# American Institute of Mathematical Sciences

April  2020, 13(4): 1243-1268. doi: 10.3934/dcdss.2020072

## Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator

 Institute for Systems Research, University of Maryland, College Park, MD 20742, USA

* Corresponding authors: Yunlong Huang and P. S. Krishnaprasad

Received  February 2018 Revised  September 2018 Published  April 2019

The field of sub-Riemannian geometry has flourished in the past four decades through the strong interactions between problems arising in applied science (in areas such as robotics) and questions of a pure mathematical character about the nature of space. Methods of control theory, such as controllability properties determined by Lie brackets of vector fields, the Hamilton equations associated to the Maximum Principle of optimal control, Hamilton-Jacobi-Bellman equation etc. have all been found to be basic tools for answering such questions. In this paper, we find a useful role for the vantage point of sub-Riemannian geometry in attacking a problem of interest in non-equilibrium statistical mechanics: how does one create rules for operation of micro- and nano-scale systems (heat engines) that are subject to fluctuations from the surroundings, so as to be able to do useful things such as converting heat into work over a cycle of operation? We exploit geometric optimal control theory to produce such rules selected for maximal efficiency. This is done by working concretely with a model problem, the stochastic oscillator. Essential to our work is a separation of time scales used with great efficacy by physicists and justified in the linear response regime.

Citation: Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072
##### References:
 [1] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar [2] A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, 2014. Available from: https://webusers.imj-prg.fr/~davide.barilari/ABB-SRnotes-110715.pdf. Google Scholar [3] P. Bamberg and S. Sternberg, A Course in Mathematics for Students of Physics: 2, Cambridge University Press, Cambridge, 1991.   Google Scholar [4] A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. Google Scholar [5] M. Born, Natural Philosophy of Cause and Chance, Dover, New York, 1964. Google Scholar [6] R. W. Brockett, Control Theory and Singular Riemannian Geometry, New Directions in Applied Mathematics (eds. P. J. Hilton and G. S. Young), Springer-Verlag, (1982), New York, 11–27.  Google Scholar [7] R. W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians (eds. Z. Ciesielski and C. Olech), Polish Scientific Publishers, (1984), Warszawa, 1357–1368.  Google Scholar [8] R. W. Brockett, Control of stochastic ensembles, The Astrom Symposium on Control(eds. B. Wittenmark and A. Rantzer), Studentlitteretur, (1999), Lund, 199–216. Google Scholar [9] R. W. Brockett, Thermodynamics with time: Exergy and passivity, Systems and Control Letters, 101 (2017), 44-49.  doi: 10.1016/j.sysconle.2016.06.009.  Google Scholar [10] R. W. Brockett and J. C. Willems, Stochastic Control and the Second Law of Thermodynamics, Proceedings of the 17th IEEE Conference on Decision and Control, IEEE, (1978), New York, 1007–1011. doi: 10.1109/CDC.1978.268083.  Google Scholar [11] C. Bustamante, J. Liphardt and F. Ritort, The non-equilibrium thermodynamics of small systems, Physics Today, 58, 7, 43 (2005). Google Scholar [12] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Mathematische Annalen, 67 (1909), 355-386.  doi: 10.1007/BF01450409.  Google Scholar [13] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York, N. Y. 1957.  Google Scholar [14] M. Chen and C. J. Tomlin, Hamilton-Jacobi reachability: Some recent theoretical advances and applications in unmanned airspace management, Annual Review of Control, Robotics, and Autonomous Systems, 1 (2018), 333-358.  doi: 10.1146/annurev-control-060117-104941.  Google Scholar [15] W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Mathematische Annalen, 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar [16] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.  Google Scholar [17] M. Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Prog. Math.(eds, A. Bellaiche and J-J. Risler), Birkhäuser, Basel, 144 (1996), 79–323.  Google Scholar [18] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on Structures Metriques des Varietes Riemanniennes (eds. J. LaFontaine and P. Pansu), 1981, English Translation by Sean M. Bates, Birkhäuser, Boston. Google Scholar [19] R. Hermann, Differential Geometry and the Calculus of Variations, Series: Mathematics in Science and Engineering, 49, Academic Press, New York, 1968.  Google Scholar [20] C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78 (1997), 2690. Google Scholar [21] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, UK, 1997.   Google Scholar [22] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton and Oxford, 2012.   Google Scholar [23] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr and C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality, Science, 296 (2002), 1832-1835.  doi: 10.1126/science.1071152.  Google Scholar [24] I. Mitchell, The flexible, extensible and efficient toolbox of level set methods, Journal of Scientific Computing, 35 (2008), 300-329.  doi: 10.1007/s10915-007-9174-4.  Google Scholar [25] R. Montgomery, Review of M. Gromov, Carnot-Carathéodory Spaces Seen from Within, Mathematical Reviews, 53C17 (53C23) featured review, 2000, MathSciNet, American Mathematical Society. Google Scholar [26] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, RI., 2002.  Google Scholar [27] K. C. Neuman and S. M. Block, Optical trapping, Review of Scientific Instruments, 75 (2004), 2787. doi: 10.1063/1.1785844.  Google Scholar [28] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2003. doi: 10.1007/b98879.  Google Scholar [29] S. Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM Journal of Mathematical Analysis, 24 (1993), 1145-1152.  doi: 10.1137/0524066.  Google Scholar [30] B. Øksendal, Stochastic Differential Equations, Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar [31] R. K. Pathria and P. D. Beale, Statistical Mechanics, 3$^{rd}$ edition, Elsevier, Burlington MA, 2011. Google Scholar [32] P. K. Rashevskii, About connecting two points of complete non-holonomic space by admissible curve (in Russian), Uch. Zapiski Ped. Inst. Libknexta, 2 (1938), 83-94.   Google Scholar [33] D. A. Sivak and G. E. Crooks, Thermodynamic metric and optimal paths, Physical Review Letters, 108 (2012), 190602.  doi: 10.1103/PhysRevLett.108.190602.  Google Scholar [34] J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351.  doi: 10.1007/BF00276493.  Google Scholar [35] P. R. Zulkowski, The Geometry of Thermodynamic Control, Ph.D thesis, University of California, Berkeley, 2014. Google Scholar [36] P. R. Zulkowski, D. A. Sivak, G. E. Crooks and M. R. DeWeese, Geometry of thermodynamic control, Physical Review E, 86 (2012), 041148. doi: 10.1103/PhysRevE.86.041148.  Google Scholar [37] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.   Google Scholar

show all references

##### References:
 [1] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar [2] A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, 2014. Available from: https://webusers.imj-prg.fr/~davide.barilari/ABB-SRnotes-110715.pdf. Google Scholar [3] P. Bamberg and S. Sternberg, A Course in Mathematics for Students of Physics: 2, Cambridge University Press, Cambridge, 1991.   Google Scholar [4] A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. Google Scholar [5] M. Born, Natural Philosophy of Cause and Chance, Dover, New York, 1964. Google Scholar [6] R. W. Brockett, Control Theory and Singular Riemannian Geometry, New Directions in Applied Mathematics (eds. P. J. Hilton and G. S. Young), Springer-Verlag, (1982), New York, 11–27.  Google Scholar [7] R. W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians (eds. Z. Ciesielski and C. Olech), Polish Scientific Publishers, (1984), Warszawa, 1357–1368.  Google Scholar [8] R. W. Brockett, Control of stochastic ensembles, The Astrom Symposium on Control(eds. B. Wittenmark and A. Rantzer), Studentlitteretur, (1999), Lund, 199–216. Google Scholar [9] R. W. Brockett, Thermodynamics with time: Exergy and passivity, Systems and Control Letters, 101 (2017), 44-49.  doi: 10.1016/j.sysconle.2016.06.009.  Google Scholar [10] R. W. Brockett and J. C. Willems, Stochastic Control and the Second Law of Thermodynamics, Proceedings of the 17th IEEE Conference on Decision and Control, IEEE, (1978), New York, 1007–1011. doi: 10.1109/CDC.1978.268083.  Google Scholar [11] C. Bustamante, J. Liphardt and F. Ritort, The non-equilibrium thermodynamics of small systems, Physics Today, 58, 7, 43 (2005). Google Scholar [12] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Mathematische Annalen, 67 (1909), 355-386.  doi: 10.1007/BF01450409.  Google Scholar [13] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York, N. Y. 1957.  Google Scholar [14] M. Chen and C. J. Tomlin, Hamilton-Jacobi reachability: Some recent theoretical advances and applications in unmanned airspace management, Annual Review of Control, Robotics, and Autonomous Systems, 1 (2018), 333-358.  doi: 10.1146/annurev-control-060117-104941.  Google Scholar [15] W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Mathematische Annalen, 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar [16] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.  Google Scholar [17] M. Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Prog. Math.(eds, A. Bellaiche and J-J. Risler), Birkhäuser, Basel, 144 (1996), 79–323.  Google Scholar [18] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on Structures Metriques des Varietes Riemanniennes (eds. J. LaFontaine and P. Pansu), 1981, English Translation by Sean M. Bates, Birkhäuser, Boston. Google Scholar [19] R. Hermann, Differential Geometry and the Calculus of Variations, Series: Mathematics in Science and Engineering, 49, Academic Press, New York, 1968.  Google Scholar [20] C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78 (1997), 2690. Google Scholar [21] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, UK, 1997.   Google Scholar [22] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton and Oxford, 2012.   Google Scholar [23] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr and C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality, Science, 296 (2002), 1832-1835.  doi: 10.1126/science.1071152.  Google Scholar [24] I. Mitchell, The flexible, extensible and efficient toolbox of level set methods, Journal of Scientific Computing, 35 (2008), 300-329.  doi: 10.1007/s10915-007-9174-4.  Google Scholar [25] R. Montgomery, Review of M. Gromov, Carnot-Carathéodory Spaces Seen from Within, Mathematical Reviews, 53C17 (53C23) featured review, 2000, MathSciNet, American Mathematical Society. Google Scholar [26] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, RI., 2002.  Google Scholar [27] K. C. Neuman and S. M. Block, Optical trapping, Review of Scientific Instruments, 75 (2004), 2787. doi: 10.1063/1.1785844.  Google Scholar [28] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2003. doi: 10.1007/b98879.  Google Scholar [29] S. Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM Journal of Mathematical Analysis, 24 (1993), 1145-1152.  doi: 10.1137/0524066.  Google Scholar [30] B. Øksendal, Stochastic Differential Equations, Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar [31] R. K. Pathria and P. D. Beale, Statistical Mechanics, 3$^{rd}$ edition, Elsevier, Burlington MA, 2011. Google Scholar [32] P. K. Rashevskii, About connecting two points of complete non-holonomic space by admissible curve (in Russian), Uch. Zapiski Ped. Inst. Libknexta, 2 (1938), 83-94.   Google Scholar [33] D. A. Sivak and G. E. Crooks, Thermodynamic metric and optimal paths, Physical Review Letters, 108 (2012), 190602.  doi: 10.1103/PhysRevLett.108.190602.  Google Scholar [34] J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Archive for Rational Mechanics and Analysis, 45 (1972), 321-351.  doi: 10.1007/BF00276493.  Google Scholar [35] P. R. Zulkowski, The Geometry of Thermodynamic Control, Ph.D thesis, University of California, Berkeley, 2014. Google Scholar [36] P. R. Zulkowski, D. A. Sivak, G. E. Crooks and M. R. DeWeese, Geometry of thermodynamic control, Physical Review E, 86 (2012), 041148. doi: 10.1103/PhysRevE.86.041148.  Google Scholar [37] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.   Google Scholar
Reachable set of a stochastic oscillator in 3D
Reconstruction of a working loop
Efficiencies of the engine along the maximum efficiency working loops
 Point number Extracted mechanical work Heat supply Dissipation $\eta$ 1 0.1207 0.8319 1.1126 0.1280 2 0.1991 1.2085 1.5302 0.1462 3 0.2775 1.5055 1.8331 0.1643 4 0.3560 1.7363 2.0515 0.1834 5 0.4344 1.9179 2.2134 0.2031 6 0.5128 2.0771 2.3471 0.2218 7 0.5913 2.2568 2.4982 0.2359 8 0.6697 2.4464 2.6546 0.2470 9 0.7481 2.6362 2.8082 0.2565 10 0.8266 2.8185 2.9525 0.2655
 Point number Extracted mechanical work Heat supply Dissipation $\eta$ 1 0.1207 0.8319 1.1126 0.1280 2 0.1991 1.2085 1.5302 0.1462 3 0.2775 1.5055 1.8331 0.1643 4 0.3560 1.7363 2.0515 0.1834 5 0.4344 1.9179 2.2134 0.2031 6 0.5128 2.0771 2.3471 0.2218 7 0.5913 2.2568 2.4982 0.2359 8 0.6697 2.4464 2.6546 0.2470 9 0.7481 2.6362 2.8082 0.2565 10 0.8266 2.8185 2.9525 0.2655
Information from the reachable set of the stochastic oscillator
 Point number distance $\tilde{\psi}-coordinate$ 1 1.1126 0.1207 2 1.5302 0.1991 3 1.8331 0.2775 4 2.0515 0.356 5 2.2134 0.4344 6 2.3471 0.5128 7 2.4982 0.5913 8 2.6546 0.6697 9 2.8082 0.7481 10 2.9525 0.8266
 Point number distance $\tilde{\psi}-coordinate$ 1 1.1126 0.1207 2 1.5302 0.1991 3 1.8331 0.2775 4 2.0515 0.356 5 2.2134 0.4344 6 2.3471 0.5128 7 2.4982 0.5913 8 2.6546 0.6697 9 2.8082 0.7481 10 2.9525 0.8266
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