Equivalence of ensembles

One last point little consideration is given to the co-ordination number requirements of active ensembles in bimetallic catalysts. Statistical calculations have assumed an infinite flat surface, although the Monte Carlo treatment by King places active atoms preferentially in high co-ordination sites, because in those positions they minimise the surface energy. The non-equivalence of ensemble-size sensitivity and particle-size sensitivity must be kept clearly in mind. 

The basic idea is to develop expressions for common thermodynamic quantities in terms of fluctuations in a system open to all species. The key lies in the fact that the fluctuating quantities characteristic of the grand canonical ensemble can then be transformed into expressions, which provide properties representative of the isothermal isobaric ensemble. Using an equivalence of ensembles argument, one can then consider these fluctuations to represent properties of small microscopic local regions of the solution of interest. This approach can be used to understand many properties of isobaric, isochoric, or osmotic systems in terms of particle number (and energy) fluctuations. 

At this point it is worth pausing and reminding ourselves what we have accomplished here. The above expressions correspond to thermodynamic properties of solutions that are amenable to experiment. The expressions involve fluctuating quantities for an equivalent open system in which the average particle numbers are equal to the fixed number of particles used in the experiment. The same argument holds for the chemical potentials, except in the opposite direction, and the volume and the pressure, that is, we are invoking the equivalence of ensembles approach. In... 

The equivalence of ensembles becomes apparent at the thermodynamic limit. Regardless of the ensemble constraints, a system at equilibrium will attain one observable value for each thermodynamic state variable. 

This behaviour is characteristic of thennodynamic fluctuations. This behaviour also implies the equivalence of various ensembles in the thermodynamic limit. Specifically, as A —> oo tire energy fluctuations vanish, the partition of energy between the system and the reservoir becomes uniquely defined and the thennodynamic properties m microcanonical and canonical ensembles become identical. 

Another way of looking at it is that Shannon information is a formal equivalent of thermodynamic entroi)y, or the degree of disorder in a physical system. As such it essentially measures how much information is missing about the individual constituents of a system. In contrast, a measure of complexity ought to (1) refer to individual states and not ensembles, and (2) reflect how mnc h is known about a system vice what is not. One approach that satisfies both of these requirements is algorithmic complexity theory. 

Fet us now consider the 3D equivalent of the aforementioned example an ensemble of uncorrelated homogeneous spheres - with polydispersity, meaning that the observed CLD... 

The quantum-mechanical equivalent of phase density is known as the density matrix or density operator. It is best understood in the case of a mixed ensemble whose systems are not all in the same quantum state, as for a pure ensemble. 

Thirdly it is easy to see that the condition that the X are independent is important. If one takes for all r variables one and the same X the result cannot be true. On the other hand, a sufficiently weak dependence does not harm. This is apparent from the calculation of the Maxwell velocity distribution from the microcanonical ensemble for an ideal gas, see the Exercise in 3. The microcanonical distribution in phase space is a joint distribution that does not factorize, but in the limit r -> oo the velocity distribution of each molecule is Gaussian. The equivalence of the various ensembles in statistical mechanics is based on this fact. 

In the words of Lancet and colleagues, a fundamentally different approach has envisaged primordial selfreplication as the collective property of ensembles of relatively simple molecules, interconnected by networks of mutually catalytic interactions. 38 The hereditary information in this case would be represented by the identity and concentration of its components. The term compositional genome has been used to describe this system, in which genetic information is not stored in a list, as in DNA, but is represented by the presence or absence of organic components.39,40 As an analogy, consider DNA to be the equivalent of a class list that records the full possible enrollment in a course. The information in a compositional genome would be represented by the presence of students who have turned up on a particular day. 

It is a top-down approach in providing explicit recipes for all these geometrical stmctures, and it is presumably equivalent to the bottom-up approach of working through their manifestations in terms of ensembles of trajectories [5]. 

The ergodic theorem states the equivalence of the average of a molecular ensemble with the time average of a single molecule. We have, therefore, compared the values for the conformational transition of our standard molecule. 

Now we see that if we choose an ensemble of a damped oscillator in which q)equ=0, (q)equ =0, the required equivalence of the ensemble averages is fulfilled. To prove the increase in entropy, first we introduce quantum entropy. The kBpsh ps operator is an operator whose eigenvalues are the terms of Shannon entropy kB pi(t) r pi(t). Thus, the Shannon entropy is the minus trace of that operator [26]... 

To generalize the zero- or the finite-temperature string methods to work in collective variables (Sects. 6.4 and 6.5), it suffices to do the following. At each iteration, it is now required to estimate the tensor A fiO) defined in (94) and determine the gradient of the density M 9) defined in (93) to be used in the equivalent of (125) (in the zero-temperature version) or sample with respect to this density to identify the equivalent of (127) (in the finite-temperature version). This requires to introduce an additional loop of sampling of blue moon type, now in the original ensemble with density m(x) under the additional constraint that 0 x) = 6. For details, see [23]. 

Generally, MD is performed to simulate a continuous phase trajectory in the microcanonical N,E,V) ensemble, while in MC method, individual phase points of an (N,V,T) ensemble are simulated. As far as the equilibrium properties are concerned, the trajectory average of MD and the configurational average of MC are equivalent (Allen and Tildesley, 1987). Recently, MD simulation of other types of ensembles also have been achieved. In the N,E,V) simulations, volume and total energy are held constant and the temperature and pressure are allowed to fluctuate. Andersen (1980) suggested methods for simulation of isobaric-isoenthalpic and isobaric-isothermal (A, P,7) ensembles. A... 

In the losartan molecule, the substituted imidazole moiety is attached to the typical tetrazolyl-diphenyl unit (Figure 19.15). In practicing analog synthesis, the Novartis scientists conserved unchanged this latter part of the molecule but tried to prepare a bioisosteric equivalent of the substituted imidazole possessing similar interaction possibilities. The lipophilic n-butyl chain was maintained, the CN dipole was replaced by a CO dipole, and the ensemble chlorine substituent plus two imidazolic carbon atoms was replaced... 

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