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Approach to a Stationary State in an External Field

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Abstract

We study relaxation towards a stationary out-of-equilibrium state by analyzing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within one-dimensional formulation of Boltzmann’s kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution towards the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by the Green-Kubo autocorrelation formula generalized to the stationary state.

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References

  1. Gervois, A., Piasecki, J.: J. Stat. Phys. 42, 1091–1102 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  2. White, R.D., Robson, R.E., Dujko, S., Nicoletopoulos, P., Li, B.: J. Phys. D, Appl. Phys. 42(19), 194001 (2009)

    Article  ADS  Google Scholar 

  3. Lorentz, H.A.: In: Collected Papers, vol. III. Martinus Nijhoff, The Hague (1936)

    Google Scholar 

  4. Piasecki, J., Wajnryb, E.: J. Stat. Phys. 21, 549 (1979)

    Article  ADS  Google Scholar 

  5. Resibois, P.: Physica A 90, 273 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  6. Piasecki, J.: J. Stat. Phys. 30, 185 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  7. Piasecki, J.: Phys. Lett. A 114, 245–249 (1986)

    Article  ADS  Google Scholar 

  8. Piasecki, J., Soto, R.: Physica A 369, 379–386 (2006)

    Article  ADS  Google Scholar 

  9. Bobylev, A.V., Carillo, J.A., Gamba, I.M.: J. Stat. Phys. 98, 743–773 (2000)

    Article  MATH  Google Scholar 

  10. Bobylev, A.V., Cercignani, C.: J. Stat. Phys. 106, 1019 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Martin, Ph.A., Piasecki, J.: J. Phys. A, Math. Theor. 40, 361–369 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Uhlenbeck, G.E., Ford, G.W., Montroll, E.W.: Lectures in Statistical Physics. American Mathematical Society, Providence (1963)

    Google Scholar 

  13. Ernst, M.H., Dorfman, J.R.: J. Stat. Phys. 12, 311–359 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  14. Ernst, M.H.: J. Stat. Phys. 34(5, 6) (1984)

  15. Coppex, F., Droz, M., Trizac, E.: Phys. Rev. E 72, 021105 (2005)

    ADS  Google Scholar 

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Authors and Affiliations

  1. ENS Lyon, CNRS, Lyon, France

    A. Alastuey

  2. Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00 681, Warsaw, Poland

    J. Piasecki

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  1. A. Alastuey
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  2. J. Piasecki
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Correspondence to A. Alastuey.

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Alastuey, A., Piasecki, J. Approach to a Stationary State in an External Field. J Stat Phys 139, 991–1012 (2010). https://doi.org/10.1007/s10955-010-9976-x

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  • DOI: https://doi.org/10.1007/s10955-010-9976-x

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