Boltzmann equilibrium distribution

In impact theory the result of a collision is described by the probability /(/, /)dJ of finding angular momentum J after the collision, if it was equal to / before. The probability is normalized to 1, i.e. / /(/, /)d/=l. The equilibrium Boltzmann distribution over J is... 

This identity [3, 15] between a weighted average of nonequilibrium trajectories (r.h.s.) and the equilibrium Boltzmann distribution (l.h.s.) is implicit in the work of Jarzynski [2], and is given explicitly by Crooks [16]. The average (... is over an ensemble of trajectories starting from the equilibrium distribution at / 0 and... 

The simulations were started from an equilibrium Boltzmann distribution on the free energy surface for A = 0. During a time t = 1, A was changed linearly in time from 0 to 1. We also performed simulations in the backward direction. However, because of the symmetry of V with respect to A, backward transformations are equivalent to forward transformations. Along the resulting trajectories, the work ftW was accumulated. Figure 5.2 shows the probability distributions of the work on the forward direction, and on the backward direction multiplied by exp(-fiW). As expected from (5.35) for AA = 0, the two distributions agree nicely. 

Readout of the ligand information by a substrate is achieved at the rates with which L and S associate and dissociate it is thus determined by the complexation dynamics. In a mixture of ligands Li, L2. .. L and substrates Si, S2. . - S , information readout may assume a relaxation behaviour towards the thermodynamically most stable state of the system. At the absolute zero temperature this state would contain only complementary LiSi, L2S2. .. L S pairs at any higher temperature this optimum complementarity state (with zero readout errors) will be scrambled into an equilibrium Boltzmann distribution, containing the corresponding readout errors (LWS , n n ), by the noise due to thermal agitation. 

Equation (1.23b) is the equilibrium Boltzmann distribution in a potential field. Substitution of (1.23b) into (1.9c) yields the Poisson-Boltzmann equation for equilibrium electric potential of the form... 

At high pressure, the excitation/de-excitation step (reaction 10.148) is essentially at equilibrium. That is, despite the fact that the reaction step 10.149 depletes the population of C (n), the concentration of third bodies is so large at high pressures, an equilibrium (Boltzmann) distribution in C (n) is maintained through the rapid excitation/de-excitation steps. Consider the steady-state population of C n) just from reaction 10.148 ... 

To obtain the statement of detailed balance for complete equilibrium, with both translational and internal degrees of freedom in thermal equilibrium, we must sum over the rate constants in Eqs (B.32) and (B.33), weighting each by its equilibrium Boltzmann distribution that is (as in Eq. (2.18)),... 

Furthermore, in the absence of reaction the system must approach the equilibrium Boltzmann distribution, f E), that is the system must obey detailed balance. 

When y is very small, the thermal relaxation in the well is not fast relative to the escape rate, and the assumption that the distribution within the well can be represented by the equilibrium Boltzmann distribution no longer holds. On the other hand we can make use of the fact that the total energy E varies on a time scale much longer than either x or u (it is conserved for y = 0). Thus changing variables from (x, v) to ( , ) and eliminating the fast phase variable (f> leads to a Smoluchowski (diffusion) equation for E. [Kramers gave the equivalent equation in terms of the action variable J( ).]... 

Similar in spirit is the Milestoning [90] method by Fiber and coworkers, who assume that the diffusion of interest occurs through a tube in configuration space, and translate the rare process into a non-Markovian hopping between configuration space hyperplanes, the so-called milestones (which are in fact rather similar to the TIS interfaces, except that they do not form a foliation). The kinetics is obtained from starting an equilibrium ensemble on a milestone, and measuring the time distribution needed to reach the next milestone. The distribution can subsequently be used to construct the kinetics. The assumption is that there is an equilibrium (Boltzmann) distribution on each milestone. 

Here z is the valence of the ions, e the proton charge, and n the number craicentration (ion number per unit volume) of the cations and anions, respectively. Assuming the equilibrium Boltzmann distribution for the ion concentration, we can relate the number concentration to the potential as follows ... 

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