Journal of Statistical Physics

, Volume 141, Issue 3, pp 589–606 | Cite as

Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

  • Yuri P. Kalmykov
  • Bernard P. J. MulliganEmail author
  • Serguey V. Titov
  • William T. Coffey


The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.


Spins Quasi-probability distributions Wigner distributions Master equation Fokker-Planck equation 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Yuri P. Kalmykov
    • 1
  • Bernard P. J. Mulligan
    • 2
    Email author
  • Serguey V. Titov
    • 3
  • William T. Coffey
    • 4
  1. 1.Laboratoire de Mathématiques, Physique et SystèmesUniversité de PerpignanPerpignan CedexFrance
  2. 2.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany
  3. 3.Institute of Radio Engineering and ElectronicsRussian Acad. Sci.FryazinoRussia
  4. 4.Department of Electronic and Electrical EngineeringTrinity CollegeDublinIreland

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