Article Contents
Article Contents

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

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

Mathematics Subject Classification: Primary: 49K15, 93E20; Secondary: 82C05.

 Citation:

Figure 2.  Reachable set of a stochastic oscillator in 3D

Figure 3.  Reconstruction of a working loop

Table 2.  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

Table 1.  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
•  [1] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [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. [3] P. Bamberg and  S. Sternberg,  A Course in Mathematics for Students of Physics: 2, Cambridge University Press, Cambridge, 1991. [4] A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. [5] M. Born, Natural Philosophy of Cause and Chance, Dover, New York, 1964. [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. [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. [8] R. W. Brockett, Control of stochastic ensembles, The Astrom Symposium on Control(eds. B. Wittenmark and A. Rantzer), Studentlitteretur, (1999), Lund, 199–216. [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. [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. [11] C. Bustamante, J. Liphardt and F. Ritort, The non-equilibrium thermodynamics of small systems, Physics Today, 58, 7, 43 (2005). [12] C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Mathematische Annalen, 67 (1909), 355-386.  doi: 10.1007/BF01450409. [13] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York, N. Y. 1957. [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. [15] W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Mathematische Annalen, 117 (1939), 98-105.  doi: 10.1007/BF01450011. [16] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992. [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. [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. [19] R. Hermann, Differential Geometry and the Calculus of Variations, Series: Mathematics in Science and Engineering, 49, Academic Press, New York, 1968. [20] C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78 (1997), 2690. [21] V. Jurdjevic,  Geometric Control Theory, Cambridge University Press, Cambridge, UK, 1997. [22] D. Liberzon,  Calculus of Variations and Optimal Control Theory, Princeton University Press, Princeton and Oxford, 2012. [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. [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. [25] R. Montgomery, Review of M. Gromov, Carnot-Carathéodory Spaces Seen from Within, Mathematical Reviews, 53C17 (53C23) featured review, 2000, MathSciNet, American Mathematical Society. [26] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, RI., 2002. [27] K. C. Neuman and S. M. Block, Optical trapping, Review of Scientific Instruments, 75 (2004), 2787. doi: 10.1063/1.1785844. [28] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2003. doi: 10.1007/b98879. [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. [30] B. Øksendal, Stochastic Differential Equations, Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4. [31] R. K. Pathria and P. D. Beale, Statistical Mechanics, 3$^{rd}$ edition, Elsevier, Burlington MA, 2011. [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. [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. [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. [35] P. R. Zulkowski, The Geometry of Thermodynamic Control, Ph.D thesis, University of California, Berkeley, 2014. [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. [37] R. Zwanzig,  Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.

Figures(3)

Tables(2)