Probability distribution canonical

In the q = l limit, the effective temperature equals the standard temperature. Otherwise, adding a constant shift to the potential energy is equivalent to rescaling the temperature at which the canonical probability distribution is computed. 

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 canonical probability distribution of potential energy Pnvt( o T) is then given by the product of the density of states n(E) and the Boltzmann weight factor Wb(E T) ... 

Nm(Eo,V) and nm be, respectively, the potential-energy histogram and the total number of samples obtained for the mth parameter set Am. The WHAM equations that yield the canonical probability distribution Pt, Eq,V) = n(Eo, V) exp(—/3E ) with any potential-energy parameter value A at any temperature T = l/k-Qp are then given by [80]... 

Fig. 4.10. The canonical probability distributions of the total potential energy of protein G obtained from the REMD simulation with 224 temperatures. They are all bell-shaped with sufficient overlaps with the neighboring ones...

Using 112 nodes of the Earth Simulator, we performed a REMD simulation of this system with 224 replicas. The REMD simulation was successful in the sense that we observed a random walk in potential energy space, which suggests that a wide conformational space was sampled. In Fig. 4.10 we show the canonical probability distributions of the total potential energy at the corresponding 224 temperatures ranging from 250 to 700 K. 

Although a similar relation - but pointwise - follows from Eq. (3.39), Eq. (3.40) is analogous with the standard relations establishing, e.g., the Helmholtz free energy in terms of a full canonical probability distribution, such as of p. 42. For an example see Reiss (1972, see Eq. (9)),... 

Upon thermalization at temperature T, a cluster can be described by the canonical probability distribution function of total energy, Pt E), which specifies the probability that the system will be found in the energy interval [E,E + A ] at the specified temperature T. The distribution function corresponding to this temperature, within the canonical ensemble description, is (see Fanourgakis et al. 1997 Schmidt et al. 1997 and references therein) ... 

Equation (27.1 -17) defines the sum Z, which is called the canonicalpartitionfunction. In German, a partition function is called a Zustandsuntme (sum over states), a name that better represents the sum and is our motivation for using Z for its symbol. The canonical probability distribution can now be written as... 

Third, generalized equations of motion have been proposed to sample arbitrary (i.e., not necessarily canonical) probability distributions [134, 135, 136, 137]. Such methods can be used, e.g., to optimize the efficiency of conformational searches [134, 135, 137] or for generating Tsallis distributions of microstates [136]. 

Because of the dominance of a certain restricted space of microstates in ordered phases, it is obviously a good idea to primarily concentrate in a simulation on a precise sampling of the microstates that form the macrostate under given external parameters such as, for example, the temperature. The canonical probability distribution functions clearly show that within the certain stable phases, only a limited energetic space of microstates is noticeably populated, whereas the probability densities drop off rapidly in the tails. Thus, an efficient sampling of this state space should yield the relevant information within comparatively short Markov chain Monte Carlo runs. This strategy is called importance sampling. 

Perspective (left) and top (right) view of the unnormalized canonical probability distribution pa E, q) for S3 at r = 300K. 

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... 

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... 

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]... 

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. 

Notice that this equation allows us to calculate ft(U) from a probability distribution measured from a simulation at temperature T0. We do not know the value of Z(T0), but it is a constant independent of U. Furthermore, since 12 has no dependence on T, measurement of p at any temperature should in principle permit its complete determination. In practice, however, the potential energies in a canonical simulation are sharply distributed around their average, away from which the statistical quality of p and hence 12 in (3.3) becomes extremely poor. 

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. 

A very useful criterion in this respect is given by the maximum entropy principle in the sense of Jaynes." The ingredients of the maximum entropy principle are (i) some reference probability distribution on the pure states and (ii) a way to estimate the quality of some given probability distribution p. on the pure states with respect to the reference distribution. As our reference probability distribution, we shall take the equidistribution defined in Eq. (30), for a two-level system (this definition of equipartition can be generalized to arbitrary dxd matrices, being the canonical measure on the d-dimensional complex projective plane - " ). The relative entropy of some probability distribution pf [see Eq. (35)] with respect to yXgqp is defined as... 

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