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

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

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


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  • Figure 1.  Bead-in-trap illustration: (a) realization of stochastic oscillator – an optical trap is deployed by focusing a laser beam with the objective lens. Due to the transfer of momentum from the scattering of incident photons, a colloidal bead near the trap focus will experience a force. When the bead is under stable trapping, the force can be approximated as a gradient force which is in the direction of the spatial light gradient. It is proportional to the optical intensity at the focus and pulls the particle towards the focal region. If the bead is at small displacement away from the focus, the gradient force is also proportional to the displacement. Thus, a bead-in-trap system can be modeled as a spring-mass system. As the bead in the optical trap is immersed in a fluid and is subject to fluctuation (Brownian motion), the optical trap system can be viewed as a heat bath. The temperature of the solution and the stiffness of the potential well (governed by the intensities of the beams) are two controllable parameters. (text adapted from [27]) (b) apparatus for biophysical measurement – in the famous experiment [11] [23] to verify the Jarzynski equality [20], an optical trap is deployed to measure the force exerted on a molecule of RNA which connects two beads. The RNA is subject to irreversible and reversible cycles of folding and unfolding. The actuator controls the position of the right bead and it will stretch the RNA. The optical trap will determine the force exerted on the molecule. The distance between the beads is the end-to-end length of the molecule. The length of the molecule and the exerted force on the molecule give the measurement of the work done on the molecule along different stochastic trajectories which is the essence of the experiment. (text and figure adapted from [11])

    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
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    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
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