1,596 research outputs found
Discrete-time thermodynamic uncertainty relation
We generalize the thermodynamic uncertainty relation, providing an entropic
upper bound for average fluxes in time-continuous steady-state systems
(Gingrich et al., Phys. Rev. Lett. 116, 120601 (2016)), to time-discrete Markov
chains and to systems under time-symmetric, periodic driving
Integrable dissipative exclusion process: Correlation functions and physical properties
We study a one-parameter generalization of the symmetric simple exclusion
process on a one dimensional lattice. In addition to the usual dynamics (where
particles can hop with equal rates to the left or to the right with an
exclusion constraint), annihilation and creation of pairs can occur. The system
is driven out of equilibrium by two reservoirs at the boundaries. In this
setting the model is still integrable: it is related to the open XXZ spin chain
through a gauge transformation. This allows us to compute the full spectrum of
the Markov matrix using Bethe equations. Then, we derive the spectral gap in
the thermodynamical limit. We also show that the stationary state can be
expressed in a matrix product form permitting to compute the multi-points
correlation functions as well as the mean value of the lattice current and of
the creation-annihilation current. Finally the variance of the lattice current
is exactly computed for a finite size system. In the thermodynamical limit, it
matches perfectly the value obtained from the associated macroscopic
fluctuation theory. It provides a confirmation of the macroscopic fluctuation
theory for dissipative system from a microscopic point of view.Comment: 31 pages, 7 figures ; introduction expanded, typos corrected and
title change
Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks
Stochastic gradient descent (SGD) is widely believed to perform implicit
regularization when used to train deep neural networks, but the precise manner
in which this occurs has thus far been elusive. We prove that SGD minimizes an
average potential over the posterior distribution of weights along with an
entropic regularization term. This potential is however not the original loss
function in general. So SGD does perform variational inference, but for a
different loss than the one used to compute the gradients. Even more
surprisingly, SGD does not even converge in the classical sense: we show that
the most likely trajectories of SGD for deep networks do not behave like
Brownian motion around critical points. Instead, they resemble closed loops
with deterministic components. We prove that such "out-of-equilibrium" behavior
is a consequence of highly non-isotropic gradient noise in SGD; the covariance
matrix of mini-batch gradients for deep networks has a rank as small as 1% of
its dimension. We provide extensive empirical validation of these claims,
proven in the appendix
Macroscopic fluctuation theory
Stationary non-equilibrium states describe steady flows through macroscopic
systems. Although they represent the simplest generalization of equilibrium
states, they exhibit a variety of new phenomena. Within a statistical mechanics
approach, these states have been the subject of several theoretical
investigations, both analytic and numerical. The macroscopic fluctuation
theory, based on a formula for the probability of joint space-time fluctuations
of thermodynamic variables and currents, provides a unified macroscopic
treatment of such states for driven diffusive systems. We give a detailed
review of this theory including its main predictions and most relevant
applications.Comment: Review article. Revised extended versio
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