ABSTRACT
Reversible dynamics is well-known to obey variational principles based on the action being the time integral of a Lagrangian with time-reversal symmetry. The purpose of the present paper is to find dissipative Lagrangians giving variational principles in dissipative dynamics with broken time-reversal symmetry. Conceptually the present theory insists on new Least Dissipation & Work Principles (LDWP) based on variational integrals weighted in time, in order to unify variational principles of physics including irreversible processes on many-body systems. This is also closely related to the present author's theory of entropy production in Kubo's scheme of transport phenomena and all nonlinear responses beyond. Through these investigations, we can understand the meaning (as a variational integral) of the action of reversible dynamics.
Subject(s)
Quantum Theory , Entropy , Nonlinear DynamicsABSTRACT
In the present study, we demonstrate how to perform, using quantum annealing, the singular value decomposition and the principal component analysis. Quantum annealing gives a way to find a ground state of a system, while the singular value decomposition requires the maximum eigenstate. The key idea is to transform the sign of the final Hamiltonian, and the maximum eigenstate is obtained by quantum annealing. Furthermore, the adiabatic time scale is obtained by the approximation focusing on the maximum eigenvalue.
ABSTRACT
Even if an original time series exhibits nonlinearity, it is not always effective to approximate the time series by a nonlinear model because such nonlinear models have high complexity from the viewpoint of information criteria. Therefore, we propose two measures to evaluate both the nonlinearity of a time series and validity of nonlinear modeling applied to it by nonlinear predictability and information criteria. Through numerical simulations, we confirm that the proposed measures effectively detect the nonlinearity of an observed time series and evaluate the validity of the nonlinear model. The measures are also robust against observational noises. We also analyze some real time series: the difference of the number of chickenpox and measles patients, the number of sunspots, five Japanese vowels, and the chaotic laser. We can confirm that the nonlinear model is effective for the Japanese vowel /a/, the difference of the number of measles patients, and the chaotic laser.
ABSTRACT
We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to accelerate the convergence of quantum annealing of the random-field Ising model. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.
ABSTRACT
To generate surrogate data in nonlinear time series analysis, the Fourier transform is generally used. In the calculation of the Fourier transform, the time series is assumed to be periodic. Because such an assumption does not always hold true, the estimation accuracy of the Fourier transformed data and thus the power spectra is reduced. Due to such an estimation error, it is also possible that the surrogate test will lead to a false conclusion; for example, that a linear time series is nonlinear. In this paper, we experimentally evaluated the effects of data windows from the viewpoint of false rejections with several types of surrogate data. Our results indicate that if the data length becomes shorter, the false rejections by the data windows are reduced to a greater extent. However, if the data length is sufficient, the use of data windows is not a viable option. In the worst possible case wherein the linear memory of the original data is very long as in the nonstationary case, the critical length of the data for which the data windows were effective was approximately 1000.
