Since fo is assumed to be a known function, moz,t is. Wolf department of physics and astronomy, university of rochester, rochester, n. The weyl procedure associates a function of two ordinary variables, called the cfunction or symbol, with an operator, called the weyl operator of the symbol. In the last few decades, however, photon, xray and electron optics have found interesting applications.
Usually systems where the mep applies are simple systems, such as gases and independent processes. In principle, the kappa distribution is a probability distribution function which, mathematically, is identical to the. Moreover, this is achieved via a direct sampling of the measured. Radial distribution functions and their role in modeling of mixtures behavior g. Pdf in contrast to classical physics, the language of quantum mechanics involves. He is a physicist and professor at shahid beheshti university, iran after obtaining his b. A generalized kinetic equation for the phasespace distribution function is derived, from which we obtain an exact expression for the number density. The phase space distribution fx, p of a quantum state is a quasiprobability distribution. The fractional fourier transform and the wigner distribution. India f center for particle physics and department of physics, university of texas, austin. Statistical distribution function and phase space physics. The properties of generalized parton distributions are described. Printed in the uk complex measures and amplitudes, generalized stochastic processes and their applications to quantum mechanics s k srinivasant and e c g sudarshant i department of mathematics, indian institute of technology, madras 600036.
It was shown that, depending on the values of qparameter within the condition 0 phase space as a usual nonrelativistic harmonic oscillator if q. The theory of distribution tries to remedy this by imbedding classical functions in a larger class of objects, the so called distributions or general functions. This cited by count includes citations to the following articles in scholar. Generalized phase space physics 230a, spring 2007, hitoshi murayama 1 symplectic structure in usual particle mechanics, the phase space is given by the coordinates qi and their conjugate momenta pi, and its volume by v z yn i1 dp idq. The simple harmonic oscillator wigner functions for n 2 left and n 3 right. Generalized phasespace distribution functions nasaads. The advantage in the gsp is that they can be measured in terms of the electric correlation matrix which. Determining the population of individual fastion orbits. It is noted that phasespace stokes parameters can be a useful tool for wigner distribution function measurements. It was some time later that moyal 60 saw the connection between the weyl rule and the wigner distribution.
Because the wigner function can be negative, it is known as a quasiprobability distribution. A point in this phase space is correspondingly called a macrostate. A general relationship between the phase space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. That is, armed with a measure of the structure of phasespace distribution functions, it becomes possible to quantitatively investigate the similarities and differences between quantal and classical distribution functions in a broad class of smooth, conservative, and strongly chaotic hamiltonian systems.
The maximum entropy principle mep states that for many statistical systems the entropy that is associated with an observed distribution function is a maximum, given that prior information is taken into account appropriately. Fractional calculus, completely monotonic functions, a. The wigner distribution and many other members of the cohen class of generalized phasespace distributions of a signal all share certain translation properties and the property that their two. For instance, one may view the pressurevolume diagram or entropytemperature diagrams as describing part of this phase space. Thus, instead of the pfunction for the harmonic oscillator for a thermal state we have a qgaussian function p q for a system that.
Velocity space weight functions can be generalized to the full 6d phase space consider a fast ion with generalized phase space coordinate x p,q. Dustcharge fluctuations with nonmaxwellian distribution. Recently i had gone through a statistical physics course and i learned about phase space. We noted that if, then f f h is a pure function of the hamiltonian which defined the general class of distribution functions valid for equilibrium ensembles. The phase space distribution function or phase space density is supposed to be the probability density of finding a particle around a given phase space point. As a main result, we provide a characterization for the cohens class l. One distribution that has been particularly useful both in nonrelativistic quantum mechanics and in paraxial optics is the wigner distribution function. For such functions, the average of the coordinate equals zero if the frame of reference coincides with the point of origin.
The wigner distribution and the fractional fourier transform the wigner timefrequency distribution has been discussed recently along with others from th e point of view of their membershi of cohenp s generalized class of phase space distributions and the constraints imposed on thi is clasf the resultins g. The concept of phase space was developed in the late 19th century by ludwig boltzmann. Consider a vector u x,x,y,y, in a generalized ndimensional phase space. In the weak gradient limit, n is shown to satisfy a generalized diffusion equation, valid for any, which can be applied to either classical or traplimited transport. That is, armed with a measure of the structure of phase space distribution functions, it becomes possible to quantitatively investigate the similarities and differences between quantal and classical distribution functions in a broad class of smooth, conservative, and strongly chaotic hamiltonian systems. Exact generalized fokkerplanck equations are derived from the linear morikubo generalized langevin equation for the case of gaussian but nonmarkovian noise. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.
Generalized phasespace kinetic and diffusion equations. A set of quasiprobability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The ones marked may be different from the article in the profile. A general relationship between the phasespace distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. Appendix fundamental properties of generalized functions. Also given is the generalized diffusion equation for the free brownian particle pdf. Relationships among generalized phasespace distributions g. In the phase space formulation, the phase space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.
The socalled wigner distribution function wdf is a phasespace mathematical tool, originally designed in the 1920s for dealing with quantum mechanics problems 3. Relationships among generalized phasespace distributions. How multiplicity determines entropy and the derivation of. Pdf introducing asymmetry into the weyl representation of operators. Relationships among generalized phase space distributions g. Perturbation theory in terms of a generalized phasespace quantization procedure.
It was shown that, depending on the values of qparameter within the condition 0 completely monotonic functions, a generalized mittagleffler function and phasespace consistency of separable augmented densities. Phase space volume scaling of generalized entropies and. The sum can be expressed in terms of the frequency in which x occurs. The wigner distribution and many other members of the cohen class of generalized phase space distributions of a signal all share certain translation properties and the property that their two.
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. Their combined citations are counted only for the first article. In the last few decades, however, photon, xray and electron optics have found interesting applications of this and other related phasespace functions 38. A general relationship between the phasespace distribution functions and the rule of associating classical quantities. Next, we treat various other distribution functions. For simplicity we will use a 2d distribution, but the same exact results apply to the more general 6d case.
Introduction it is well known that the uncertainty principle makes the concept of phase space in quantum mechanics problematic. The phasespace distribution fx, p of a quantum state is a quasiprobability distribution. Sep 12, 2017 the weyl procedure associates a function of two ordinary variables, called the cfunction or symbol, with an operator, called the weyl operator of the symbol. Cohen, generalized phasespace distribution functions, j. The phase space distribution does not have to be bilinear in the state function. Appendix fundamental properties of generalized functions a. Pdf perturbation theory in terms of a generalized phase. Zweifel t department of nuclear engineering, the university of michigan, ann arbor received 22 may 1968 the generalized phase spae distributions, inc1udin. Similarly, the electrostatic potential qjx,t can be found by choosing mz,z gx,x iiqh, where gx,x is the greens function for the poisson equation. Recall that the phase space distribution function satisfied a liouville equation. The dynamics and other properties of the generalized phasespace distribution are considered. A generalized kinetic equation for the phase space distribution function is derived, from which we obtain an exact expression for the number density. Why arent classical phase space distribution functions. Jan 30, 2018 phase space volume scaling of generalized entropies and anomalous diffusion scaling governed by corresponding nonlinear fokkerplanck equations skip to main content thank you for visiting.
Fractional calculus, completely monotonic functions, a generalized mittagleffler function and phasespace consistency of separable augmented densities. The phase space can also refer to the space that is parameterized by the macroscopic states of the system, such as pressure, temperature, etc. But, classically, through hamiltons equations, the systems time evolution is completely determined once the initial conditions are specified. Generalized uncertainty principle corrections to the simple. However, we can use it to compute the expectation value of any phasespace function qx.
In this paper we have studied a possible generalization of quantum distributions in phase space to a nonextensive scenario by proposing, as a first step in such a direction, a qgaussian distribution as a feasible pfunction, that we call p q. Generalized phasespace kinetic and diffusion equations for. Generalized phasespace distribution functions aip publishing. The distribution function as we have seen before the distribution function or phasespace density fx. A set of quasiprobability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is. The beam modeled by phase space distribution function. Reality of the wigner functions and quantization, research letters in physics, vol. Gup corrections to the simple harmonic oscillator in phase space 4 figure 1.
Fokkerplanck equations for simple nonmarkovian systems. It is noted that phase space stokes parameters can be a useful tool for wigner distribution function measurements. Pdf generalized phase space representation of operators. Just as the wavefunctions in quantum mechanics, the. The wigner distribution and the fractional fourier transform the wigner timefrequency distribution has been discussed recently along with others from th e point of view of their membershi of cohenp s generalized class of phasespace distributions and the constraints imposed on thi is clasf the resultins g. Unfortunally not all functions are di erentiable or has a fourier transform. The phase space distribution also known as the phase space portrait, or the score of a function, in analogy to the musical example mentioned earlier is a useful guide for understanding the singularities of a function, and how that function will behave. The resulting phase space functions directly show negativities even for quantum e ciencies signi cantly below 50%.
Given the weyl operator, what is the matching symbol. Velocityspace weight functions can be generalized to the full 6d phase space consider a fast ion with generalized phasespace coordinate x p,q. A general relationship between the phase space distribution functions and the rule of associating classical quantities. Theoretically, it has been shown that the velocity distribution function obeys a power law at energies higher than the critical energy when a superthermal radiation. In this chapter, we develop a formulation where all distributions may be studied in a unified way. Generalized weighting scheme for 6f particlesimulation method. In optics, a class of phase space distributions that are bilinear in the field may be constructed for paraxial, stationary, partially coherent optical fields. One generally formulates this association by defining the operator corresponding to a given symbol. Then quantum phasespace distributions and their offspring are discussed. The arcsine distribution on a,b, which is a special case of the beta distribution if. Nonetheless, functions which bear some resemblance to phase space distribution functions, quasiprobability distribution functions, have proven to be of great use in the study of quantum mechanical systems.
We introduce a family of phasespace representations of wigner type associated with invertible matrices and explore their general properties. Nov 30, 2012 they devised a distribution function different ones aimed to calculate quantum averages by way of phase space averaging. Ordering theorems and generalized phase space distributions. Almi ansoori chemical engineering department, university of illinois at chicago mc 063 chicago, il 6060usa77052 keywords. Let us now use the continuity equation with our phase space. Our distribution functions are consistently normalized such that f dz fz,t v, so they are not true probability density functions. Chaos and quantumclassical correspondence via phasespace. Incomplete detection of nonclassical phasespace distributions m.
Let sx be the expected signal produced by the fast ion. Radial distribution functions and their role in modeling. This allows the writing of correspondence rules at will, of. The generalized stokes parameters gsp are studied under the theory of phase space. In the phasespace formulation, the phasespace distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices. Fokkerplanck equations which generate the momentum and phase space probability distribution functions pdf s for free brownian particles and the phase space pdf for brownian oscillators are presented. The phasespace distribution does not have to be bilinear in the state function. The phase space density for a hamiltonian system is an invariant of the motion. The total signal then the sum of the all the individual signals. It starts with the physics of form factors and parton distributions. But if our system is hamiltonian liouville theorem. Oconnell, quantum distribution functions in nonequilibrium statistical mechanics, in frontiers of nonequilibrium statistical physics 8395, plenum publishing corporation, 1986.
Then quantum phase space distributions and their offspring are discussed. Chaos and quantumclassical correspondence via phase. Kinetic corrections from analytic nonmaxwellian distribution functions in magnetized plasmas. Electromagnetic wigner functions are introduced by use of the twopoint statistics of gsp. Generalized uncertainty principle corrections to the. Vogel1 1arbeitsgruppe theoretische quantenoptik, institut fur physik, universit at rostock, d18051 rostock, germany 2integrated quantum optics group, applied physics, university of paderborn, 33098 paderborn, germany 3clarendon laboratory, university of. A phase space formulation of quantum state functions, international journal of modern physics b, 7 1993, pp. Also given is the generalized diffusion equation for the free brownian particle pdf in the zero inertia limit. They devised a distribution function different ones aimed to calculate quantum averages by way of phasespace averaging. Zweifel t department of nuclear engineering, the university of michigan, ann arbor received 22 may 1968 the generalized phasespae distributions, inc1udin. If each point in the phase space is considered as a random quantity with a particular probability ascribed to every possible state i.
Pdf joint distribution functions and the generalized. Complex measures and amplitudes, generalized stochastic. Other distributions such as kappa 5 or generalized r,q distribution may. Phase space volume scaling of generalized entropies and anomalous diffusion scaling governed by corresponding nonlinear fokkerplanck equations skip to main content thank you for visiting. Plasmas 23, 082504 that stands for interpreted nonmaxwellian distribution function. Treumann1,a and wolfgang baumjohann2 1department of geophysics and environmental sciences, munich university, munich, germany 2space research institute, austrian academy of sciences, graz, austria acurrently at. One of the main differences with the literature is our analytic representation of the distribution function in the velocity phase space with as few. The best ellipse fitting the beam distribution is the beam twiss functions can be defined through the rms emittance.
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