Experimental validation of the 1/τ -scaling entropy generation in finite-time thermodynamics with dry air
EExperimental validation of the /τ -scaling entropy generation in finite-timethermodynamics with dry air Yu-Han Ma,
1, 2
Ruo-Xun Zhai,
3, 2
Chang-Pu Sun,
1, 2 and Hui Dong ∗ Beijing Computational Science Research Center, Beijing 100193, China Graduate School of China Academy of Engineering Physics,No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China Beijing Normal University, Beijing 100875, China
The second law of thermodynamics can be described as the non-decreasing of the entropy inthe irreversible thermodynamic process. Such phenomenon can be quantitatively evaluated withthe irreversible entropy generation (IEG), which was recently found to follow a /τ scaling for thesystem under a long contact time τ with the thermal bath. This scaling, predicted in many finite-time thermodynamic models, is of great potential in the optimization of heat engines, yet remainslack of direct experimental validation. In this letter, we design an experimental apparatus to testsuch scaling by compressing dry air in a temperature-controlled water bath. More importantly, wequantitatively verify the optimized control protocol to reduce the IEG. Such optimization shall bringnew insight to the practical design of heat engine cycles. Introduction - Heat engines, converting heat into usefulwork, have important practical applications and attracta wide range of research interests in both classical andquantum thermodynamics [1–6]. In classical thermody-namics, the Carnot theorem [1] limits the maximum ef-ficiency of heat engines with the well-know Carnot ef-ficiency η C = 1 − T c /T h , where T c ( T h ) is temperaturefor the cold (hot) bath. Unfortunately, achieving suchefficiency is typically accompanied by a vanishing out-put power due to the infinite long operation time in aquasi-static thermodynamic process [1, 7–10]. The futil-ity of such heat engine with vanishing power has pushedto design finite-time cycle to achieve high efficiency whilemaintaining the output power [11–16]. For such design,the quantitative evaluation of the irreversibility is the keyfor optimization [6, 11, 14–20]. The trade-off relation be-tween power and efficiency [21–24] are significantly deter-mined by the relation of irreversible entropy generation(IEG) on the control time τ . In the near-equilibrium re-gion, the IEG in a finite-time isothermal process is foundinversely proportional to the process time τ , namely the C /τ scaling with the coefficient C . Such scaling has beendiscovered in different finite-time thermodynamic models[25], such as endo-reversible model [11, 12, 26, 27], lin-ear model [14, 19, 28, 29], stochastic model [30–34], andlow-dissipation model [16, 24, 35]. Moreover, the scalingrelation has been established not only for the classicalworking substance [12], but also for quantum workingsubstance [23, 24, 36]. The coefficient C is determinedby the statistical properties of the working substance andthe heat bath, and has recently been proved to be relatedto the way that the working substance being manipulated[36, 37].To our best knowledge, the direct verification of the C /τ scaling was rarely explored, although the behaviorof finite-time heat engines has been studied in several ex-perimental platforms [39–46]. In this letter, we focus onexperimental measuring IEG of dry air via the work done in the finite-time isothermal process with designed appa-ratus, and reveal the impact of control scheme on theIEG quantitatively. In order to validate the C /τ scaling,a controlled apparatus in Fig. 1(a) is designed to measurethe state of the dry air, which is sealed in a compressiblecylinder (A) and three buffer cylinders (B, C, D), im-mersed in a temperature-controlled water bath. A pistonis installed in the cylinder A to compress the air with acomputer-controlled stepper motor M. By setting differ-ent push programs, a controllable change in the volumeof the gas over time is achieved, i.e., V ( t ) = V − A L ( t ) ,where V = 2 . × − m is the initial volume of thegas, and A = 1 . × − m is the cross sectional areaof the cylinder A. The current setup allows us to realizethe finite-time isothermal process with different processtime τ .We firstly sketch the origin of the C /τ scaling for ageneral classic system, which contacts with a heat bathof constant temperature T e . A control parameter, e.g.,the volume V of the gas, is tuned from t = 0 to the endof the process t = τ . In this process, with the endo-reversible assumption [11, 12, 26, 27], IEG is written as[12] ∆ S (ir) = ˆ τ (cid:18) dQ s T s + dQ e T e (cid:19) , (1)where dQ s = − dQ e is the heat absorbed by the systemfrom the heat bath. The effective temperature T s of thesystem generally varies with time in the control process.In the condition of the quasi-static process with infinitecontrol time ( τ → ∞ ), the system is always in the ther-mal equilibrium with T s = T e . For the long time τ incomparison to the relaxation time t r between the gas andthe heat bath, the system is in the linear irreversible re-gion, such that T s is slightly deviated from the bath tem-perature, namely | T s − T e | /T e (cid:28) . The heat exchangerate between the system and bath follows the Newton’slaw of cooling as a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Figure 1. Experimental setup for measuring irreversible en-tropy generation in the finite-time isothermal process. (a)Experimental setup. The dry air is sealed in four connectedcylinders A, B, C, and D. The piston of A is propelled bythe computer-controlled stepper motor M to achieve the con-trolled compression of the gas. Three pressure sensors S1, S2,and S3 are connected to the top of the three cylinders B, C,and D respectively to measure the air pressure in the cylinders P ( t ) . And the displacement of the piston L ( t ) is detected bya position sensor S4 to reveal the gas volume V . The cylindersare immersed in the water bath with adjustable temperature.(b) P − V diagram of the gas under the bath temperature T e = 313 . K is illustrated in figure (b). The green diamonds,blue triangles, and yellow circles are obtained for the pistonspeeds mm/s, mm/s, and mm/s, respectively. The redsolid line shows the theoretical quasi-static isothermal pro-cess, namely, P V = const , and the black solid line representsthe adiabatic process with
P V γ = const . Here, γ = 1 . is theheat capacity ratio of the dry air [38]. dQ s dt = − κ ( T s − T e ) , (2)where κ is the thermal conductance. Combining Eqs. (1)and (2), we obtain IEG as ∆ S (ir) = ´ J d e tκT τ , (3)where J = dQ s /d e t is the heat flux, and e t = t/τ is thenormalized time. The above equation shows the origin of /τ scaling for the IEG.For the current dry air system with volume compressedfrom V to V f , the IEG is found proportional to the ir-reversible work W (ir) (IW) in the process under the longtime limit as follow W (ir) = T e ∆ S (ir) = P ( V f − V ) κT e τ , (4) where P is the initial pressure of the dry air. Theirreversible work W (ir) ( τ ) = W ( τ ) − W q is obtainedby subtracting the work W q = P V ln ( V /V f ) in thequasi-static isothermal process from the work W ( τ ) = − ´ τ P dV in the finite-time isothermal process. The de-tail of the current derivation is shown in the Supplemen-tary Materials. We will characterize the irreversibility ofthe current system via the irreversible work, which is adirectly measurable quantity [12, 17] in our current setup.
Verification of /τ scaling - To measure the work W ( τ ) , we monitor the pressure P = P ( t ) with three sen-sors, numbered S1, S2 and S3 (range 0-0.15Mpa, accu-racy 0.1%) on the top of the three cylinders B, C, and Drespectively. The volume change dV = A dL ( t ) is mea-sured through the piston position L ( t ) with the sensorS4 (range 0-0.3m, accuracy 0.1%).In the whole compression process, the four cylindersare immersed in a large water bath ( volumn L) withcontrollable temperature (accuracy 0.5K). The internalequilibrium time of the gas is much smaller than the re-laxation time t r that the gas is always in the equilibriumstate with temperature T s , known as the endo-reversible[11, 12, 26, 27]. In the current setup, t r = 1 . s ismeasured in the experiment with details explained in theSupplementary Materials.The state of the dry air is illustrated via the P-V di-agram in Fig. 1(b). The pressure P ( t ) is obtained fromthe sensor S1 and the volume is measured by V ( t ) = V − A L ( t ) with L ( t ) from the sensor S4. The total dis-placement of the piston is ∆ L = L ( τ ) = 240 mm. Thesample frequency for all sensors is set as . In theplot, we show the P-V diagram for the different pistonspeeds v , mm/s (green diamond), mm/s (blue tri-angle), and mm/s (yellow circle). It can be seen fromFig. 1(b) that the slower the pushing speed, the closer thecurve is to the quasi-static isothermal process, as shownby the red solid line. Conversely, the less heat exchangebetween the gas and the heat bath for the fast push, andthe P − V curve is closer to the adiabatic process markedwith the black solid line ( P V γ = const ). Here γ = 1 . is the heat capacity ratio for dry air [38]. The data frompressure sensors S2 and S3 are illustrated in Supplemen-tary Materials.By integrating the P − V curve, we obtain the workdone by the piston to the gas as W ( τ ) = − ˆ τ P ( t ) ˙ V ( t ) dt. (5)The work as the function of the process time τ is il-lustrated in Fig. 2(a), where the red circle and bluediamond are obtained by setting the bath temperature T e =323.15K and T e =313.15K respectively. Fig. 2(a)shows that the work approaches a stable value, whichmatches the work in the quasi-static isothermal process W q (the dash-dotted line). As shown with the log-log plot Figure 2. Work in the finite time isothermal process. (a)Work done by the piston on the gas as the function of the pro-cess time τ . The experimental results are illustrated by thered circled and blue diamond with the corresponding bathtemperature T e = 323 . K and T e = 313 . K respectively.The work W q in the quasi-static process is marked by thered (blue) dash-dotted line for T e = 323 . K (313 . K ) . Thelog-log plot of the irreversible work W (ir) as the function of di-mensionless time τ /t r is illustrated in (b) with T e = 323 . Kand (c) with T e = 313 . K. The corresponding theoreticalresult of Eq. (4) is represented by the black solid line. of the IW in Fig. 2(b) and (c), in the long time region of τ (cid:29) t r , the experimental obtained IW is in good agree-ment with the theoretical prediction of Eq. (3),which isrepresented by the black solid line. Therefore, we vali-date the behavior that the IEG is inversely proportionalto the process time in the long-time region. Effect of the control scheme - With the above compres-sion process at the constant speed, we have validated the /τ scaling of the IEG via the measurement of the IW.As predicted in the previous study [12, 36], the coefficientin the /τ scaling relation of IW not only is determinedby the system parameters of the working substance andheat bath, but also relates to the specific way how thestate of the working substance is tuned. In the following Figure 3. The volume change and the P − V diagram inthe discrete isothermal process with the 3-step case as anexample. (a) Volume changes with time in the 3-step DIPwith different push modes, where the step time is δτ = 8 s.The piston is pushed with L i = ( i/ α ∆ L, ( i = 1 , , , where L i is the displacement of the piston after the end of the i -th step. The gas volume V i = V − A L i being tuned sub-linearly ( α = 0 . ), linearly ( α = 1 ), and super-linearly ( α =3 ) are illustrated by the green dashed line, red dotted line,and blue dash-dotted line respectively. (b) P-V diagram ofthe 3-step DIP. Series of adiabatic (black dashed line) andisochoric (blue dotted line) processes are used to approacha finite-time isothermal process (red solid line). In the i -th( i = 1 , , ) step, the gas volume is firstly compressed from V i to V i +1 adiabatically, then the gas isochornically relaxes tothe thermal equilibrium state with the same temperature T e as the water bath. The experimental P − V diagram is shownin the Supplementary Materials. experiment, we will show the impact of different controlschemes on IW with our setup via a discrete isothermalprocess [17].The discrete isothermal process, introduced by An-dresen et al. in Ref. [47], is an effective approach tooptimize the finite-time Carnot engine. Since then, thediscrete step thermodynamic process have also been usedto study of different thermodynamic issues, such as workdistribution [48], thermodynamic length [49], and opti-mization of quantum heat engines [36, 50]. The basic ideaof the discrete step isothermal process (DIP) is to use aseries of adiabatic and isochoric processes to constructa finite-time isothermal process. The discrete isother-mal process has two obvious advantages, theoretically thestate of the working substance can be analytically solvedand experimentally the work and the heat exchange areseparated for direct measurement.In our setup, the piston is rapidly pushed in the i -th step to the position L i ( i = 1 , , ..., M ) to form anadiabatic process, and then relaxes to thermal equilib-rium through the isochoric process with duration δτ ,as shown in Fig. 3(a). The initial (final) piston po-sition is L = 0 ( L M = ∆ L ). For clarity, we showthree control schemes with different α with total stepnumber M = 3 and duration time δτ = 8 s as an ex-ample. At the beginning of the each adiabatic process,the gas maintains the same temperature as the waterbath, since δτ is larger than the relaxation time t r that exp[ − δτ /t r ] (cid:28) . We define the average speed of thepiston in one step as v i = ( L i − L i − ) /δτ . The steppermotor can be set to push the piston L i with a powerfunction L i = ( i/M ) α ∆ L in the i -th step.For the discrete isothemal process involving M (cid:29) steps, the IW of the system is explicitly written as [SeeSupplementary Materials for detailed derivation] W (ir) = ΛΘ M , (6)where
Θ = ( γ − P ( V f − V ) / (2 V ) relates to the ini-tial and final state of the system. And Λ = (cid:10) v (cid:11) / h v i ,characterizing the speed fluctuation of the piston, is de-termined by the control scheme of the stepper motor with (cid:10) v (cid:11) ≡ P Mi =1 v i /M and h v i = P Mi =1 v i /M . The currentgeneral formula in Eq. ( 6) recovers the result for pis-ton compressed with the constant speed noticing Λ = 1 .With the fixed process time τ = M δτ , any control schemeunder power function [36] results in the larger Λ > ,which in turn induces the larger IW W (ir) than that withthe constant speed.With the current setup, we can experimentally demon-strate the effect of the control function on the IW. Thecontrol functions are realized by the different power in-dexes α . The volume change of the gas in a 3-step DIP isillustrated in Fig. 3(a), where the green dashed line, reddotted line, and blue dash-dotted line relate to the pis-ton been pushed sub-linearly, linearly, and super-linearly,respectively. The schematic P − V curve for the DIP isillustrated in Fig. 3(b).The irreversible work done in DIP is obtained by in-tegrating area under the P-V curve, and illustrated inFig. 4(a) as a function of the total step number M forthree different power indexes α = 0 . (green triangle), . (red circle) and . (blue diamond). Each data pointshave been averaged with 20 repeats. The correspondingdashed lines show the fitting with the theoretical resultin Eq. (6). At the large- M region, the IW is inverselyproportional to M , namely, inversely proportional to thetotal time τ .To show the dependence of the IW on the control func-tion, we plot the coefficient Λ of the /M scaling in Eq.(6) as a function of the index α in Fig. 4(b). The experi-mental data for coefficient Λ , shown as diamonds in Fig.4(b), is obtained by fitting curves in 4(a) with Eq. (6)for different α at large step number M . The theoreticalresult of Eq. (6) is shown as the green circle in Fig. 4.The figure shows the agreement between the theoretical Figure 4. Irreversible work with different piston push schemesof the discrete isothermal process. The temperature of thewater bath is T e = 313 . K . (a) log-log plot of irreversiblework as the function of step number M . We demonstrate the /M scaling for three control functions with α = 3 (the bluediamond), . (green triangle), and (red circle). (b) The ob-tained parameter Λ in Eq. (6) as the function of power index α . The experiment results, represented by the red diamond,are obtained by fitting the relation of W (ir) ∼ /M . The the-oretical curve in Eq. (6) is plotted with the green dash-dottedline as a comparison. result and the experimental data. The experimental datashows a minimum irreversible work at α = 1 . We con-clude that within the set of power function, the minimalIW is achieved with the linearly control function [36],namely α = 1 as shown in Fig. 4.With the dependence of the control function Λ , wecan control the IW of the system by different schemes ofcompression to adjust the power and efficiency of the heatengine [36]. Experimentally, such tunning of irreversibleentropy generation via adjusting the mode of operation ismeaningful for the design of heat engine with high outputpower and efficiency. Conclusion-
We have designed the apparatus with thecylinder-gas system to validate the theoretically pre-dicted /τ scaling of irreversible entropy generation inthe finite-time thermodynamics. Our experiment for thefirst time directly shows that the irreversible entropy gen-eration, obtained by measuring the irreversible work, isinversely proportional to the process time τ in the long-time region [Fig. 2(b)], namely, ∆ S (ir) ∝ /τ . Moreimportantly, we demonstrated the proportional relation-ship between IEG and the speed fluctuation of the pis-ton in different gas compression schemes for the discreteisothermal process. Specifically, we verified the minimalIEG can be achieved by pushing the piston linearly withinthe set of the power control functions. This providesa feasible and convenient solution for the optimizationof the actual heat engine by applying different controlschemes to the work substance in different processes ofthe thermodynamic cycle.The similar detection of the irreversible work can alsobe realized in quantum system, such as trapped ions[40, 45, 51], NMR system [52] and superconducting cir-cuit systems [53, 54]. The generalization of the currentmeasurement in quantum regime could potentially showsthe influence of coherence on these thermodynamic quan-tities [55–57].Yu-Han Ma is grateful to Hong Yuan and Jin-Fu Chenfor helpful discussions. This work is supported by theNSFC (Grants No. 11534002 and No. 11875049), theNSAF (Grant No. U1730449 and No. U1530401), andthe National Basic Research Program of China (GrantsNo. 2016YFA0301201 and No. 2014CB921403). H.D.also thanks The Recruitment Program of Global YouthExperts of China. ∗ [email protected][1] K. Huang, Introduction To Statistical Physics, 2Nd Edi-tion (T&F/Crc Press, 2013), ISBN 978-1-4200-7902-9.[2] M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod.Phys. , 1665 (2009).[3] M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. , 771 (2011).[4] J. P. Pekola, Nat. Phys. , 118 (2015).[5] S. Vinjanampathy and J. Anders, Contemp. Phys. ,545 (2016).[6] F. Binder, L. A. Correa, C. Gogolin, J. Anders, andG. Adesso, eds., Thermodynamics in the QuantumRegime (Springer International Publishing, 2018).[7] C. M. Bender, D. C. Brody, and B. K. Meister, J. Phys.A: Math. Theor. , 4427 (2000).[8] T. E. Humphrey, R. Newbury, R. P. Taylor, and H. Linke,Phys. Rev. Lett. , 116801 (2002).[9] H. T. Quan, Y. xi Liu, C. P. Sun, and F. Nori, Phys.Rev. E , 031105 (2007).[10] Y.-H. Ma, S.-H. Su, and C.-P. Sun, Phys. Rev. E ,022143 (2017).[11] F. L. Curzon and B. Ahlborn, Am. J. Phys. , 22 (1975).[12] P. Salamon, A. Nitzan, B. Andresen, and R. S. Berry,Phys. Rev. A , 2115 (1980).[13] K. Sekimoto and S. ichi Sasa, J. Phys. Soc. Jpn , 3326(1997). [14] C. V. den Broeck, Phys. Rev. Lett. , 190602 (2005).[15] Z. C. Tu, J. Phys. A: Math. Theor. , 312003 (2008).[16] M. Esposito, R. Kawai, K. Lindenberg, and C. V. denBroeck, Phys. Rev. Lett. , 150603 (2010).[17] B. Andresen, P. Salamon, and R. S. Berry, Physics todayp. 63 (1984).[18] J. Chen, J. Phys. D Appl. Phys. , 1144 (1994).[19] A. Ryabov and V. Holubec, Phys. Rev. E , 050101(2016).[20] V. Holubec and A. Ryabov, Phys. Rev. E , 062107(2017).[21] V. Holubec and A. Ryabov, J. Stat. Mech.: Theory E. , 073204 (2016).[22] N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett. , 190601 (2016).[23] V. Cavina, A. Mari, and V. Giovannetti, Phys. Rev. Lett. , 050601 (2017).[24] Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Phys. Rev. E , 042112 (2018).[25] Z.-C. Tu, Chin. Phys. B , 020513 (2012).[26] M. H. Rubin, Phys. Rev. A , 1272 (1979).[27] B. Sahin, A. Kodal, and H. Yavuz, Energy , 1219(1996).[28] Y. Wang and Z. C. Tu, Phys. Rev. E , 011127 (2012).[29] J. Stark, K. Brandner, K. Saito, and U. Seifert, Phys.Rev. Lett. , 140601 (2014).[30] I. Derényi and R. Astumian, Phys. Rev. A , R6219(1999).[31] T. Schmiedl and U. Seifert, EPL (Europhysics Letters) , 20003 (2007).[32] Y. Izumida and K. Okuda, EPL (Europhysics Letters) , 60003 (2008).[33] U. Seifert, Rep. Prog. Phys , 126001 (2012).[34] A. Dechant, N. Kiesel, and E. Lutz, Phys. Rev. Lett. , 183602 (2015).[35] C. de Tomás, A. C. Hernández, and J. M. M. Roco, Phys.Rev. E , 010104 (2012).[36] Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Phys. Rev. E , 022133 (2018).[37] Z. Gong, Y. Lan, and H. T. Quan, Physical Review Let-ters , 180603 (2016).[38] D. E. Krause and W. J. Keeley, Phys. Teach. , 481(2004).[39] V. Blickle and C. Bechinger, Nat. Phys. , 143 (2011).[40] O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Phys. Rev. Lett. ,203006 (2012).[41] J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler,S. Krinner, C. Kollath, T. Esslinger, and A. Georges,Science , 713 (2013).[42] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, andE. Lutz, Phys. Rev. Lett. , 030602 (2014).[43] J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Nat.Nanotechnol , 358 (2014).[44] I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R.Parrondo, and R. A. Rica, Nat. Phys. , 67 (2015).[45] J. Rossnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah,E. Lutz, F. Schmidt-Kaler, and K. Singer, Science ,325 (2016).[46] S. Deng, A. Chenu, P. Diao, F. Li, S. Yu, I. Coulamy,A. del Campo, and H. Wu, Sci. Adv. , eaar5909 (2018).[47] B. Andresen, R. S. Berry, A. Nitzan, and P. Salamon,Phys. Rev. A , 2086 (1977).[48] H. T. Quan, S. Yang, and C. P. Sun, Phys. Rev. E , , 100602 (2007).[50] E. Geva and R. Kosloff, J. Chem. Phys. , 3054 (1992).[51] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q.Yin, H. Quan, and K. Kim, Nat. Phys. , 193 (2015).[52] R. J. de Assis, T. M. de Mendonça, C. J. Villas-Boas,A. M. de Souza, R. S. Sarthour, I. S. Oliveira, and N. G.de Almeida, Phys. Rev. Lett. , 240602 (2019). [53] F. Giazotto, T. T. Heikkila, A. Luukanen, A. M. Savin,and J. P. Pekola, Rev. Mod. Phys. , 217 (2006).[54] R. Uzdin and N. Katz, arXiv:1908.08968 (2019).[55] K. Brandner, M. Bauer, and U. Seifert, Phys. Rev. Lett. , 170602 (2017).[56] S. Su, J. Chen, Y. Ma, J. Chen, and C. Sun, Chin. Phys.B , 060502 (2018).[57] P. A. Camati, J. F. G. Santos, and R. M. Serra, Phys.Rev. A , 062103 (2019). upplementary Materials to “Experimental validation of the /τ -scaling entropygeneration in finite-time thermodynamics with dry air” Yu-Han Ma,
1, 2
Ruo-Xun Zhai,
3, 2
Chang-Pu Sun,
1, 2 and Hui Dong ∗ Beijing Computational Science Research Center, Beijing 100193, China Graduate School of China Academy of Engineering Physics,No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China Beijing Normal University, Beijing 100875, China
This document is devoted to providing the detailed derivations and the supporting discussions to the main contentin the Letter.
I. IRREVERSIBLE ENTROPY GENERATION AND IRREVERSIBLE WORK OF DRY AIR
In this section we derive the relation between the irreversible work and the irreversible entropy generation for thecurrent system with dry air. The irreversible entropy generation is generally defined as ∆ S (ir) = ˆ τ (cid:18) dQT s − dQT e (cid:19) (S1) = ˆ τ dU + P dVT s − T e ˆ τ ( dU + P dV ) (S2) = ˆ τ C V dT + nRT s V dVT s − T e ˆ τ ( C V dT + P dV ) (S3) = C V ln (cid:18) T s ( τ ) T e (cid:19) − C V T s ( τ ) − T e T e − ´ τ P dVT e + nR ln (cid:18) V f V (cid:19) . (S4)Under the long time limit with | T s − T e | /T e (cid:28) , the irreversible entropy generation ∆ S (ir) is simplified to the firstorder of ( T s − T e ) /T e as ∆ S (ir) = W − W q T e , (S5)where W ( τ ) = − ´ τ P dV is the total work done by the piston to the gas during the finite-time process and W q = nRT e ln( V /V f ) is the work done for quasi-static process. Here, n is the number of moles and R is the ideal gasconstant. The irreversible work is defined as W (ir) = W − W q , which is connected to the entropy generation via W (ir) = T e ∆ S (ir) . (S6)This simplification allows the direct measurement of the irreversibility via irreversible work W (ir) in the current setup. II. MEASURE THE RELAXATION TIME t r In the data analysis, the relaxation time t r can be directly determined via the relaxation process. During theisochornic process with the pressure relaxation, the change of the internal energy of the gas is caused by the heatexchange dUdt = dQ s dt = − κ ( T s − T e ) . (S7)Combining with internal energy equation dU = C V dT , we have the explicit evolution of the temperature as T s ( t ) = T e + [ T s (0) − T e ] e − t/t r , (S8) a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Figure S1. The pressure change after the fast compression. The blue triangles show the experimental data from pressure sensorS1, and the red line illustrates the fitting with the curve in Eq. (S9). The relaxation time t r = 1 . s is obtained by fitting theexperimental data.Figure S2. P − V diagram of the gas in discrete isothermal process. P − V diagram with different step number M . The bluedotted line, green dash-dotted line, and yellow dashed line relates to M = 1 , M = 3 , and M = 6 , respectively. The isothermalline of the gas is illustrated by the red solid line. where, t r = C V /κ is the relaxation time. The dynamical change of the temperature is directly reflected through thechange of the pressure via the ideal gas equation as P ( t ) = nRT s ( t ) . To measure the relaxation time, we compressthe sealed gas with the maximum speed to the final volume V f , and measure the pressure change P ( t ) . The measureddata, shown in Fig. S1, is fitted with the curve P ( t ) = nRV [ T e + [ T s (0) − T e ] e − t/t r ] . (S9)The experimental fitting results in the relaxation time t r = 1 . . III. IRREVERSIBLE ENTROPY GENERATION IN DISCRETE ISOTHERMAL PROCESS
In this section, we provide the detailed derivation of Equation (6) in the letter. The discrete isothermal process (DIP)consisting of a series of adiabatic processes and isochoric processes is used to approach the quasi-static isothermalprocess. The experimental obtained P − V diagram of the gas in DIP with different step number M are illustratedin Fig. S2.The work done by the piston to the gas in the i -th step comes only from the i -th adiabatic process as W i = − ˆ V i V i − P i ( V ) dV, (S10)where the pressure follows the adiabatic equation of idea gas as P i ( V ) = P i V γi − V γ , V ∈ [ V i − , V i ] , (S11)where γ = C P /C V is the heat capacity ratio of the gas. Initially at i -th adiabatic step, the gas is in equilibrium withthe bath i.g., T i = T e . And the pressure is P i = nk B T i /V i − . The work done at the i -th adiabatic process is obtainedexplicitly by integration, W i = − ˆ V i V i − P i V γi − V γ dV = nRT e − γ "(cid:18) V i V i − (cid:19) − γ − . (S12)And the total work is the summation over all the adiabatic process ∆ W = M X i =1 W i = − nRT e − γ M X i =1 "(cid:18) V i V i − (cid:19) − γ − . (S13)A similar result was reported in Ref. [S1], where the authors used isobaric processes instead of adiabatic processes.The volume of the sealed gas is controlled by the piston via the function V i = V − A L i , (S14)where A is the cross section of the piston. With the control, the work for each adiabatic process of Eq. (S12) becomes W i = − nRT e − γ "(cid:18) V − A L i V − A L i − (cid:19) − γ − (S15) = − nRT e − γ "(cid:18) L i − − L i V / A − L i − (cid:19) − γ − (S16) ≈ − nRT e − γ " (1 − γ ) L i − − L i V / A − L i − − γ (1 − γ )2 (cid:18) L i − − L i V / A − L i − (cid:19) (S17) = nRT e v i δτV / A − L i − + γnRT e (cid:18) v i δτV / A − L i − (cid:19) , (S18)where v i ≡ ( L i − L i − ) /δτ is the average speed of the piston in the i -th step. Then, by keeping up to the secondorder of A L i /V , we obtain the work done in the whole process as W = M X i =1 " nRT e v i δτV / A − L i + γnRT e (cid:18) v i δτV / A − L i (cid:19) (S19) ≈ nRT e A V M X i =1 v i δτ (cid:18) A L i V (cid:19) + γnRT e V M X i =1 ( v i δτ ) (S20) = nRT e A V M X i =1 v i δτ + nRT e A V M X i =1 v i δτ i X j =1 v j δτ + γnRT e A V M X i =1 ( v i δτ ) (S21) = nRT e A V M X i =1 v i δτ + nRT e A V M X i =1 v i δτ ! − M X i =1 ( v i δτ ) + γnRT e A V M X i =1 ( v i δτ ) (S22) = nRT e A L M V + nRT e A L M V + ( γ − nRT e A V M X i =1 ( v i δτ ) (S23) = nRT e ( V f − V ) V + nRT e ( V f − V ) V + ( γ − nRT e ( V f − V ) V P Mi =1 ( v i ) (cid:16)P Mi =1 v i (cid:17) . (S24)Here, L M = δτ P Mi =1 v i and V f − V = A L M are respectively the total displacement of the piston and the change ofthe gas volume in the whole process. Note that the first two terms of Eq. (S24) is just the Taylor’s expansion of thework done in quasi-static isothemal process W q = − nRT e ln( V f /V ) up to (∆ V /V ) . Consequently, the irreversiblework W (ir) = W − W q is given by W (ir) = ( γ − nRT e ( V f − V ) V P Mi =1 ( v i ) (cid:16)P Mi =1 v i (cid:17) (S25) = ( γ − P ( V f − V ) M V (cid:10) v (cid:11) h v i (S26)where (cid:10) v (cid:11) = hP Mi =1 v i i /M , and h v i = (cid:16)P Mi =1 v i (cid:17) /M . With the definitions Λ ≡ (cid:10) v (cid:11) / h v i , and Θ ≡ P ( V f − V ) / (2 V ) , the irreversible work in discrete isothermal process of Eq. (6) in the main context is obtained. IV. ADDITIONAL RESULTS FROM PRESSURE SENSORS S2 AND S3
In addition to the experimental data from pressure sensor S1 in the main context, we show the results fromthe pressure sensor S2 and S3 as following in Figures S3, and S4. These figures illustrate mainly the data at thetemperature at T e = 313 . . Similar figures for T e = 323 . can be obtained upon request. ∗ [email protected][S1] B. Andresen, R. S. Berry, A. Nitzan, and P. Salamon, Phys. Rev. A , 2086 (1977). Figure S3. The irreversible work measured in the linear control scheme with sensors S2 (left panel) and S3 (right panel).
Figure S4. The irreversible work for the discrete process and the coefficient Λ estimated from experimental data with sensorsS2 (left panel) and S3 (right panel) at temperature T e = 313 .15K