Spectral diffusion kernel

J. Baier, M.F. Richter, R.J. Cogdell, S. Oellerich, J. K5hler, Determination of the spectral diffusion kernel of a protein by single molecule spectroscopy. Phys. Rev. Lett. 100, 8108-1-4 (2008)... 

The organization of this chapter is as follows. Section 1.5.2 introduces the TLS model and discusses the dynamics of a single TLS. In Section 1.5.3 general results for the spectral diffusion kernel and the absorption line shape when the chromophore is coupled to an arbitrary number of TLSs are presented. Section 1.5.4 will discuss how the experimental observables (single molecule line shapes, spectral diffusion tra-... 

A related quantity, the conditional probability density (also called the spectral diffusion kernel) P co, /jojo) is defined by... 

P (o, t (Oo) is the probability density that the chromophore has transition frequency (o at time t given that it had frequency coo at time 0. While the functional form of this spectral diffusion kernel is quite complicated in general, at short times certain simplifications occur. In particular, if the positions of the TLSs occupy a regular lattice in three-dimensional space, all of the relaxation rates Kj are the same, all of the occupation probabilities pj are the same and equal to 1/2 (the high-temperature limit), and the perturbations vj are dipolar, then it was shown by Klauder and Anderson [29] and more recently by Zumofen and Klafter [30] that the spectral diffusion kernel is Lorentzian ... 

In the second type of experiment that measures single molecule spectral dynamics one performs repeated fluorescence excitation scans of the same molecule. In each scan the line shape is described as above, but now there is the possibility that the center frequency of the line will change from scan to scan because of slow fluctuations. Thus one can measure the center frequency as a function of time, producing what has been called a spectral diffusion trajectory. This trajectory can, in principle, be characterized completely by the spectral diffusion kernel of Eqs. (16) and (19), but of course it must be understood that only the slow Kj < 1 /t) TLSs contribute. In fact, the experimental trajectories are really too short to be analyzed with this spectral diffusion kernel. Instead, it is useful [11, 12] to consider three simpler characterizations of the spectral diffusion trajectories the frequency-frequency correlation function in Eq. (14), the distribution of frequencies from Eq. (15), and the distribution of spectral jumps from Eq. (21). For this application of the theoretical results, in all three of these formulas j should be replaced by s, the labels for the slow TLSs. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

Specular reflection occurs when the angle of reflection equals the angle of incidence, whereas diffuse reflection is spread out evenly over a full hemisphere. The main difficulty in measuring the spectral transmittance properties of a kernel is the problem of collecting sufficient energy from the transmitted signal. To collect the maximum amount of light under such conditions the phototube should surround the entire sample. The nearest approach to this condition can be... 

As with other spectral dimensionality reduction techniques, Diffusion Maps has a large computational overhead due to the eigendecomposition of a large square kernel matrix. One way to overcome this computational cost is to approximate the final embedding based on a known set of learnt points. Such a method lies at the heart of /u.-isometric Difussion Maps [27] (/u,-DM). The term /u-isometric... 

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