30 research outputs found
Continuous time random walk with correlated waiting times
Based on the Langevin description of the Continuous Time Random Walk (CTRW),
we consider a generalization of CTRW in which the waiting times between the
subsequent jumps are correlated. We discuss the cases of exponential and slowly
decaying persistent power-law correlations between the waiting times as two
generic examples and obtain the corresponding mean squared displacements as
functions of time. In the case of exponential-type correlations the
(sub)diffusion at short times is slower than in the absence of correlations. At
long times the behavior of the mean squared displacement is the same as in
uncorrelated CTRW. For power-law correlations we find subdiffusion
characterized by the same exponent at all times, which appears to be smaller
than the one in uncorrelated CTRW. Interestingly, in the limiting case of an
extremely long power-law correlations, the (sub)diffusion exponent does not
tend to zero, but is bounded from below by the subdiffusion exponent
corresponding to a short time behavior in the case of exponential correlations
Theory of Systematic Computational Error in Free Energy Differences
Systematic inaccuracy is inherent in any computational estimate of a
non-linear average, due to the availability of only a finite number of data
values, N. Free energy differences (DF) between two states or systems are
critically important examples of such averages in physical, chemical and
biological settings. Previous work has demonstrated, empirically, that the
``finite-sampling error'' can be very large -- many times kT -- in DF estimates
for simple molecular systems. Here, we present a theoretical description of the
inaccuracy, including the exact solution of a sample problem, the precise
asymptotic behavior in terms of 1/N for large N, the identification of
universal law, and numerical illustrations. The theory relies on corrections to
the central and other limit theorems, and thus a role is played by stable
(Levy) probability distributions.Comment: 5 pages, 4 figure
Chance and stability: stable distributions and their applications
An introduction to the theory of stable distributions and their applications. It contains a modern outlook on the mathematical aspects of the theory. The authors explain numerous peculiarities of stable distributions and describe the principle concept of probability theory and function analysis. A significant part of the book is devoted to applications of stable distributions. Another notable feature is the material on the interconnection of stable laws with fractals, chaos and anomalous transport processes