Microcanonical constant

Dynamics simulations employing conventional periodic boundary conditions are usually carried out in the microcanonical (constant N, constant volume, constant energy) ensemble. However, techniques have been introduced that allow NVT (constant N, constant volume, constant temperature), NPH... 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

RRKM fit to microcanonical rate constants of isolated tran.s-stilbene and the solid curve a fit that uses a reaction barrier height reduced by solute-solvent interaction [46],... 

RRKM theory assumes a microcanonical ensemble of A vibrational/rotational states within the energy interval E E + dE, so that each of these states is populated statistically with an equal probability [4]. This assumption of a microcanonical distribution means that the unimolecular rate constant for A only depends on energy, and not on the maimer in which A is energized. If N(0) is the number of A molecules excited at / =... 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Song K and Hase W L 1999 Fitting classical microcanonical unimolecular rate constants to a modified RRK expression anharmonic and variational effects J. Chem. Phys. 110 6198-207... 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

In a canorrical ensemble the total temperature is constant. In the microcanonical ensemble, however, the temperature will fluctuate. The temperature is directly related to the kinetic energy of the system as follows ... 

If the coupling parameter (the Bath relaxation constant in HyperChem), t, is too tight (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcanonical) ensemble. The trajectory is then neither canonical or microcanon-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for these simulations ... 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

What about a microcanonical ensemble based approach The simplest prescription would be to define a rule that flips the spins only at a constant energy that is, for 6H = 0 or Ai = 2. The reader may recall that we have already studied some of the behaviors of this rule in section 3.4 it is, in fact, equal to the Pbedkin-reversible rulq Q2R ([vich84], [vich86aj). Prom the comments made in our earlier discussion, it... 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

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