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Register a new userFluctuation-dissipation Type Theorem in Stochastic Linear Learning
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The fluctuation-dissipation theorem (FDT) is a simple yet powerful consequence of the first-order differential equation governing the dynamics of systems subject simultaneously to dissipative and stochastic forces. The linear learning dynamics, in which the input vector maps to the output vector by a linear matrix whose elements are the subject of learning, has a stochastic version closely mimicking the Langevin dynamics when a full-batch gradient descent scheme is replaced by that of stochastic gradient descent. We derive a generalized FDT for the stochastic linear learning dynamics and verify its validity among the well-known machine learning data sets such as MNIST, CIFAR-10 and EMNIST.
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In this work, a physical system described by Hamiltonian $mathbf{H}_omega = mathbf{H}_0 + mathbf{V}_omega(mathbf{x},t)$ consisted of a solvable model $mathbf{H}$ and external random and time-dependent potential $mathbf{V}_omega(mathbf{x},t)$ is investigated. Under the conditions that the average external potential with respect to the configuration $omega$ is constant in time, and, for each configuration, the potential changes smoothly that the evolution of the system follows Schrodinger dynamics, the mean-dynamics can be derived from taking average of the equation with respect to configuration parameter $omega$. It provides extra contributions from the deviations of the Hamiltonian and evolved state along the time to the Heisenberg and Liouville-von Neumann equations. Consequently, the Kubos formula and the fluctuation-dissipation relation obtained from the construction is modified in the sense that the contribution from the information of randomness and memory effect from time-dependence are present.
We introduce a simple prescription for calculating the spectra of thermal fluctuations of temperature-dependent quantities of the form $hat{delta T}(t)=int d^3vec{r} delta T(vec{r},t) q(vec{r})$. Here $T(vec{r}, t)$ is the local temperature at location $vec{r}$ and time $t$, and $q(vec{r})$ is an arbitrary function. As an example of a possible application, we compute the spectrum of thermo-refractive coating noise in LIGO, and find a complete agreement with the previous calculation of Braginsky, Gorodetsky and Vyatchanin. Our method has computational advantage, especially for non-regular or non-symmetric geometries, and for the cases where $q(vec{r})$ is non-negligible in a significant fraction of the total volume.
We use a relationship between response and correlation function in nonequilibrium systems to establish a connection between the heat production and the deviations from the equilibrium fluctuation-dissipation theorem. This scheme extends the Harada-Sasa formulation [Phys. Rev. Lett. 95, 130602 (2005)], obtained for Langevin equations in steady states, as it also holds for transient regimes and for discrete jump processes involving small entropic changes. Moreover, a general formulation includes two times and the new concepts of two-time work, kinetic energy, and of a two-time heat exchange that can be related to a nonequilibrium effective temperature. Numerical simulations of a chain of anharmonic oscillators and of a model for a molecular motor driven by ATP hydrolysis illustrate these points.
The fluctuation dissipation theorem (FDT) is the basis for a microscopic description of the interaction between electromagnetic radiation and matter.By assuming the electromagnetic radiation in thermal equilibrium and the interaction in the linear response regime, the theorem interrelates the spontaneous fluctuations of microscopic variables with the kinetic coefficients that are responsible for energy dissipation.In the quantum form provided by Callen and Welton in their pioneer paper of 1951 for the case of conductors, electrical noise detected at the terminals of a conductor was given in terms of the spectral density of voltage fluctuations, $S_V({omega})$, and was related to the real part of its impedance, $Re[Z({omega})]$, by a simple relation.The drawbacks of this relation concern with: (I) the appearance of a zero point contribution which implies a divergence of the spectrum at increasing frequencies; (ii) the lack of detailing the appropriate equivalent-circuit of the impedance, (iii) the neglect of the Casimir effect associated with the quantum interaction between zero-point energy and boundaries of the considered physical system; (iv) the lack of identification of the microscopic noise sources beyond the temperature model. These drawbacks do not allow to validate the relation with experiments. By revisiting the FDT within a brief historical survey, we shed new light on the existing drawbacks by providing further properties of the theorem, focusing on the electrical noise of a two-terminal sample under equilibrium conditions. Accordingly, we will discuss the duality and reciprocity properties of the theorem, its applications to the ballistic transport regime, to the case of vacuum and to the case of a photon gas.
We examine the Hall conductivity of macroscopic two-dimensional quantum system, and show that the observed quantities can sometimes violate the fluctuation dissipation theorem (FDT), even in the linear response (LR) regime infinitesimally close to equilibrium. The violation can be an order of magnitude larger than the Hall conductivity itself at low temperature and in strong magnetic field, which are accessible in experiments. We further extend the results to general systems and give a necessary condition for such large-scale violation to happen. This violation is a genuine quantum phenomenon that appears on a macroscopic scale. Our results are not only bound to the development of the fundamental issues of nonequilibrium physics, but the idea is also meaningful for practical applications, since the FDT is widely used for the estimation of noises from the LRs.
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