Harmonic Oscillator with Fluctuating Mass

Author(s)
Moshe Gitterman

ABSTRACT

We generalize the previously considered cases of a harmonic oscillator subject to a random force (Brownian motion), or having random frequency, or random damping. We consider here a random mass which corresponds to an oscillator where the particles of the surrounding medium adhere to the oscillator for some (random) time after collision, thereby changing the oscillator mass. Such a model is appropriate to chemical and biological solutions as well as to some nano-technological devices. The first moment and stability conditions for white and dichotomous noise are analyzed.

We generalize the previously considered cases of a harmonic oscillator subject to a random force (Brownian motion), or having random frequency, or random damping. We consider here a random mass which corresponds to an oscillator where the particles of the surrounding medium adhere to the oscillator for some (random) time after collision, thereby changing the oscillator mass. Such a model is appropriate to chemical and biological solutions as well as to some nano-technological devices. The first moment and stability conditions for white and dichotomous noise are analyzed.

Cite this paper

nullM. Gitterman, "Harmonic Oscillator with Fluctuating Mass,"*Journal of Modern Physics*, Vol. 2 No. 10, 2011, pp. 1136-1140. doi: 10.4236/jmp.2011.210140.

nullM. Gitterman, "Harmonic Oscillator with Fluctuating Mass,"

References

[1] M. Gitterman, “The Noisy Oscillator: the First Hundred Years, from Einstein until Now,” World Scientific, 2005.

[2] M. Gitterman, “New Stochastic Equation for a Harmonic Oscillator : Brownian Motion with Adhesion,” Journal of Physics C, Vol. 280, 2010, p. 012049

[3] J. Luczka, P. Hanggi and A. Gadomski, “Diffusion of Clusters with Randomly Growing Masses,” Physical Review E, Vol. 51, No. 6, 1995, pp. 5762-5769.

[4] M. S. Abdalla, “Time-Dependent Harmonic Oscillator with Variable Mass under the Action of a Driving Force,” Physical Review A, Vol. 34, No. 6, 1986, pp. 4598-4605. doi:10.1103/PhysRevA.34.4598

[5] J. Portman, M. Khasin, S. W. Shaw and M. I. Dykman, “The Spectrum of an Oscillator with Fluctuating Mass and Nanomechanical Mass Sensing,” Bulletin of the American Physical Society, March Meeting 2010, Vol. 55, No. 2, Portland, 15-19 March 2010, Abstract V14.00010.

[6] V. E. Shapiro and V. M. Loginov, “Formula of Differentiation and Their Use for Solving Stochastic Equation,” Physica A, Vol. 91, No. 3-4, 1978, pp. 563-574. doi:10.1016/0378-4371(78)90198-X

[7] R. C. Bourret, U. Frish and A Pouquet, “Brownian Motion of Harmonic Oscillator with Stochastic Frequency,” Physica A, Vol. 65, No. 2, 1973, pp. 303-320.

[8] L. Arnold, “Random Dynamic Systems,” Springer, Berlin, 1998.

[9] N. G. van Kampen, “Stochastic Processes in Physics and Chemistry,” Elsevier, North-Holland, 1992.

[10] N. Leprovost, S. Aumaitre and K. Mallick, “Stability of a Nonlinear Oscillator with Random Damping,” European Physical Journal, Vol. 49, 2006, pp. 453-458.

[11] K. Kitahara, W. Horsthemke and R. Lefever, “Colored- Noise Induced Transitions: Exact Results for External Dichotomous Markovian Noise,” Physics Letters. A, Vol. 70, 1979, pp. 377-380.

[12] K. Kitahara, W. Horsthemke and R. Lefever, “Phase Diagrams of Noise Induced Transitions,” Progress Theoretical Physics, Vol. 64, 1980, pp. 1233-1247. doi:10.1143/PTP.64.1233

[13] V. I. Klyatskin, “Dynamic Systems with Parameter Fluctuations of the Telegraphic Process Types,” Radiophysics and Quantum Electronics, Vol. 20, 1977, pp. 384-393. doi:10.1007/BF01033925

[14] V. Berdichevsky and M. Gitterman, “Stochastic Resonance in Linear Systems Subject to Multiplicative and Additive Noise,” Physical Review E, Vol. 60, No. 2, 1999, pp. 1494-1499.

[15] F. Sasagawa, “Critical Slowing Dawn in Random Growing-Rate Models with General Two Level Markovian Noise,” Progress Theoretical Physics, Vol. 69, No. 3, 1983, pp. 790-800. doi:10.1143/PTP.69.790

[16] K. Ouchi, T. Horita and H. Fujisaka, “Critical Dynamics of Phase Transition Driven by Dichotomous Markov Noise,” Physical Review E, Vol. 74, 2006, p. 031106. doi:10.1103/PhysRevE.74.031106

[17] Y. Jia, X.-P. Zheng, X.-M. Hu and J.-R. Li, “Effect of Colored Noise in Stochastic Resonance in a Bistable System Subject to Multiplicative and Additive Noise,” Physical Review E, Vol. 63, 2001, p. 031107. doi:10.1103/PhysRevE.63.031107

[18] S. Z. Ke, D. J. Wu and L. Cao, “Phase Transitions in a Bistable System Driven by Two Colored Noise,” European Physical Journal B, Vol. 12, 1999, pp. 119-122. doi:10.1007/s100510050985

[1] M. Gitterman, “The Noisy Oscillator: the First Hundred Years, from Einstein until Now,” World Scientific, 2005.

[2] M. Gitterman, “New Stochastic Equation for a Harmonic Oscillator : Brownian Motion with Adhesion,” Journal of Physics C, Vol. 280, 2010, p. 012049

[3] J. Luczka, P. Hanggi and A. Gadomski, “Diffusion of Clusters with Randomly Growing Masses,” Physical Review E, Vol. 51, No. 6, 1995, pp. 5762-5769.

[4] M. S. Abdalla, “Time-Dependent Harmonic Oscillator with Variable Mass under the Action of a Driving Force,” Physical Review A, Vol. 34, No. 6, 1986, pp. 4598-4605. doi:10.1103/PhysRevA.34.4598

[5] J. Portman, M. Khasin, S. W. Shaw and M. I. Dykman, “The Spectrum of an Oscillator with Fluctuating Mass and Nanomechanical Mass Sensing,” Bulletin of the American Physical Society, March Meeting 2010, Vol. 55, No. 2, Portland, 15-19 March 2010, Abstract V14.00010.

[6] V. E. Shapiro and V. M. Loginov, “Formula of Differentiation and Their Use for Solving Stochastic Equation,” Physica A, Vol. 91, No. 3-4, 1978, pp. 563-574. doi:10.1016/0378-4371(78)90198-X

[7] R. C. Bourret, U. Frish and A Pouquet, “Brownian Motion of Harmonic Oscillator with Stochastic Frequency,” Physica A, Vol. 65, No. 2, 1973, pp. 303-320.

[8] L. Arnold, “Random Dynamic Systems,” Springer, Berlin, 1998.

[9] N. G. van Kampen, “Stochastic Processes in Physics and Chemistry,” Elsevier, North-Holland, 1992.

[10] N. Leprovost, S. Aumaitre and K. Mallick, “Stability of a Nonlinear Oscillator with Random Damping,” European Physical Journal, Vol. 49, 2006, pp. 453-458.

[11] K. Kitahara, W. Horsthemke and R. Lefever, “Colored- Noise Induced Transitions: Exact Results for External Dichotomous Markovian Noise,” Physics Letters. A, Vol. 70, 1979, pp. 377-380.

[12] K. Kitahara, W. Horsthemke and R. Lefever, “Phase Diagrams of Noise Induced Transitions,” Progress Theoretical Physics, Vol. 64, 1980, pp. 1233-1247. doi:10.1143/PTP.64.1233

[13] V. I. Klyatskin, “Dynamic Systems with Parameter Fluctuations of the Telegraphic Process Types,” Radiophysics and Quantum Electronics, Vol. 20, 1977, pp. 384-393. doi:10.1007/BF01033925

[14] V. Berdichevsky and M. Gitterman, “Stochastic Resonance in Linear Systems Subject to Multiplicative and Additive Noise,” Physical Review E, Vol. 60, No. 2, 1999, pp. 1494-1499.

[15] F. Sasagawa, “Critical Slowing Dawn in Random Growing-Rate Models with General Two Level Markovian Noise,” Progress Theoretical Physics, Vol. 69, No. 3, 1983, pp. 790-800. doi:10.1143/PTP.69.790

[16] K. Ouchi, T. Horita and H. Fujisaka, “Critical Dynamics of Phase Transition Driven by Dichotomous Markov Noise,” Physical Review E, Vol. 74, 2006, p. 031106. doi:10.1103/PhysRevE.74.031106

[17] Y. Jia, X.-P. Zheng, X.-M. Hu and J.-R. Li, “Effect of Colored Noise in Stochastic Resonance in a Bistable System Subject to Multiplicative and Additive Noise,” Physical Review E, Vol. 63, 2001, p. 031107. doi:10.1103/PhysRevE.63.031107

[18] S. Z. Ke, D. J. Wu and L. Cao, “Phase Transitions in a Bistable System Driven by Two Colored Noise,” European Physical Journal B, Vol. 12, 1999, pp. 119-122. doi:10.1007/s100510050985