Variables time independent

If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation... 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

In this section I will write h for the time-dependent states and ijr for the time-independent ones. The jri may themselves depend on the space and spin variables of all the particles present. 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

Values of all the variables, stored by the PREPARE statement can be plotted at the end of the simulation or after an interrupt, using the GRAPH command. The GRAPH T,A,B,C,D, command, for example, plots a combined post-mortem graph of all the concentrations with respect to the independent variable time... 

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

Thus the problem involves the two independent variables, time t and length Z. The distance variable can be eliminated by finite-differencing the reactor length into N equal-sized segments of length AZ such that N AZ equals L, where L is the total reactor length. 

It is assumed that both state variables x, and x2 are measured with respect to time and that the standard experimental error (oe) is 0.1 (g/L) for both variables. The independent variables that determine a particular experiment are (i) the inoculation density (initial biomass concentration in the bioreactor), Xq i, with range 1 to 10 g/L, (ii) the dilution factor, D, with range 0.05 to 0.20 h 1 and (iii) the substrate concentration in the feed, cF, with range 5 to 35 g/L. 

In the following program, my table is called table(nent, nvar), in which the first column is the independent variable, time, which must increase monotonically down the table, and the other columns are one or more tabulated variables. In this example there is only one dependent variable,... 

Time is a variable in kinetics but not in thermodynamics rates dealt with in the latter are with respect to temperature, pressure, etc., but not with respect to time equilibrium is a time-independent state. 

The effective potential has a form similar to that for the time-independent case with the density argument replaced by the TD density variable. Also some of the functionals may be current density dependent. 

The unperturbed Hamiltonian 3 is the same for all systems and is time-independent. The time-dependent perturbation G(t), different for each system, is considered as a stationary stochastic variable. We may, without loss of generality, suppose that the mean value of G(t) over the ensemble is equal to zero. We denote by a,p,y,. . . the eigenstates of supposed to be non-degenerate, and by fix, the corresponding energies. 

This equation is a partial differential equation whose order depends on the exact form of/ and F. Its solution is usually not straightforward and integral transform methods (Laplace or Fourier) are necessary. The method of separation of variables rarely works. Nevertheless, useful information of practical geological importance is apparent in the form taken by this equation. The only density distributions that are time independent must obey... 

Note that, in spite of the time variable being t, surface conditions now depend only on parameter t. Duhamel s principle states that if C(x,t,t) can be calculated, which should usually be an easy task due to time-independent surface conditions, the solution writes... 

We will discuss in this book only deterministic systems that can be described by ordinary or partial differential equations. Most of the emphasis will be on lumped systems (with one independent variable, time, described by ordinary differential equations). Both English and SI units will be used. You need to be familiar with both. 

Density and velocity can change as the fluid flows along the axial or z direction. There are now two independent variables time t and position z. Density and... 

Note that independent variable, time, disappears from this problem. While final time constraints in (16) appear naturally in (17), other constraints that need to be enforced over the time domain are difficult to handle. For example, Sargent and Sullivan (1977) converted these to final time constraints by integrating the square of the constraint violations and forcing these to be less than a tolerance at final time this however, leads to degeneracies in solving the nonlinear program. 

We number the steps with i = 1, 2,. .., with i = and i = N being steps at the beginning and end of the step train. Let hi(x, f) describe the random motion of the i step in the train about its center of mass, which is assumed to be fixed - direct interaction terms are needed to produce center of mass dynamics fi(x, t) is the local chemical potential and di, with d, = °° = dn is the average distance between the centers of mass of adjacent steps (see Fig. 3). d, are time-independent in this analysis. In terms of these variables, the Langevin Eq. for the i step is ... 

Note that the steady-state solution, Eq. (60), depends explicitly on time through x (f). To obtain a time-independent solution we must change variables x —> x— x (f) and describe the motion of the bead in the reference frame that is solid and moves with the trap. We will come back to this problem in Section IV.A.3. 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

To facilitate the discussion it is helpful to specify three of the numerous meanings of the word state . We shall call a site any value of the stochastic variable X or n. We shall call a macrostate any value of the macroscopic variable . A time-dependent macrostate is a solution of the macroscopic equation (X.3.1), a stationary macrostate is a solution of (X.3.3). We shall call a mesostate any probability distribution P. A time-dependent meso-state is a solution of the master equation, the stationary mesostate is the time-independent solution PS(X). 

To illustrate what is meant by scaling, let us return to the problem of the dissolving sphere and try to make sure that all the important dependent variables are in the interval [0, 1]. The independent variable, time, must be allowed to run its course. We have certainly done this with the radius of the sphere, for r can only diminish so, if the initial radius is R,x = r/R is obviously the correct choice. With an eye to extending the model later, we define U as the terminal velocity of a sphere of radius R, and because this decreases with decreasing radius, v = u/U is certainly in [0,1]. The very simple relationship v = x2 holds as long as our assumption of the validity of Stokes law is true. 

Let N(t) be the random variable representing at time t, for instance, the number of reactants in a reversible chemical reaction. Each reactive act is followed by a decrease or an increase of one reactant. Furthermore, let X(t) = N(t)/V denote the concentration variable, where V is the volume of the chemical system and consider e = 1IV. Thus, one can envisage that per reactive act X(t) changes by e. The process A (t), t > 0 may be interpreted as a one-step process characterized by the following time-independent transition rate densities ... 

---