Reversibility and irreversibility from an initial value formulation

From a time evolution equation for the single particle distribution function derived from the N-particle distribution function (A. Muriel, M. Dresden, Physica D 101 (1997) 297), an exact solution for the 3D Navier–Stokes equation – an old problem – has been found (A. Muriel, Results Phys. 1 (2011) 2). In this Letter, a second exact conclusion from the above-mentioned work is presented. We analyze the time symmetry properties of a formal, exact solution for the single-particle distribution function contracted from the many-body Liouville equation. This analysis must be done because group theoretic results on time reversal symmetry of the full Liouville equation (E.C.G. Sudarshan, N. Mukunda, Classical Mechanics: A Modern Perspective, Wiley, 1974). no longer applies automatically to the single particle distribution function contracted from the formal solution of the N-body Liouville equation. We find the following result: if the initial momentum distribution is even in the momentum, the single particle distribution is reversible. If there is any asymmetry in the initial momentum distribution, no matter how small, the system is irreversible.     

Thermodynamics of the Early Universe

The relationship between relativistic thermodynamics of the early Universe with the Logunov metric and a gravitational analog of statistical mechanics is examined. An equation of state for gravitational atoms is derived. These atoms can be the medium that gave rise to the contents of our Universe or miniUniverses. A gravitational analog of the first law of thermodynamics is obtained. It is also found that the symmetrical in time Liouville equation can have a partial solution with a broken symmetry in time.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 7–17, March, 2005.     

Time-fractional extensions of the Liouville and Zwanzig equations

This paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.     

Classical Liouville action on the sphere with three hyperbolic singularities

The classical solution to the Liouville equation in the case of three hyperbolic singularities of its energy–momentum tensor is derived and analyzed. The recently proposed classical Liouville action is explicitly calculated in this case. The result agrees with the classical limit of the three-point function in the DOZZ solution of the quantum Liouville theory.     

A propagator theory applied to wave mechanics in phase space

The fact that the classical Liouville equation can be analyzed as a dynamical equation in Hilbert-Koopman (HK.) space is used in order to develop a perturbative method for the wave mechanics in phase space: an explicit solution of the Liouville equation inqp representation is exhibited. The connection between the solution obtained and the dynamics of correlations is established by computing theqp-kp transformation function in HK space. To elucidate the method, an application is presented and the result compared to that available in the literature.     

Exact quantum three-point function of Liouville highest weight states

In their earlier works on the quantum Liouville theory, Gervais and Neveu derived the exact spectrum of highest weight states of the conformal algebra. In the present paper, we determine the interaction between three of these states exactly, in the weak coupling regime of the quantum Liouville dynamics. It is first studied, at the classical level, by computing the time delays from an appropriate classical solution. The result is then extended to the quantum case, by following a path taken some time ago by Faddeev, Kulish, and Korepin, for the sine-Gordon theory: the coupling constant is replaced by the renormalized one, and the classical action that takes the form of a three-dimensional line integral is replaced by a discrete sum running over the exact quantum spectrum of the three asymptotic states that forms a three-dimensional lattice. At the quantum level, the classical S-matrix, that is the exponential of the action, becomes a product to be computed along a line on this lattice. It must only depend upon the end points and this completely determines the three-point function at the quantum level. Its structure is reminiscent of the other exact S-matrices that have been discovered earlier.     

Light-cone quantisation of the Liouville and Toda field theories

The Toda field is a multicomponent field in two space-time dimensions satisfying a generalisation of the Liouville equation ?²? + exp ? = 0. We define the quantum field theory, and solve for the fields in terms of their initial values on a forward light-cone, demonstrating that our solution is regular. We give an explicit result for the Liouville equation which is the quantum version of the well-known classical solution. We also discuss the energy-momentum spectrum, and the conformal properties of the theory.     

Formal structure of kinetic theory

We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL _o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL _o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.Work supported by the National Science Foundation.     

Stochastic electrodynamics. III. Statistics of the perturbed harmonic oscillator-zero-point field system

In this third paper in a series on stochastic electrodynamics (SED), the nonrelativistic dipole approximation harmonic oscillator-zero-point field system is subjected to an arbitrary classical electromagnetic radiation field. The ensemble-averaged phase-space distribution and the two independent ensemble-averaged Liouville or Fokker-Planck equations that it satisfies are derived in closed form without furtner approximation. One of these Liouville equations is shown to be exactly equivalent to the usual Schrödinger equation supplemented by small radiative corrections and an explicit radiation reaction (RR) vector potential that is similar to the Crisp-Jaynes semiclassical theory (SCT) RR potential. The wave function in this SED Schrödinger equation is shown to have thea priori significance of position probability amplitude. The other Liouville equation has no counterpart in ordinary quantum mechanics, and is shown to restrict initial conditions such that (i) The Wigner-type phase-space distribution is always positive, (ii) in the absence of an applied field, the only allowed solution of both equations is the quantum ground state, and (iii) if a previously applied field is suddenly turned off, then spontaneous transitions occur, with no need for a triggering perturbation as in SCT, until the system is in the ground state. It is also shown that the oscillator energy is a fluctuating quantity that must take on a continuum of values, with average value equal to the quantum ground-state energy plus a contribution due to the applied classical field.     

Shear flow in the two-body Boltzmann gas

We present the analytic solution of the two-particle Boltzmann equation for hard disks undergoing isothermal shear, within the relaxation time approximation. This system has a steady state for all values of the strain rate. The fluid is shear thinning and exhibits normal stress differences. This solution is compared with the collision free solution of the Liouville equation. The non-equilibrium entropy is also calculated.     

On Kaup and Newell＇s Method for Solving DNLS Equation

Kaup and Newell＇s revised inverse scattering transform for the derivative nonlinear Schrodinger （DNLS） equation is investigated. We compared it with a more reasonable approach proposed recently, which is rigorously proven by the Liouville theorem. It is conduded that Kanp and Newell＇s revision is only suitable for giving single-soliton solution and can not be generalized to multi-soliton case.     

On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics

It is shown that time and entropy operators may exist as superoperators in the framework of the Liouville space provided that the Hamiltonian has an unbounded absolutely continuous spectrum. In this case the Liouville operator has uniform infinite multiplicity and thus the time operator may exist. A general proof of the Heisenberg uncertainty relation between time and energy is derived from the existence of this time operator.     

Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker-Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε, the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O(ε²). Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O(ε²) solution of the Liouville equation yields an equation for W to O(ε³). An augmented Smoluchowski equation for W is obtained from the O(ε²) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.     

Generalized fractional operators on time scales with application to dynamic equations

We introduce more general concepts of Riemann–Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value problem described by generalized fractional order differential equations on time scales are proved.     

Conformal bootstrap in Liouville field theory

We consider the recently proposed analytic expression for the three-point function in the Liouville field theory on a sphere. It is verified that in the classical limit this expression reduces to what the classical Liouville theory predicts. Using the suggested three-point function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and check numerically that it satisfies the conformal bootstrap equations. The Liouville reflection amplitude which follows explicitly from the structure constants is also considered and compared with the results of the Bethe ansatz technique.     

Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases

Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with 0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations, anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping.     

超Liouville模型精确解

运用Leznov-Saveliev代数分析方法和Drinfeld-Sokolov构造分别给出超协变形式和分量形式超Liouville模型精确解.     

Integrability of Liouville System on High Genus Riemann Surface.Ⅰ. Classical Case

By using the theory of uniformization of Riemann surfaces,we study properties of the Liouville equation and its general solution on a Riemann surface of genus g>1.After obtaining Hamiltonian formalism in terms of free fields and calculating classical exchange matrices,we prove the classical integrability of Liouville system on high genus Riemann surface.     

Algebraic integrability and generalized symmetries of dynamical systems

It has been shown that when an n-dimensional dynamical system admits a generalized symmetry vector field which involves a divergence-free Liouville vector field, then it possesses n−1 independent first integrals (i.e., it is algebraically integrable). Furthermore, the Liouville vector field can be employed for the classification of algebraically integrable dynamical systems. The results have been discussed on examples which arise in physics.