Microstates

We have already met this concept in Section 3.10 where we identified microstates or complexions as the number of ways the overall state of a system can be constructed. The number of ways a state of the system can arise determines the probability of its occurrence. If we toss two coins the probability of getting one head and one tail is twice that of getting two heads. If we write out all the possibilities (HT, TH, HH, TT) we see that this is because two microstates give rise to the mixed configuration. Consider a fixed number of molecules distributed over energy levels. As a concrete example let us take three molecules distributed between equally spaced energy levels (Fig. 9.3) with the total energy fixed at three units. We see that three distinct distributions, I, II, and III are possible. But not all these distributions are equally probable. We can see why this is so if we assume the molecules can be individually identified and label each one so as to enumerate the microstates which go to make up each of the three different distributions. We find that six times as many microstates make up II as III, and thus it is six times as probable (Fig. 9.3). 

The general formula for the number of microstates, W, which make up a distribution is the same as that for the number of ways N objects can be distributed between a number of boxes so that there are in the first box, n2 in the second, and so on, 

The total number of possible arrangements is Nl but we must divide this by ttjnj etc. as we cannot distinguish between states that differ only by an exchange of particles within a given energy level. Thus for distribution I we 

In all studies involving methods based on absorption or scattering of light, X rays, or neutrons, the characteristic time scales on which radiation interacts with the substance are many orders of magnitude shorter than those of atomic motions. Therefore, it is not the motions themselves but the disordering which arises due to molecular dynamics that should be investigated. 

The distribution of microstates may be defined as the distribution of spatial dislocations, orientations, and interactions of groups of the main chain and side groups with respect to their most probable values. 

If microstates lead to the existence of a distribution of energies of interaction between aromatic groups and neighboring groups of atoms, then the individual spectra of these groups in different microstates shift differently, which results in an inhomogeneous contour of the absorption band. The application of selective photoexcitation permits specific effects of the distribution of microstates on spectral, temporal, and polarization fluorescence properties to be observed. 221 Such effects have been observed in studies of proteins, 1,8) and, as we show below, they may be used to obtain important information on dynamics. 

The reverse process, the gathering of all the molecules into one bulb, Is nonspontzneous. 

If we start with a single molecule in the left bulb, there are two possible arrangements when we open the stopcock. The molecule can stay in the left bulb, or it can move into the right bulb. 

Note thm vvith each addHional moleoule in this system, the number of microstates doubles. 

Therefore, the probability is i (one chance in two) that the molecule will be found in the left bulb after the stopcock is opened. Each of the two possible arrangements is called a microscopic state or microstate. 

If we have two molecules, there are four possible arrangements or four microstates. (Different molecules are distinguished by different colors.) 

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This is the density of microstates for one free particle in volume V= L. ... 

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

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... 

For a microcanonical ensemble, p = [F( )] for each of the allowed F( ) microstates. Thns for an isolated system in eqnilibrinm, represented by a microcanonical ensemble. 

Consider the microstates with energy Ej snch that Ej < E. The total number of such microstates is given by... 

Consider two systems in thennal contact as discussed above. Let the system II (with volume and particles N ) correspond to a reservoir R which is much larger than the system I (with volume F and particles N) of interest. In order to find the canonical ensemble distribution one needs to obtain the probability that the system I is in a specific microstate v which has an energy E, . When the system is in this microstate, the reservoir will have the energy E = Ej.- E due to the constraint that the total energy of the isolated composite system H-II is fixed and denoted by Ej, but the reservoir can be in any one of the R( r possible states that the mechanics within the reservoir dictates. Given that the microstate of the system of... 

The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

I hcre arc two types of Cl calculations im piemen ted in Hyper-Ch ern sin gly exciled Cl an d in icroslate Cl. I hc sin gly excited C which is available for both ah initio and sem i-etn pirical calculations may be used to generate CV spectra and the microstate Cl available only for the semi-empirical methods in HyperChern is used to improve the wave function and energies including the electron ic correlation. On ly sin gle point calculation s can he perform cd in HyperChetn using Cl. 

The Microstate Cl Method lowers the energy of the tin correlated groun d state as well as excited states, fhe Sngly Eixcitecl Cl Method is particularly appropriate for calculating LIV visible spectra, and does not affect the energy of the ground state (Brillouin s fheo-rem). 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

For the extension to two dimensions we consider a square lattice with nearest-neighbor interactions on a strip with sites in one direction and M sites in the second so that, with cyclic boundary conditions in the second dimension as well, we get a toroidal lattice with of microstates. The occupation numbers at site i in the 1-D case now become a set = ( ,i, /25 5 /m) of occupation numbers of M sites along the second dimension, and the transfer matrix elements are generalized to... 

Entropy is often described as a measure of disorder or randomness. While useful, these terms are subjective and should be used cautiously. It is better to think about entropic changes in terms of the change in the number of microstates of the system. Microstates are different ways in which molecules can be distributed. An increase in the number of possible microstates (i.e., disorder) results in an increase of entropy. Entropy treats tine randomness factor quantitatively. Rudolf Clausius gave it the symbol S for no particular reason. In general, the more random the state, the larger the number of its possible microstates, the more probable the state, thus the greater its entropy. 

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