Time derivative

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. 

For each collision there is an inverse one, so we can also express the time derivative of the //-fiinction in temis of the inverse collisions as... 

The time derivative of tiiis expression together with A3.1.19) unplies... 

Using the Heisenberg equation of motion, (AS,2,40). the connnutator in the last expression may be replaced by the time-derivative operator... 

There are tliree steps in the calculation first, solve the frill nonlinear set of hydrodynamic equations in the steady state, where the time derivatives of all quantities are zero second, linearize about the steady-state solutions third, postulate a non-equilibrium ensemble through a generalized fluctuation dissipation relation. 

If we write the time-dependent Sclnodinger equation as 5i i/(T = -(i/li)f7i i, then, after replacing the time derivative by a central difference, we obtain... 

A more powerfiil method for evaluating the time derivative of the wavefiinction is the split-operator method [39]. Flere we start by fomially solving ihd ild. = /7 / with the solution V fD = e Note that //is... 

The observable NMR signal is the imaginary part of the sum of the two steady-state magnetizations, and Mg. The steady state implies that the time derivatives are zero and a little fiirther calculation (and neglect of T2 tenns) gives the NMR spectrum of an exchanging system as equation (B2.4.6)). 

The steady-state solution without saturation to this equation is obtained by setting the time derivatives to zero and taking the tenns linear in as in equation (B2.4.11). 

Here dots over symbols designate time derivatives. If now the modulus a and phase ( ) are introduced through... 

Unlike classical systems in which the Lagrangean is quadratic in the time derivatives of the degrees of freedom, the Lagrangeans of both quantum and fluid dynamics are linear in the time derivatives of the degrees of freedom. 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

Time derivatives in expansion (2.113) can now be substituted using the differential equation (2.112) (Donea, 1984 The first order time derivative in expansion (2.113) is substituted using Equation (2.112) as... 

Repeated differentiation of Equation (2.114) with respect to the time variable also gives the higlier-order time derivatives of the unknown, for example... 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Governing flow equations, originally written in an Eulerian framework, should hence be modified to take into account the movement of the mesh. The time derivative of a variable / in a moving framework is found as... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

Using a similar procedure the second order time derivative of pressure is found as... 

After substitution of the first- and second-order time derivatives of the unknowns in Equations (4.132) to (4.134) from Equations (4.139) to (4.141) and spatial discretization of the resulting equations in the usual manner the working equations of the scheme are derived. In these equations, fimctions given at time level n+aAt can be interpolated as... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

We have used a common notation from mechanics in Eq. (5-4) by denoting velocity, the first time derivative of a , x, and acceleration, the second time derivative, x. In a conservative system (one having no frictional loss), potential energy is dependent only on the location and the force on a particle = —f, hence, by differentiating Eq. (5-3),... 

The stress—relaxation process is governed by a number of different molecular motions. To resolve them, the thermally stimulated creep (TSCr) method was developed, which consists of the following steps. (/) The specimen is subjected to a given stress at a temperature T for a time /, both chosen to allow complete orientation of the mobile units that one wishes to consider. (2) The temperature is then lowered to Tq T, where any molecular motion is completely hindered then the stress is removed. (3) The specimen is subsequendy heated at a controlled rate. The mobile units reorient according to the available relaxation modes. The strain, its time derivative, and the temperature are recorded versus time. By mnning a series of experiments at different orientation temperatures and plotting the time derivative of the strain rate observed on heating versus the temperature, various relaxational processes are revealed as peaks (243). 

Each physical quantity is expressed in one and only one unit, eg, the meter for length, the kilogram for mass, and the second for time. Derived units are defined by simple equations relating two or more base units. Some are given special names, such as newton for force and joule for work and energy. 

It has been estimated that using available neutron intensities such as 10 neutrons/(cm -s) concentrations of B from 10—30 lg/g of tumor with a tumor cell to normal cell selectivity of at least five are necessary for BNCT to be practical. Hence the challenge of BNCT ties in the development of practical means for the selective deUvery of approximately 10 B atoms to each tumor cell for effective therapy using short neutron irradiation times. Derivatives of B-enriched /oj o-borane anions and carboranes appear to be especially suitable for BNCT because of their high concentration of B and favorable hydrolytic stabiUties under physiological conditions. 

Implicit methods can also be used. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. 

The terms may be quantities or rates of flow of material or enthalpy. Inputs and outputs are streams that cross the vessel boundaries. A heat of reaction within the vessel is a. source. A depletion of reactant in the vessel is a. sink. Accumulation is the time derivative of the content of the reference quantity in the vessel of the volume times the concentration, 3V C /df, or of the total enthalpy of the vessel contents, d[WCfT-T,i)]/dt. 

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