Liouville equation differential equations

A New Topological invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

In closing this section, we note that the stochastic master equation, Eq. (16), can be used to study the effect of boundary conditions on transport equations. If a(x, y, t) is sufficiently peaked as a function of x — y, that is if transitions occur from y to states in the near neighborhood of y, only, then the master equation can be approximated by a Fokker-Planck equation. The effects of the boundary on the master equation all appear in the properties of a(x, y, t). However, in the transition to the Fokker-Planck differential equation, these boundary effects appear as boundary conditions on the differential equation.7 These effects are prototypes for the study of how molecular boundary conditions imposed on the Liouville equation are reflected in the macroscopic boundary conditions imposed on the hydrodynamic equations. 

In the cause of logical clarity, it is unfortunate that the equations of transport are so often derived from the Boltzmann integro-differential equation. Their derivation from the Liouville equation is a straightforward exercise in -dimensional calculus,... 

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

Having treated in the previous two sections linear stochastic differential equations we now return to the general case (1.1). Just as normal differential equations, it can be translated into a linear equation by going to the associated Liouville equation. To do this we temporarily take a single realization y(t) of Y(t) and consider the non-stochastic equation... 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

Sturm-Liouville differential equations, resulting from a separation of variables have been known since the middle of the 19th century. Separation constants, subject to boundary conditions, yield sets of characteristic, or eigenvalue, solutions. 

Volumes have been written about the red herring known as Schrodinger s cat. Without science writers looking for sensation, it is difficult to see how such nonsense could ever become a topic for serious scientific discussion. Any linear differential equation has an infinity of solutions and a linear combination of any two of these is another solution. To describe situations of physical interest such an equation is correctly prepared by the specification of appropriate boundary conditions, which eliminate the bulk of all possible solutions as irrelevant. Schrodinger s equation is a linear differential equation of the Sturm-Liouville type. It has solutions, known as eigenfunctions, the sum total of which constitutes a state function or wave function, which carries... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

Analytical expressions for coherence- and polarization-transfer functions can be derived using various approaches. These approaches are based on the Liouville-von Neumann equation [Eq. (47)]. If relaxation is neglected, the evolution of the density operator under an effective Hamiltonian is governed by the differential equation... 

A generalization of Liouville s equation was presented in [10]. Applying modern techniques in differential geometry [13], as they are applied to dynamical systems [14-16], gives rise to the generalized Liouville equation of the form [17] ... 

The boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

By applying the Sturm-Liouville theorem, the coefficient A for partial differential equations with nonhomogeneous boundary conditions is obtained as ... 

For the solution of this differential equation it is advantageous to transform the matrix equation of Hilbert space into a vector equation in Liouville space ( /, is the identity matrix of the four-dimensional subspace) ... 

The conservation of density of a mechanical system in phase space (Liouville s theorem) implies a rigorous functional relation between g(ri2) and 53(> i2,> i3,> 23)- Starting from this rigorous functional relation (an integro-differential equation ) an approximate closed equation can be obtained " by using the superposition approximation which asserts that... 

---