Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature Variations

We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This set-up covers, in particular, periodic micro and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck type dynamics, we show that these coefficients can be expressed as a sum of an adiabatic contribution and one reminiscent of a Green-Kubo expression that contains deviations from adiabaticity. Furthermore, we show that the generalized kinetic coefficients fulfill an Onsager-Casimir type symmetry tracing back to microscopic reversibility. This symmetry allows for non-identical off-diagonal coefficients if the driving protocols are not symmetric under time-reversal. We then derive a novel constraint on the kinetic coefficients that is sharper than the second law and provides an efficiency-dependent bound on power. As one consequence, we can prove that the power vanishes at least linearly when approaching Carnot efficiency. We illustrate our general framework by explicitly working out the paradigmatic case of a Brownian heat engine realized by a colloidal particle in a time-dependent harmonic trap subject to a periodic temperature profile. This case study reveals inter alia that our new general bound on power is asymptotically tight.

[1]  K. Okuda,et al.  Onsager coefficients of a Brownian Carnot cycle , 2010, 1006.2589.

[2]  Z. Tu.  Stochastic heat engine with the consideration of inertial effects and shortcuts to adiabaticity. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  C. Broeck,et al.  Thermodynamic efficiency at maximum power. , 2005 .

[4]  Udo Seifert,et al.  Multi-terminal thermoelectric transport in a magnetic field: bounds on Onsager coefficients and efficiency , 2013, 1308.2179.

[5]  Robert S. Whitney,et al.  Most efficient quantum thermoelectric at finite power output. , 2013, Physical review letters.

[6]  T. Prosen,et al.  Fundamental aspects of steady state heat to work conversion , 2013, 1311.4430.

[7]  U. Seifert,et al.  Extracting work from a single heat bath through feedback , 2011, 1102.3826.

[8]  U. Seifert.  Stochastic thermodynamics, fluctuation theorems and molecular machines , 2012, Reports on progress in physics. Physical Society.

[9]  L. Onsager.  Reciprocal Relations in Irreversible Processes. II. , 1931 .

[10]  Udo Seifert,et al.  Bound on thermoelectric power in a magnetic field within linear response. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Peter Jung,et al.  Periodically driven stochastic systems , 1993 .

[12]  U. Seifert,et al.  Optimal potentials for temperature ratchets. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  U. Seifert,et al.  Classical Nernst engine. , 2013, Physical review letters.

[14]  Rafael Sánchez,et al.  Thermoelectric energy harvesting with quantum dots , 2014, Nanotechnology.

[15]  Massimiliano Esposito,et al.  Quantum-dot Carnot engine at maximum power. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  C. Van den Broeck,et al.  Efficiency of isothermal molecular machines at maximum power. , 2012, Physical review letters.

[17]  Armen E Allahverdyan,et al.  Carnot cycle at finite power: attainability of maximal efficiency. , 2013, Physical review letters.

[18]  A. Kociszewski,et al.  On the Fokker-Planck equation with time-dependent drift and diffusion coefficients and its exponential solutions , 1985 .

[19]  Clemens Bechinger,et al.  Realization of a micrometre-sized stochastic heat engine , 2011, Nature Physics.

[20]  Massimiliano Esposito,et al.  Universality of efficiency at maximum power. , 2009, Physical review letters.

[21]  H. Risken.  Fokker-Planck Equation , 1996 .

[22]  Udo Seifert,et al.  Efficiency of a Brownian information machine , 2012, 1203.0184.

[23]  H. Linke,et al.  Reversible thermoelectric nanomaterials. , 2004, Physical Review Letters.

[24]  Udo Seifert,et al.  Efficiency at maximum power: An analytically solvable model for stochastic heat engines , 2007, 0710.4097.

[25]  F. Ritort,et al.  Free-energy inference from partial work measurements in small systems , 2014, Proceedings of the National Academy of Sciences.

[26]  Giuliano Benenti,et al.  Thermodynamic bounds on efficiency for systems with broken time-reversal symmetry. , 2011, Physical review letters.

[27]  F. Curzon,et al.  Efficiency of a Carnot engine at maximum power output , 1975 .

[28]  S. Sharma,et al.  The Fokker-Planck Equation , 2010 .

[29]  J. Pekola.  Towards quantum thermodynamics in electronic circuits , 2015, Nature Physics.

[30]  V. Holubec.  An exactly solvable model of a stochastic heat engine: optimization of power, power fluctuations and efficiency , 2014, 1404.2030.

[31]  J. Koski,et al.  Experimental realization of a Szilard engine with a single electron , 2014, Proceedings of the National Academy of Sciences.

[32]  U. Seifert,et al.  Optimal finite-time processes in stochastic thermodynamics. , 2007, Physical review letters.

[33]  H. Callen.  Thermodynamics and an Introduction to Thermostatistics , 1988 .

[34]  Quantum, cyclic, and particle-exchange heat engines , 2004, cond-mat/0407508.

[35]  K. Okuda,et al.  Linear irreversible heat engines based on local equilibrium assumptions , 2015, 1501.03987.

[36]  Dmitri Petrov,et al.  Adiabatic processes realized with a trapped Brownian particle. , 2014, Physical review letters.

[37]  J. Rossnagel,et al.  Single-ion heat engine at maximum power. , 2012, Physical review letters.

[38]  U. Seifert.  Efficiency of autonomous soft nanomachines at maximum power. , 2010, Physical review letters.

[40]  T. Prosen,et al.  Thermopower with broken time-reversal symmetry , 2011, 1107.1431.

[41]  R. Whitney.  Finding the quantum thermoelectric with maximal efficiency and minimal entropy production at given power output , 2014, 1408.3348.

[42]  P. Hänggi,et al.  Optimal strategy for controlling transport in inertial Brownian motors , 2006, Journal of physics. Condensed matter : an Institute of Physics journal.

[43]  Koji Okuda,et al.  Onsager coefficients of a finite-time Carnot cycle. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  J.T.M. van Beek,et al.  Piezoresistive heat engine and refrigerator , 2010, 1001.3170.

[45]  H Linke,et al.  Reversible quantum brownian heat engines for electrons. , 2002, Physical review letters.

[46]  David Sánchez,et al.  Thermoelectric transport of mesoscopic conductors coupled to voltage and thermal probes , 2011, 1107.3576.

[47]  Massimiliano Esposito,et al.  Thermoelectric efficiency at maximum power in a quantum dot , 2008, 0808.0216.

[48]  Jingning Zhang,et al.  Experimental test of the quantum Jarzynski equality with a trapped-ion system , 2014, Nature Physics.

[49]  Sumiyoshi Abe.  Maximum-power quantum-mechanical Carnot engine. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Udo Seifert,et al.  Strong bounds on Onsager coefficients and efficiency for three-terminal thermoelectric transport in a magnetic field. , 2013, Physical review letters.