Momentum-jump

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

The approximation (58) resembles the usual momentum-jump ansatz employed in various surface-hopping methods [55, 57, 58, 61, 66, 67, 82]. In order to determine the momentum shift of a trajectory, however, the latter formulations typically require the conservation of nuclear energy ... 

Within the momentum-jump approximation (58), the algorithm provides an in principle exact stochastic realization of the QCL equation. Apart from practical problems to be discussed below, the trajectory implementation therefore represents a well-defined computational scheme. 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

The presence of the momentum derivatives in J makes the action of this operator difficult to simulate, because it acts on all functions to its right. This will generate a branching tree of trajectories. This difficulty is avoided by making the momentum-jump approximation. To see how this approximation is obtained, the following change of variables is made ... 

The approximations surrounding the definition of the J operator comprise the momentum-jump approximation. This translation or shift of the momentum corresponds precisely to the amount of energy transferred during a transition... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

The propagator e JS)ss, is responsible for quantum transitions and bath momentum changes. In order to compute its action, we use the momentum-jump approximation [12, 23] that replaces the small continuous momentum changes with momentum jumps that accompany each quantum transition. In this approximation, the matrix elements of e can be written in terms of a matrix M to 0(S2),... 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

Since the Hugoniot represents the locus of all possible states behind the shock front, then a line joining the initial and final states on the P-v Hugoniot represents the jump condition. This line is called the Raleigh line and is shown in Figure 17.3. If we eliminate the particle velocity term u by manipulating the mass- and momentum-jump equations, and let mq = 0, we get... 

Here we show the steps leading to this momentum-jump approximation . Since S / = AEai3dai3 with = Ea — we may write... 

In the first line of this equation we made the main assumption that the first two terms on the left hand side could be approximated by the exponential of the operator. In the second line we wrote the momentum vector as a sum of its components along dap and perpendicular to 4/3) nd in the penultimate line we used the fact that the exponential operator is a translation operator in the variable (P dap). In the last line the momentum jump AP is given by... 

If the transition is accepted, then, using the momentum jump approximation, we translate the momentum PAt to Pat = Pat + AP where AP is defined in (84). We then write... 

In the jump-condition formulation the physical problem is generally decomposed into k bulk phase domains where the continuity and momentum equations for isothermal incompressible flows holds, and at the interface between these domains boundary conditions are specified using the interface jump conditions. That is, across the interface some quantities are required to be continuous, while others are required to have specific jumps. The discontinuous (singular) momentum jump condition can be derived by use of the surface divergence theorem (see e.g., [63] p 51 [26]). A rigorous derivation of the jump balances for the multi-fluid model is given in sect 3.3. 

The interface boundary conditions, i.e., the momentum jump conditions, are expressed as (i.e., no surface tension gradients are considered) ... 

For dispersed flows containing very small fluid particles the interfacial tension effects have occasionally been considered significant [129]. In this case, the average momentum jump condition (3.147) becomes [112] [54] [129] ... 

In this case the averaged momentum jump condition can be expressed by ... 

In practice, however, it is difficult to parameterize the interfacial area averaged velocity so in the momentum equation it s often set equal to the bulk velocity and in the momentum jump condition the mass transfer terms are simply neglected enabling a closure relation for the interfacial drag terms which are in agreement with Newton s 3. law. Hence,... 

It is noted that the momentum derivatives of the coupling elements (51) represent one of the main obstacles of a practical trajectory-based evaluation of the QCL equation, because these terms require the knowledge of the function in question not only at a particular point in phase space but at the same time also at nearby points. As a remedy, we may restrict ourselves to the limit of small momentum changes Snm/P At 1 and approximate 1 - - SnmP exp(S m5/dP). Since eP / Pf p) = /(p + S ), the approximation reduces the action of the differential operator to a simple shift of momenta. We note that this approximation resembles the usual momentum-jump ansatz employed in various surface-hopping methods. ... 

The momentum jump prescribed by this equation corresponds to the energy transfer in a transition and satisfies energy conservation. If there is insufficient kinetic energy available in the bath for the quantum subsystem to make the transition, the transition is not allowed. 

The pressure distribution inside the collision complex at t = 0.2 ms is shown in Fig. 1.23. The maximum pressure appears inside the low viscous liquid next to the inner interface. Three positions exhibiting a pressure jump can be observed, two at the outer surface of the droplet and one at the inner interface, where surface tension is not present. The momentum jump condition at the interface is given by Eq. (1.12). Extracting the normal component of the jump condition in Eq. (1.12) yields... 

Another expression for V — Up follows from the momentum jump condition, eqn (14.10) ... 

---