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Stochastic Thermodynamics

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Non-equilibrium Energy Transformation Processes

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Abstract

Stochastic thermodynamics is an advancing field with many applications to small systems of current interest

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Notes

  1. 1.

    Note that the generalization of this definition for the discrete models is not very natural, because, in such case, one has to substitute a finite difference for the position derivative in the force definition.

  2. 2.

    In this section, we use the notation introduced in Sect. 2.2 for the continuous models, nevertheless the results remains valid also for the discrete models described in Sect. 2.1, indeed.

  3. 3.

    Without loose of generality we assume that \(T_{+} > T_{-}\).

  4. 4.

    It is important to note that the matrix \(\mathbb {R}_\mathrm{p}(t) \equiv \mathbb {R}_\mathrm{p}(t\,|\,0)\) is, contrary to the driving \({{\varvec{Y}}}(t)\), always continuous, i.e., \(\mathbb {R}_\mathrm{p}(t_+^-) = \mathbb {R}_\mathrm{p}(t_+^+)\) and \(\mathbb {R}_\mathrm{p}(t_{{\mathrm {p}}}^-) = \mathbb {R}_\mathrm{p}(t_{{\mathrm {p}}})\).

References

  1. Jarzynski, C. (1997). Nonequilibrium equality for free energy differences. Physical Review Letters, 78, 2690–2693, doi:10.1103/PhysRevLett.78.2690.

  2. Carberry, D. M., Reid, J. C., Wang, G. M., et. al. (2004). Fluctuations and irreversibility: An experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap. Physical Review Letters, 92, 140601, doi:10.1103/PhysRevLett.92.140601.

  3. Hänggi, P., & Marchesoni, F. (2009). Artificial Brownian motors: Controlling transport on the nanoscale. Reviews of Modern Physics, 81, 387–442, doi:10.1103/RevModPhys.81.387.

  4. Speck, T. (2011). Work distribution for the driven harmonic oscillator with time-dependent strength: Exact solution and slow driving. Journal of Physics A: Mathematical and Theoretical, 44(30), 305001, http://stacks.iop.org/1751-812144/i=30/a=305001.

  5. Bressloff, P. C., & Newby, J. M. (2013). Stochastic models of intracellular transport. Reviews of Modern Physics, 85, 135–196, doi:10.1103/RevModPhys.85.135.

  6. Jarzynski, C. (1997). Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Physical Review E, 56, 5018–5035, doi:10.1103/PhysRevE.56.5018.

  7. Crooks, G. (1998). Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. Journal of Statistical Physics, 90(5–6), 1481–1487. ISSN 0022–4715, doi:10.1023/A:1023208217925.

  8. Crooks, G. E. (1999). Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Physical Review E, 60, 2721–2726, doi:10.1103/PhysRevE.60.2721.

  9. Crooks, G. E. (2000). Path-ensemble averages in systems driven far from equilibrium. Physical Review E, 61, 2361–2366, doi:10.1103/PhysRevE.61.2361.

  10. Van Kampen, N. (2011). Stochastic processes in physics and chemistry. North-Holland Personal Library: Elsevier Science. ISBN 9780080475363, http://books.google.cz/books?id=N6II-6HlPxEC.

  11. Chvosta, P., Reineker, P., & Schulz, M. (2007). Probability distribution of work done on a two-level system during a nonequilibrium isothermal process. Physical Review E, 75, 041124, doi:10.1103/PhysRevE.75.041124.

  12. Chvosta, P., & Reineker, P. (1999) Dynamics under the influence of semi-Markov noise. Physica A: Statistical Mechanics and its Applications, 268, 103–120, ISSN 0378–4371. doi:10.1016/S0378-4371(99)00021-7, http://www.sciencedirect.com/science/article/pii/S0378437199000217.

  13. Motl, L., & Zahradník, M. (1995). Pěstujeme lineární algebru. Karolinum. ISBN 9788071841869, http://books.google.cz/books?id=0shyAAAACAAJ.

  14. Gillespie, D. T. (1992). Markov processes: An introduction for physical scientist. San Diego: Academic press, Inc.

    Google Scholar 

  15. Seifert, U. (2008). Stochastic thermodynamics: Principles and perspectives. The European Physical Journal B: Condensed Matter and Complex Systems, 64, 423–431. ISSN 1434–6028, doi:10.1140/epjb/e2008-00001-9.

  16. Imparato, A., & Peliti, L. (2005). Work probability distribution in single-molecule experiments. EPL (Europhysics Letters), 69(4), 643, http://stacks.iop.org/0295-5075/69/i=4/a=643.

  17. Imparato, A., & Peliti, L. (2005) Work distribution and path integrals in general mean-field systems. EPL (Europhysics Letters), 70(6), 740, http://stacks.iop.org/0295-5075/70/i=6/a=740.

  18. Å ubrt, E., & Chvosta, P. (2007). Exact analysis of work fluctuations in two-level systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(09), P09019. http://stacks.iop.org/1742-5468/2007/i=09/a=P09019.

  19. Imparato, A., & Peliti, L. (2005). Work-probability distribution in systems driven out of equilibrium. Physical Review E, 72, 046114, doi:10.1103/PhysRevE.72.046114.

  20. Chatelain, C., & Karevski, D. (2006). Probability distributions of the work in the two-dimensional Ising model. Journal of Statistical Mechanics: Theory and Experiment, 2006(06), P06005, http://stacks.iop.org/1742-5468/2006/i=06/a=P06005.

  21. Híjar,H., Quitana-H, J., & Sutmann, G. (2007). Non-equilibrium work theorems for the two-dimensional using model. Journal of Statistical Mechanics: Theory and Experiment, 2007(04), P04010, http://stacks.iop.org/1742-5468/2007/i=04/a=P04010.

  22. Einax, M., & Maass, P. (2009). Work distributions for Ising chains in a time-dependent magnetic field. Physical Review E, 80, 020101. doi:10.1103/PhysRevE.80.020101, http://link.aps.org/doi/10.1103/PhysRevE.80.020101.

  23. Chvosta, P., Einax, M., Holubec, V., et al., et al., (2010). Energetics and performance of a microscopic heat engine based on exact calculations of work and heat distributions. Journal of Statistical Mechanics, 2010(03), P03002, http://stacks.iop.org/1742-5468/2010i=03/a=P03002.

  24. Verley, G., Van den Broeck, C., & Esposito, M. (2013). Modulated two-level system: Exact work statistics. Physical Review E, 88, 032137, doi:10.1103/PhysRevE.88.032137.

  25. Ritort, F. (2004). Work and heat fluctuations in two-state systems. Journal of Statistical Mechanics: Theory and Experiment, 2004(10), P10016, http://stacks.iop.org/1742-5468/2004/i=10/a=P10016.

  26. Manosas, M., Mossa, A., Forns, N., et al. (2009). Dynamic force spectroscopy of DNA hairpins: II. Irreversibility and dissipation. Journal of Statistical Mechanics: Theory and Experiment, 2009(02), P02061, http://stacks.iop.org/1742-5468/2009/i=02/a=P02061.

  27. Risken, H. (1985). The Fokker-Planck equation: Methods of solution and applications. Berlin: Springer.

    Google Scholar 

  28. Sekimoto, K. (1997). Kinetic characterization of heat bath and the energetics of thermal ratchet models. Journal of the Physical Society of Japan, 66(5), 1234–1237, doi:10.1143/JPSJ.66.1234.

  29. Vilar, J. M. G., & Rubi, J. M. (2008). Failure of the Work-Hamiltonian connection for free-energy calculations. Physical Review Letters, 100, 020601, doi:10.1103/PhysRevLett.100.020601.

  30. Peliti, L. (2008). On the work-Hammiltonian connection in manipulated systems. Journal of Statistical Mechanics: Theory and Experiment, 2008(05), P05002, http://stacks.iop.org/1742-5468/2008/i=05/a=P005002.

  31. Zimanyi, E. N., & Silbey, R. J. (2009). The work-Hamiltonian connection and the usefulness of the Jarzynski equality for free energy calculations. Journal of Chemical Physics, 130(17), 171102, doi:10.1063/1.3132747,0902.3681.

  32. Vilar, J. M. G., & Rubi, J. M., (2011). Work-Hamiltonian connection for anisoparametric processes in manipulated microsystems. Journal of Non-Equilibrium Thermodynamics, 36, 123–130. doi:10.1515/jnetdy.2011.008, http://www.degruyter.com/view/j/jnet.2011.36.issue-2/jnetdy.2011.008/jnetdy.2011.008.xml.

  33. Mazonka, O., & Jarzynski, C. (1999). Exactly solvable model illustrating far-from-equilibrium predictions. eprint arXiv:cond-mat/9912121, arXiv:cond-mat/9912121, http://arxiv.org/abs/cond-mat/9912121.

  34. Baule, A., & Cohen, E. G. D. (2009). Fluctuation properties of an effective nonlinear system subject to Poisson noise. Physical Review E, 79, 030103, doi:10.1103/PhysRevE.79.030103.

  35. Engel, A. (2009). Asymptotics of work distributions in nonequilibrium systems. Physical Review E, 80, 021120. doi:10.1103/PhysRevE.80.021120, http://link.aps.org/doi/10.1103/PhysRevE.80.021120.

  36. Ryabov, A., Dierl, M., Chvosta, P., et. al. (2013). Work distribution in a time-dependent logarithmic-harmonic potential: Exact results and asymptotic analysis. Journal of Physics A: Mathematical and Theoretical, 46(7), 075002, http://stacks.iop.org/1751-8121/46/i=7/a=075002.

  37. Nickelsen, D., & Engel, A. (2011). Asymptotics of work distributions: the pre-exponential factor. The European Physical Journal B, 82(2011), 207–218. ISSN 1434–6028, doi:10.1140/epjb/e2011-20133-y.

  38. van Zon, R., & Cohen, E. G. D. (2003). Stationary and transient work-fluctuation theorems for a dragged Brownian particle. Physical Review E, 67, 046102, doi:10.1103/PhysRevE.67.046102.

  39. van Zon, R., & Cohen, E. G. D. (2004). Extended heat-fluctuation theorems for a system with deterministic and stochastic forces. Physical Review E, 69, 056121, doi:10.1103/PhysRevE.69.056121.

  40. Cohen, E. G. D. (2008). Properties of nonequilibrium steady state: A path integral approach. Journal of Statistical Mechanics: Theory and Experiment, 2008(07), P07014, http://sacks.iop.org/1742-5468/2008/i-07/a=P07014.

  41. Campisi, M., Hänggi, P., & Talkner, P. (2011). Colloquium: Quantum fluctuation relations: Foundations and applications. Reviews of Modern Physics, 83, 771–791, doi:10.1103/RevModPhys.83.771.

  42. Gibbs, J. (2010). Elementary principles in statistical mechanics: Developed with especial reference to the rational foundation of thermodynamics. Cambridge University Press: Cambridge Library Collection—Mathematics. ISBN 9781108017022, http://www.google.cz/books?id=7VbC-15f0SkC.

  43. Schurr, J. M., & Fujimoto, B. S. (2003). Equalities for the nonequilibrium work transferred from an external potential to a molecular system. Analysis of single-molecule extension experiments. The Journal of Physical Chemistry B, 107(50), 14007–14019, doi:10.1021/jp0306803.

  44. Jarzynski, C. (2007). Comparison of far-from-equilibrium work relations. Comptes Rendus Physique, 8, 495–506. doi:10.1016/j.crhy.2007.04.010, arXiv:cond-mat/0612305.

  45. Bustamante, C., Liphardt, J., & Felix, R. (2005). The nonequilibrium thermodynamics of small systems. Physics Today, 58, 43, http://dx.doi.org/10.1063/1.2012462.

  46. Ritort, F. (2006). Single-molecule experiments in biological physics: Methods and applications. Journal of Physics: Condensed Matter, 18(32), R531, http://stacks.iop.org/0953-8984/18/i=32/a=R01.

  47. Schuler, S., Speck, T., Tietz, C., et. al. (2005). Experimental test of the fluctuationtheorem for a driven two-level system with time-dependent rates. Physical Review Letters, 94, 180602, doi:10.1103/PhysRevLett.94.180602.

  48. Ritort, F. (2008). Nonequilibrium fluctuations in small systems: From physics to biology. In Advances in chemical physics (Vol.137, pp. 31-123). John Wiley & Sons Inc., ISBN 9780470238080, doi:10.1002/9780470238080.ch2.

  49. Harris, R. J., & Schü, G. M. (2007). Fluctuation theorem for stochastic dynamics. Journal of Statistical Mechanics: Theory and Experiment, 2007(07), P07020, http://stacks.iop.org/1742-5468/2007/i=07/a=P07020.

  50. Esposito, M., & Van den Broeck, C. (2010). Three detailed fluctuation theorems. Physical Review Letters, 104, 090601, doi:10.1103/PhysRevLett.104.090601.

  51. Sagawa, T., & Ueda, M. (2010). Generalized jarzynski equality under nonequilibrium feedback control. Physical Review Letters, 104, 090602, doi:10.1103/PhysRevLett.104.090602.

  52. Sagawa, T., & Ueda, M. (2012). Nonequilibrium thermodynamics of feedback control. Physical Review E, 85, 021104, doi:10.1103/PhysRevE.85.021104.

  53. Kurchan, J. (1998). Fluctuation theorem for stochastic dynamics. Journal of Physics A: Mathematical and General, 31(16), 3719, http://stacks.iop.org/0305-4470/31/i=16/a=003.

  54. Searles, D. J., & Evans, D. J. (1999). Fluctuation theorem for stochastic systems. Physical Review E, 60, 159–164, doi:10.1103/PhysRevE.60.159.

  55. Hatano, T., & Sasa, S.-I. (2001). Steady-state thermodynamics of Langevin systems. Physical Review Letters, 86, 3463–3466, doi:10.1103/PhysRevLett.86.3463.

  56. Seifert, U. (2005). Entropy production along a stochastic trajectory and an integral fluctuation theorem. Physical Review Letters, 95, 040602, doi:10.1103/PhysRevLett.95.040602.

  57. Esposito, M., & Van den Broeck, C. (2010). Three faces of the second law. I. Master equation formulation. Physical Review E, 82, 011143, doi:10.1103/PhysRevE.82.011143.

  58. Van den Broeck, C., & Esposito, M. (2010). Three faces of the second law. II. Fokker-Planck formulation. Physical Review E, 82, 011144, doi:10.1103/PhysRevE.82.011144.

  59. García-García, R., Domínguez, D., Lecomte, V., et. al. (2010). Unifying approach for fluctuation theorems from joint probability distributions. Physical Review E, 82, 030104, doi:10.1103/PhysRevE.82.030104.

  60. Kurchan, J. (2007). Non-equilibrium work relations. Journal of Statistical Mechanics: Theory and Experiments, 2007(07), P07005, http://stacks.iop.org/1742-5468/2007/i=07/a=P07005.

  61. Baule, A., & Cohen, E. G. D. (2009). Steady-state work fluctuations of a dragged particle under external and thermal noise. Physical Review E, 80, 011110, doi:10.1103/PhysRevE.80.011110.

  62. Bochkov, G., & Kuzovlev, Y. (1981). Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem. Physica A: Statistical Mechanics and its Applications, 106(3), 443–479, ISSN 0378–4371, doi:10.1016/0378-4371(81)90122-9.

  63. Bochkov, G., & Kuzovlev, Y. (1981). Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes. Physica A: Statistical Mechanics and its Applications, 106(3), 480–520, ISSN 0378–4371, doi:10.1016/0378-4371(81)90123-0.

  64. Hummer, G., & Szabo, A. (2001). Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proceedings of the National Academy of Sciences, 98(7), 3658–3661. doi:10.1073/pnas.071034098, http://www.pnas.org/content/98/7/3658.full.pdf+html, http://www.pnas.org/content/98/7/3658.abstract.

  65. Mossa, A., Manosas, M., Forns, N., et al. (2009). Dynamic forces spectroscopy of DNA hairpins: I. Force Kinetics and free energy landscapes. Journal of Statistical Mechanics: Theory and Experiment, 2009(02), P02060, http://stacks.iop.org/1742-5468/2009/i=02/a=P02060.

  66. Nostheide, S., Holubec, V., Chvosta, P., et. al. (2013). Unfolding kinetics of periodic DNA hairpins. arXiv preprint arXiv:1312.4146, arxiv.org/abs/1312.4146.

  67. Crooks, G. E., & Jarzynski, C. (2007). Work distribution for the adiabatic compression of a dilute and interacting classical gas. Physical Review E, 75, 021116, doi:10.1103/PhysRevE.75.021116.

  68. Liphardt, J., Dumont, S., Smith, S. B., et. al. (2002). Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science, 296(5574), 1832–1835. doi:10.1126/science.1071152, http://www.sciencemag.org/content/296/5574/1832.full.pdf, http://www.sciencemag.org/content/296/5574/1832.abstract.

  69. Collin, D., Ritort, F., Jarzynski, C., et al. (2005). Verification of the crooks fluctuation theorem and recovery of RNA folding free energies. Nature, 473, 231, http://dx.doi.org/10.1038/nature04061M3.

  70. Blickle, V., Speck, T., Helden, L., et. al. (2006). Thermodynamics of a colloidal particle in a time-dependent nonharmonic potential. Physical Review Letters, 96, 070603, doi:10.1103/PhysRevLett.96.070603.

  71. Gomez-Solano, J. R., Bellon, L., Petrosyan, A. et. al. (2010). Steady-state fluctuation relations for systems driven by an external random force. EPL (Europhysics Letters), 89(6), 60003, http://stacks.iop.org/0295-5075/89/i=6/a=60003.

  72. Xiao, T. J., Hou, Z., & Xin, H. (2008). Entropy production and fluctuation theorem along a stochastic limit cycle. The Journal of Chemical Physics, 129(11), 114506. doi:10.1063/1.2978179, http://link.aip.org/link/?JCP/129/114506/1.

  73. Holubec, V., Chvosta, P., Einax, M., et al. (2011). Attempt time Monte Carlo: An alternative for simulation of stochastic jump processes with time-dependent transition rates. EPL (Europhysics Letters), 93(4), 40003, http://stacks.iop.org/0295-5075/93/i=4/a=40003.

  74. Sevick, E., Prabhakar, R., Williams, S. R., et. al. (2008). Fluctuation theorems. Annual Review of Physical Chemistry, 59(1), 603–633. doi:10.1146/annurev.physchem.58.032806.104555, http://www.annualreviews.org/doi/pdf/10.1146/annurev.physchem.58.032806.104555.

  75. Manosas, M., & Ritort, F. (2005). Thermodynamic and kinetic aspects of RNA pulling experiments. Biophysical Journal, 88(5), 3224–3242, ISSN 0006–3495. doi:10.1529/biophysj.104.045344, http://www.sciencedirect.com/science/article/pii/S0006349505733749.

  76. Maragakis, P., Ritort, F., Bustamante, C., et. al. (2008). Bayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise. The Journal of Chemical Physics, 129(2), 024102. doi:10.1063/1.2937892, http://link.aip.org/link/?JCP/129/024102/1.

  77. Calderon, C. P., Harris, N. C., Kiang, C.-H., et. al. (2009). Quantifying multiscale noise sources in single-molecule time series. The Journal of Physical Chemistry B, 113(1), 138–148, doi:10.1021/jp807908c.

  78. Alemany, A., Ribezzi, M., Ritort, F. (2011). Recent progress in fluctuation theorems and free energy recovery. AIP Conference Proceedings, 1332(1), 96–110. doi:10.1063/1.3569489, http://link.aip.org/link/?APC/1332/96/1.

  79. Takagi, F., & Hondou, T. (1999). Thermal noise can facilitate energy conversion by a ratchet system. Physical Review E, 60, 4954–4957, doi:10.1103/PhysRevE.60.4954.

  80. Sekimoto, K., Takagi, F., & Hondou, T. (2000). Carnot’s cycle for small systems: Irreversibility and cost of operations. Physical Review E, 62, 7759–7768, doi:10.1103/PhysRevE.62.7759.

  81. Van den Broeck, C., Kawai, R., & Meurs, P. (2004). Microscopic analysis of a thermal Brownian motor. Physical Review Letters, 93, 090601, doi:10.1103/PhysRevLett.93.090601.

  82. Schmiedl, T., & Seifert, U. (2008). Efficiency at maximum power: An analytically solvable model for stochastic heat engines. EPL (Europhysics Letters), 81(2), 20003, http://stacks.iop.org/0295-5075/81/i=2/a=20003.

  83. Esposito, M., Lindenberg, K., & Van den Broeck, C. (2009). Universality of Efficiency at Maximum Power. Physical Review Letters, 102, 130602, doi:10.1103/PhysRevLett.102.130602.

  84. Esposito, M., Kawai, R., Lindenberg, K., et. al. (2010). Efficiency at maximum power of low-dissipation carnot engines. Physical Review Letters, 105, 150603, doi:10.1103/PhysRevLett.105.150603.

  85. Blickle, V., & Bechinger, C. (2011). Realization of a micrometre-sized stochastic heat engine. Nature Physics, 8(2), 143–146, doi:10.1038/nphys2163.

  86. Seifert, U. (2012). Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on Progress in Physics, 75(12), 126001, http://stacks.iop.org/0034-4885/75/i=12/a=126001.

  87. Zhan-Chun, T. (2012). Recent advance on the efficiency at maximum power of heat engines. Chinese Physics B, 21(2), 020513, http://stacks.iop.org/1674-1056/21/i=2/a=020513.

  88. Arnaud, J., Chusseau, L., & Philippe, F. (2010). A simple model for Carnot heat engines. American Journal of Physics, 78(1), 106–110. doi:10.1119/1.3247983, http://link.aip.org/link/?AJP/78/106/1.

  89. Esposito, M., Kawai, R., Lindenberg, K., et. al. (2010). Quantum-dot carnot engine at maximum power. Physical Review E, 81, 041106, doi:10.1103/PhysRevE.81.041106.

  90. Rezek, Y., & Kosloff, R. (2006). Irreversible performance of a quantum harmonic heat engine. New Journal of Physics, 8(5), 83, http://stacks.iop.org/1367-2630/8/i=5/a=083.

  91. Henrich, M. J., Rempp, F., & Mahler, G. (2007). Quantum thermodynamic Otto machines: A spin-system approach. The European Physical Journal Special Topics, 151(1), 157–165, ISSN 1951–6355. doi:10.1140/epjst/e2007-00371-8. http://dx.doi.org/10.1140/epjst/e2007-00371-8.

  92. Allahverdyan, A. E., Johal, R. S., & Mahler, G. (2008). Work extremum principle: Structure and function of quantum heat engines. Physicak Review E, 77, 041118, doi:10.1103/PhysRevE.77.041118.

  93. Abah, O., Roßnagel, J., Jacob, G., et. al. (2012). Single-Ion heat engine at maximum power. Physical Review Letters, 109, 203006, doi:10.1103/PhysRevLett.109.203006.

  94. Sinitsyn, N. A. (2011). Fluctuation relation for heat engines. Journal of Physics A: Mathematical and Theoretical, 44(40), 405001, http://stacks.iop.org/1751-8121/44/i=40/a=405001.

  95. Lahiri, S., Rana, S., & Jayannavar, A. M. (2012). Fluctuation relations for heat engines in time-periodic steady states. Journal of Physics A: Mathematical and Theoretical, 45(46), 465001, http://stacks.iop.org/1751-8121/45/i=46/a=465001.

  96. Chambadal, P. (1957). Les Centrales nucléaires. Colin: Collection Armand Colin, http://books.google.cz/books?id=TX8KAAAAMAAJ.

  97. Novikov, I. I. (1958). The efficiency of atomic power stations. Journal of Nuclear Energy II, 7, 125.

    Google Scholar 

  98. Curzon, F. L., & Ahlborn, B. (1975). Efficiency of a Carnot engine at maximum power output. American Journal of Physics, 43(1), 22–24. doi:10.1119/1.10023, http://link.aip.org/link/?AJP/43/22/1.

  99. Durmayaz, A., Sogut, O. S., Sahin, B., et. al. (2004). Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics. Progress in Energy and Combustion Science, 30(2), 175–217, ISSN 0360–1285. doi:10.1016/j.pecs.2003.10.003, http://www.sciencedirect.com/science/article/pii/S0360128503000777.

  100. Tu, Z. C.(2008). Efficiency at maximum power of Feynman’s ratchet as a heat engine. Journal of Physics A: Mathematical and Theoretical, 41(31), 312003, http://stacks.iop.org/1751-8121/41/i=31/a=312003.

  101. Callen, H. (2006). Thermodynamics & an intro to thermostatistics. (Student ed.), New York: Wiley India Pvt. Limited, ISBN 9788126508129. http://books.google.cz/books?id=uOiZB_2y5pIC.

  102. Holubec, V. (2009). Nonequilibrium thermodynamics of small systems. Diploma Thesis, Charles University in Prague, Faculty of Mathematics and Physics.

    Google Scholar 

  103. Chvosta, P., Holubec, V., Ryabov, A., et. al. (2010). Thermodynamics of two-stroke engine based on periodically driven two-level system. Physica E: Low-dimensional Systems and Nanostructures, 42(3), 472–476, ISSN 1386–9477. http://dx.doi.org/10.1016/j.physe.2009.06.031; Proceedings of the International Conference Frontiers of Quantum and Mesoscopic Thermodynamics FQMT ’08, http://www.sciencedirect.com/science/article/pii/S13869 47709002380.

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Holubec, V. (2014). Stochastic Thermodynamics. In: Non-equilibrium Energy Transformation Processes. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07091-9_2

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