Work, entropy production, and thermodynamics of information under protocol constraints

We consider a physical system with state space $X$, which can be either discrete or continuous ($X=\mathbb{R}^{n}$). The term “probability distribution” will refer to a probability mass function over $X$ in the discrete case and to a probability density function over $X$ in the continuous case. We interchangeably use notation like $p(x)$ and $p_{x}$ (as will be clear from context) to indicate the probability of state $x$. We use $\mathcal{P}$ to refer to the set of all probability distributions over $X$.

The system evolves in a stochastic manner during a driving protocol over time $t\in[0,1]$. We will write $p(t)$ to indicate the distribution at time $t$ corresponding to some initial distribution $p(0)=p$, and $p(1)=p^{\prime}$ to indicate the distribution at the end of the protocol. For a discrete-state system, the distribution at time $t$ evolves according to the time-dependent master equation,

$\displaystyle{\textstyle\partial_{t}}p_{x}(t)=\sum_{x^{\prime}}\left[L_{xx^{\prime}}(t)p_{x^{\prime}}(t)-L_{x^{\prime}x}(t)p_{x}(t)\right],$

(10)

where $L_{x^{\prime}x}(t)$ is the transition rate from state $x$ to state $x^{\prime}$. We assume that the system is coupled to a heat bath at inverse temperature $\beta$, and so each $L(t)$ obeys local detailed balance (see Section IX for a generalization of this assumption). Formally, this means that $\pi^{L(t)}_{x^{\prime}}L_{xx^{\prime}}(t)=\pi^{L(t)}_{x}L_{x^{\prime}x}(t)$ for all $x$,$x^{\prime}$, and $t$, where $\pi^{L(t)}$ is the stationary distribution of rate matrix $L(t)$, which we assume is unique (though this latter assumption can be relaxed ³³3The assumption of unique stationary distributions can be relaxed as long as the operator $\phi$ (as discussed in Section III) satisfies the following weak technical condition: for all $p\in\mathcal{P}$ and each stationary distribution $\pi$ of each $L\in\Lambda$, $D(p\|\phi(\pi))<\infty$ whenever $D(p\|\pi)<\infty$. Note that $\phi(\pi)$ is also a stationary distribution of $L$ by Lemma 1 in Appendix A, so this condition is automatically satisfied when the generators have unique stationary distributions (since in that case $\pi=\phi(\pi)$). Note also that if some $L\in\Lambda$ have multiple stationary distributions $\pi$, the corresponding EP rate in Eq. 11 can be equivalently defined using any $\pi$ such that $D(p\|\pi)<\infty$.).

The rate of entropy production (EP rate) incurred at time $t$ can be written as (Eq. 33 in esposito2010three )

$\displaystyle\dot{\Sigma}(p(t),L(t))=-\sum_{x}{\textstyle\partial_{t}}p_{x}(t)\ln\frac{p_{x}(t)}{\pi^{L(t)}_{x}}\geq 0,$

(11)

where ${\textstyle\partial_{t}}p_{x}(t)$ is defined in Eq. 10. Note that the right side of Eq. 11 is sometimes called the “nonadiabatic EP rate” in stochastic thermodynamics, and it is equal to the overall EP rate for a system coupled to a single bath and obeying detailed balance esposito2010three . The total EP incurred by a time-extended protocol over $t\in[0,1]$ that carries out the transformation $p\!\shortrightarrow\!p^{\prime}$ is given by the integral of the EP rate,

$\displaystyle\Sigma(p\!\shortrightarrow\!p^{\prime})=\int_{0}^{1}\dot{\Sigma}(p(t),L(t))\,dt.$

(12)

The work extracted during a protocol can be calculated by using Eqs. 12 and 2, once the initial and final nonequilibrium free energies, $F_{E}(p)$ and $F_{E^{\prime}}(p^{\prime})$, are specified. To define these free energies, we assume that there is some fixed pair of energy functions, $E$ and $E^{\prime}$, which specify the Boltzmann equilibrium distributions of $L(0)$ and $L(1)$ respectively.

For a continuous-state system evolving under a continuous master equation (van1992stochastic, ; risken1996fokker, ), the sums in Eqs. 10 and 11 should be replaced by integrals (see Eq. 31 in van2010three ). A prototypical example of a continuous master equation, which we will use below, is a Fokker-Planck equation (ermak1978brownian, ; van1992stochastic, ),

$\displaystyle{\textstyle\partial_{t}}p(x,t)=-\nabla\cdot(\mathsf{A}(x,t)p(x,t)-\mathsf{D}(x,t)\nabla p(x,t)),$

(13)

where $\mathsf{A}$ and $\mathsf{D}$ are drift and diffusion terms.

We will often write dynamical equations like Eqs. 10 and 13 using the notation ${\textstyle\partial_{t}}p(t)=L(t)p(t)$, where $L(t)$ is a bounded linear operator that is called the (infinitesimal) generator of the dynamics at time $t$. Note that for a continuous-state system in phase space, it may be that the system is isolated from the bath for some $t\in[0,1]$, in which case ${\textstyle\partial_{t}}p(t)=L(t)p(t)$ should be understood in terms of the Liouville equation. (For example, if a system is first isolated and evolves in a Hamiltonian manner, and is then brought in contact with a bath at inverse temperature $\beta$ and allowed to equilibrate).

We begin by presenting our general mathematical framework. The application of this framework to concrete situations is described in latter sections.

A driving protocol $\{L(t):t\in[0,1]\}$ is said to be constrained if there is some restricted set of generators $\Lambda$ such that $L(t)\in\Lambda$ at all $t$. For a given set of allowed generators $\Lambda$, we consider an associated operator $\phi:\mathcal{P}\to\mathcal{P}$ which satisfies two conditions. The first condition states that

$\displaystyle D(p\|q)=D(p\|\phi(p))+D(\phi(p)\|q)$

(14)

for all $p\in\mathcal{P}$ and $q\in\mathrm{img}\;\phi$ with $D(p\|q)<\infty$ (where $\mathrm{img}\;\phi=\{\phi(p):p\in\mathcal{P}\}$ is the image of the operator $\phi$). Eq. 14 is sometimes called the Pythagorean identity of KL divergence in information geometry amari2016information . Any $\phi$ that obeys Eq. 14 can be written in terms of the following projection ⁴⁴4This is because $D(p\|q)\geq D(p\|\phi(p))$ for any $q\in\mathrm{img}\;\phi$, which follows from Eq. 14 and the non-negativity of KL divergence.

$\displaystyle\phi(p)=\operatorname*{\arg\,\min}_{q\in\mathrm{img}\;\phi}D(p\|q),$

(15)

which shows that $D(p\|\phi(p))$ is the minimal information-theoretic distance from $p$ to the set of distributions $\mathrm{img}\;\phi$.

The second condition is that $\phi$ obeys the following commutativity relation for all $L\in\Lambda$:

$\displaystyle e^{\tau L}\phi(p)=\phi(e^{\tau L}p)\quad\forall\tau\geq 0,p\in\mathcal{P}.$

(16)

In other words, given any initial distribution $p$, the same final distribution is reached regardless of whether $p$ first relaxes under $L$ for time $\tau$ and then undergoes $\phi$, or instead first undergoes $\phi$ and then relaxes under $L$ for time $\tau$.

Note that the Pythagorean identity in Eq. 14 concerns only the operator $\phi$, while the commutativity relation in Eq. 16 concerns the relationship between $\phi$ and the generators in $\Lambda$ (and therefore all of the generators $L(t)$ in the driving protocol, since $L(t)\in\Lambda$ at all $t$ by assumption). Beyond these two conditions, the operator $\phi$ can be arbitrary, and may be linear or nonlinear. In the following sections of this paper, will show how to choose $\phi$ for various types of constrained protocols.

Importantly, any $\phi$ that satisfies the two conditions above maps any distribution $p$ to a corresponding “accessible” distribution $\phi(p)$, which controls the amount of work that can be extracted from $p$ by a constrained driving protocol. To prove this, we first show that for any $L\in\Lambda$ that obeys Eq. 16, the equilibrium distribution $\pi^{L}$ satisfies (Lemma 1 in Appendix A)

$\displaystyle\pi^{L}\in\mathrm{img}\;\phi.$

(17)

We also derive the following mathematical result, will be central to much of what follows: if $\phi$ obeys Eq. 14 and Eq. 16 for some generator $L$, then the EP rate incurred by any distribution $p$ under $L$ can be written as the sum of two non-negative terms: the EP rate incurred by $\phi(p)$ under $L$, and the instantaneous contraction of the KL divergence between $p$ and $\phi(p)$.

We sketch the proof of this theorem in terms of a discrete-time relaxation over interval $\tau$, as shown in Fig. 2 (see Appendix A for details). Consider some distribution $p$ that relaxes for time $\tau$ under the generator $L$, thereby reaching the distribution $e^{\tau L}p$ (solid gray line). The EP incurred by this relaxation is given by the contraction of KL divergence to the equilibrium distribution $\pi$, $\Sigma(p\!\shortrightarrow\!e^{\tau L}p)=D(p\|\pi)-D(e^{\tau L}p\|\pi)$ (contraction of purple lines) esposito2010three ; van2010three . Given Eq. 17, we can apply the Pythagorean identity, Eq. 14, to both $D(p\|\pi)$ and $D(e^{\tau L}p\|\pi)$, which lets us rewrite $\Sigma(p\!\shortrightarrow\!e^{\tau L}p)$ as the sum of two terms: $D(p\|\phi(p))-D(e^{\tau L}p\|\phi(e^{\tau L}p)$ (green lines) and $D(\phi(p)\|\pi)-D(\phi(e^{\tau L}p)\|\pi)$ (red lines). Applying the commutativity relation, Eq. 16, shows that the first term is non-negative by the data-processing inequality and that the second term is equal to $\Sigma(\phi(p)\!\shortrightarrow\!e^{\tau L}\phi(p))$, the EP incurred by letting $\phi(p)$ relax freely under $L$. The continuous-time statement found in Theorem 1 follows by taking the appropriate $\tau\to 0$ limit, while noting that the EP rate, Eq. 11, can be rewritten in terms of the limit $\lim_{\tau\to 0}\frac{1}{\tau}[D(p\|\pi)-D(e^{\tau L}p\|\pi)]$.

Now suppose that Eq. 16 holds, so that the assumptions of Theorem 1 are satisfied during the entire protocol. In that case, as we show in Lemma 3 in Appendix A, any constrained protocol that carries out the transformation $p\!\shortrightarrow\!p^{\prime}$ must also transform the initial distribution $\phi(p)$ to the final distribution $\phi(p^{\prime})$. We can then, in essence, integrate Theorem 1 over time and derive the following result about total EP.

We use Theorem 2 to derive several useful bounds on EP and work. First, since $\Sigma(\phi(p)\!\shortrightarrow\!\phi(p^{\prime}))\geq 0$ by the non-negativity of EP, the contraction of KL divergence between $p$ and $\phi(p)$ bounds the EP incurred by a constrained driving protocol that carries out the transformation $p\!\shortrightarrow\!p^{\prime}$,

$\displaystyle\Sigma(p\!\shortrightarrow\!p^{\prime})\geq D(p\|\phi(p))-D(p^{\prime}\|\phi(p^{\prime}))\geq 0,$

(18)

which appeared as Eq. 7 in the introduction. Furthermore, $D(p\|\phi(p))-D(p^{\prime}\|\phi(p^{\prime}))\geq 0$ immediately implies that

$\displaystyle\Sigma(p\!\shortrightarrow\!p^{\prime})\geq\Sigma(\phi(p)\!\shortrightarrow\!\phi(p^{\prime})).$

(19)

We can also derive the decomposition of free energy and the bound on extractable work, which appeared as Eqs. 5 and 6 in the introduction. Consider some transformation $p\!\shortrightarrow\!p^{\prime}$, and write the initial nonequilibrium free energy as

$$F_{E}(p)=F_{E}(\pi)+D(p\|\pi)/\beta,$$

(20)

where $\pi\propto e^{-\beta E}$ is the Boltzmann distribution for the initial energy function $E$, and $F_{E}(\pi)$ is the equilibrium free energy (esposito2011second, ). Using Eq. 17 and the Pythagorean identity, Eq. 14, we decompose the nonequilibrium free energy into a sum of the accessible free energy and the inaccessible free energy,

	$\displaystyle F_{E}(p)$	$\displaystyle=F_{E}(\pi)+[D(p\|\phi(p))+D(\phi(p)\|\pi)]/\beta$
		$\displaystyle=F_{E}(\phi(p))+D(p\|\phi(p))/\beta.$		(21)

Using a similar derivation, we can write the nonequilibrium free energy at the end of the protocol as

$\displaystyle F_{E^{\prime}}(p^{\prime})=F_{E^{\prime}}(\phi(p^{\prime}))+D(p^{\prime}\|\phi(p^{\prime}))/\beta.$

(22)

Subtracting Eq. 22 from Eq. 21 shows that the drop in the nonequilibrium free energy during $p\!\shortrightarrow\!p^{\prime}$ is given by

$$F_{E}(p)-F_{E^{\prime}}(p^{\prime})=F_{E}(\phi(p))-F_{E^{\prime}}(\phi(p^{\prime}))+\\ \left[D(p\|\phi(p))-D(p^{\prime}\|\phi(p^{\prime}))\right]/\beta.$$

(23)

Combining this result with Theorem 2 and Eq. 2, and then rearranging, shows that the work involved in carrying out $p\!\shortrightarrow\!p^{\prime}$ is equal to the work involved in carrying out the accessible transformation $\phi(p)\!\shortrightarrow\!\phi(p^{\prime})$:

$\displaystyle W(p\!\shortrightarrow\!p^{\prime})=W(\phi(p)\!\shortrightarrow\!\phi(p^{\prime})).$

(24)

Finally, by combining with Eq. 1, we arrive at an upper bound on work that can be extracted by a constrained protocol:

$\displaystyle W(p\!\shortrightarrow\!p^{\prime})\leq F_{E}(\phi(p))-F_{E^{\prime}}(\phi(p^{\prime})),$

(25)

which is tighter than the bound given by the second law, Eq. 1.

The bounds in Eqs. 18 and 25, as well as the decomposition of free energy in Eq. 21, are the main theoretical results arising from our general framework. Fig. 3 provides a schematic way of understanding these results. Theorem 2 states that, for a constrained protocol that carries out the map $p\!\shortrightarrow\!p^{\prime}$, the EP incurred during the system’s actual trajectory (solid gray line) is given by the EP that would incurred by a “projected trajectory” that carries out the transformation $\phi(p)\!\shortrightarrow\!\phi(p^{\prime})$ (dashed gray line), plus the drop in the KL divergence from the system’s distribution to the set $\mathrm{img}\;\phi$ over the course of the protocol (contraction of green lines). Since the EP of the projected trajectory must be non-negative, the drop in the distance from the system’s distribution to $\mathrm{img}\;\phi$ serves as a lower bound on EP, as in Eq. 18. In addition, Theorem 2 states that this decrease in the KL divergence must be positive, meaning that the system’s distribution must get closer to $\mathrm{img}\;\phi$ over the course of the protocol.

Following Fig. 3, it can be helpful to think of the trajectory $p\!\shortrightarrow\!p^{\prime}$ as composed of three segment: (1) from $p$ down to $\phi(p)$, (2) from $\phi(p)$ to $\phi(p^{\prime})$ while staying within $\mathrm{img}\;\phi$, and (3) from $\phi(p^{\prime})$ up to $p^{\prime}$ (note that this decomposition is useful for accounting purposes, but does not generally reflect the actual trajectory the system takes in going from $p$ to $p^{\prime}$). The first and third segments contribute (positively and negatively, respectively) only to EP, while the projected second segment $\phi(p)\!\shortrightarrow\!\phi(p^{\prime})$ contributes both to EP and to work. Thus, the work involved in $p\!\shortrightarrow\!p^{\prime}$ is determined entirely by the work involved in the second segment, as stated in Eq. 24.

Note also the formal similarity between our decomposition of the drop in free energy, Eq. 23, and the decompositions of EP in Theorem 2. Indeed, like Theorem 2, the result Eq. 23 can be illustrated with Fig. 3: during the transformation $p\!\shortrightarrow\!p^{\prime}$ (solid gray line), the drop in free energy is given by the drop in free energy incurred by the transformation $\phi(p)\!\shortrightarrow\!\phi(p^{\prime})$ (dotted gray line), plus the contraction of the KL divergence from the system’s distribution to the set $\mathrm{img}\;\phi$ (green lines).

In general, our bounds on EP and work will not always be achievable. Suppose, however, that the final distribution $p^{\prime}$ is in equilibrium, so $p^{\prime}=\phi(p^{\prime})$ by Eq. 17. Eq. 18 then gives

$\displaystyle\Sigma(p\!\shortrightarrow\!p^{\prime})\geq D(p\|\phi(p)).$

(26)

This bound is achievable if the generators in $\Lambda$ have a continuous curve of equilibrium distributions from $\phi(p)$ to $p^{\prime}=\phi(p^{\prime})$. Imagine a protocol in which the initial distribution $p$ first relaxes to the equilibrium distribution $\phi(p)$, and then undergoes quasistatic driving from $\phi(p)$ to $\phi(p^{\prime})$ while remaining in equilibrium throughout (in terms of Fig. 3, the system first relaxes along the green arrow connecting $p$ to $\phi(p)$, then follows the dashed line to $\phi(p^{\prime})$ quasistatically). The relaxation step incurs $D(p\|\phi(p))$ of EP, while the quasistatic step incurs a vanishing amount of EP, so the bound in Eq. 26 will be achieved.

The framework introduced in the previous section has implications for the thermodynamics of information under constraints. Consider the type of feedback control setup described in the introduction: first an observation apparatus $M$ measures some system observable, then the system undergoes a driving protocol that depends on the measurement outcome $m$. Let $L^{(m)}(t)$ indicate the driving protocol conditioned on $m$, and $p_{X|m}$ and $p_{X^{\prime}|m}^{\prime}$ indicate the distributions over system states at the beginning and end of the corresponding driving protocol. As standard in the literature parrondo2015thermodynamics , for simplicity we assume that all protocols start and end with the same energy functions, $E$ and $E^{\prime}$, and that during the protocols, the measurement apparatus $M$ and the system $X$ are energetically decoupled and that $M$ does not change state.

Given the above assumptions, it is straightforward to show that the EP incurred by the joint “supersystem” $X\times M$ obeys

$\displaystyle\Sigma_{XM}=\sum_{m}p(m)\Sigma_{m},$

(31)

where $\Sigma_{m}$ is the EP incurred by protocol $L^{(m)}(t)$ in carrying out the transformation $p_{X|m}\!\shortrightarrow\!p_{X^{\prime}|m}^{\prime}$. Similarly, by taking expectations of Eq. 2 and rearranging (see derivation of Eq. 4), the average extracted work under feedback control can be written as

$\displaystyle\langle W\rangle=\Delta F+[I(X;M)\!-\!I(X^{\prime};M)]-\!\sum_{m}p(m)\Sigma_{m},$

(32)

where for notational convenience we’ve used $\Delta F=F_{E}(p)-F_{E^{\prime}}(p^{\prime})$ to indicate the drop of marginal free energy. Thus, any lower bounds on $\Sigma_{m}$ (the EP values incurred by the individual protocols $L^{(m)}(t)$) can be translated into bounds on the overall EP and average extractable work for a feedback control setup.

For example, suppose that there is some single set of constraints that applies to all of the driving protocols, in that there is some set of generators $\Lambda$ such that $L^{(m)}(t)\in\Lambda$ for all $t$ and $m$, as well as an operator $\phi$ that obeys the Pythagorean identity, Eq. 14, and the commutativity relation, Eq. 16, for all $L\in\Lambda$. In that case, the framework described in Section III leads to bounds on each $\Sigma_{m}$ term. In particular, using Eqs. 18 and 31 gives the bound

$$\Sigma_{XM}\geq\\ D(p_{X|M}\|\phi(p_{X|M}))-D(p_{X^{\prime}|M}^{\prime}\|\phi(p_{X^{\prime}|M}^{\prime}))\geq 0,$$

(33)

where we’ve defined the conditional KL divergence $D(p_{X|M}\|\phi(p_{X|M}))=\sum_{m}p(m)D(p_{X|m}\|\phi(p_{X|m}))$, and similarly for $D(p_{X^{\prime}|M}^{\prime}\|\phi(p_{X^{\prime}|M}^{\prime}))$. Plugging into Eq. 32 gives the following bound on average extractable work:

$\displaystyle\left\langle W\right\rangle\leq\Delta F+[I_{\mathrm{acc}}^{\phi}(X;M)-I_{\mathrm{acc}}^{\phi}(X^{\prime};M)]/\beta,$

(34)

where $I_{\mathrm{acc}}^{\phi}(X;M)$ is given by

$\displaystyle I_{\mathrm{acc}}^{\phi}(X;M)$

$\displaystyle=I(X;M)-D(p_{X|M}\|\phi(p_{X|M})),$

(35)

and similarly for $I_{\mathrm{acc}}^{\phi}(X^{\prime};M)$.

We refer to $I_{\mathrm{acc}}^{\phi}(X;M)$ as the accessible information in measurement $M$, since any decrease in accessible information can contribute to work extraction (Eq. 34). We refer to the conditional KL divergence $D(p_{X|M}\|\phi(p_{X|M}))$ as the inaccessible information, since any decrease in inaccessible information must be dissipated as EP, and not extracted as work (Eq. 33). The inaccessible information is non-negative by properties of KL divergence, so $I_{\mathrm{acc}}^{\phi}(X;M)\leq I(X;M)$. In addition, whenever $p\in\mathrm{img}\;\phi$ (e.g., when $p$ is an equilibrium distribution, by Eq. 17), the accessible information can be rewritten in simpler form as

$\displaystyle I_{\mathrm{acc}}^{\phi}(X;M)=D(\phi(p_{X|M})\|p),$

(36)

as follows from Eq. 35 by writing $I(X;M)=D(p_{X|M}\|p)$ and applying the Pythagorean theorem, Eq. 14.

In general, measurements of different observables on the same system will give rise to different amounts of accessible and inaccessible information. At a high level, one should choose measurements that maximize the accessible information $I_{\mathrm{acc}}^{\phi}(X;M)$, or alternatively the “efficiency” quantified as bits of accessible information per bit of measured information, $I_{\mathrm{acc}}^{\phi}(X;M)/I(X;M)\leq 1$. Optimal measurements satisfy $I_{\mathrm{acc}}^{\phi}(X;M)=I(X;M)$, which happens when the conditional distributions over system states $p_{X|m}$ are invariant under the action of $\phi$ (i.e., when $\phi(p_{X|m})=p_{X|m}$ for each $m$).

Note that similar results can also be derived using other kinds of bounds on $\Sigma_{m}$ (e.g., when the individual protocols obey a combination of constraints, so that Eq. 28 holds).

We now use the general framework introduced above to derive bounds on EP under symmetry constraints.

Consider a compact group $\mathcal{G}$ that has a measurable action over $X$, such that each ${g}\in\mathcal{G}$ is a bijection $X\to X$ ⁵⁵5A compact group $\mathcal{G}$ has a measurable action over $X$ if the action $\mathcal{G}\times X\to X$ is a measurable function, where we assume $\mathcal{G}$ and $X$ are endowed with their respective Borel algebras.. For continuous $X$, we assume that each $g\in\mathcal{G}$ is a rigid transformation. For notational convenience, for each $g\in\mathcal{G}$ we define the composition operator $\Phi_{g}$, so that for any function $f:X\to\mathbb{R}$,

$\displaystyle\Phi_{g}(f)(x)=f(g(x)).$

(37)

We say that a set of generators $\Lambda$ obeys symmetry constraints (with respect to the action of group $\mathcal{G}$) if the following commutativity relation holds for all $L\in\Lambda$:

$\displaystyle\Phi_{g}L=L\Phi_{g}.\qquad\forall{g}\in\mathcal{G}.$

(38)

In other words, $\Lambda$ obey symmetry constraints when, for each $L\in\Lambda$ and ${g}\in\mathcal{G}$, it does not matter whether one first applies the generator $L$ and then the bijection ${g}$ over the state space, or first applies the bijection ${g}$ over the state space and then the generator $L$. In more concrete terms, for a (continuous or discrete) master equation $L$, Eq. 38 holds if the transition rates are invariant under the action of $\mathcal{G}$:

$\displaystyle L_{xx^{\prime}}=L_{{g}(x){g}(x^{\prime})}\qquad\forall x,x^{\prime}\in X,g\in\mathcal{G}.$

(39)

We can also derive simple sufficient conditions for potential-driven Fokker-Planck equations of the type

$\displaystyle Lp=\nabla\cdot(\nabla E_{L})p+\beta^{-1}\Delta p,$

(40)

where $E_{L}$ is the energy function of generator $L$. Then, Eq. 38 holds if all available energy functions are invariant under the action of $\mathcal{G}$,

$\displaystyle E_{L}(x)=E_{L}({g}(x))\quad\forall x\in X,g\in\mathcal{G},L\in\Lambda\,.$

(41)

(Eq. 38 is derived from Eqs. 39 and 41 in Appendix B).

We now define a linear operator $\phi_{\mathcal{G}}$ which satisfies the Pythagorean identity and the commutativity relation, Eqs. 14 and 16, for symmetry constraints. Let $\phi_{\mathcal{G}}$ map each $p\in\mathcal{P}$ to its average under the action of $\mathcal{G}$,

$\displaystyle\phi_{\mathcal{G}}(p)(x):=\int_{\mathcal{G}}p(g(x))\,d\mu(g),$

(42)

where $\mu$ is the uniform (normalized Haar) measure over $\mathcal{G}$ ⁶⁶6 Technically, the definition of the twirling operator in Eq. 42 applies only when $p$ is a finite-valued probability density function (which excludes things such as the Dirac delta “function”). A more general formulation of our results can be developed in terms of probability measures rather than probability densities (see Ch. 3 in eaton_group_1989 for a version of Eq. 42 defined in terms of probability measures).. For a finite group, the integral in Eq. 42 should be replaced by a summation. Following the terminology in quantum physics, we sometimes refer to $\phi_{\mathcal{G}}$ as a twirling operator vollbrecht2001entanglement ; vaccaro_tradeoff_2008 . Intuitively, $\phi_{\mathcal{G}}(p)$ symmetrizes $p$, removing all information in $p$ concerning the state of the system along the “coordinates” specified by the symmetry constraints.

In Appendix B, we show that $\phi_{\mathcal{G}}$ obeys the Pythagorean identity and, as long as Eq. 38 holds, the commutativity relation of Eq. 16. Thus, any protocol that carries out the transformation $p\!\shortrightarrow\!p^{\prime}$ while obeying symmetry constraints with respect to $\mathcal{G}$ permits the decomposition of EP found in Theorem 2, with $\phi=\phi_{\mathcal{G}}$, and satisfies all the bounds on work and EP that follow from that result.

In particular, using Eq. 21, we can decompose the free energy $F_{E}(p)$ of any distribution $p$ into the accessible free energy $F_{E}(\phi_{\mathcal{G}}(p))$, which is the free energy in the twirled (and therefore symmetric) version of $p$, and the inaccessible free energy $D(p\|\phi_{\mathcal{G}}(p))/\beta$. Note that $D(p\|\phi_{\mathcal{G}}(p))$ is a non-negative measure of the asymmetry in distribution $p$ with respect to the symmetry group $\mathcal{G}$, which vanishes when $p$ is invariant under $\phi_{\mathcal{G}}$. Thus, for any protocol that obeys symmetry constraints, the first inequality in Eq. 18 states that any “drop in asymmetry” must be dissipated as EP, and not turned into work. The second inequality in Eq. 18 states that the asymmetry in the system’s distribution can only decrease during the protocol. (Some of the above results for symmetry constraints have been previously uncovered in quantum thermodynamics vaccaro_tradeoff_2008 ; janzing_quantum_2006 ; see Section VIII.)