Thermodynamics of collisional models for Brownian particles: General properties and efficiency

We introduce the idea of {\it collisional models} for Brownian particles, in which a particle is sequentially placed in contact with distinct thermal environments and external forces. Thermodynamic properties are exactly obtained, irrespective the number of reservoirs involved. In the presence of external forces, entropy production presents a bilinear form in which Onsager coefficients are exactly calculated. Analysis of Brownian engines based on sequential thermal switchings is proposed and considerations about their efficiencies are investigated taking into account distinct external forces protocols. Our results shed light to a new (and alternative) route for obtaining efficient thermal engines based on finite times Brownian machines.


I. INTRODUCTION
Stochastic thermodynamics has proposed a general and unified scheme for addressing central issues in thermodynamics [1][2][3][4][5]. It includes not only an extension of concepts from equilibrium to nonequilibrium systems but also it deals with the existence of new definitions and bounds [6][7][8][9], general considerations about the efficiency of engines at finite time operations [1][2][3] and others aspects. In all cases, the concept of entropy production [1,4,10] plays a central role, being a quantity continuously produced in nonequilibrium steady states (NESS), whose main properties and features have been extensively studied in the last years, including their usage for typifying phase transitions [11][12][13][14].
Basically, a NESS can be generated under two fundamental ways: from fixed thermodynamic forces [15,16] or from time-periodic variation of external parameters [17][18][19][20]. In this contribution, we address a different kind of periodic driving, suitable for the description of engineered reservoirs, at which a system interacts sequentially and repeatedly with distinct environments [21][22][23]. Commonly referred as collisional models, they have been inspired by the assumption that in many cases (e.g. the original Brownian motion) a particle collides only with few molecules of the environment and then the subsequent collision will occur with another fraction of uncorrelated molecules. Collisional models have been viewed as more realistic frameworks in certain cases, encompassing not only particles interacting with a small fraction of the environment, but also those presenting distinct drivings over each member of system [24][25][26][27] or even species yielding a weak coupling with the reservoir. More recently, they have been (broadly) extended for quantum systems for mimicking the environment, represented by a weak interaction between the system and a sequential collection of uncorrelated particles [28][29][30].
With the above in mind, we introduce the concept of repeated interactions for Brownian particles. More specifically, a particle under the influence of a given external force is placed in contact with a reservoir during the time interval τ /N and afterwards it is replaced by an entirely different (and independent) set of interactions. Exact expressions for thermodynamic properties are derived and the entropy production presents a bilinear form, in which Onsager coefficients are obtained as function of period. Considerations about the efficiency are undertaken and a suited regime for the system operating as an efficient thermal machine is investigated.
The present study sheds light for fresh perspectives in nonequilibrium thermodynamics, including the possibility of experimental buildings of heat engines based on Brownian dynamics [31][32][33][34][35][36] with sequential reservoirs. Also, they provide us the extension and validation of recent bounds between currents and entropy production, the so called thermodynamic uncertainty relations (TURs) [8,9,[37][38][39][40][41], which has aroused a recent and great interest. This paper is organized as follows: Secs. II and III present the model description and its exact thermodynamic properties. In Sec. IV we extend analysis for external forces and considerations about efficiency are performed in Sec. V. Conclusions and perspectives are drawn in Sec. VI.

II. MODEL AND FOKKER-PLANCK EQUATION
We are dealing with a Brownian particle with mass m placed in sequential contact with N different thermal reservoirs. Each contact has a duration of τ /N and occurs during the intervals τ i−1 ≤ t < τ i , where τ i = iτ /N for i = 1, .., N , in which the particle evolves in time according to the following Langevin equation where quantities v i α i and F i (t) account for the particle velocity, the viscous constant and external force, respectively. From now on, we shall express them in terms of reduced quantities: γ i = α i /m and f i (t) = F i (t)/m. The stochastic force ζ i (t) = B i (t)/m accounts for the interaction between particle and the i-th environment and satisfies the properties and respectively, where T i is the bath temperature. Let P i (v, t) be the velocity probability distribution at time t, its time evolution is described by the Fokker-Planck (FP) equation [3,16,42] It is worth mentioning that above equations are formally identical to description of the overdamped harmonic oscillator subject to the harmonic force f h = −kx just by replacing x → v,k/α → γ i , 1/α → γ i /m. If external forces are null and the particle is placed in contact to a single reservoir, the probability distribution approaches for large times the Gibbs (equilibrium) distribution P eq i (v) = e −E/kBTi /Z, being E = mv 2 /2 its kinetic energy and Z the partition function. On the other hand, this will not be the case when the system is placed in contact with sequential and distinct reservoirs. In such case, the system dissipates heat and continuously produce entropy. The solution of Eq. (4) is also a Gaussian type P NESS From the FP equation, the time variation of the energy system U i = E i in contact with the i-th reservoir fulfills the first law of thermodynamics dU i /dt = −(Ẇ i +Q i ), whereẆ i andQ i are the work per unity of time and heat flux from the system to the environment (thermal bath) given bẏ respectively. In the absence of external forces, all heat flux comes from/goes to the thermal bath. Conversely, by assuming the system entropy S is given by S i (t) = −k B P i (t) ln[P i (t)]dv i , one finds that its time derivative has the form of [16,42], where Π i (t) and Φ i (t) are identified as the entropy production and the flux of entropy, respectively, and given by respectively. The latter term can also be rewritten in a more convenient form As stated before, Π eq = Φ eq = 0, whereas Π NESS = Φ NESS > 0. From now on, quantities will be expressed in terms of the "reduced temperature" Γ i = 2γ i k B T i /m and k B = 1.

III. EXACT SOLUTION FOR ARBITRARY SET OF SEQUENTIAL RESERVOIRS
Instead of the solving FP equation, thermodynamic properties can be alternatively calculated from the averages v i (t) and variances b i (t). Their time evolutions are calculated from Eq. (4) and read and respectively, where appropriate partial integrations were performed. Their solutions are given by the following expressions and coefficients A i 's are evaluated by taking into account the set of continuity relations for the averages and variances respectively. Since the system returns to the initial state after a complete period, v , all coefficients can be solely calculated in terms of model parameters, temperature reservoirs and the period. In other words, above conditions state that the probability density at each point returns to the same value after every period. For simplicity, we shall assume the same viscous constant γ i = γ in all i's and taking into account that v i 's vanish in the absence of external forces, the entropy production only depends on the coefficients A i 's and Γ i 's. Hence, the coefficient A i becomes where x = e −2γτ /N and all of them can be found from a linear recurrence relation for i = 2, ....N . As the particle returns to the initial configuration the after a complete period, A N then reads By equaling Eqs. (13) and (14) for i = N , all coefficients A i 's can be finally calculated and are given by for i = 1, and (16) respectively for i > 1. The entropy flux Φ i (t) is given by Eq. (7), whose contribution averaged over a period τ reads From Eqs. (15) and (16), it follows that and we arrive at an expression for Π solely dependent on the model parameters In order to show that Π ≥ 0, we resort to the inequal- in consistency with the second law of thermodynamics.
As an concrete example, we derive explicit results for the two sequential reservoirs case. From Eqs. (10) and (11), coefficients A 1 and A 2 reduce to the following expressions where A 2 = −A 1 and hence for 0 ≤ t < τ /2 and τ /2 ≤ t < τ , respectively whose mean entropy production reads Note that Π ≥ 0 and it vanishes when Γ 1 = Γ 2 . In the limit of slow (τ >> 1) and fast (τ << 1) oscillations, Π approaches to the following asymptotic expressions respectively and such a latter expression is independent on the period. The entropy production can be conveniently written down as a flux-times-force expression Π = J T f T where f T and J T attempt for the thermodynamic force f T = (1/Γ 1 − 1/Γ 2 ) and its associated flux, respectively. J T can also be rewritten as J T = L T T f T , where L T T is the Onsager coefficient given by Note that L T T ≥ 0 (as expected). Fig. 1 depicts the average entropy production Π versus τ for distinct values of Γ 2 and Γ 1 = 1, γ = 1. Note that it is monotonically increasing with f T and reproduces above asymptotic limits.

IV. FORCED BROWNIAN AND SEQUENTIAL RESERVOIRS
Next, we extend analysis for the case of a Brownian particle in contact with sequential reservoirs and external forces. We shall focus on the two stage case and two simplest external forces protocols: constant and linear drivings. More specifically, the former is given by where f 1 and f 2 denote their strengths in the first and second half period, respectively, whereas the latter case accounts for forces evolving linearly over the time according to the slopes: with λ 1 and λ 2 being their associated forces. It has been considered in Ref. [41] in order to compare the performance of distinct bounds between currents and the entropy production (TURs). In the presence of external forces, FP equation has the same form of Eq. (11), but now v i (t)'s will be different from zero.

A. Constant external forces
In the presence of constant external forces, the mean velocities v i (t)'s are given by for the first or second half of each period, respectively.
In the same way as before, the steady entropy production per period Π can be evaluated from Eq. (7) (by taking k B = 1) and reads and we arrive at the following expression Since γτ ≥ 0 and 1 − tanh(x)/x ≥ 0, it follows that Π ≥ 0. Note that Π reduces to Eq. (24) as f 1 = f 2 = 0.

Bilinear form and Onsager coefficients
The shape of Eq. (33) reveals that the entropy production can also be written down as flux-times-force expression where forces f T = (1/Γ 1 −1/Γ 2 ) and f 1(2) have associated fluxes J T , J 1 and J 2 given by J T = L T T f T (identical to Eq. (26)), respectively, where L 11 , L 12 , L 21 and L 22 denote their Onsager coefficients given by and L 12 = L 21 = 1 γτ respectively. Coefficients L 22 and L 21 have the same shape of L 11 and L 12 by replacing 1 ↔ 2, respectively. Besides, L 11 and L 22 ≥ 0 and they satisfy the inequality 4L 11 L 22 − (L 12 + L 21 ) 2 ≥ 0, in consistency with the positivity of the entropy production.

Bilinear form and Onsager coefficients
As for the constant force case, the entropy production has also the shape of Eqs. (34)- (35) given by Π = J T f T + J 1 λ 1 + J 2 λ 2 , being L T T the same to Eq. (26), whereas the other Onsager coefficients read and respectively, and coefficients L 22 and L 21 are identical to L 11 and L 12 by exchanging 1 ↔ 2. As in the constant driving, it is straightforward to verify that L 11 and L 22 are strictly positive and 4L 11 L 22 − (L 12 + L 21 ) 2 ≥ 0.

V. EFFICIENCY
Distinct works have tackled the conditions in which periodically driven systems can operate as thermal machines [43][44][45][46][47][48]. The conversion of a given type of energy into another one requires the existence of a generic force X 1 operating against its flux J 1 X 1 ≤ 0 counterbalancing with driving forces X 2 and X T in which J 2 X 2 + J T X T ≥ 0. A measure of efficiency η is given by where in such case X T = f T and we have taken into account Eq. (34) for relating fluxes and Onsager coefficients. Let us consider the case of a particle in contact with a hot and cold reservoirs, but with temperatures close to each other Γ 1 ≈ Γ 2 = Γ. In such case ∆Γ = Γ 1 − Γ 2 << 1 and the thermodynamic force f T approaches to f T ≈ ∆Γ/Γ 2 . Taking into account that the best machine aims at maximizing the efficiency and minimizing the dissipation Π for a given power output P = −Γ 1 J 1 X 1 , it is important to analyze the role of three load forces, X 1mP , X 1mE and X 1mS , in which the power output and efficiency are maximum and the dissipation is minimum, respectively [46]. Their values can be obtained straightforwardly from expressions for P and Eq. (43), respectively. Due to the present symmetric relation between Onsager coefficients L 12 = L 21 (in both cases), they acquire simpler forms and read 2X 1mP = −L 12 X 2 /L 11 , with A(X 2 , X T ) being given by and X 1mS = −L 12 X 2 /L 11 = 2X 1mP , respectively, where X i = f i and λ i for the constant and linear drivings, respectively. The efficiencies at minimum dissipation, maximum power and its maximum value become η mS = 0, and respectively, and finally their associated power outputs read P mS = 0, P mP = Γ 1 L 2 12 X 2 2 /4L 11 and respectively. We pause to make a few comments: First, above expressions extend the findings from Ref. [46] for the case of two reservoirs and a couple of driving forces. Second, both efficiency and power vanish when X 1 = X 1mS and X 1 = 0 and are strictly positive between those limits. Hence the physical regime in which the system can operate as an engine is bounded by the lowest entropy production Π mS = L T T X 2 T + (L 22 − L 2 12 /L 11 )X 2 2 and the value Π * = L T T X 2 T + L 22 X 2 2 . Third, despite the long expressions for Eqs. (47) and (48), powers P mP , P mE and efficiencies η mP , η mE are linked through a couple of simple expressions (in similarity with Refs. [45,46]): and they imply that 0 ≤ η mP < η mE (with 0 ≤ η mE ≤ 1 and 0 ≤ η mP ≤ 1/2) and 0 ≤ P mE ≤ P mP . Fourth and last, the achievement of most efficient machine η mE = 1 implies that the system has to be operated at null power P mE = 0 and hence the projection of a machine operating for a finite (maximum) power will imply at a loss of its efficiency. Our purpose here aims at not only extending relevant concepts about efficiency for Brownian particles in contact with sequential reservoirs, but also to show that a desired compromise between maximum power and maximum efficiency can be achieved by adjusting conveniently the model parameters (such as the period and the driving). This is appraised in Figs. 2 and 3 in which quantities are depicted for distinct periods τ and temperature differences ∆Γ's for constant and linear drivings, respectively. In both cases, quantities follow theoretical predictions and exhibit similar portraits, in which efficiencies and power outputs present maximum values at f 1mE and f 1mP (λ 1mE and λ 1mP ), respectively. The loss of efficiency from the maximum η mE as f 1 (λ 1 ) goes up (down) is signed by the increase of dissipation (as expected) until vanishing when Π = Π * . Absolute values of forces and efficiencies increase as the period τ (see e.g. panels (a)) and/or temperature differences (see e.g. panels (b)) are lowered. In the limit of fast switchings, τ → 0, Onsager coefficients become simpler and L 11 , L 22 approach to (Γ 1 + Γ 2 )/(4Γ 1 Γ 2 ), whereas L 12 = L 11 (constant driving) and L 12 = −L 11 (linear driving). Some remarkable quantities then approach to the asymptotic values f 1mS → −f 2 = 2f 1mP (constant forces) and λ 1mS → λ 2 = 2λ 1mP (time dependent ones) and η mP respectively. For Γ 1 ≈ Γ 2 , η mP → 1/2, η mE → 1 and P mP reads P mP → f 2 2 /8 (P mP → λ 2 2 /8), respectively and thereby the limit of an ideal machine is achieved for low periods and equal temperatures.
Despite the similarities between external forces protocols, there are some differences between them, as compared in Fig. 4. The linear driving is more efficient than the constant one for short periods and their power outputs are also superior. Although both drivings provide lower efficiencies and powers when worked at larger periods, constant drivings are somewhat more efficient in this case.
We close this section by remarking that although short periods indicates a general route for optimizing the efficiency of thermal machines in contact to sequential reservoirs, the present description provides to properly tune the period and forces in order to obtain the desirable compromise between maximum efficiency and power.

VI. CONCLUSIONS
The thermodynamics of a Brownian particle periodically placed in contact with sequential thermal reservoirs is introduced. We have obtained explicit (exact) expressions for relevant quantities, such as heat, work and entropy production. Generalization for an arbitrary number of sequential reservoirs and the influence of external forces were considered. Considerations about the efficiency were undertaken, in which Brownian machines can be properly operated ensuring the reliable compromise between efficiency and power for small switching periods.
As a final comment, we mention the several new perspectives to be addressed. First, it might be very interesting to extend such study for other external forces protocols (e.g. sinusoidal time dependent ones) as well as for time asymmetric switchings, in order to compare their efficiencies. Finally, it would be very remarkable to verify the validity of recent proposed uncertainties relations (TURs) for Fokker-Planck equations [39,41], in such class of systems.

VII. ACKNOWLEDGMENT
We acknowledge Karel Proesmans for a careful reading of the manuscript and useful suggestions. C. E. F acknowledges the financial support from FAPESP under grant 2018/02405-1.