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arXiv:cond-mat/0612318 · Progress of Theoretical Physics

Deformed Fokker-Planck Equations

C.-L. Ho, Ryu Sasaki B

Published Oct 1, 2007• 7 citations
arXivHTMLPDF

Abstract

Based on the well-known relation between Fokker-Planck equations and Schrödinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schrödinger equations of “discrete” QM. The latter is a natural discretization of QM and its Schrödinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable “discrete”QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.

DOI
10.1143/ptp.118.667

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•Jan 1, 2000•0 citations
Based on the well-known relation between Fokker-Planck equations and Schrödinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schrödinger equations of “discrete” QM. The latter is a natural discretization of QM and its Schrödinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable “discrete”QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.
We again note that Langevin equations written for observables (– operators in quantum mechanics) conform to the Heisenberg view. On the other hand, an entity that embodies the wave function of the Schrödinger picture is the von Neumann density operator, usually denoted by t( )ρ , whose classical counterpart is the probability distribution function (pdf) discussed earlier. It is helpful to introduce a formalism for the pdf completely at par with the Langevin description. The resultant Fokker-Planck (FP) equations serve as the subjects of this chapter. The simplest FP equation can be derived from the master equation (2.25) for a stationary Markov process, as shown below (Agarwal 1983, Risken 1996).
Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-) Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.
The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.
The Fokker_Planck equation can be derived in a consistent manner through a microscopic approach based on a unified scheme of classical and quantum mechanics. Here we shall derive it through a purely quantum mechanical approach based on the reversible Schrodinger dynamics. We also give a brief discussion of the path integral representation of the Fokker_Planck equation in light of our derivation. We conclude that, because of the use of the representation of eigenstates of the time-independent Hamiltonian in our derivation, the thermodynamical entropy in this case must correspond to a coarse-graining of the quantum entropy.
It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.