Ensemble time-dependent properties

If the coiiplin g parameter (the Bath relaxation constan t in IlyperChem), t, is loo Tight" (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcan on leal) ensemble. The trajectory is then neither canonical or microcan on-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for Ih CSC sim ii lalion s ... 

Electrons switch between levels characterized by Ms values. Let us examine now an ensemble of n molecules, each with an unpaired electron, in a magnetic field at a given temperature. The bulk system is at constant energy but at the molecular level electrons move, molecules rotate, there are concerted atomic motions (vibrations) within the molecules and, in solution, molecular collisions. Is it possible to have information on these dynamics on a system which is at equilibrium The answer is yes, through the correlation function. The correlation function is a product of the value of any time-dependent property at time zero with the value at time t, summed up to a large number n of particles. It is a function of time. In this case the property can be the Ms value of an unpaired electron and the particles are the molecules. The correlation function has its maximum value at t = 0 since each molecule has one unpaired electron, the product of the... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

Ensembles generated by MC techniques are naturally of the constant NVT type, while MD methods naturally generate a constant NVE ensemble. Both MC and MD methods, however, may be modified to simulate other ensembles, as described in Sections 14.1.1 and 14.2.2. Of special importance is the constant NPT condition, which directly relates to most experimental conditions. The primary advantage of MD methods is that time appears explicitly, i.e. such methods are natural for simulating time-dependent properties, such as correlation functions, and for calculating properties that depend on particle velocities. Furthermore, if the relaxation time for a given process is (approximately) known, the required simulation time can be estimated beforehand (i.e. it must be at least several multiples of the relaxation time). 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

These two methods are different and are usually employed to calculate different properties. Molecular dynamics has a time-dependent component, and is better at calculating transport properties, such as viscosity, heat conductivity, and difftisivity. Monte Carlo methods do not contain information on kinetic energy. It is used more in the lattice model of polymers, protein stmcture conformation, and in the Gibbs ensemble for phase equilibrium. 

Kinetics deals with many-particle systems (thermodynamic ensembles). The properties measured as a function of time depend on the scale of observation, and this scale is chosen in relation to the question we wish to ask. The smaller the scale, the more inhomogeneous and fluctuating the homogeneous systems appear to be. For example, we describe the activated atomic jump frequency v as... 

We show how the response of a molecule to an external oscillating electric field can be described in terms of intrinsic properties of the molecules, namely the (hyper)polarizabilities. We outline how these properties are described in the case of exact states by considering the time-development of the exact state in the presence of a time-dependent electric field. Approximations introduced in theoretical studies of nonlinear optical properties are introduced, in particular the separation of electronic and nuclear degrees of freedom which gives rise to the partitioning of the (hyper)polarizabilities into electronic and vibrational contributions. Different approaches for calculating (hyper)polarizabilities are discussed, with a special focus on the electronic contributions in most cases. We end with a brief discussion of the connection between the microscopic responses of an individual molecule to the experimentally observed responses from a molecular ensemble... 

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

We will analyze the SM spectra and their fluctuations semiclassically using the stochastic Bloch equation in the limit of a weak laser field. The Kubo-Anderson sudden jump approach [58-61] is used to describe the spectral diffusion process. For several decades, this model has been a useful tool for understanding line shape phenomena, namely, of the average number of counts < > per measurement time T, and has found many applications mostly in ensemble measurements, for example, NMR [60], and nonlinear spectroscopy [62]. More recently, it was applied to model SMS in low-temperature glass systems in order to describe the static properties of line shapes [14-16, 63] and also to model the time-dependent fluctuations of SMS [64-66]. 

Molecular dynamics simulations have generally a great advantage of allowing the study of time-dependent phenomena. However, if thermodynamic and structural properties alone are of interest, Monte Carlo methods might be more useful. On the other hand, with the availability of ready-to-use computer simulation packages (e g.. Molecular Simulations Inc. 1999), the implementation of particular statistical ensembles in molecular dynamics simulations becomes nowadays much less problematic even for an end user without deep knowledge of statistical mechanics. 

It is certainly conceivable that the statistical properties of a turbulent flow can change at a rate that is comparable with the time scale of the turbulent motion so that proper ensemble averages are required, i.e., repetitions under identical conditions, which of course cannot normally be done in environmental flows. Indeed, often the time-dependent nature of environmental flows, such as on-shore, offshore diurnal wind cycles or tidal flows, can be the most important feature. The main point is that the mean concentration requires fewer realizations (or record length for steady circumstances) than higher moments for an adequate approximation of an ensemble average and that the k = 0 result of Eq. (25.18) is also correct for time-dependent flows over complex geometry. 

In order to use statistical mechanics to calculate the macroscopic properties introduced only by spatial averaging one needs to perform enses le averages over states. For present purposes classical mechanics will suffice using the phase space of conjugate coordinates and momenta qi and pi with a time dependent probablity density function z (qi p t). The meao expectation value of a function A (qi. p. t) is denoted by A(t) (q pj t) f(qj Pj t)> where the angle brackets now refer to the ensemble average over the qj and pj. 

Their methods borrow results from the theory of the dynamical properties of inhomogeneous fluids. It should be possible to reformulate the proof without this use of time-dependent functions but since this has not yet been done we must first make a digression to collect the auxiliary results we need. Naturally these results involve only ensemble averages that are stationary with respect to time. 

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