The publication of this article unfortunately contained mistakes. Proof of Lemma 1, Propositions 19 and 20, corresponding proofs, and Remark 28 contained mistakes. In proof of Lemma 1, there are errors in the evaluations of neglecting higher order terms. In Remark 28, the terminology of the Laplace-Beltrami operator is inappropriate, and we rewrite a statement by using a generalization of the divergence operator. In Proposition 19, the eta coordinate system \(\eta _{P_t}({\varvec{x}})= \ln P_t({\varvec{x}}) - \ln P^{\textrm{pcan}}_t ({\varvec{x}})\) should be the theta coordinate system \(\theta _{P_t}({\varvec{x}})= \ln P_t({\varvec{x}}) - \ln P^{\textrm{pcan}}_t ({\varvec{x}}).\) In Proposition 20, the Kullback–Leibler divergence in Eq. (131) should be \(D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }}\Vert {\mathbb {P}}^1_{{\varvec{0}}} ),\) not \(D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }}\Vert {\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }} ).\) Accordingly, the corresponding proofs should be corrected. Some typos should be fixed.
The corrected statements are given below.
Lemma 1
The entropy production rate \(\sigma _{\tau }\) is given by
$$\begin{aligned} \sigma _{\tau } = \lim _{dt \rightarrow 0}\frac{D_{\textrm{KL}}({\mathbb {P}} \Vert {\mathbb {P}}^{\dagger } )}{dt}. \end{aligned}$$
(12)
Proof
We rewrite the Kullback–Leibler divergence \(D_{\textrm{KL}}({\mathbb {P}} \Vert {\mathbb {P}}^{\dagger })\) as
$$\begin{aligned} D_{\textrm{KL}}({\mathbb {P}} \Vert {\mathbb {P}}^{\dagger })&= \int d{\varvec{x}}_{\tau } d {\varvec{x}}_{\tau +dt} {\mathbb {P}} ({\varvec{x}}_{\tau +dt}, {\varvec{x}}_{\tau }) \ln \frac{{\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) P_{\tau }({\varvec{x}}_{\tau }) }{ {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau } \mid {\varvec{x}}_{\tau +dt}) P_{\tau }({\varvec{x}}_{\tau +dt}) } \nonumber \\&\quad + \int d {\varvec{x}}_{\tau +dt} P_{\tau +dt} ({\varvec{x}}_{\tau +dt})\ln \frac{ P_{\tau }({\varvec{x}}_{\tau +dt}) }{P_{\tau +dt}({\varvec{x}}_{\tau +dt}) }\nonumber \\&= \int d{\varvec{x}}_{\tau } d {\varvec{x}}_{\tau +dt} {\mathbb {P}} ({\varvec{x}}_{\tau +dt}, {\varvec{x}}_{\tau }) \ln \frac{{\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) P_{\tau }({\varvec{x}}_{\tau }) }{ {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau } \mid {\varvec{x}}_{\tau +dt}) P_{\tau }({\varvec{x}}_{\tau +dt}) } \nonumber \\&\quad - dt \int d {\varvec{x}}_{\tau +dt} \partial _{\tau '} P_{\tau '} ({\varvec{x}}_{\tau +dt}) \mid _{\tau ' =\tau +dt} + O(dt^2) \nonumber \\&= \int d{\varvec{x}}_{\tau } d {\varvec{x}}_{\tau +dt} {\mathbb {P}} ({\varvec{x}}_{\tau +dt}, {\varvec{x}}_{\tau }) \ln \frac{{\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) P_{\tau }({\varvec{x}}_{\tau }) }{ {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau } \mid {\varvec{x}}_{\tau +dt}) P_{\tau }({\varvec{x}}_{\tau +dt}) } + O(dt^2), \end{aligned}$$
(13)
where \(O(dt^2)\) means the term that satisfies \(\lim _{dt \rightarrow 0} O(dt^2)/dt=0\) and we used \(\int d{\varvec{x}}_{\tau } {\mathbb {P}} ({\varvec{x}}_{\tau +dt}, {\varvec{x}}_{\tau }) = P_{\tau +dt} ({\varvec{x}}_{\tau +dt})\) and \(\int d {\varvec{x}}_{\tau +dt} \partial _{\tau '} P_{\tau '} ({\varvec{x}}_{\tau +dt}) \mid _{\tau ' =\tau +dt} =0.\) From Eq. (3), we obtain
$$\begin{aligned}&\ln \frac{{\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) P_{\tau }({\varvec{x}}_{\tau }) }{ {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau } \mid {\varvec{x}}_{\tau +dt}) P_{\tau }({\varvec{x}}_{\tau +dt}) } \nonumber \\&\quad = ({\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } ) \circ \left[ \frac{ {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) }{\mu T} + O(dt) \right] + O( \Vert {\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } \Vert ^2 ), \end{aligned}$$
(14)
where \(\circ \) stands for the Stratonovich integral defined as \(({\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } ) \circ {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) =({\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } ) \cdot {\varvec{\nu }}_{\tau } ([{\varvec{x}}_{\tau } + {\varvec{x}}_{\tau +dt}]/2),\) O(dt) means the term that satisfies \(\lim _{dt \rightarrow 0} O(dt)=0\) and \(O( \Vert {\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } \Vert ^2 )\) is the higher order term which satisfies \(\int d{\varvec{x}}_{\tau +dt } {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) O( \Vert {\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } \Vert ^2 )= O(dt).\) Because the Gaussian integral provides
$$\begin{aligned}&\frac{1}{dt} \int {d \varvec{x}_{\tau }P_{\tau }(\varvec{x}_{\tau })} \int d{\varvec{x}}_{\tau +dt } {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) \ln \frac{{\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) P_{\tau }({\varvec{x}}_{\tau }) }{ {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau } \mid {\varvec{x}}_{\tau +dt}) P_{\tau }({\varvec{x}}_{\tau +dt}) } \nonumber \\&\quad = \int {d \varvec{x}_{\tau }P_{\tau }(\varvec{x}_{\tau })} \int d{\varvec{x}}_{\tau +dt} {\mathbb {T}}_{\tau }({\varvec{x}}_{\tau +dt} \mid {\varvec{x}}_{\tau }) \frac{{\varvec{x}}_{\tau +dt} - {\varvec{x}}_{\tau } }{dt} \circ \left[ \frac{{\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau })}{\mu T}+O(dt) \right] + O(dt)\nonumber \\&\quad = \frac{1}{\mu T} \int {d \varvec{x}_{\tau }P_{\tau }(\varvec{x}_{\tau })} \Vert {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) \Vert ^2 + O(dt), \end{aligned}$$
(15)
we obtain
$$\begin{aligned} \lim _{dt \rightarrow 0}\frac{D_{\textrm{KL}}({\mathbb {P}} \Vert {\mathbb {P}}^{\dagger })}{dt} = \frac{1}{\mu T} \int d {\varvec{x}}_{\tau } \Vert {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) \Vert ^2 P_{\tau } ({\varvec{x}}_{\tau }) + \lim _{dt \rightarrow 0}O(dt) = \sigma _{\tau }, \end{aligned}$$
(16)
from Eqs. (13) and (15). \(\square \)
FormalPara
Remark 28
If a pseudo canonical distribution is given by a time-independent distribution \(P^\textrm{pcan}_t (\varvec{x}) =P^\textrm{st} (\varvec{x})\), the Fokker–Planck equation can be rewritten as the heat equation,
$$\begin{aligned} \partial _{t} [ \partial _{P_t(\varvec{x})} D_\textrm{KL} (P_t \Vert P^\textrm{st}) ] = \mu T \textrm{div}_{P_t} \left( \nabla [ \partial _{P_t(\varvec{x})} D_\textrm{KL} (P_t \Vert P^\textrm{st}) ] \right) , \end{aligned}$$
(100)
or equivalently,
$$\begin{aligned} \partial _{t} \ln \frac{P_t(\varvec{x})}{ P^\textrm{st} (\varvec{x})}= \mu T \textrm{div}_{P_t} \left( \nabla \ln \frac{P_t(\varvec{x})}{ P^\textrm{st} (\varvec{x})} \right) , \end{aligned}$$
(101)
where \(\textrm{div}_{P_t} \) is the operator defined as
$$\begin{aligned} \textrm{div}_{\sqrt{| g |}} (\cdots )= (\sqrt{|g|})^{-1} \nabla \cdot [ \sqrt{|g|} (\cdots )], \end{aligned}$$
(102)
with \(\sqrt{|g|}= P_t (\varvec{x})\). Because the operator \(\textrm{div}_{\sqrt{| g |}}\) is a generalization of the divergence operator for non-Euclidean space with the absolute value of the determinant of the metric tensor |g|, the Fokker–Planck equation may be regarded as a kind of the diffusion equation for \(\partial _{P_t(\varvec{x})} D_\textrm{KL} (P_t \Vert P^\textrm{st})= \ln P_t(\varvec{x}) - \ln P^\textrm{st} (\varvec{x})\). As a consequence of the diffusion process, we may obtain \(\partial _{P_t(\varvec{x})} D_\textrm{KL} (P_t \Vert P^\textrm{st}) = \ln P_t(\varvec{x}) - \ln P^\textrm{st} (\varvec{x}) \rightarrow 0\) in the limit \(t \rightarrow \infty \), which implies the relaxation to a steady state \(P_t(\varvec{x}) \rightarrow P^\textrm{st}(\varvec{x})\).
FormalPara
Proposition 19
The excess entropy production rate \(\sigma ^{\textrm{ex}}_t\) is bounded as follows.
$$\begin{aligned} v_\textrm{info}(t) \sqrt{ {\textrm{Var}}_{P_t}[\theta _{P_t}]} \ge \sigma ^{\textrm{ex}}_t, \end{aligned}$$
(125)
where \(\theta _{P_t}({\varvec{x}})\) is the theta coordinate system defined as \(\theta _{P_t}({\varvec{x}}) = \ln P_t({\varvec{x}}) - \ln P^{\textrm{pcan}}_t ({\varvec{x}}).\)
FormalPara
Proof
The excess entropy production is given by
$$\begin{aligned} \sigma ^{\textrm{ex}}_t&= - \int d {\varvec{x}} P_t({\varvec{x}}) [\partial _t \ln P_t ({\varvec{x}})] \theta _{P_t} ({\varvec{x}}) \end{aligned}$$
(126)
$$\begin{aligned}&= - \int d {\varvec{x}} P_t({\varvec{x}}) [\partial _t \ln P_t ({\varvec{x}})] [ \theta _{P_t} ({\varvec{x}}) - {\mathbb {E}}_{P_t} [\theta _{P_t}]], \end{aligned}$$
(127)
where we used \(\int d{\varvec{x}} \partial _t P_t({\varvec{x}})=0.\) From the Cauchy–Schwartz inequality, we obtain
$$\begin{aligned} (\sigma ^{\textrm{ex}}_t)^2&= \left( \int d {\varvec{x}} P_t({\varvec{x}}) [\partial _t \ln P_t ({\varvec{x}})] (\theta _{P_t} ({\varvec{x}}) - {\mathbb {E}}_{P_t} [\theta _{P_t}] ) \right) ^2 \nonumber \\&\le \left( \int d {\varvec{x}} P_t({\varvec{x}}) (\partial _t \ln P_t ({\varvec{x}}))^2 \right) \left( \int d {\varvec{x}} P_t({\varvec{x}}) (\theta _{P_t} ({\varvec{x}}) - {\mathbb {E}}_{P_t} [\theta _{P_t}] )^2 \right) \nonumber \\&= [v_\textrm{info}(t)]^2 {\textrm{Var}}_{P_t}[\theta _{P_t}]. \end{aligned}$$
(128)
By taking the square root of each side, we obtain Eq. (125). \(\square \)
FormalPara
Proposition 20
The entropy production rate \(\sigma _{\tau },\) the excess entropy production rate \(\sigma _{\tau }^{\textrm{ex}}\) and the housekeeping entropy production rate \(\sigma _{\tau }^{\textrm{hk}}\) for original Fokker–Planck dynamics (1) are given by
$$\begin{aligned} \sigma _{\tau }&= \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}_{{\varvec{\nu }}_{\tau }}^1 \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt}, \end{aligned}$$
(129)
$$\begin{aligned} \sigma _{\tau }^{\textrm{ex}}&= \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }} \Vert {\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }})}{dt} = \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }^*} \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt}, \end{aligned}$$
(130)
$$\begin{aligned} \sigma _{\tau }^{\textrm{hk}}&= \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }} \Vert {\mathbb {P}}^1_{ {\varvec{\nu }}^*_{\tau }})}{dt} = \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }} \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt}, \end{aligned}$$
(131)
where \({\varvec{\nu }}^*_{\tau }({\varvec{x}})\) is the optimal mean local velocity defined as \({\varvec{\nu }}^*_{\tau }({\varvec{x}}) = \nabla \phi _{\tau }({\varvec{x}}),\) and \({\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau })\) is the mean local velocity \({\varvec{\nu }}_{\tau } ({\varvec{x}}) = \mu ( {\varvec{F}}_{\tau } ({\varvec{x}}) - T \nabla \ln P_{\tau }({\varvec{x}}) )\) for the original Fokker–Planck equation \(\partial _{\tau } P_{\tau }({\varvec{x}}) =- \nabla \cdot ( {\varvec{\nu }}_{\tau } ({\varvec{x}}) P_{\tau } ({\varvec{x}}) ).\)
FormalPara
Proof
For any \({\varvec{v}} ({\varvec{x}}) \in {\mathbb {R}}^d\) and \({\varvec{v}}' ({\varvec{x}}) \in {\mathbb {R}}^d,\) we obtain
$$\begin{aligned} \ln \frac{{\mathbb {T}}^1_{\tau ;{\varvec{v}}} ({\varvec{x}}_{\tau + dt} \mid {\varvec{x}}_{\tau } )}{{\mathbb {T}}^1_{\tau ;{\varvec{v}}'} ({\varvec{x}}_{\tau + dt} \mid {\varvec{x}}_{\tau } )}&= - \frac{\Vert {\varvec{x}}_{\tau +dt}- {\varvec{x}}_{\tau } - \mu {\varvec{F}}_{\tau }({\varvec{x}}_{\tau })dt - {\varvec{v}}({\varvec{x}})dt - {\varvec{\nu }}_{\tau } ( {\varvec{x}}_{\tau })dt \Vert ^2}{4 \mu T dt} \nonumber \\&\quad + \frac{\Vert {\varvec{x}}_{\tau +dt}- {\varvec{x}}_{\tau } - \mu {\varvec{F}}_{\tau }({\varvec{x}}_{\tau })dt - {\varvec{v}}'({\varvec{x}})dt - {\varvec{\nu }}_{\tau } ( {\varvec{x}}_{\tau })dt \Vert ^2}{4 \mu T dt} \nonumber \\&= \frac{({\varvec{x}}_{\tau +dt}- {\varvec{x}}_{\tau } - \mu {\varvec{F}}_{\tau }({\varvec{x}}_{\tau })dt - {\varvec{\nu }}_{\tau } ( {\varvec{x}}_{\tau })dt ) \cdot ({\varvec{v}}({\varvec{x}}_{\tau }) - {\varvec{v}}({\varvec{x}}_{\tau })) }{2 \mu T } \nonumber \\&\quad + \frac{ \Vert {\varvec{v}}'^2({\varvec{x}}_{\tau }) - {\varvec{v}}^2({\varvec{x}}_{\tau }) \Vert dt}{4 \mu T }. \end{aligned}$$
(132)
Thus, the Kullback–Leibler divergence is calculated as
$$\begin{aligned} D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{v}}} \Vert {\mathbb {P}}^1_{{\varvec{v}}'})&= \int d {\varvec{x}}_{\tau } P_{\tau } ({\varvec{x}}_{\tau }) \int d {\varvec{x}}_{\tau +dt} {\mathbb {T}}^1_{\tau ;{\varvec{v}}} ({\varvec{x}}_{\tau + dt} \mid {\varvec{x}}_{\tau } ) \ln \frac{{\mathbb {T}}^1_{\tau ;{\varvec{v}}} ({\varvec{x}}_{\tau + dt} \mid {\varvec{x}}_{\tau } )}{{\mathbb {T}}^1_{\tau ;{\varvec{v}}'} ({\varvec{x}}_{\tau + dt} \mid {\varvec{x}}_{\tau } )} \nonumber \\&= dt \frac{1}{4\mu T}\int d {\varvec{x}}_{\tau } P_{\tau } ({\varvec{x}}_{\tau }) \Vert {\varvec{v}}({\varvec{x}}_{\tau }) - {\varvec{v}}({\varvec{x}}_{\tau }) \Vert ^2. \end{aligned}$$
(133)
Therefore, by plugging \(({\varvec{v}}, {\varvec{v}}')=({\varvec{\nu }}_{\tau }, {\varvec{0}}),\) \(({\varvec{v}}, {\varvec{v}}')=({\varvec{\nu }}_{\tau }, {\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }),\) \(({\varvec{v}}, {\varvec{v}}')=({\varvec{\nu }}_{\tau }^*, {\varvec{0}}),\) \(({\varvec{v}}, {\varvec{v}}')=({\varvec{\nu }}_{\tau }, {\varvec{\nu }}^*_{\tau }),\) \(({\varvec{v}}, {\varvec{v}}')=({\varvec{\nu }}_{\tau }- {\varvec{\nu }}_{\tau }^*, {\varvec{0}})\) into Eq. (133), we obtain
$$\begin{aligned}&\lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}_{{\varvec{\nu }}_{\tau }}^1 \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt} = \frac{1}{\mu T}\int d {\varvec{x}}_{\tau } P_{\tau }({\varvec{x}}_{\tau }) \Vert {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) \Vert ^2 = \sigma _{\tau } , \end{aligned}$$
(134)
$$\begin{aligned}&\lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }} \Vert {\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }})}{dt} = \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }^*} \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt} \nonumber \\&\quad =\frac{1}{\mu T}\int d {\varvec{x}}_{\tau } P_{\tau }({\varvec{x}}_{\tau }) \Vert {\varvec{\nu }}_{\tau }^* ({\varvec{x}}_{\tau }) \Vert ^2 = \sigma ^{\textrm{ex}}_{\tau } , \end{aligned}$$
(135)
$$\begin{aligned}&\lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau }} \Vert {\mathbb {P}}^1_{ {\varvec{\nu }}^*_{\tau }})}{dt} = \lim _{dt \rightarrow 0} \frac{4 D_{\textrm{KL}}({\mathbb {P}}^1_{{\varvec{\nu }}_{\tau } - {\varvec{\nu }}^*_{\tau }} \Vert {\mathbb {P}}^1_{{\varvec{0}}})}{dt} \nonumber \\&\quad =\frac{1}{\mu T}\int d {\varvec{x}}_{\tau } P_{\tau }({\varvec{x}}_{\tau }) \Vert {\varvec{\nu }}_{\tau } ({\varvec{x}}_{\tau }) -{\varvec{\nu }}_{\tau }^* ({\varvec{x}}_{\tau }) \Vert ^2 = \sigma ^{\textrm{hk}}_{\tau }. \end{aligned}$$
(136)
\(\square \)
The original article has been corrected.
