3 research outputs found
Magnetization dynamics: path-integral formalism for the stochastic Landau-Lifshitz-Gilbert equation
We construct a path-integral representation of the generating functional for
the dissipative dynamics of a classical magnetic moment as described by the
stochastic generalization of the Landau-Lifshitz-Gilbert equation proposed by
Brown, with the possible addition of spin-torque terms. In the process of
constructing this functional in the Cartesian coordinate system, we critically
revisit this stochastic equation. We present it in a form that accommodates for
any discretization scheme thanks to the inclusion of a drift term. The
generalized equation ensures the conservation of the magnetization modulus and
the approach to the Gibbs-Boltzmann equilibrium in the absence of non-potential
and time-dependent forces. The drift term vanishes only if the mid-point
Stratonovich prescription is used. We next reset the problem in the more
natural spherical coordinate system. We show that the noise transforms
non-trivially to spherical coordinates acquiring a non-vanishing mean value in
this coordinate system, a fact that has been often overlooked in the
literature. We next construct the generating functional formalism in this
system of coordinates for any discretization prescription. The functional
formalism in Cartesian or spherical coordinates should serve as a starting
point to study different aspects of the out-of-equilibrium dynamics of magnets.
Extensions to colored noise, micro-magnetism and disordered problems are
straightforward.Comment: 47 pages + appendix, published versio
Dynamical symmetries of Markov processes with multiplicative white noise
We analyse various properties of stochastic Markov processes with
multiplicative white noise. We take a single-variable problem as a simple
example, and we later extend the analysis to the Landau-Lifshitz-Gilbert
equation for the stochastic dynamics of a magnetic moment. In particular, we
focus on the non-equilibrium transfer of angular momentum to the magnetization
from a spin-polarised current of electrons, a technique which is widely used in
the context of spintronics to manipulate magnetic moments. We unveil two hidden
dynamical symmetries of the generating functionals of these Markovian
multiplicative white-noise processes. One symmetry only holds in equilibrium
and we use it to prove generic relations such as the fluctuation-dissipation
theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to
prove fluctuation theorems. The other symmetry yields strong dynamical
relations between correlation and response functions which can notably simplify
the numerical analysis of these problems. Our construction allows us to clarify
some misconceptions on multiplicative white-noise stochastic processes that can
be found in the literature. In particular, we show that a first-order
differential equation with multiplicative white noise can be transformed into
an additive-noise equation, but that the latter keeps a non-trivial memory of
the discretisation prescription used to define the former.Comment: 44 page
Nonextensive statistical mechanics and high energy physics
The use of the celebrated Boltzmann-Gibbs entropy and statistical mechanics is justified for ergodic-like systems. In contrast, complex systems typically require more powerful theories. We will provide a brief introduction to nonadditive entropies (characterized by indices like q, which, in the q → 1 limit, recovers the standard Boltzmann-Gibbs entropy) and associated nonextensive statistical mechanics. We then present somerecent applications to systems such as high-energy collisions, black holes and others. In addition to that, we clarify and illustrate the neat distinction that exists between Lévy distributions and q-exponential ones, a point which occasionally causes some confusion in the literature, very particularly in the LHC literatur