Characteristic packet equations

Figure 1. Temperature-dependent macrostate dissection of a two-dimensional potential-energy landscape, (a) Potential V as a function of two coordinates, (b) Gibbs-Boltzmann distribution p at low (left), medium (middle), and high (right) temperatures, (c) Corresponding p at each temperature constructed from solutions to the characteristic packet equations, (d) Characteristic packet solution parameters R° and 0 for each macrostate (labeled with indices a, / , 6, y, and s). (e) Trajectory diagram of macrostate conformational free energies Fa as a function of temperature. (Reproduced from Church et al. [17] with permission obtained.)...

Characteristic Packet Equations The first step is to approximate pB by p, a sum of Gaussians... 

The solution to the problem of finding initial conditions for solving the characteristic packet equations is to use an annealing procedure. The unique high-temperature characteristic packet can be found by solving Eqs. 

Adaptive Temperature Jump. The size of a temperature jump is restricted only by the requirement that the initial conditions for solving the characteristic packet equations (provided by the higher-temperature characteristic packet) be within the attractor region of the lower temperature solution. This will be so until temperature is lowered below a branchpoint when one solution will trap the solver. As long as we have not jumped to a temperature too far below the branchpoint, we can find the other solution by looking for the missing probability mass that is present in the parent but not in the child... 

Equations (2.11b) and (2.11c) can be solved by numerical iteration [16] to find a single solution in the vicinity of initial conditions <0)R , <0)f °. Convergence can be assessed using [(< )< )2, (("+1) )2] as a dimensionless indicator of change, where < )< ° is the characteristic packet at the nth iteration. However, finding appropriate initial conditions is a nontrivial problem whose solution is postponed to Section II.E. Once R° and A have been computed, Eq. (2.11a) explicitly determines V° as... 

In this beam-sweeping scheme the effective spatial distribution of the ions sampled is defined by the characteristics of the sweeping action and the detector slit parameters [20]. Maintaining the fast rise time of the deflection pulse is critical in maintaining spatially small ion packets at the detector surface, and thus adequate resolution. The overall resolution for the differential impulse-sweeping mode in Fig. 12.3 can be estimated with the following equation developed by Bakker [20] ... 

In addition to possessing wavelength characteristics, light also has properties that indicate it is composed of discrete energy packets called photons. The relationship between the energy of photons and their frequency is given by the equation ... 

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