Deterministic integral equation

Most methods of correction for instrumental broadening in SEC (or hydrodynamic chromatography) are based on the deterministic integral equation due to Tung ( ) ... 

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

The computer simulations employed the molecular dynamics technique, in which particles are moved deterministically by integrating their equations of motion. The system size was 864 Lennard-Jones atoms, of which one was the solute (see Table II for potential parameters). There were no solute-solute interactions. Periodic boundary conditions and the minimum image criterion were used (76). The cutoff radius for binary interactions was 3.5 G (see Table II). Potentials were truncated beyond the cutoff. 

The various fields that make up a science tend naturally to integrate, and with time such a process can lead to a true synthesis. Physics was the first science to achieve a synthesis of its disciplines, and it may be useful to compare that experience with its biological counterpart. The first unification occurred between mechanics and thermodynamics, in the first half of the nineteenth century, and the second came shortly afterwards, with the integration of electromagnetism. The result was the imposing edifice of classical physics, a conceptual system that described all reality in terms of particles and waves, with equations that seemed perfect because they were perfectly deterministic. The common denominator of all branches of classical physics was in fact the concept of determinism, and nodoby doubted, in the nineteenth century, that that was the true logic of the universe. 

Using stochastic differential equations can also represent the stochastic models. A stochastic differential equation keeps the deterministic mathematical model but accepts a random behaviour for the model coefficients. In these cases, the problems of integration are the main difficulties encountered. The integration of stochastic differential equations is known to be carried out through working methods that are completely different from those used for the normal differential equations... 

In a hybrid method, molecules are displaced in time according to conventional molecular dynamics (MD) algorithms, specifically, by integrating Newton s equations of motion for the system of interest. Once the initial coordinates and momenta of the particles are specified, motion is deterministic (i.e., one can determine with machine precision where the system will be in the near future). In the context of Eq. (2.1), the probability of proposing a transition from a state 0 to a state 1 is determined by the probability with which the initial velocities of the particles are assigned from that point on, motion is deterministic (it occurs with probability one). If initial velocities are sampled at random from a Maxwellian distribution at the temperature of interest, then the transition probability function required by Eq. 

For deterministic dynamics, a trial trajectory is obtained by integrating the equations of motion from the shooting point. The forward segment of... 

Figure 1.6. In a shooting move for deterministic trajectories, a time slice xjf = qjf, pj on the old path (solid line) is selected at random and the corresponding momenta are changed by a small random amount 5p. Integration of the equations of motion backward to time 0 and forward to time t starting from the modified state xjf = pj yields the new...

Molecular dynamics, like MC, is a dynamical procedure, but of a deterministic rather than stochastic nature. One starts from an arbitrary configuration and an initial set of particle velocities, and Newton s equations of motion of the system are integrated numerically as a function of time this time (unlike the MC time) corresponds to the real time. For fluids, MC and MD have comparable efficiency. For dense materials like proteins, MD is the more efficient because the random MC trial moves are rejected with high probability unless the moves are very small. The difficulty in obtaining the entropy with MC (discussed above) applies also to MD. 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

A major difference between quantum and classical mechanics is that classical mechanics is deterministic while quantum mechanics is probabilistic (more correctly, quantum mechanics is also deterministic, but the interpretation is probabilistic). Deterministic means that Newton s equation can be integrated over time (forward or backward) and can predict where the particles are at a certain time. This, for example, allows prediction of where and when solar eclipses will occur many thousands of years in advance, with an accuracy of meters and seconds. Quantum mechanics, on the other hand, only allows calculation of the probability of a particle being at a certain place at a certain time. The probability function is given as the square of a wave function, P t,i) = P (r,f), where the wave function T is obtained by solving either the Schrodinger (non-relativistic) or Dirac (relativistic) equation. Although they appear to be the same in Figure 1.2, they differ considerably in the form of the operator H. 

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