{'fraction_non_alphanumeric': 0.042272449602932194, 'fraction_numerical': 0.012176746080228059, 'mean_word_length': 4.605112987902306, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Unattainability of Carnot efficiency in thermal motors: Coarse-graining and entropy production of Feynman-Smoluchowski ratchet

We revisit and analyze the thermodynamic efficiency of the Feynman-Smoluchowski (FS) ratchet, a classical thought experiment describing an autonomous heat-work converter. Starting from the full kinetics of the FS ratchet and deriving the exact forms of the hidden dissipations resulting from coarse-graining, we restate the historical controversy over its thermodynamic efficiency. The existence of hidden entropy productions implies that the standard framework of stochastic thermodynamics applied to the coarse-grained descriptions fails in capturing the dissipative feature of the system. In response to this problem, we explore an extended framework of stochastic thermodynamics to reconstruct the hidden entropy production from the coarse-grained dynamics. The approach serves as a key example of how we can systematically address the problem of thermodynamic efficiency in a multi-variable fluctuating non-equilibrium system.

Coarse-graining the description of a fluctuating heat engine can change its apparent dissipative feature and affect the estimated efficiency. We here explicitly show that the Feynman-Smoluchowski (FS) ratchet, a classical thought experiment describing an autonomous heat-work converter, suffers from this very problem of hidden dissipation. Through singular perturbation analysis of the original FS ratchet model, we obtain the exact form of the coarse-grained dynamics, which turns out to be the Büttiker-Landauer motor, another classical model of Brownian heat engine. Explicit forms of the hidden entropy productions provide a unified understanding on how the original argument by Feynman led to the overestimated efficiency. The approach serves as a key example of how we can systematically address the problem of thermodynamic efficiency in a multi-variable fluctuating system.
Introduction.-The framework of stochastic thermodynamics has not only allowed experimental characterization of small thermodynamic systems [1][2][3], but has also established a unified scheme to address fundamental questions in thermodynamics. Identities and inequalities formulated for general stochastic dynamics have been given thermodynamical interpretations such as the second law [4][5][6], role of information feedback [7,8], bound on efficiencies of engines at finite time operations [9,10], and laws extended to nonequilibrium setups [11][12][13].
The crucial concept behind the developments in stochastic thermodynamics is the entropy production, which is typically defined through local detailed balance using transition probabilities [14]. It has been pointed out, however, that this entropy production is only an effective quantity, meaning that it does not always correspond to the physical entropy production that appears in conventional thermodynamics [15][16][17]. This is clearly illustrated in the situation where the system may be modeled at different scales of description; the entropy production calculated from the fine-grained transition probabilities can be different from that obtained using coarse-grained transition probabilities. In this case, even if the original system correctly described the thermodynamic nature of the system, the coarse-graining step can "hide" a part of the entropy production, leaving only an entropy production-like quantity in the coarse-grained system.
In the classical example of the Feynman-Smoluchowski (FS) ratchet, we notice that a similar issue to this scaledependent entropy production has been discussed. The FS ratchet is a thought experiment where the ratchet and pawl attached to two different heat baths mechanically interact (see FIG. 1a). Smoluchowski argued that there is no rotation and extracted work if the device is placed in an isothermal environment [18]. Feynman revisited this problem in his famous textbook [19] to consider the case where the temperatures are different in the two baths, and claimed that the ratchet can operate as a Carnot-efficient engine at the stalled state, where the driving force balances with the load. Feynman's argument was based on the evaluation of exchanged energy with the heat baths at forward/backward steps of rotation, which is essentially simplifying the original dynamics to only focus on the discrete relative position of the ratchet. Parrondo and Español found, however, that there is a finite heat flow through the kinetic energy even at the stalled state [20], meaning that the momentum variable overlooked by Feynman's analysis plays a significant role in the dissipation. Numerous studies based on specific simplified models [21][22][23] followed up on the existence of such heat leak, essentially proving that the FS ratchet cannot operate at Carnot efficiency.
In this Letter, we show explicitly that the classical discussions in the efficiency of the FS ratchet can be restated as a problem of hidden entropy production, i.e., the discrepancy of irreversible entropy production in the different scales of descriptions. We derive two coarse-grained description of the FS ratchet (Model-2 and Model-3), and exact relations between the dissipative features of different descriptions [Eqs. (7,8)] through singular perturbation analysis. Numerical simulations clarify the impact of hidden entropy production on the thermodynamic efficiencies (FIG. 4). Our results provide a unified view on the efficiency of the FS ratchet model, and illustrate the utility of the controlled perturbation method in addressing thermodynamic behaviors of multi-variable fluctuating engines.
Setup and Coarse-graining.-As shown in FIG. 1a, the FS ratchet consists of a vane and a gear connected by a rigid axle, and a pawl meshing with the gear. An external load couples with the axle, and applies a constant torque, f . The equations of motion for the angle of the coaxial vane and gear θ and the height of the pawl x readṡ with U(θ, x) = λ[x − φ(θ)] 2 /2 + kx 2 /2 for simplicity (see [25]). Here, p is the angular momentum conjugated to θ, and m is the corresponding moment of inertia. The effect of two heat baths are modeled as Langevin heat baths characterized by the temperatures, T h and T c , and the viscous frictional coefficients, Γ and γ. ξ and ζ are independent white Gaussian noises with zero means and unit variances. We have already eliminated the momentum variable of the pawl; this overdamped limit coarse-graining is trivial and does not involve hidden entropy production [25]. The first and second terms of mechanical potential U(θ, x) are attributed to the spring and the interaction between the tip of pawl and the surface of gear, respectively. The periodic function φ(θ) describes the shape of gear. k and λ are the spring coefficient and the strength of interaction, respectively.
We first consider the limit where the tip of pawl is tightly confined on the surface of gear: λ/γ → ∞. From singular perturbation analysis [25], we obtain the coarse-grained equation of motioṅ where Ξ represents another white Gaussian noise with zero mean and unit variance.
From Model-2, we can further take the overdamped limit m/Γ → 0, while keeping the ratio γ/Γ constant, again through singular perturbation theory. We obtain overdamped Langevin equation [1,25] The symbols · and ⊙ indicate the product in the sense of Itô and anti-Itô, respectively, which specify the interpretation of factor on the left: We show in [25] that we arrive at a different overdamped model when switching the order of the tightly confined and overdamped limits. Through these coarse-grained dynamics, we can observe how the FS ratchet operates as an autonomous heat engine. Firstly, it is clear from Eq. (2) that in the case of T h = T c the effective temperature will become constant T eff = T h = T c leading to isothermal dynamics producing no positive work W = − f θ ≤ 0. Here, W is the steady-state work rate, and · represents the steady-state ensemble average. According to Model-3, in the case where T h T c , the θ-dependent effective temperature can produce work, W > 0, if the generalized potential difference − dθU ′ eff (θ)/T eff (θ) has sufficiently large magnitude and an opposite sign to f . This phenomena is known as the Büttiker-Landauer (BL) motor [24]. Coarsegraining and the thermodynamic efficiency of the BL motor (i.e., from underdamped to overdamped) has been discussed previously [15,26,27], and the details of the efficiency will be discussed in the next section. The coarse-grained descriptions #2 and #3 are the first main results of this Letter.
Stochastic Thermodynamics of the FS ratchet.-The standard prescription of stochastic thermodynamics [2] gives the effective entropy production rates in the heat baths through the transition probabilities of the models. We introduce the heat fluxes Q h where • represents the product in the sense of Stratonovich. Then, we can write the entropy productions for each model as In contrast to Q h 1 + Q c 1 and Q 2 which are equal to the rate of change in the kinetic energy and the (effective) potential energy, Q 3 cannot be interpreted as the change in the system's energy. Thus, it is already clear that σ 3 is different to the conventional entropy production, and is only an effective quantity.
Although irreversibility in nonequilibrium steady states is characterized by the ensemble averages of the entropy production rates, we find in the FS ratchet model that there are finite differences in the averaged entropy production between different models even in the coarse-grained limit, which we call the hidden entropy production. First, from the formula for σ 1 and σ 2 derived in [25], we obtain the explicit form for the hidden entropy production rate between the Models-1 and 2: The hidden entropy production rate is positive when T h T c , which indicates that the coarse-graining by the tightly confined limit makes the dissipation seemingly decrease. Next, we find the hidden entropy production between Models-2 and 3, which generalizes the previous results obtained in the coarse-graining limit of the BL motor [15]. Following a similar method [25], we obtain in the limit of m/Γ → 0: which is again positive unless T eff (θ) = const. The condition for the positivity of hidden entropy production has been discussed in [16]. The explicit forms of the two hidden entropy production rates Eqs (7,8) are the second main results. Numerical Simulation.-We performed numerical simulations of the FS ratchet with φ(θ) = sin(2πθ) + 0.25 sin(4πθ) Here, ǫ and λ 0 are parameters introduced to control the separation of time scales. The limit of λ 0 → ∞ represents the tight confinement of the pawl to the ratchet, and ǫ → 0 realizes the overdamped limit while keeping the ratio of (λ/γ) −1 to m/Γ proportional to λ 0 . The functional forms of Numerical results of the steady-state entropy production rates are plotted in FIG. 3. In the tightly confined limit which is essentially achieved at λ 0 10, the entropy production rate at Model-1, σ 1 , converges to the sum of σ 2 and the analytically obtained first hidden entropy production rate [r.h.s. of Eq. (7)]. Next, fixing the parameter at tightly confined regime, λ 0 = 20, we see the convergence of σ 2 to the sum of σ 3 and the second hidden entropy production rate [r.h.s. of Eq. (8)] in the limit of ǫ → 0. Furthermore, the entropy production rate at Model-1 σ 1 diverges with ǫ −1 as shown in FIG. 3. This divergence could be understood by further taking the overdamped limit in Eq. (7): To understand better the impact of hidden entropy production, we calculated the thermodynamic efficiencies of the models, η 1 , η 2 and η 3 . In Model-1, we have the natural thermodynamic efficiency η 1 : For η 2 and η 3 , we adopted a generalized definition of efficiency [28] to account for the non-uniform continuous temperature. In this definition, the average rates of heat release to the heat bath at a given effective temperature, Q 2,3 (T eff ) , are introduced by averaging the heat flux over the angle θ with the same effective temperature. The averaged heat release and absorption rates, Q rel 2,3 and Q abs 2,3 are defined as the weighted averages of the positive/negative Q 2,3 (T eff ) , respectively (see [25]). From these we obtain the generalized efficiencies η 2,3 := 1 − Q rel 2,3 / Q abs 2,3 . We show the result of the efficiencies in FIG. 4. A major difference can be found between η 1,2 and η 3 at f → f stall , where f stall is the force that sets the stalled state, θ = 0. The non-vanishing η 3 indicates that there is no heat flow from the hot heat bath to cold upon zero work, which corresponds to σ 3 = 0. In contrast, η 1 and η 2 vanish at f → f stall , even in the limit of ǫ → 0. This is due to a finite hidden heat flow producing entropy. Moreover, η 1 approaches to 0 regardless of f at ǫ → 0 corresponding to the divergence σ 1 → ∞ [Eq. (9)]. These results highlight the effects of coarse-graining on the qualitative behaviors of thermodynamic efficiency; one may assume a significantly higher efficiency of an engine by neglecting the dissipative contributions of the fast variables.
Decomposition of Langevin Dynamics.-Here, we consider if it is possible to reconstruct the thermodynamic irreversibility defined at the fine-grained description from the observation at the coarse-grained scale. In a system where the time scales of variables are well-separated, it is challenging to probe the dynamics of the fast variable, meaning that the hidden entropy production and the real thermodynamic efficiency are almost impossible to measure [29]. Although there is no general workaround to the problem of inaccessible fast variables, we find in Model-2 of the FS ratchet that by using the notion of two heat baths we may split the coarse-grained dynamics and evaluate the entropy production at the fine-grained scale (Model-1). This is obtained by considering the dynamics as a mixture of two Langevin dynamics with different temperatures and frictions corresponding to the two heat baths: instead of a single set of effective temperature and friction [Eqs. (1,2)]. Here, Γ h (θ) = Γ, Γ c (θ) = γφ(θ) 2 and we consider that i is switched between h and c with a sufficiently fast switching rate Λ. This stochastic process not only reproduces the dynamics of Model-2 at the sufficiently longer time scale than Λ −1 [25], but also gives the entropy production rate, σ 4 , that satisfies [25] lim λ/γ→∞ Equation (10) is useful when we know the original temperatures of the heat baths but can only observe the dynamics at the coarse-grained scale. Since G(θ) and T eff (θ) can be measured at the coarse-grained scale, we may solve Eqs.(1, 2) using T h and T c to obtain Γ i (θ) in such situation, which allows the evaluation of σ 4 as opposed to σ 2 which misses out on a finite dissipation [Eq. (7)]. The formulation of entropy production based on the decomposition of the stochastic transition into contributions from different environments has been previously discussed [1,20,30]. The approach here is a natural extension of these strategies to the case of a continuousvariable heat engine.
Discussion and conclusion.-Based on the explicit forms of hidden entropy productions and the calculated efficiencies, we here revisit the original discussions by Feynman [19], Parrondo, and Español [20]. As pointed out by Parrondo and Español, Feynman's claim that Carnot efficiency is achievable is equivalent to assuming that the ratchet may work reversibly. In our framework, the time-reversible situation arises at the stalled state in the overdamped scale, where we have σ 3 = 0. The coarse-grained model matches with Feynman's viewpoint, as he considered the discretized transitions of the ratchets motion excluding the momentum to phenomenologically obtain the thermodynamic efficiency. Parrondo and Español claimed that Feynman's argument failed to count the additional entropy produced in a fine-grained description of the ratchet model. Indeed, if we set U(θ, x) = λ(θ − x) 2 /2 and f = 0, their model may be considered as an underdamped version of Model-1, and the additional entropy production in their model is equivalent with the hidden entropy production between Model-1 and Model-2.
In this Letter, we clarified how stochastic thermodynamics combined with explicit coarse-graining can shed new light on the classical problem of the FS ratchet. Through deriving the forms of entropy production and numerically obtaining the efficiencies of the FS ratchet, we highlighted how the thermodynamic aspect of a non-equilibrium system can appear different upon coarse-graining. Such difference is crucial not only because focusing on a small number of variables is a natural theoretical procedure as taken by Feynman [19], but also because the measurement in experiment is typically restricted to a small set of slow variables. We believe that the explicit approach on the fine-grained theoretical models will prove useful in the understanding of the efficiencies of nanoscale energy transducers such as bio-molecular motors, since separation of time-scale is a typical issue. In this context, the decomposition of stochastic transitions points to a general workaround to connect stochastic thermodynamics between the different scales. This work is supported by the Grant-in-Aids for JSPS

S1. DERIVATION OF COARSE-GRAINED DYNAMICS
In the main text, we start from Model-1 where θ represents the angle of the coaxial vane and gear, p is the angular momentum conjugated to θ, m is the corresponding moment of inertia, x represents the height of the pawl, f is the torque exerted on the axle, and the two Langevin heat baths are described by independent white Gaussian noises with zero means and unit variances, ξ and ζ, together with the temperatures, T h and T c , and the viscous frictional coefficients, Γ and γ. Model-2 and Model-3 are obtained by taking the tightly confined limit and the overdamped limit successively. The calculations describing the two coarse-graining are discussed in the following subsections A and C.
In FIG. S1, a complete picture of a series of coarse-graining is shown. Model-1 has an underdamped version, which we call Model-0 where p and m are the momentum and the mass of pawl, respectively. Starting from Model-0, it is possible to take the overdamped limit for the vane and gear first, which results in Model-5. Either through Model-1 or 5, we can obtain Model-6, which is the overdamped model of the Feynman-Smoluchowski ratchet. Note that the coarse-graining of momentum degrees of freedom attached to the homogeneous heat bath (green arrows in FIG. S1) is trivial and does not involve hidden entropy production. Finally, by taking the tightly-confined limit in Model-6, we obtain a single-variable overdamped dynamics, Model-7 (subsection D). The discrepancy between Model-3 and 7, arises due to the difference in order of the two limits.

A. Derivation of Model-2 based on Singular Perturbation Theory
In this subsection, we derive the coarse-grained dynamics corresponding to Model-2 from a generalized version of Model-1. The generalized model differs from the original Model-1 in the choice of U (θ, x). We here require U (θ, x) to be decomposed as where U I is a trapping potential function, and ε is a dimensionless small parameter. If we choose U I (d) = λ 0 d 2 /2 and U 0 (x) = k x 2 /2, the original Model-1 is reproduced, where λ = λ 0 /ε. We employ the singular perturbation theory of Kramers-Fokker-Planck equation to derive the coarse-grained dynamics. The Kramers-Fokker-Planck equation corresponding to Model-1, may be written as where P(θ, p, x) is the joint probability density of θ, p, and x. Our goal here is to obtain the coarse-grained dynamics, that is, the differential equation for the joint probability density of θ and p, P(θ, p), which is obtained by integrating out x from P(θ, p, x): Throughout this Supplemental Material, the integrals with respect to θ, p, x, and rescaled variables are taken over the domain of integrand. Although the time derivative of P(θ, p) is formally written as this does not give the coarse-grained dynamics, since the right hand side of Eq. (S4) still depends on P(θ, p, x). The heart of the singular perturbation theory is to decompose the time-dependence of P(θ, p, x) into the explicit part and the implicit part through P(θ, p). In an appropriate limit, the explicit part decays quickly and the right hand side of Eq. (S4) essentially turns into a functional of P(θ, p). The problem of the singular perturbation theory is mapped onto that of the ordinary perturbation theory by introducing M and Ω which describe P(θ, p, x) and the dynamics of P(θ, p), respectively. For this purpose, we first switch the variables from (t, x) to the scaled time and distance as The explicit and implicit dependence of P(θ, p, s) on T is implemented by describing P(θ, p, s) as an output of a T -dependent operator, M, on P(θ, p): Furthermore, we represent the time-evolution of P(θ, p) by a T -dependent operator, Ω, on P(θ, p): Since M depends on T explicitly and implicitly [through P(θ, p)], the substitution of M into the left hand side of Eq. (S6) gives according to the chain rule. Applying Eq. (S7) also on the right hand side of Eq. (S6), we obtain (S10) The remaining task is to apply the standard procedure of perturbation theory. We expand M and Ω into series of ε 1/2 : Here, the difference in the lowest order for M and Ω is due to Eqs. (S8). The leading order of Eq. (S10) gives from which we obtain where [exponentially decaying terms] depend on T explicitly and decay exponentially at the time scale of O(1), and Z = ds exp (−U I (s)/T c ).
Since we have interest in the longer time scale than O(1), we neglect [exponentially decaying terms] hereafter. Under this assumption of time scale, Ω (0) [P(θ ′ , p ′ ); T ](θ, p) vanishes, since the last term in the right hand side of Eq. (S8) is zero in the leading order. The sub-leading order of Eq. (S10) is which has a particular solution By substituting Eq. (S16) into Eq. (S8), The Kramers equation immediately follows from the relation, ∂P(θ, p)/∂T = Ω[P(θ ′ , p ′ ); T ]: which corresponds to the underdamped Langevin equation of Model-2 with U eff (θ) = U 0 (φ(θ)), (S19) is solved as where we explicitly show the time-dependence of x, θ and ζ. By substituting Eq. (S23) into the equation of motion of the gear and the vane, we obtaiṅ holds for an arbitrary function ϕ, we finally reach the underdamped Langevin equation corresponding to Model-2, Although this derivation is much simpler than the general method presented in the previous subsection, it does not explicitly give the asymptotic form of the probability density, which is essential in evaluating the entropy production rate [see §S2].

C. Derivation of Overdamped Dynamics
The coarse-graining from Model-2 to Model-3 may be also formulated in the framework of §S1-A. The Kramers equation corresponding to Model-2 [Eq. (S18)] may be rewritten as in terms of the effective friction coefficient and temperature. By introducing the scaled time and momentum where m 0 is the reference point of mass, we obtain an equation to which we apply the singular perturbation theory, Here, m/Γ acts as a small parameter. Following the procedure in §S1-A, we definẽ and apply the standard perturbation theory to Eq. (S29) expressed in terms ofM andΩ. The leading order gives, which has a solutionM SinceΩ (0) [P(θ ′ ); τ](θ) again vanishes, we should proceed to the sub-leading order of Eq. (S29), which has a particular solutioñ By substituting Eq. (S36) into Eq. (S31) , We finally reach In order to obtain the corresponding overdamped Langevin equation, we make use of the correspondence between the Fokker-Planck and Langevin descriptions [N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, 2007)]. A multiplicative Langevin equation with the product in the sense of Stratonovicḣ can be mapped to an additive Langevin equationẊ where Ξ is white Gaussian noise with zero mean and unit variance, andX = X dX/C(X ),Ā(X ) = A(X )/C(X ). Since Eq. (S40) trivially corresponds to the Fokker-Planck equation, Now, we apply the standard perturbation theory to Eq. (S48) expressed in terms ofM andΩ. Since the leading order gives we obtainM where It immediately follows that the O(ǫ −1/2 ) term ofΩ vanishes. The sub-leading order of Eq. (S48) becomes which givesM which gives the Fokker-Planck equation: corresponding to Langevin equation