Physics Page A Path Integral Approach to the Theory of Heliospheric Cosmic-Ray Modulation. This paper introduces the path integral method, which has been widely used in quantum mechanics and statistical mechanics, into the field of cosmic-ray modulation theory to solve the Fokker-Planck equation for cosmic-ray transport. In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The path integral approach recognizes that the motion of cosmic rays is a Markov stochastic process. The derivation of the path integral yields a Lagrangian, L, consisting of parameters characterizing particle diffusion, drift, convection, adiabatic energy change, and Fermi acceleration. When its action functional integral ∫ is minimized, it yields an Euler-Lagrange equation that describes the most probable trajectory of charged particles randomly walking in heliospheric magnetic fields. The most probable trajectory is equivalent to the classical trajectory of particles in quantum mechanics. A general solution to the cosmic-ray modulation equation with an initial boundary value problem is also formulated in this paper. The path integral has been applied to an example case of steady-state, one-dimensional, spherically symmetric modulation with a boundary at 100 AU. The modulated cosmic-ray spectra obtained with the path integral method agree very well with those from other methods, even though a simple semiclassical approximation is used in the evaluation of the path integral in this calculation. In addition to being able to calculate the modulated spectrum, the path integral method reveals new information about the average behavior of individual particles during their transit through the heliosphere, such as the particle trajectories, energy-loss behavior, and source-particledistribution, all of which are normally not available through simply solving the Fokker-Planck equation. It is expected that more complex modulation problems can also be dealt with by this method, since with the path integral approach, the mathematical problem of cosmic-ray modulation can be treated as a problem of quantum mechanics, for which many mathematical tools have been developed in the past five decades. Galactic cosmic rays are energetic charged particles originating outside the heliosphere of the sun. When these particles transit through the heliosphere, their fluxes are modulated by the expanding solar wind with embedded magnetic fields. In the past several decades, numerous experiments, both space-borne and ground-based, have gathered a wealth of information regarding the heliospheric modulation of cosmic rays. These experiments include, for example, the Voyager and Pioneer missions near the equatorial of the outer heliosphere, and recently Ulysses, which was sent to the polar regions of the heliosphere. The distribution function of cosmic rays (number of particles in phase space d^3 x d^3 p) is is nearly isotropic when there is sufficient stochastic particle scattering by the magnetic field in the interplanetary medium. The behavior of the cosmic-ray omnidirectionaldistribution function, f(p, x, t), where x is the spatial position, p is the particle momentum, and t is the time, can be described by a Fokker-Planck equation. ∂f/∂t = ∇· (k· ∇f--vf-vdf) + 1/3 (∇·v)1/p^2 ∂/∂p (p^3f) + 1/p^2 ∂/∂p(D p P^2 ∂/∂p f) This equation reflects all five distinct particle transport effects: diffusion, convection, gradient and curvature drift, adiabatic energy change, and Fermi acceleration. The coefficient ϰ is the particle-diffusiontensor in the spatial coordinates x, V is the convective velocity of the solar wind plasma, and V d is the gradient and curvature drift velocity of cosmic-ray particles in inhomogeneous large-scale magnetic fields. The term containing D p represents the classical second-order Fermi acceleration and the term containing 1/3 ∇ v describes the effect of adiabatic energy change when the particles convect with a divergent magnetic plasma. Although in many cases the effect of second-order Fermi acceleration is not significant, we have left it in the equation for the purpose of maintaining a uniform format with regard to all the variables. The particle transport in the equation is remarkably general in application. In the past three decades, it has been used as the fundamental equation, similar to the Schrödinger equation in quantum mechanics, to study cosmic-ray modulation by the solar wind and diffusive cosmic-ray acceleration. Shocks can also be discussed within the framework of this equation, in which case the divergence of the plasma flow velocity simply becomes a delta function at the shock. Numerous results obtained by solving the Fokker-Planck equation above with various methods (analytical, numerical, approximation). We note that the Fokker-Planck equation and the Schrödinger equation of quantum mechanics are very similar. First, they are both equations for describing the behavior of the probability density distribution of particles undergoing Markov stochastic processes. In quantum mechanics, the modulus of wave functions, |ψ|2 represents the probability density of a particle, while the distribution function, f, of cosmic rays is proportional to the probability density, subject only to multiplying a constant for the total number of particles. Furthermore, the Fokker-Planck and Schrödinger equations have similar mathematical formats. Let us define a four-dimensional configuration space, {q μ, μ=1....4,} to be the combination of spatial coordinations x and momentum p (the momentum space is reduced to one dimension because we restrict ourself to the isotropic distribution case). The q-space is no longer a phase space in this representation.When the spatial position x is expressed in Cartesian coordinates, the q-space is a Euclidean space, in which the commutation rule between the differential operator and the spatial coordinate is simple and safe to use, as often discussed in quantum mechanics. To see more about this check out the website in the SOURCES section Vocabulary words: *Heliosphere: the region of space, encompassing the solar system in which the solar wind has a significant influence. *Partial Differential Equation: an equation containing one or more partial derivatives. *Derivative: the rate of change when looking at real life applications *Integral: a function of which a given function is the derivative which yields that function when differentiated,and which may express the area under the curve of a graph of the function. *Path Integral Formulation: The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. *Action: an attribute of the dynamics of a physical system. Action is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. The action takes different values for different paths. Action has the dimensions of [energy]·[time], and its SI unit is joule-second. This is the same unit as that of angular momentum. *Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. SOURCES: iopscience.iop.o rg/0004-637X/ 510/2/715/ fulltext/
Posted on: Sun, 16 Mar 2014 14:32:51 +0000
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