Particle, test

In these equations the electrostatic potential i might be thought to be the potential at the actual electrodes, the platinum on the left and the silver on the right. However, electrons are not the hypothetical test particles of physics, and the electrostatic potential difference at a junction between two metals is nnmeasurable. Wliat is measurable is the difference in the electrochemical potential p of the electron, which at equilibrium must be the same in any two wires that are in electrical contact. One assumes that the electrochemical potential can be written as the combination of two tenns, a chemical potential minus the electrical potential (- / because of the negative charge on the electron). Wlien two copper wires are connected to the two electrodes, the... 

A unifonn monoenergetic beam of test or projectile particles A with nnmber density and velocity is incident on a single field or target particle B of velocity Vg. The direction of the relative velocity m = v -Vg is along the Z-axis of a Cartesian TTZ frame of reference. The incident current (or intensity) is then = A v, which is tire number of test particles crossing unit area nonnal to the beam in unit time. The differential cross section for scattering of the test particles into unit solid angle dO = d(cos vji) d( ) abont the direction ( )) of the final relative motion is... 

The integral cross section is therefore the effective area presented by each field particle B for scattering of the test particles A into all directions. The probability that the test particles are scattered into a given direction v ( /, ([)) is the ratio... 

An electron or atomic beam of (projectile or test) particles A with density N, of particles per cm travels with speed V and energy E tln-ongh an infinitesimal thickness dv of (target or fielc0 gas particles B at rest with... 

A distributiony (v ) of NM) test particles (cm of species A in a beam collisionally interacts with a distribution of N () field particles of species B. Collisions with B will scatter A out of the beam at the... 

Finally, by considering increasing the number of particles by one in the canonical ensemble (looking at the excess, non-ideal, part), it is easy to derive the Widom [34] test-particle fomuila... 

Here we have separated temis in the potential energy which involve tlie extra test particle, n Hesf... 

Simulations in the Gibbs ensemble attempt to combine features of Widom s test particle method with the direct simulation of two-phase coexistence in a box. The method of Panagiotopoulos et al [162. 163] uses two fiilly-periodic boxes, I and II. 

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

Atoms not explicitly included in the trajectory must be generated. The position at which an atom may be placed is in some sense arbitrary, the approach being analogous to the insertion of a test particle. Chemically meaningful end states may be generated by placing atoms based on internal coordinates. It is required, however, that an atom be sampled in the same relative location in every configuration. An isolated molecule can, for example, be inserted into... 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

The chemical potential of particles belonging to species a and (3 is measured by using the classical test particle method (as proposed by Fischer and Heinbuch [166]) in parts II and IV of the system i.e., we calculate the average value of (e) = Qxp[—U/kT]), where U denotes the potential energy of the inserted particles. 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Fig. 20 shows the density profiles in the reactive and nonreactive parts of the system. The number density in the reactive part is very high (a one-component density at the center of this part is 0.596, so the number density of two components is twice as high). However, the density in the nonreactive part is much lower and equal to 0.404. The application of the test particle methods is therefore easy. There is a well-established density plateau in the nonreactive part consequently, the determination of the bulk density in this part is straightforward and accurate. 

As shown in Fig. 7, let <5denote the distance between the wall and a test particle, Ohe the angle from the particle velocity vector V to the Z axis, and fi be the angle from the part of the V in XY plane to the Z axis, then we can write the mean free path of the test particle as... 

It is clear that if S> Xj, particle-particle collision happens most probably and X(9,P) = Xh. If Sparticle-wall collision happens within the travel distance of <5/cos 9. Then, the local free path of the test particle, X, is the average oiX 9,0) over the whole ranges of 6 and j3. That is. 

Probe methods like particle insertion and test particle methods (29-32) are quite useful for computing chemical potentials of constituent particles in systems with low densities. Test particles are randomly inserted the average Boltzmann factor of the insertion energy yields the free energy. For dense systems these methods work poorly because of the poor statistics obtained. 

Tanner, R. I., A test particle approach to flow classification for viscoelastic fluids. AIChE J. 22, 910-918 (1976). 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The diffusion equation describes the evolution of the mean test particle density h(r,t) at point r in the fluid at time t. Denoting the Fourier transform of the local density field by hk(t), in Fourier space the diffusion equation takes the form... 

For small wavevectors the test particle density is a nearly conserved variable and will vary slowly in time. The correlation function in the memory term in the above equation involves evolution, where this slow mode is projected... 

Terminal velocity, linear thermodynamics intermediate regimes and maximum flux, 25-27 regression theorem, 18-20 Test particle density, multiparticle collision dynamics, macroscopic laws and transport coefficients, 100-104 Thermodynamic variables heat flow, 58-60... 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

We note that the calculation of At/ will depend primarily on local information about solute-solvent interactions i.c., the magnitude of A U is of molecular order. An accurate determination of this partition function is therefore possible based on the molecular details of the solution in the vicinity of the solute. The success of the test-particle method can be attributed to this property. A second feature of these relations, apparent in Eq. (4), is the evaluation of solute conformational stability in solution by separately calculating the equilibrium distribution of solute conformations for an isolated molecule and the solvent response to this distribution. This evaluation will likewise depend on primarily local interactions between the solute and solvent. For macromolecular solutes, simple physical approximations involving only partially hydrated solutes might be sufficient. 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Beck, T. L., Quantum path integral extension of Widom s test particle method for chemical potentials with application to isotope effects on hydrogen solubilities in model solids, J. Chem. Phys. 1992, 96, 7175-7177... 

Potential distribution methods are conventionally called test particle methods. Because the assertions above outline a general and basic position for the potential distribution theorem, it is appropriate that the discussion below states the potential... 

As noted in the Introduction, the PDT is widely recognized with the moniker test particle method. This name reflects a view of how calculations of ((e l3AU° ))0 might be tried solute conformations are sampled, solvent configurations are sampled, and then the two systems are superposed the energy change is calculated, and... 

Direct test particle methods are expected to be inefficient, compared to other possibilities, for molecular systems described with moderate realism. Successful placements of a test particle may be complicated, and placements with favorable Boltzmann factor scores may be rare. Fortunately, the tools noted above are generally available to design more-specific approaches for realistic cases. 

Nevertheless, direct test particle calculations have been of great conceptual importance, particularly in cases where there is a consensus on the relevance of simplified model solutes [2-4, 6, 9, 10,41 15]. The related particle insertion techniques are used for simulating phase equilibria, as discussed in Chap. 10. 

Most free energy and phase-equilibrium calculations by simulation up to the late 1980s were performed with the Widom test particle method [7]. The method is still appealing in its simplicity and generality - for example, it can be applied directly to MD calculations without disturbing the time evolution of a system. The potential distribution theorem on which the test particle method is based as well as its applications are discussed in Chap. 9. 

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