Phase space reconstruction using inputoutput time series data david m. Wigners quasiprobability distribution function in phasespace is a special weyl. These scar functions are then superpositions of gaussian states with different degrees and directions of squeezing, i. Nov 30, 2012 they devised a distribution function different ones aimed to calculate quantum averages by way of phase space averaging. The dynamical evolution of the phase space distribution function for hamiltonian systems is described by the. Generalized estimating equations gee for glmtype data. 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. Evolution of phase space probabilities introduction to. Walker, nicholas tufillaro integrated solutions laboratory hp laboratories palo alto hpl199924 february, 1999 phase space, embedding, blackbox model, nonlinear timeseries in this letter we suggest that a method recently proposed by wayland et al. Generalized semiinvariant distributions on padic spaces. Phase space geometry in classical and quantum mechanics john r. They devised a distribution function different ones aimed to calculate quantum averages by way of phasespace averaging. This orthogonal set is called the generalized discrete fourier transform gdft.
Phase space structure of generalized gaussian cat states. The time evolution of the generalized gibbs entropy associated with a given choice of volume. 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. Why arent classical phase space distribution functions. The theory of distributions, or generalized functions, provides a uni ed framework for performing standard calculus operations on nonsmooth functions, measures such as the dirac delta function, and even more general measurelike objects in the same way as they are done for smooth functions. In the theory of stochastic processes the phase space is the measurable space often with an additional topological, differentiable or vector structure in which the process takes values. To this end only the quasiprobability distribution functions are helpful, because phase space in the quantum approach is constructed through them. Estimating the parameters of the generalized lambda. Optimal design of phase function in generalized dft. Klauder y departments of physics and mathematics university of florida gainesville, fl 32611 abstract phase space is the state space of classical mechanics, and this manifold is normally endowed only with a symplectic form. With this choice of orthogonal set 9 becomes identical to the projection operator introduced by zwanzig 1960. Phase space geometry in classical and quantum mechanics.
Next, we treat various other distribution functions. Geometric constructions and tomography demosthenes ellinas1 and anthony j. A distribution is a continuous linear functional on the set of in. On the statistical mechanics of nonhamiltonian systems. Measurement of the phase space distribution in a complex plasma. For instance, one may view the pressurevolume diagram or entropytemperature diagrams as describing part of this phase space. The distribution function i we have seen that the dynamics of our discrete system of npoint masses is given by 6nequations of motion, which allow us to compute 6nunknowns x. Instead of relaxing to steady states described by the usual thermodynamical ensembles, a generalized gibbs ensemble gge was proposed to describe the corresponding steady states via the manybody density matrix 1 3, 11, 14, where. Bracken2, 1technical university of crete, department of sciences, division of mathematics, gr 731 00 chania, crete, greece. The kinetic equation is derived without using the so called rotatingwave approximation. The space of test functions consists of the functions that have compact support in see support of a generalized function. Experimental observation of a generalized gibbs ensemble. The relationship between quantum mechanics and classical mechanics is studied in terms of the phase space descriptions.
Here one especially talks of a phase space when it is in some sense nontrivial. Phase space, statistical distributions august 28, 2017 1 9. A general relationship between the phase space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. Motif distributions in phasespace networks for characterizing experimental twophase. Parameter estimation methods that attempt to match moments or quantiles of the data require minimizing a bivariate nonlinear function. Emergent phase space description of unitary matrix model suvankar dutta indian institute of science education and research bhopal tata institute of fundamental research mumbai march, 2017 ongoing work, jhep 1604, 104 2016 and arxiv. This cited by count includes citations to the following articles in scholar. Emergent phase space description of unitary matrix model suvankar dutta indian institute of science education and research bhopal school of physical science jnu, new delhi april, 2017 ongoing work, jhep 1604, 104 2016 and arxiv. Phasespace representation of a nonhermitian system with. Pdf the statistical mechanical partition function can be used to construct different forms of phase space distributions not restricted to the. Quantum collision theory with phasespace distributions core. A generalized enhanced fourier law and underlying connections. Phase space reconstruction using inputoutput time series data. Nonetheless, the physics of a phasespace distribution is very rich and one can try to select certain situations where a semiclassical interpretation is still possible.
Equations 6 and 7 state that this set is uncountable, and there are infinitely many constant modulus sets with nonlinear and linear phase functions whereas dft is the unique set with linear phase for the case of. The generalized lambda distribution, while extremely versatile in this regard, is also a difficult distribution to fit. In other words, one has n vectorvalued functions x j x jq 1,q 2,q d,t. The ones marked may be different from the article in the profile. Cdf generalized poisson distribution function tree level 3. In mathematics, generalized functions, or distributions, are objects extending the notion of functions. It is endowed with the topology of the strong inductive limit of the increasing sequence of spaces of type fs, where is a strictlyincreasing sequence of open sets that exhausts, compact.
Pdf general approach to sun quasidistribution functions. But my question is how to determine the statistics like mean, variance, pdf in phase space. Equilibrium phasespace distribution functions for spins. Phase space of partially coherent light with discontinuous. Phase space concepts, uspas, june 2005 8 we would like to determine the transformation of the beam enclosed by an ellipse through the accelerator consider a vector u x,x,y,y, in a generalized ndimensional phase space. This book begins with background formalism in estimation theory to establish the necessary prerequisites. Ordering theorems and generalized phase space distributions. Generalized coordinates if there are m constraints of the form f. Way back in chapter 2, i glossed over a subtle but very important issue. It was some time later that moyal 60 saw the connection between the weyl rule and the wigner distribution.
We study a general class of phase space distribution functions on. Cdf conwaymaxwellpoisson distribution function tree level 3. As we have seen before the distribution function or phasespace density fx. Measurement of the phase space distribution in a complex. Let us definelet us define space as phase space of one particle atom or molecule theas phase space of one particle atom or molecule. 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. Phase space formulation of the quantum manybody problem. A general relationship between the phasespace distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. Why arent classical phase space distribution functions always delta functions. Landau damping in space plamas with generalized r, q distribution function article pdf available in physics of plasmas 1212. For an arbitrary complex potential, we derive a generalized continuity equation for the wigner function. Generalized moyal structures in phase space, kinetic. Quantumstate estimation is an important field in quantum information theory that deals with the characterization of states of affairs for quantum sources.
How multiplicity determines entropy and the derivation of. The dynamics and other properties of the generalized phasespace distribution. The time has come to ease my guilty conscience by dealing with it properly. We noted that if, then ffh is a pure function of the hamiltonian which defined the general class of distribution functions valid for equilibrium ensembles. 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. Then quantum phasespace distributions and their offspring are discussed. The analysis of with respect to time provides information regarding the path taken by the. Your reason as to why the distribution must change is understood. The phase space distribution fx, p of a quantum state is a quasiprobability distribution. Generalized hamilton function in the phase space of coordinates and their multiple derivatives article pdf available october 2002 with 11 reads how we measure reads. The phasespace distribution fx, p of a quantum state is a quasiprobability distribution. The nonlinear term f is a continuous function which is superlinear at the origin and infinity.
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. Quark wigner distributions andorbital angular momentum. Generalized su2 covariant wigner functions and some of. We present a phasespace study of a nonhermitian hamiltonian with pt symmetry based on the wigner distribution function. We are able to determine the timedependent distribution functions of specific processes, either through minimization of the relative entropy or through maximization of the generalized entropy under nonlinear constraints. Generalized phasespace distribution functions nasaads. In contrast to classical physics, the language of quantummechanics involves operatorsand wave functionsor, more generally, density operators. A point in this phase space is correspondingly called a macrostate. 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 dynamics and other properties of the generalized phasespace distribution are considered. Introduction it is well known that the uncertainty principle makes the concept of phase space in quantum mechanics problematic. General approach to sun quasidistribution functions. Recall that the phase space distribution function satisfied a liouville equation. Observables in quantum mechanics can be described by operators or by functions and distributions in phase space 1 23456789.
The phase space can also refer to the space that is parameterized by the macroscopic states of the system, such as pressure, temperature, etc. In fact, the most famous generalized function was discovered in physics by dirac before the analysts cottoned on, and generalized functions are often known as distributions, as a nod to the charge distribution example which inspired them. The work described below demonstrates the importance of using a generalized form of the standard maxwellian probability distribution function to model the velocity space portion of phase space. But, classically, through hamiltons equations, the systems time evolution is completely determined once the initial conditions are specified. It starts with the physics of form factors and parton distributions. A system has generalized linear phaseif its frequency response can be written as hej. Generalized hyperbolic distribution function r documentation. Zhong and gross journal of the european optical societyrapid publications phase space of partially coherent light with discontinuous surfaces minyi zhong 0 herbert gross 0 0 friedrichschilleruniversity jena, institute of applied physics, jena 07743, germany we propagate partially coherent light through discontinuous surfaces and analyze the optical effects in phase space. This yields particularly simple formulas in dimension d 2. M, then all the coordinates of the n particles are given in terms of generalized coordinates q i where 1. 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.
Emergent phase space description of unitary matrix model. The phase space distribution does not have to be bilinear in the state function. Then it is shown that the classical limit of the corresponding phasespace equation is independent of the ordering of operators corresponding to the inverse of the generalized wigner transformation, provided the latter is involutive. Since t is a function of conditions 0, can write the time correlation function as h j 0 a t s. Phase space dynamics introduction a mechanical system can be described completely by the hamiltonian of the system, where is the generalized coordinates of the system and is the generalized conjugate momenta of the system.
General approach to quantum mechanics as a statistical theory. Simple derivations of generalized linear and nonlinear. Phase space formulation of the quantum mechanical particleinabox problem nicholas wheeler, reed college physics department december 2000 introduction. The space d0 of generalized functions distributions by l. The c hoice actually arbitrary, and for con v enience, w e c ho ose the initial conditions. The construction of the correct phase space and the computation of an explicit form of a certain quasiprobability distribution function enables one to understand what is the. By treating h as a small perturbation to unperturbed hamiltonian ho, the solution of the. The total crosssection is obtained by integrating over all possible nal state momenta using the lorentz invariant measure.
One of the main differences with the literature is our analytic representation of the distribution function in the velocity phase space with as few. Calculates the probability pdf and cumulative cdf distribution functions of the pareto distributions generalized, type i, type ii given the shape, scale and location parameters. The the macrosystem phase space spacespace is equal to the sum of spaces spaces. The approach here is generalized estimating equations gee. Phasespaceregion operators and the wigner function. This tutorial gives an overview of the use of the wigner function as a tool for.
Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. Weyl quantization, phase space distributions, wigner distribution, moyal bracket. In this chapter, we develop a formulation where all distributions may be studied in a unified way. Generalized phase space tomography for intense beams a. A generalized enhanced fourier law and underlying connections to major frameworks for quasiballistic phonon transport 3 at the l2 order in angular momentum.
Pdf landau damping in space plamas with generalized r, q. Estimates the shape, scale and location parameters for the generalized extremevalue gev distribution using maximumlikelihood estimation mle. Digital signal processing fir filters with generalized. Appendix fundamental properties of generalized functions. Phasespace representation of a nonhermitian system with pt. Since the heatflux is a physically more accessible quantity than the distribution function, this formalism has certain advantages over others, some of which. Summary of 2nd lecture university of maryland department of. Weyl quantization and wigner distributions on phase space. The fractional fourier transform and the wigner distribution. In quantum physics, phase space methods have been used for state identification and characterization by plotting symbols of the density matrix as a distribution function on the sphere or in the q. How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems.
Pdf generalized phasespace tomography for intense beams. In physics and geometry, there are two closely related vector spaces, usually threedimensional but in general could be any finite number of dimensions position space also real space or coordinate space is the set of all position vectors r in space, and has dimensions of length. Pdf generalized hamilton function in the phase space of. Phase space volume scaling of generalized entropies and. A set of quasiprobability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. Thus, as advertised above, the volume of a shell of constant energy in phase space is virtually the same as the volume of the entire space for ee phase space fall 2007, physics 233b, hitoshi murayama 1 twobody phase space the twobody phase is the basis of computing higher body phase spaces. Wolf department of physics and astronomy, university of rochester, rochester, n. The necessity of extending the phase space has already been observed in the case of a single crystal 7, 17, nite unions 20 and in quasicrystals 19, 21, where the collision kernel is independent of x. Generalized functions, space of encyclopedia of mathematics. To answer this, consider the problem of a simple harmonic. The generalized phase space descriptions of a quantum system are constructed as special linear representations of the space of the linear operators, acting on the state vector space of the system. This allows the writing of correspondence rules at will, of which the ones presently known are particular cases. Generalized phasespace tomography for intense beams. The suitability of the resulting fitted distribution must be evaluated.