Canonical ensembles probability distribution

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

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... 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

Let us illustrate this procedure with the grand-canonical ensemble, and take the scenario in which we desire to achieve a uniform distribution in particle number N at a given temperature. In the weights formalism, we introduce the weighting factor r/(/V) into the microstate probabilities from (3.31) so that... 

In the canonical example, we could estimate the free energy difference between two runs by examining the overlap in their probability distributions. Similarly, in the grand canonical ensemble, we can estimate the pressure difference between the two runs. If the conditions for run I arc f//1. V. > ) and for run 2 (po, VjK), then... 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

The canonical ensemble is often stated to describe a system in contact with a thermal reservoir. States of all energies, from zero to arbitrarily large values are available to the system, but all states no longer have equal probabilities. The system does not spend the same fraction of time in each state. To determine the probability distribution among the available microstates it is important to understand that the system plus reservoir constitute a closed system, to which the principle of equal probability applies once more. 

The second generic strategy [61] utilizes two single-ensemble-averages— that is, averages with respect to the separate ensembles defined by 2 A and / In particular, one may, in principle, measure the canonical probability distributions of the order parameter in each ensemble separately and exploit the relationship between them [63],... 

The same idea can be used to interpolate data generated from multiple simulations [3]. Consider a series of canonical ensemble Monte-Carlo simulations conducted at r different temperatures. The simulation is performed at pn, and the resulting data are stored and sorted in A (F) histograms, where the total number of entries is n . The probability distribution corresponding to an arbitrary temperature p is given by... 

Fig. 13. Illustration of the grand-canonical simulation technique for temperature UbT/e = 1.68 and p = pcoex- A cuboidal system geometry 13.8cr x 13.8cr x 27.6cr is used with periodic boundary conditions in all three directions. The solid line corresponds to the negative logarithm of the probability distribution, P p), in the grand canonical ensemble. The two minima correspond to the coexisting phases and the arrows on the p axis mark their densities. The height of the plateau yields an accurate estimate for the interfacial tension, yLV- The dashed line is a parabohc fit in the vicinity of the liquid phase employed to determine the compressibihty. Representative system configurations are sketched schematically. From [62]...

The corresponding ensemble is therefore called canonical. The constant A is chosen such that Tr(p) = 1, from which it follows that p can be interpreted as defining a probability distribution over the eigenstates of the Hamiltonian H. It is easy to demonstrate that... 

In contrast, a system in contact with a thermal bath (constant-temperature, constant-volume ensemble) can be in a state of all energies, from zero to arbitrary large energies however, the state probability is different. The distribution of the probabilities is obtained under the assumption that the system plus the bath constimte a closed system. The imposed temperature varies linearly from start-temp to end-temp. The main techniques used to keep the system at a given temperature are velocity rescaling. Nose, and Nos Hoover-based thermostats. In general, the Nose-Hoover-based thermostat is known to perform better than other temperature control schemes and produces accurate canonical distributions. The Nose-Hoover chain thermostat has been found to perform better than the single thermostat, since the former provides a more flexible and broader frequency domain for the thermostat [29]. The canonical ensemble is the appropriate choice when conformational searches of molecules are carried out in vacuum without periodic boundary conditions. 

An important partition function can be derived by starting from Q (T, V, N) and replacing the constant variable AT by fi. To do that, we start with the canonical ensemble and replace the impermeable boundaries by permeable boundaries. The new ensemble is referred to as the grand ensemble or the T, V, fi ensemble. Note that the volume of each system is still constant. However, by removing the constraint on constant N, we permit fluctuations in the number of particles. We know from thermodynamics that a pair of systems between which there exists a free exchange of particles at equilibrium with respect to material flow is characterized by a constant chemical potential fi. The variable N can now attain any value with the probability distribution... 

As mentioned earlier, the limiting probabilities occur only in the ratio pjpm and the value of the denominator, Qc, is therefore not required. Hence, in the canonical ensemble, the acceptance of a trial move depends only on the Boltzmann factor of the energy difference, AU, between the states m and n. If the system looses energy, then the trial move is always accepted if the move goes uphill, that is, if the system gains energy, then we have to play a game of chance on exp(—fi AU) [6]. In practical terms, we have to calculate a random number, uniformly distributed in the interval [0,1]. If < exp(—p AU), then we accept the move, and reject otherwise. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

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