Canonical equilibrium probability

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

The summation includes all possible stationary states with energy , that are occupied by the molecule. The equilibrium probability Pi that the molecule populates each of these states is given by a Boltzmann distribution for the canonical ensemble. In other words. 

Equation [7] expresses the balance between the flux of all other states X toward X (the second term on the right-hand side of eqn [7]), leading to an inaease of P(X, t), and the flux out of the StateX (the first term on the right-hand side of eqn [7]), leading to a decrease of P(X, t). Now, for the application of the importance sampling MC method in statistical physics, one requires that the transition probability W(X X ) satisfy the detailed balance principle with the (canonic) equilibrium distribution Peq(X) = Z" exp(-V(X)/feB r), Z being the partition ftinaion... 

The grand canonical ensemble describes a system of constant volume, but capable of exchanging both energy and particles with its environment. Simulations of open systems under these conditions are particularly useful in the study of adsorption equilibria, surface segregation effects, and nanoscopically confined fluids and polymers. Under these conditions, the temperature and the chemical potentials jti,- of the freely exchanged species are specified, while the system energy and composition are variable. This ensemble is also called the jx VT ensemble. In the case of a one-component system it is described by the equilibrium probability density... 

It is a well-established fact that, within the context of the canonical ensemble, the equilibrium probability for finding expectation values is determined by the equilibrium density operator... 

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]... 

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

Or one can say that even if the equilibrium constant for the above reaction is negligibly small, the unsolvated oxycarbenium ion (XII) is a part of a relatively unimportant canonical form of the tert.-oxonium ion, which implies a certain probability that the oxycarbenium ion may react as such by detaching itself from the ether and finding some other basic site. The important point, however, is that under the normal conditions under which DCA are polymerised the attempts to distinguish experimentally between oxycarbenium ions and tert.-oxonium ions as the active species appear at present to be quite pointless. 

Definition of Critical and Rate-Limiting Bottlenecks" The hypothesis of local equilibrium within the reservoirs means that the set of transitions from reservoir to reservoir can be described as a Markov process without memory, with the transition probabilities given by eq. 4. Assuming the canonical ensemble and microscopic reversibility, the rate constant Wji, for transitions from reservoir i to reservoir j can be written... 

When N2 and Ni increase, the situation of the classical equilibrium is reached for which the most probable canonical state is that of the highest P(i). The successive values of P(i) and P(i)-1 come closer and closer and the limiting value of the ratio given by Equation 10 is unity. 

If the oscillator is weakly coupled to the bath, in canonical thermal equilibrium the probability of finding the oscillator in the nth state is of course P q = e / En/Zq, where ft = I/kT and the oscillator s canonical partition function is Zq = e In addition, the oscillator s off-diagonal (in this energy representation) density matrix elements are zero. The average oscillator energy (in thermal equilibrium) is Eeq = n13nPnq-... 

How do we calculate the probability of a fluctuation about an equilibrium state Consider a system characterized by a classical Hamiltonian H r, p ) where p and denote the momenta and positions of all particles. The phase space probability distribution isf (r, p ) = Q exp(—/i22(r, p )), where Q is the canonical partition function. 

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... 

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