t^{\prime})\to 0$ because $p(x,t|x^{\prime},t^{\prime})\to p(x,t)$ and time correlations are “instantaneously” killed.

E.2 Fluctuations of the energy

increment

Since the increment of kinetic energy is for a given protocol fixed and equal to $\Delta\theta/2$ (in dimensionless variables), we focus on the fluctuations of the increment of the potential energy

[TABLE]

The average value is straightforward, $\overline{\Delta u}=\Delta\theta/2$. Fluctuations are also easy to calculate, since

[TABLE]

and the variance is readily written in terms of the correlation function as

[TABLE]

Employing Eq. (91), we get

[TABLE]

For slow driving, the last term on the rhs is exponentially small in the connecting time $t_{\text{f}}$, since

[TABLE]

Neglecting this exponentially decreasing term (EDT), we have that

[TABLE]

In conclusion, $\sigma_{\Delta u}$ coincides with that of the quasi-static limit, except for EDT. In fact, the whole distribution $\mathcal{P}(\Delta u)$

[TABLE]

coincides with that for the quasi-static limit except for EDT, because

[TABLE]

Then,

[TABLE]

This integration has been carried out in Ref. Martínez et al. (2015),

[TABLE]

where $K_{0}$ is the zero-th order modified Bessel function of the second kind. The above results are valid for slow–not necessarily adiabatic–driving between two equilibrium states. In fact, adiabaticity does not play any role here.

E.3 Fluctuations of the work

In dimensionless variables, work is given by

[TABLE]

so that

[TABLE]

Therefore, the work variance is

[TABLE]

where we have used that $C(t,t^{\prime})$ and thus the integrand is symmetric under the exchange $t\leftrightarrow t^{\prime}$.

Now, we insert Eq. (91) into (E.3) and go to the slow $\tau$ variable to write

[TABLE]

For $t_{\text{f}}\gg 1$, only a narrow region of width $t_{\text{f}}^{-1}$ contributes to the second integral. Thus, to the lowest order we can (i) substitute $\tau^{\prime}$ with $\tau$ in both $\dot{\kappa}(\tau^{\prime})$ and $\sigma_{x,\text{eq}}(\tau^{\prime})$ (ii) approximate $\int_{\tau^{\prime}}^{\tau}d\tau^{\prime\prime}\kappa(\tau^{\prime\prime})\sim\kappa(\tau)\Delta$, with $\Delta=\tau-\tau^{\prime}$, and (iii) extend the integral over $\Delta$ to the interval $[0,+\infty)$. Then, we have that

[TABLE]

and finally the variance for the work reads

[TABLE]

Corrections of the order of $t_{\text{f}}^{-2}$ have been neglected.

In the quasi-static limit $t_{\text{f}}\to\infty$, the variance vanishes and work becomes delta-distributed around its mean. For long $t_{\text{f}}$ but not infinite, work becomes Gaussian-distributed with the variance given by Eq. (107) to the lowest order. Adiabaticity only enters the picture by restricting the stiffness and temperature profiles in Eq. (107). In addition, for adiabatic processes $\overline{Q}=0$ and the mean work $\overline{W}=\Delta E=\Delta\theta$.

E.4 Fluctuations of the heat

We now turn our attention to the fluctuations of the heat. Since the kinetic contribution is fixed in the overdamped description, this is equivalent to consider the fluctuations of its configurational contribution $Q_{x}=\Delta u-W$. Therefore, the deviation from the mean value is given by

[TABLE]

The $n$-th central moment is defined by

[TABLE]

To calculate such moments, we will need to evaluate $n$-times correlation functions. For the variance, this means that it suffices to know the two-time correlations introduced in Eq. (87), as has already been the case for the work fluctuations. More specifically,

[TABLE]

and we focus in the following on the last term, i.e on the calculation of the energy-work cross-correlation.

From the definitions of $\Delta u$ and $W$, it is straightforward that

[TABLE]

and making use of Eq. (91) we have that

[TABLE]

Again, going to the slow variable $\tau$,

[TABLE]

Similarly to the calculation for $\sigma_{W}^{2}$, both terms contribute in a narrow $\tau$ interval, namely that close to $\tau=1$ ($\tau=0$) for the first (second) one. Thus we obtain

[TABLE]

neglecting once more $O(t_{\text{f}}^{-2})$ corrections. Finally, we get for the variance of the heat

[TABLE]

Integration by parts simplifies this into

[TABLE]

This expression is also valid for slow driving, regardless of being adiabatic or not. Similarly to the case of the work distribution, adiabaticity only enters the picture by restricting the stiffness and temperature profiles that can be substituted into Eq. (116).

Therefore, in the limit of slow driving we find a small change in the variance of the heat–recall that it is non-zero and equal to $\sigma_{\Delta u}^{2}$ for the quasi-static case. For slow but not quasi-static driving, work is no longer delta-distributed around the mean and then the fluctuations of heat and energy increment are not equivalent: the corrections are of the order of $t_{\text{f}}^{-1}$ for the former but exponentially small for the latter. A relevant question thus arises: whether or not the heat distribution conserves its shape, i.e if all the change of the distribution can be encoded in the change of the variance. Although the calculation of the whole heat distribution seems to be a challenging mathematical problem–even for the harmonic case, we show that the situation is more complex in the following, by obtaining the third central moment of the distribution–recall that the distribution of the heat is asymmetric around its mean.

Consistently with the comments above, we consider the third central moment $\mu_{Q,3}$. Making use of Eq. (108), we have that

[TABLE]

In order to obtain $\mu_{Q,3}$, we need to evaluate three-time correlations of the kind

[TABLE]

In the same approximation employed throughout these appendices, i.e. that given by Eqs. (82) and (86), this correlation has the asymptotic behaviour

[TABLE]

In the following, we repeatedly use this expression for $A(t_{1},t_{2},t_{3})$ to calculate all contributions to $\mu_{Q,3}$.

We start by considering

[TABLE]

In this case, there is no integration and any term that mixes $(x_{\text{f}}^{2}-\sigma_{x,\text{f}}^{2})$ and $(x_{\text{i}}^{2}-\sigma_{x,\text{i}}^{2})$ contains an EDT of the form $\exp[-2\varphi(t_{\text{f}},0)]=\exp[-2\hat{\kappa}t_{\text{f}}]$. Thus, we recover the quasi-static situation in which $(x_{\text{f}}^{2}-\sigma_{x,\text{f}}^{2})$ and $(x_{\text{i}}^{2}-\sigma_{x,\text{i}}^{2})$ are uncorrelated, except for EDT, i.e.

[TABLE]

If $\theta_{\text{f}}\neq\theta_{\text{i}}$, this contribution is different from zero, in accordance with the heat fluctuations being asymmetric around its mean in the quasi-static limit.

Now, we turn our attention to

[TABLE]

i.e.

[TABLE]

The term containing the correlation $A(t_{\text{f}},t,0)$ is exponentially decreasing, $A(t_{\text{f}},t,0)\sim 8\sigma_{x,\text{eq}}^{2}(t)\sigma_{x,\text{i}}^{4}\exp[-2\varphi(t_{\text{f}},0)]$. Then, we only need to consider the other two terms. We start with the analysis of the first one, specifically

[TABLE]

In the limit $t_{\text{f}}\gg 1$, we can once more use Watson’s lemma to estimate the integral to the lowest order, with the result

[TABLE]

An analogous calculation yields

[TABLE]

Then, up to order $t_{\text{f}}^{-1}$ we have that

[TABLE]

Let us analyse the following contribution

[TABLE]

Note that $t\geq t^{\prime}$ in the last two lines, making use of the symmetry of the integrand under the exchange $t\leftrightarrow t^{\prime}$.

Again, we have to calculate three-times correlation functions. We start by analysing the term stemming from

[TABLE]

More specifically, we have to find the lowest order contribution to the integral

[TABLE]

Once more, the asymptotic estimate of this integral to the lowest order can be calculated by applying Watson’s lemma. First, we integrate over $\tau^{\prime}$ at given $\tau$, and this yields

[TABLE]

because the exponential reaches its maximum value for $\tau^{\prime}=\tau$, i.e. when $\tau^{\prime}$ is closest to unity. The integral over $\tau$ is now dominated by the contribution of a narrow interval close to $\tau=1$, i.e by applying again Watson’s lemma we get

[TABLE]

Therefore, this contribution is of the order $t_{\text{f}}^{-2}$ and thus subdominant to that in $\mu_{Q,2}^{(2)}$, which was of the order of $t_{\text{f}}^{-1}$.

Now we look into the contribution coming from $A(t,t^{\prime},0)$. In this case, it is better to introduce the condition $t\geq t^{\prime}$ by integrating $t^{\prime}$ from [math] to $t_{\text{f}}$ and restricting $t$ to the interval $[t^{\prime},t_{\text{f}}]$. By doing so, the calculation follows completely similar lines as those above. The correlation $A(t,t^{\prime},0)$ has a term $\exp[-2\varphi(t,0)]=\exp[-2t_{\text{f}}\int_{0}^{\tau}d\tau^{\prime\prime}\kappa(\tau^{\prime\prime})]$: the first integration over $t$ gives a $t_{\text{f}}^{-1}$ factor and makes $t=t^{\prime}$ in $\varphi$, the second integration over $t^{\prime}$ gives a second $t_{\text{f}}^{-}1$ factor and makes $t^{\prime}=0$ in $\varphi$. Then, we have that this contribution is also proportional to $t_{\text{f}}^{-2}$, specifically

[TABLE]

With a similar line of reasoning, it is possible to show that the last contribution to $\mu_{Q,3}$,

[TABLE]

is also subdominant–i.e. it does not contain $t_{\text{f}}^{-1}$ terms. This can also be qualitatively understood by recalling that work fluctuations are Gaussian in the slow driving limit $t_{\text{f}}\gg 1$: $\overline{(W-\overline{W})^{3}}$ should vanish to the lowest order.

Finally, we get that

[TABLE]

and the correction to the third central moment is a pure boundary term. Interestingly, this means that the heat distribution is not simply being compressed/decompressed around its mean. The heat PDF $\mathcal{P}(Q)$ is such that

[TABLE]

even in the slow driving limit. Had we $\mathcal{P}(Q)=f\left(\frac{Q-\overline{Q}}{\sigma_{Q}}\right)$, the third central moment would be proportional to $\sigma_{Q}^{3}$. In other words, $\mu_{Q,3}/\sigma_{Q}^{3}$ would be constant, independent of $t_{\text{f}}$, to the considered order. It is quite clear that $\mu_{Q,3}/\sigma_{Q}^{3}$ does depend on $t_{\text{f}}$, i.e. the $t_{\text{f}}^{-1}$ corrections coming from $\mu_{Q,3}$ and $\sigma_{Q}^{3}$ do not cancel out. Thus, the shape of the heat distribution is not preserved when we change the connecting time.

Acknowledgements.

This work has been financially supported by UNIPD STARS Stg (CdA Rep. 40, 23.02.2018) BioReACT grant (C.A.P.), the Agence Nationale de la Recherche research funding Grant No. ANR-18-CE30-0013 (D.G.-O., E.T.), and by the Spanish Ministerio de Ciencia, Innovación y Universidades through Grant (partially financed by the ERDF) No. PGC2018-093998-B-I00 (A.P.).