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Work Distribution in Logarithmic-Harmonic Potential

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Stochastic Dynamics and Energetics of Biomolecular Systems

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Abstract

In this chapter we address a model of a particle diffusing in an harmonic potential for which the work characteristic function can be obtained exactly. Using Lie-algebraic approach, the Fokker-Planck equation for the joint PDF of work and position is reduced to the Riccati equation, which, for a specific driving protocol, is solved exactly in terms of elementary functions yielding desired information about work PDF including all its moments and the both its tails.

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Notes

  1. 1.

    This result agrees with Eq. (19) in Ref. [31], where it has been derived in connection with a diffusion problem with logarithmic factors in drift and diffusion coefficients. See also Refs. [32, 48].

  2. 2.

    For \(t=2\) and \(g=0\) we obtain \(D\xi _{0}(2) \dot{=} 1.827\), \(r(2)/\Gamma (1/2)\dot{=}1.021\), which is in perfect agreement with Eq. (5.30) in [26].

References

  1. M. Manosas, A. Mossa, N. Forns, J.M. Huguet, F. Ritort, Dynamic force spectroscopy of DNA hairpins: II. Irreversibility and dissipation. J. Stat. Mech. P02061 (2009). doi:10.1088/1742-5468/2009/02/P02061

    Google Scholar 

  2. A. Mossa, M. Manosas, N. Forns, J.M. Huguet, F. Ritort, Dynamic force spectroscopy of DNA hairpins: I. Force kinetics and free energy landscapes. J. Stat. Mech. P02060 (2009). doi:10.1088/1742-5468/2009/02/P02060

    Google Scholar 

  3. F. Ritort, Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism. J. Stat. Mech. P10016 (2004). doi:10.1088/1742-5468/2004/10/P10016

    Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  5. P. Chvosta, P. Reineker, M. Schulz, Probability distribution of work done on a two-level dystem during a nonequilibrium isothermal process. Phys. Rev. E 75, 041124 (2007). doi:10.1103/PhysRevE.75.041124

    Article  MathSciNet  ADS  Google Scholar 

  6. E. Šubrt, and Chvosta, P., Exact analysis of work fluctuations in two-level systems. J. Stat. Mech. P09019 (2007). doi:10.1088/1742-5468/2007/09/P09019

    Google Scholar 

  7. P. Chvosta, M. Einax, V. Holubec, A. Ryabov, P. Maass, Energetics and performance of a microscopic heat engine based on exact calculations of work and heat distributions. J. Stat. Mech. P03002 (2010). doi:10.1088/1742-5468/2010/03/P03002

    Google Scholar 

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

  9. A. Imparato, L. Peliti, G. Pesce, G. Rusciano, A. Sasso, Work and heat probability distribution of an optically driven Brownian particle: theory and experiments. Phys. Rev. E 76, 050101 (2007). doi:10.1103/PhysRevE.76.050101

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. A. Baule, E.G.D. Cohen, Steady-state work fluctuations of a dragged particle under external and thermal noise. Phys. Rev. E 80, 011111 (2009). doi:10.1103/PhysRevE.80.011110

    Article  ADS  Google Scholar 

  12. R. van Zon, E.G.D. Cohen, Extension of the fluctuation theorem. Phys. Rev. Lett. 91, 110601 (2003). doi:10.1103/PhysRevLett.91.110601

    Article  Google Scholar 

  13. R. van Zon, S. Ciliberto, E.G.D. Cohen, Power and heat fluctuation theorems for electric circuits. Phys. Rev. Lett. 92, 130601 (2004). doi:10.1103/PhysRevLett.92.130601

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. E.G.D. Cohen, Properties of nonequilibrium steady states: a path integral approach. J. Stat. Mech. P07014 (2008). doi:10.1088/1742-5468/2008/07/P07014

    Google Scholar 

  17. T. Taniguchi, E.G.D. Cohen, Onsager-Machlup theory for nonequilibrium steady states and fluctuation theorems. J. Stat. Phys. 126, 1 (2007). doi:10.1007/s10955-006-9252-2

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. T. Taniguchi, E.G.D. Cohen, Inertial effects in nonequilibrium work fluctuations by a path integral approach. J. Stat. Phys. 130, 1 (2008). doi:10.1007/s10955-007-9398-6

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. T. Taniguchi, E.G.D. Cohen, Nonequilibrium steady state thermodynamics and fluctuations for stochastic systems. J. Stat. Phys. 130, 633 (2008). doi:10.1007/s10955-007-9471-1

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. N. Singh, Onsager-Machlup theory and work fluctuation theorem for a harmonically driven Brownian particle. J. Stat. Phys. 131, 405 (2008). doi:10.1007/s10955-008-9503-5

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. J.I. Jiménez-Aquino, R.M. Velasco, Power fluctuation theorem for a Brownian harmonic oscillator. Phys. Rev. E 87, 022112 (2013). doi:10.1103/PhysRevE.87.022112

    Article  ADS  Google Scholar 

  22. J.I. Jiménez-Aquino, R.M. Velasco, Power-fluctuation theorem for a Brownian oscillator in a thermal bath. J. Phys. A: Math. Theor. 46, 325001 (2013). doi:10.1088/1751-8113/46/32/325001

    Article  Google Scholar 

  23. T. Speck, U. Seifert, Dissipated work in driven harmonic diffusive systems: general solution and application to stretching Rouse polymers. Eur. Phys. J. B 43, 521 (2005). doi:10.1140/epjb/e2005-00086-6

    Article  ADS  Google Scholar 

  24. A. Engel, Asymptotics of work distributions in nonequilibrium systems. Phys. Rev. E 80, 021120 (2009). doi:10.1103/PhysRevE.80.021120

    Article  ADS  Google Scholar 

  25. A. Ryabov, M. Dierl, P. Chvosta, M. Einax, P. Maass, Work distribution in a time-dependent logarithmic-harmonic potential: exact results and asymptotic analysis. J. Phys. A: Math. Theor. 46, 075002 (2013). doi:10.1088/1751-8113/46/7/075002

    Article  MathSciNet  ADS  Google Scholar 

  26. D. Nickelsen, A. Engel, Asymptotics of work distributions: the pre-exponential factor. Eur. Phys. J. B 82, 207 (2011). doi:10.1140/epjb/e2011-20133-y

    Article  ADS  Google Scholar 

  27. T. Speck, Work distribution for the driven harmonic oscillator with time-dependent strength: exact solution and slow driving. J. Phys. A: Math. Theor. 44, 305001 (2011). doi:10.1088/1751-8113/44/30/305001

    Article  MathSciNet  ADS  Google Scholar 

  28. D.D.L. Minh, A.B. Adib, Path integral analysis of Jarzynski’s equality: analytical results. Phys. Rev. E 79, 021122 (2009). doi:10.1103/PhysRevE.79.021122

    Article  ADS  Google Scholar 

  29. R.R. Deza, G.G. Izús, H.S. Wio, Fluctuation theorems from non-equilibrium Onsager-Machlup theory for a Brownian particle in a time-dependent harmonic potential. Cent. Eur. J. Phys. 7, 472 (2009). doi:10.2478/s11534-009-0038-4

    Google Scholar 

  30. C. Kwon, J.D. Noh, H. Park, Work fluctuations in a time-dependent harmonic potential: rigorous results beyond the overdamped limit. Phys. Rev. E 88, 062102 (2013). doi:10.1103/PhysRevE.88.062102

    Article  ADS  Google Scholar 

  31. C.F. Lo, Exact propagator of the Fokker-Planck equation with logarithmic factors in diffusion and drift terms. Phys. Lett. A 319, 110 (2003). doi:10.1016/j.physleta.2003.10.005

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. J.A. Giampaoli, D.E. Strier, C. Batista, G. Drazer, H.S. Wio, Exact expression for the diffusion propagator in a family of time-dependent anharmonic potentials. Phys. Rev. E 60, 2540 (1999). doi:10.1103/PhysRevE.60.2540

    Article  ADS  Google Scholar 

  33. J. Wei, E. Norman, Lie algebraic solution of linear differential equations. J. Math. Phys. 4, 575 (1963). doi:10.1063/1.1703993

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. R.M. Wilcox, Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962 (1967). doi:10.1063/1.1705306

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. F. Wolf, Lie algebraic solutions of linear Fokker-Planck equations. J. Math. Phys. 29, 305 (1988). doi:10.1063/1.528067

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. S. Karlin, H.M. Taylor, A First Course in Stochastic Processes (Academic Press, New York, 1975). ISBN 0-12-398552-8

    MATH  Google Scholar 

  37. S. Karlin, H.M. Taylor, A Second Course in Stochastic Processes (Academic Press, New York, 1981). ISBN 0-12-398650-8

    MATH  Google Scholar 

  38. E. Lutz, F. Renzoni, Beyond Boltzmann-Gibbs statistical mechanics in optical lattices. Nat. Phys. 9, 615 (2013). doi:10.1038/nphys2751

    Article  Google Scholar 

  39. A.J. Bray, Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model. Phys. Rev. E 62, 103 (2000). doi:10.1103/PhysRevE.62.103

    Article  MathSciNet  ADS  Google Scholar 

  40. O. Hirschberg, D. Mukamel, G.M. Schütz, Approach to equilibrium of diffusion in a logarithmic potential. Phys. Rev. E 84, 041111 (2011). doi:10.1103/PhysRevE.84.041111

    Article  ADS  Google Scholar 

  41. O. Hirschberg, D. Mukamel, G.M. Schütz, Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium. J. Stat. Mech. P02001 (2011). doi:10.1088/1742-5468/2012/02/P02001

    Google Scholar 

  42. A. Dechant, E. Lutz, D.A. Kessler, E. Barkai, Fluctuations of time averages for Langevin dynamics in a binding force field. Phys. Rev. Lett. 107, 240603 (2011). doi:10.1103/PhysRevLett.107.240603

    Article  ADS  Google Scholar 

  43. D.A. Kessler, E. Barkai, Infinite covariant density for diffusion in logarithmic potentials and optical lattices. Phys. Rev. Lett. 105, 120602 (2010). doi:10.1103/PhysRevLett.105.120602

    Article  ADS  Google Scholar 

  44. D.A. Kessler, E. Barkai, Theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices. Phys. Rev. Lett. 108, 230602 (2012). doi:10.1103/PhysRevLett.108.230602

    Article  ADS  Google Scholar 

  45. A. Dechant, E. Lutz, D.A. Kessler, E. Barkai, Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials. Phys. Rev. E 85, 051124 (2012). doi:10.1103/PhysRevE.85.051124

    Article  ADS  Google Scholar 

  46. G. Doetsch, Introduction to the Theory and Application of The Laplace Transformation (Springer, Berlin, 1974)

    Book  MATH  Google Scholar 

  47. B.G. Wybourne, Classical Groups for Physicists, 2nd edn. (Wiley, New York, 1974)

    MATH  Google Scholar 

  48. D.E. Strier, G. Drazer, H.S. Wio, An analytical study of stochastic resonance in a monostable non-harmonic system. Physica A 283, 255 (2000). doi:10.1016/S0378-4371(00)00163-1

    Article  ADS  Google Scholar 

  49. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (Chapman & Hall/CRC, Boca Raton, 2003). ISBN 1-58488-297-2

    MATH  Google Scholar 

  50. H. Bateman, A. Erdélyi, Tables of Integral Transforms, vol. 1 (McGraw-Hill Book Company, INC., New York, 1954). ISBN 07-019549-8

    Google Scholar 

  51. A. Erdélyi (ed.), Higher Transcendental Functions, vol. 2 (Robert E. Krieger Publishing Company, INC., Malabar, Florida, 1955). ISBN 0-89874-069-X (v. II)

    MATH  Google Scholar 

  52. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1999)

    Google Scholar 

  53. A.E. Cohen, Control of nanoparticles with arbitrary two-dimensional force fields. Phys. Rev. Lett. 94, 118102 (2005). doi:10.1103/PhysRevLett.94.118102

    Article  ADS  Google Scholar 

  54. V. Blickle, C. Bechinger, Realization of a micrometre-sized stochastic heat engine. Nat. Phys. 8, 143 (2012). doi:10.1038/nphys2163

    Article  Google Scholar 

  55. D. Nickelsen, A. Engel, Asymptotic work distributions in driven bistable systems. Phys. Scr. 86, 058503 (2012). doi:10.1088/0031-8949/86/05/058503

    Article  Google Scholar 

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  1. Department of Macromolecular Physics, Charles University in Prague, Prague, Czech Republic

    Artem Ryabov

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Ryabov, A. (2016). Work Distribution in Logarithmic-Harmonic Potential. In: Stochastic Dynamics and Energetics of Biomolecular Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-27188-0_6

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