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The reversal time, superparamagnetic relaxation time, of the magnetization of fine single domain ferromagnetic nanoparticles owing to thermal fluctuations plays a fundamental role in information storage, paleomagnetism, biotechnology, etc. Here a comprehensive tutorial-style review of the achievements of fifty years of development and generalizations of the seminal work of Brown [Phys. Rev. 130, 1677] on thermal fluctuations of magnetic nanoparticles is presented. Analytical as well as numerical approaches to the estimation of the damping and temperature dependence of the reversal time based on Brown's Fokker-Planck equation for the evolution of the magnetic moment orientations on the surface of the unit sphere are critically discussed while the most promising directions for future research are emphasized. V C 2012 American Institute of Physics. [http://dx

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Classical escape rates of uniaxial spin systems are characterized by a prefactor differing from and much smaller than that of the particle problem, since the maximum of the spin energy is attained everywhere on the line of constant latitude: ϭconst, 0рр2. If a transverse field is applied, a saddle point of the energy is formed, and high, moderate, and low damping regimes ͑similar to those for particles͒ appear. Here we present the first analytical and numerical study of crossovers between the uniaxial and other regimes for spin systems. It is shown that there is one HD-Uniaxial crossover, whereas at low damping the uniaxial and LD regimes are separated by two crossovers. ͓S1063-651X͑99͒10512-9͔PACS number͑s͒: 05.40.Ϫa, 75.50.TtThe study of thermal activation escape rates of fine magnetic particles, which are usually modelled as classical spins with predominantly uniaxial anisotropy, may be traced from the early predictions of Néel ͓1͔ through the first theoretical treatments of Brown ͓2,3͔ to the recent experiments of Wernsdorfer et al. ͓4͔ on individual magnetic particles of controlled form. These experiments allow one to check the Stoner-Wohlfarth angular dependence of the switching field ͓5͔ and to make a comparison ͓6,7͔ with existing theories where the energy barrier is reduced by applying a magnetic field. The theories checked are those for the intermediate-tohigh damping ͑IHD͒ case ͓3,8,9͔, as well as for the lowdamping ͑LD͒ case ͓10͔.The IHD and LD limits for spins are similar to those for the particle problem, which were established by Kramers ͓11͔. The most significant difference is that for spins in the HD limit the prefactor ⌫ 0 in the escape rate ⌫ϭ⌫ 0 exp (Ϫ⌬U/T) behaves as ⌫ 0 ϰa, where a is the damping constant ͓if the Landau-Lifshitz ͑LL͒ equation is used͔, whereas for particles ⌫ 0 ϰ1/a. A question which has not yet been addressed, both theoretically and experimentally, and which is the subject of this paper, is how these three dampinggoverned regimes merge into the single uniaxial regime ͓2͔ if the field is removed?Let us consider the Hamiltonianwhere 0 ϭM s V is the magnetic moment and K ϭKV is the uniaxial anisotropy energy of the particle. The FokkerPlanck equation for the distribution function of the spins f (,,t), which follows from the stochastic LL equation, reads

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Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.