In the original article, there was an error. In section 2.1 the diffusion matrix D the in-line equation was defined with a factor of 2 instead of 1/2, i.e., D = 2σσT instead of D = σσT/2.
In section 2.1 in the paragraph following Equation (4), a copy-paste error occurred in the sentence “… where for reversible system (i.e., those obeying detailed balance) we have .”
In section 2.2. in the paragraph following Equation (13) there is an obvious redundant additional factor present immediately after the in-line equation: .
A correction has been made to section 2.1 [paragraph following Equation (1)]. The paragraph now reads:
“where D is the symmetric positive-definite diffusion matrix. propagates probability measures μt(x) in time, which will throughout be assumed to posses well-behaved probability density functions P(x, t), i.e., dμt(x) = P(x, t)dx [thereby posing some restrictions on F(x)]. On the level of individual trajectories Equation (1) corresponds to the It equation dxt = F(xt)dt+σdWt with Wt being a d-dimensional vector of independent Wiener processes whose increments have a Gaussian distribution with zero mean and variance dt, i.e., , and where σ is a d×d symmetric noise matrix such that D = σσT/2. Moreover, we assume that F(x) admits the following decomposition into a potential (irrotational) field −D∇φ(x) and a non-conservative component ϑ(x), F(x) = −D∇φ(x)+ϑ(x) with the two fields being mutually orthogonal ∇φ(x)·ϑ(x) = 0 [73]. By insertion into Equation (1) one can now easily check that , such that the stationary solution of the Fokker-Planck equation (also referred to as the steady state [74, 75], which is the terminology we adopt here) by construction does not depend on the non-conservative part ϑ(x).”
A correction has been made to the aforementioned sentence in section 2.1, in the paragraph following Equation (4), which now reads:
“such that the conditional probability density starting from a general initial condition |p0〉 becomes P(x, t|p0, 0) = 〈x|Û(t)|p0〉 ≡ ∫dx0p0(x0)G(x, t|x0, 0). Moreover, as F(x) is assumed to be sufficiently confining (i.e., sufficiently fast), such that corresponds to a coercive and densely defined operator on V (and on W, respectively) [76–78]. Finally, is throughout assumed to be normal, i.e., and thus henceforth V = W, where for reversible system (i.e., those obeying detailed balance) we have ”.
Finally, the redundant factor has been deleted in section 2.2 in the paragraph following Equation (13).
“can be equal to Qpss(q, t|q0, 0). As this will generally not be the case this essentially means that the projected dynamics is in general non-Markovian. The proof is established by noticing that such that .”
The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.
Summary
Keywords
Fokker-Planck equation, spectral theory, projection operator method, occupation time, single file diffusion, Bethe ansatz, free energy landscape
Citation
Lapolla A and Godec A (2020) Corrigendum: Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File. Front. Phys. 8:7. doi: 10.3389/fphy.2020.00007
Received
12 December 2019
Accepted
08 January 2020
Published
31 January 2020
Volume
8 - 2020
Edited and reviewed by
Carlos Mejía-Monasterio, Polytechnic University of Madrid, Spain
Updates
Copyright
© 2020 Lapolla and Godec.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Aljaž Godec agodec@mpibpc.mpg.de
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
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