Hierarchical Onsager symmetries in adiabatically driven linear irreversible heat engines

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

Linear irreversible thermodynamics is a universal framework that systematically describes the response of equilibrium systems under weak nonequilibrium perturbations [28,29]. Despite its importance, the application of linear irreversible thermodynamics to heat engines operating under small temperature differences has been limited, until recently [30][31][32][33][34][35][36][37][38]. This is because the identification of thermodynamic fluxes and forces is highly complex for heat engines undergoing cyclic changes. Nevertheless, such an identification is essential because the performance of heat engines depends on the response coefficients, that is, Onsager coefficients, in the linear response regime [6,11]. In particular, the linear irreversible thermodynamics for the temperature difference and the speed of adiabatically changing parameters of cyclic heat engines is limited to a few specific examples [30][31][32]. Cyclic heat engines can experience continuous equilibrium change along a cycle and be substantially perturbed from a reference equilibrium point. This makes the application of the linear response theory, which is usually defined for a response from a one-equilibrium point, difficult and obscure. Notably, the identified Onsager symmetry for these models is derived only phenomenologically, by adopting intuitive global fluxes and * izumida@k.u-tokyo.ac.jp forces per cycle, without deriving a relation to the local thermodynamic fluxes and forces defined at any instant of a cycle.
By contrast, in recent studies on quantum thermoelectrics, such a linear response for adiabatically changing parameters has been investigated as an effect of adiabatic ac driving applied to a system [39,40]. Remarkably, the Onsager coefficients defined globally for a one-cycle period of ac driving, which determine the overall performance of the thermoelectrics, are expressed in terms of locally defined Onsager coefficients at any instant during driving [39,40]. The key of this formulation is to apply the standard linear response theory to instantaneous equilibrium states instead of the usual equilibrium states by regarding that the adiabatically changing parameters have "frozen," fixed values. Considering the universal nature of linear irreversible thermodynamics, we are motivated to uncover a similar hierarchical structure for adiabatically driven linear irreversible heat engines. To this end, we focus on the simplest heat engine model. We establish a hierarchical relationship between global and local Onsager coefficients for a generic Gaussian heat engine model obeying Fokker-Planck dynamics. The adiabatic dynamics can be easily obtained based on the idea of time-scale separation [41], which is one of the advantages of this model. Moreover, based on the detailed structure of the Onsager coefficients, we derive an efficiency bound, tighter than the Carnot efficiency, under a given speed of adiabatic change.
Model-. The heat engine consists of a working substance (system) and thermal bath. The state of the system x = (x 1 , · · · , x n ) at time t is specified by a probability distribution P(x, t). The system is periodically operated based on p external parameters λ(t) = (λ 1 (t), · · · , λ p (t)) and the bath temperature T (t) with period τ cyc ; λ(t + τ cyc ) = λ(t) and T (t + τ cyc ) = T (t). The energy of the system is given by H(x, t), which is a function of λ(t). Specifically, the external parameters are expressed as λ(t) = λ 0 + g w (ǫt) using the timeindependent part λ 0 and the time-dependent part g w . Here, ǫ ≡ 1/τ cyc denotes a small parameter corresponding to the speed of the process. Thus, a long period of time t = O(1/ǫ) is required for a finite increment of g w . The bath temperature T (t) is given by where ∆T (t) ≡ γ q (ǫt)∆T , and ∆T ≡ T h − T c and γ q (ǫt) are the temperature difference and periodic function satisfying 0 ≤ γ q (ǫt) ≤ 1, respectively [33]. We define the entropy production rate per cycleσ for the system and thermal bath. Hereafter, we denote by the overdot a quantity per unit time or a quantity being time differentiated. The energy change rate becomesĖ ≡ d dt H(x, t) = d dt dx n H(x, t)P(x, t), where · refers to an ensemble average with respect to P(x, t). We decomposeĖ into the sum of the heat and work fluxesQ andẆ;Ė = dx n H(x, t) ∂P(x,t) ∂t + dx n ∂H(x,t) ∂t P(x, t) ≡Q −Ẇ. Then, we can defineσ aṡ where the prime symbol denotes the time derivative with respect to the slow time T ≡ ǫt andġ w (ǫt) = dg w (ǫt) dt = ǫg ′ w (ǫt). The dot between symbols denotes an inner product. Here, we have defined the following work and heat fluxes per cycle as thermodynamic fluxes: The corresponding thermodynamic forces are defined as We assume the global linear response relations J = LF between J ≡ (J w , J q ) T and F ≡ (F w , F q ) T defined over one cycle of the heat engine in the limit of ǫ → 0 and ∆T → 0: where L corresponds to the global Onsager coefficients. Our goal is to find a detailed expression of L in terms of its local counterpart defined at any instant of the cycle, thereby establishing a hierarchical relationship between the two. Fokker-Planck dynamics-. For further calculation of J, we need to specify the dynamics of P(x, t). In what follows, we consider generic Gaussian heat engines described based on multivariate Ornstein-Uhlenbeck processes as the simplest models. The energy of the system, which serves as a potential function, thus takes the following quadratic form: where H(t) is a positive-definite symmetric matrix. We assume that x is even variables under time reversal. The probability distribution of the system P(x, t) obeys the Fokker-Planck (FP) equation with the time-dependent drift matrix A and diffusion matrix B (i, j = 1, · · · , n) [42,43]: where J i (x, t) is a probability current. A is a symmetric matrix and B is a positive-definite symmetric matrix. B is further assumed to be invertible. The probability distribution is assumed to be the zero-mean Gaussian distribution: where the symmetric covariance matrix The equation to be solved is replaced with the dynamic equations in Eq. (10) For the stationary distribution to agree with a Boltzmann distribution at temperature T c , the following detailed balance condition is usually imposed [43]: which together with Eq. (11) yields Ξ −1 0 = H 0 /k B T c with k B being Boltzmann constant. Here, as a natural generalization of Eq. (12), we impose the detailed balance condition, including the time-dependent part: whose validation will be clarified below. We decompose A(t), B(t), and Ξ(t) into time-independent and time-dependent parts as We solve Eq. (14) perturbatively with respect to ǫ. Because a regular perturbation yields a secular term, we use a two-timing method based on time-scale separation [41]. As a result, we obtain Ξ(t) as [44] where Ξ ad (t) and δΞ nad (t) are the adiabatic solution and the lowest non-adiabatic correction to it, respectively, as From Eqs. (13) and (16), we have Thus, the probability distribution P(x, t) in the adiabatic limit ǫ → 0 agrees with an instantaneous equilibrium distribution with energy H(x, t) and temperature T (t), which validates the condition given by Eq. (13). Local and global linear response relations for speed and temperature differences-. We can now evaluate the thermodynamic fluxes in Eqs. (2) and (3) using Eqs. (15)- (17).
Note that we can rewrite Eq. (3) as J q = 1 τ cyc τ cyc 0 dtγ q (ǫt) dx n ∂H(x,t) ∂x i J i (x, t) using Eq. (8), and we can express Eq. (2) and (3) as the time average of the local thermodynamic fluxes as respectively, where we define the response vectors j = T as the local thermodynamic fluxes. We also introduce the conjugate local nonequilibrium perturbation vector The perturbations are the speed of adiabatically changing parameters and temperature difference, and the responses are the generalized pressure and instantaneous heat flux. The linear relationship between the perturbations and responses can be written as a local fluxforce form [39,40], namely j = j ad + Λf in the limit of ǫ → 0 and ∆T → 0, where j ad is an adiabatic response that remains within the limit of ǫ → 0, and Λ is the local Onsager matrix given by We can expand j w and j q with respect to f as [45] to the linear order of O(|f|). We thus identify j ad and Λ as respectively. We can confirm the Onsager symmetry Λ ww,mm ′ = Λ ww,m ′ m and anti-symmetry Λ wq,m = −Λ qw,m (m, m ′ = 1, · · · , p) at the local level. The former symmetry relates to the dissipation, while the latter anti-symmetry relates to the dissipationless cross-coupling between the heat flux and the work flux (heat engine-refrigerator symmetry).
Subsequently, we consider the global linear response relations J = LF in Eqs. (5) and (6). The global thermodynamic fluxes in Eqs. (19) and (20) can be rewritten as J w = 1 0 dT g ′ w (T ) · j w and J q = 1 0 dT γ q (T ) j q in terms of the slow time T = ǫt. We note that the contribution from j ad vanishes upon cycle averaging. Note that F w = ǫ/T c and F q ≃ ∆T/T 2 c in the linear response regime, and using Eqs. (22) and (23), we immediately arrive at the following expression for the global Onsager matrix L: The local and global Onsager matrices in Eqs. (25) and (26) constitute the first main results of this study. The global Onsager coefficients L are given as the integration over one cycle of the local Onsager coefficients Λ in Eq. (25). This yields a hierarchical relationship between L and Λ, thereby relating the different levels of symmetries. In particular, L shows Onsager anti-symmetry L wq = −L qw , reflecting the Onsager antisymmetry Λ wq,m = −Λ qw,m for Λ.
In the linear response regime, the entropy production rate per cycleσ = J w F w + J q F q in Eq. (1) takes the quadratic forṁ σ = L ww F 2 w + (L wq + L qw )F w F q + L qq F 2 q , where we have used Eqs. (5) and (6). The second law of thermodynamicsσ ≥ 0 imposes constraints on L: For the present system, we finḋ by using the explicit form of L in Eq. (26). Remarkably, we readily observe L ww ≥ 0, and thus,σ ≥ 0 from the positivedefinite quadratic form of L ww in Eq. (26). The anti-symmetric coefficients do not contribute toσ because they represent a reversible, adiabatic change in entropy. The vanishing L qq also reducesσ, which arises from nonsimultaneous contact with the thermal baths at different temperatures. This property is essentially the same as that known as the tight-coupling condition [6]. Note that we have the optional thermodynamic fluxes and forces. By switching the roles of J w and F w , that isJ w = F w andF w = J w , while maintainingJ q = J q and F q =F q , we obtain another global Onsager matrixL: assuming that L ww is nonvanishing and using L qq = 0. Thus, we can confirm the symmetric non-diagonal elements and the vanishing determinant, where the latter corresponds to the tight-coupling condition. Such a choice of fluxes and forces was adopted to identify Onsager coefficients of the finite-time Carnot cycle in [30][31][32]. As we will see below, the vanishing L qq , equivalently, the tight-coupling condition, implies the attainability of the Carnot efficiency in the adiabatic limit ǫ → 0 [39]. Thermodynamic efficiency-. Using the global linear response relations in Eqs. (5) and (6) together with Eq. (26), we formulate the power P and efficiency η of our Gaussian heat engines: where η C ≡ ∆T/T h ≃ ∆T/T c is the Carnot efficiency. In the adiabatic limit F w → 0, we recover η = η C . For small ǫ, the power behaves as P = −L wq ∆T ǫ/T 2 c + O(ǫ 2 ). It should agree with ∆T ∆S ǫ, where ∆S denotes an adiabatic entropy change of the system and ∆T ∆S is an adiabatic work per cycle. Thus, we identify L wq = −L qw = −T 2 c ∆S , which clarifies the vanishing contribution of these antisymmetric parts to the irreversible entropy production rateσ. The efficiency under a given F w , that is, the speed ǫ, is bounded by the upper side as where T c L 2 is the minimum value of L ww . Reparameterizing from T to θ (0 ≤ θ ≤ 1), we have Using the Cauchy-Schwartz inequality, we obtain L ww ≥ T c . Equation (32) constitutes our second main result. It yields a tighter bound than the Carnot efficiency imposed by the conventional second law of thermodynamics and is attained for an optimal protocol under a given cycle speed. Such a bound was obtained by virtue of the detailed structure of the global Onsager coefficients (Eq. (26)). L is equivalent to the thermodynamic length, which constrains the minimum dissipation along finite-time transformations close to equilibrium states [46][47][48][49][50][51][52][53]. An expression similar to Eq. (32) including an effect of temperature-variation speed was recently derived based on a geometric formulation of quantum heat engines [23]. Here, we derived the similar form in terms of the global linear response relations between the speed of adiabatically changing parameters and temperature difference. Example: Brownian heat engine-. We demonstrate our results by using the simplest illustrative case of a onedimensional stochastic Brownian heat engine model (n = m = 1) [8,16,46]. Let x 1 = x be the position of a Brownian particle immersed in a thermal bath. The probability P(x, t) obeys the following FP equation [54,55]: where γ is viscous friction coefficient and H(x, t) = U(x, t) = λ(t) 2 x 2 with λ(t) = λ 0 + g w (ǫt) is a harmonic potential. We identify A and B as A = A 11 = − λ(t) γ and B = B 11 = 2k B T (t) γ . Because the Boltzmann distribution with T c and λ 0 is p 0 (x) = λ 0 2πk B T c e − λ 0 x 2 2k B Tc , the variance at equilibrium is Ξ 0,11 = k B T c /λ 0 . By using the adiabatic solution Ξ ad,11 (t) = k B T (t)/λ(t). The local linear response relations j = j ad + Λf are then obtained from Eqs. (24) and (25) as up to O(|f|), which determines the local and global Onsager matrices Λ and L as (36) respectively. We can confirm the Onsager anti-symmetry in Λ and L, as expected. For a Carnot-like cycle with γ q (T ) = 1 for 0 ≤ T < T h (0 < T h < 1) and γ q (T ) = 0 for T h ≤ T < 1 [33], we have L wq = −L qw = −T 2 c ∆S = k B T 2 c 2 ln(λ 1 /λ 0 ), where λ 1 ≡ λ(T h ) and λ 0 = λ(0) = λ(1) are the minimum and maximum values of λ along the cycle, respectively. We can using the optimal protocol λ * (T ) for a given λ 0 and λ 1 [46]: λ * (T ) = The efficiency bound in Eq. (32) for the present case thus becomes A comparison of the bound Eq. (37) with that, for example, for a linear protocol connecting λ 0 and λ 1 highlights the importance of protocol optimization as a design principle.
Concluding perspective-. We developed a linear response theory for generic Gaussian heat engines as the simplest model of adiabatically driven linear irreversible heat engines. We established the hierarchical relationship between the local and global Onsager coefficients. Further, we derived the efficiency bound under a given rate of adiabatic change; the derived bound is tighter than the Carnot efficiency imposed by the second law of thermodynamics. We expect that the present results will contribute to a deeper understanding of the physical principles and optimal control of nonequilibrium heat engines.
We note complementary approaches for the formulation of the linear irreversible thermodynamics to periodically driven heat engines Ref. [33][34][35][36][37][38]. In these approaches, the other thermodynamic force (that is, in addition to the temperature difference) is the strength of periodic forcing, and not its speed, as in the present approach. Interestingly, the Onsager coefficients in these cases were found to be decomposed into adiabatic and non-adiabatic contributions. The existence of different types of linear irreversible thermodynamics implies the rich and versatile structures of periodically driven heat engines, and this deserves further investigation.