Liouvillian

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

If we subdivide the Liouvillian into the two parts by separating the force and velocity terms,... 

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. The internal vibrations of the molecules are often very fast compared to their translational and orientational motions. Although this and other systems, like proteins, have already been treated using RESPA,[17, 34, 22, 23, 24, 25, 26] another example, and the one we focus on here, is that of a system of very light particles (of mass m) dissolved in a bath of very heavy particles (mass M).[14] The positions of the heavy particles are denoted y and the positions of the light particles rire denoted by X. In this case the total Liouvillian of the system is ... 

Introdueing the above force breakup into the Liouvillian gives. 

The system defined by the Liouvillian is called the reference system. Now applying the Trotter factorization to the propagator exp iLs + Fi arising from this subdivision gives the new propagator,[17]... 

Here the Liouvillian of system Hx is determined by relation (7.23). This, in line with condition (7.5), yields... 

The Liouvillian iLo- = Ho, , where , is the Poisson bracket, describes the evolution governed by the bath Hamiltonian Hq in the held of the fixed Brownian particles. The angular brackets signify an average over a canonical equilibrium distribution of the bath particles with the two Brownian particles fixed at positions Ri and R2, ( -)0 = Z f drNdpNe liW J , where Zo is the partition function. 

An important consequence of the MPC theory is the existence of a time operator T. This operator does not commute with the Liouvillian LT — TL= i. Its average value is interpreted by Prigogine as the age of the system, closely related to entropy. More generally, any positive, monotonously decreasing function, M = M T), is a Lyapounov function. On the other hand, the transformation A appears formally as a square root A = This property leads directly to an W-theorem for intrinsically... 

MSN. 138. T. Petrosky and I. Prigogine, Complex spectral representations and singular eigenvalue problem of Liouvillian in quantum scatering, in Symposium Quantum Physics and the Universe, Waseda University, Tokyo, Japan, 1992. 

At the microscopic level of description, the hydrodynamic mode of diffusion is dehned as the Liouvillian eigenstate ... 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

Sg denotes the adjoint electron spin operator. One should notice that the expression [exp(—iLtx)Sp Do i p(f2ML)] results in the S-operators and the ml being (implicitly) time-dependent. In order to continue any further, we need to specify the lattice and its Liouvillian. 

The simplest possible physical picture of the lattice contains the electron Zeeman interaction, the axially symmetric ZFS (whose principal axis coincides with the dipole-dipole axis) and the molecular rotation. The corresponding Liouvillian is given by ... 

Li is the matrix representation of the lattice Liouvillian in the space of the basis operators, 1 is a unit (super)operator and Ci are projection vectors representing the operators of Eq. (32) in the same space. The... 

However, if the initial state is a thermal state, such as the canonical den-sity matrix p - (1/Z)exp(- 3//), the autocorrelation is no longer given by a single quantum amplitude but becomes a sum of quantum amplitudes in which quantum phases are randomized. In the classical limit ft - 0, the leading expression becomes the purely classical autocorrelation function with the dynamics being ruled by the classical Liouvillian operator ci = Hci> ... 

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