How Green Functions are used in various domains part1(Advanced Mathematics)
How Green functions work
- Accelerating Nonequilibrium Green functions simulations with embedding selfenergies(arXiv)
Author : Niclas Schlünzen, Karsten Balzer, Hannes Ohldag, Jan-Philip Joost, Michael Bonitz
Abstract : Real-time nonequilibrium Green functions (NEGF) have been very successful to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently we have introduced the G1 — G2 scheme that allows for a dramatic reduction to time-linear scaling [Schlünzen, Phys. Rev. Lett. 124, 076601 (2020); Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts or similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding selfenergy — a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1 — G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced selfenergies that can be represented by the G1 — G2 scheme [as described in Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memory-less structure of the equations and their time linear scaling.
2. The mathematical physical equations satisfied by retarded and advanced Green’s functions(arXiv)
Author : Huai-Yu Wang
Abstract : In mathematical physics, time-dependent Green’s functions (GFs) are the solutions of differential equations of the first and second time derivatives. Habitually, the time-dependent GFs are Fourier transformed into the frequency space. Then, analytical continuation of the frequency is extended to below or above the real axis. After inverse Fourier transformation, retarded and advanced GFs can be obtained, and there may be arbitrariness in such analytical continuation. In the present work, we establish the differential equations from which the retarded and advanced GFs are rigorously solved. The key point is that the derivative of the time step function is the Dirac delta function plus an infinitely small quantity, where the latter is not negligible because it embodies the meaning of time delay or time advance. The retarded and advanced GFs defined in this paper are the same as the one-body GFs defined with the help of the creation and destruction operators in many-body theory. There is no way to define the causal GF in mathematical physics, and the reason is given. This work puts the initial conditions into differential equations, thereby paving a way for solving the problem of why there are motions that are irreversible in time.
3.Principled interpolation of Green’s functions learned from data (arXiv)
Author : Harshwardhan Praveen, Nicolas Boulle, Christopher Earls
Abstract : We present a data-driven approach to mathematically model physical systems whose governing partial differential equations are unknown, by learning their associated Green’s function. The subject systems are observed by collecting input-output pairs of system responses under excitations drawn from a Gaussian process. Two methods are proposed to learn the Green’s function. In the first method, we use the proper orthogonal decomposition (POD) modes of the system as a surrogate for the eigenvectors of the Green’s function, and subsequently fit the eigenvalues, using data. In the second, we employ a generalization of the randomized singular value decomposition (SVD) to operators, in order to construct a low-rank approximation to the Green’s function. Then, we propose a manifold interpolation scheme, for use in an offline-online setting, where offline excitation-response data, taken at specific model parameter instances, are compressed into empirical eigenmodes. These eigenmodes are subsequently used within a manifold interpolation scheme, to uncover other suitable eigenmodes at unseen model parameters. The approximation and interpolation numerical techniques are demonstrated on several examples in one and two dimensions
