Corrigendum: Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

A Corrigendum on
Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

by Lapolla, A., and Godec, A. (2019). Front. Phys. 7:182. doi: 10.3389/fphy.2019.00182

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 = 2σσ^T instead of D = σσ^T/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 $\widehat{{L}}{\widehat{{L}}}^{†}={\widehat{{L}}}^{†}\widehat{{L}}=0$.”

In section 2.2. in the paragraph following Equation (13) there is an obvious redundant additional factor ${\text{Ψ}}_{00}^{-1}(\text{q})d\text{q}$ present immediately after the in-line equation: ${\langle {\text{Ψ}}_{l0}|{\text{Ψ}}_{k0}\rangle }_{{\text{Ψ}}_{00}^{-1}}\equiv {\int }_{\Xi }d{\text{q}}^{\prime }{\text{Ψ}}_{00}{({\text{q}}^{\prime })}^{-1}{\text{Ψ}}_{0l}({\text{q}}^{\prime }){\text{Ψ}}_{k0}({\text{q}}^{\prime })$.

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. $\widehat{{L}}$ propagates probability measures μ_t(x) in time, which will throughout be assumed to posses well-behaved probability density functions P(x, t), i.e., dμ_t(x) = P(x, t)dx [thereby posing some restrictions on F(x)]. On the level of individual trajectories Equation (1) corresponds to the It$\widehat{\text{o}}$ equation dx_t = F(x_t)dt+σdW_t with W_t being a d-dimensional vector of independent Wiener processes whose increments have a Gaussian distribution with zero mean and variance dt, i.e., $\langle d{W}_{t,i}d{W}_{t^{\prime },j}\rangle ={\delta }_{ij}\delta (t-t^{\prime })dt$, and where σ is a d×d symmetric noise matrix such that D = σσ^T/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 $\widehat{{L}}{\text{e}}^{-\phi (\text{x})}=0$, 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., $\underset{\text{x}\to \infty }{lim}P(\text{x},t)=0,\forall t$ sufficiently fast), such that $\widehat{{L}}$ corresponds to a coercive and densely defined operator on V (and $\widehat{{L}}†$ on W, respectively) [76–78]. Finally, $\widehat{{L}}$ is throughout assumed to be normal, i.e., ${\widehat{{L}}}^{†}\widehat{{L}}-\widehat{{L}}{\widehat{{L}}}^{†}=0$ and thus henceforth V = W, where for reversible system (i.e., those obeying detailed balance) we have $\widehat{{L}}\iff {\widehat{{L}}}^{†}$”.

Finally, the redundant factor ${\text{Ψ}}_{00}^{-1}(\text{q})d\text{q}$ has been deleted in section 2.2 in the paragraph following Equation (13).

“can be equal to Q_pss(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 ${\text{Ψ}}_{kl}({\text{q}}^{\prime })={\text{Ψ}}_{lk}^{†}({\text{q}}^{\prime })$ such that ${\langle {\text{Ψ}}_{l0}|{\text{Ψ}}_{k0}\rangle }_{{\text{Ψ}}_{00}^{-1}}\equiv {\int }_{\Xi }d{\text{q}}^{\prime }{\text{Ψ}}_{00}{({\text{q}}^{\prime })}^{-1}{\text{Ψ}}_{0l}({\text{q}}^{\prime }){\text{Ψ}}_{k0}({\text{q}}^{\prime })$.”

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.