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Comment on “Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond” [J. Chem. Phys. 144, 094109 (2016)]
W. M. Haddad and V. S. Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach (Princeton University Press, Princeton, 2008).
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A natural boundary condition is implied by the assumption that in the neighborhood of a fixed point, a linear process is valid. This suggests a gradient expansion near the fixed point:6 for each order, only linear algebra equations are solved. Hence, there is a unique solution for each order, therefore any order. Formally, this leads to the conclusion on the uniqueness of solution.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, New York, 2012), Vol. 44.
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Connections between a “SDE decomposition” to other frameworks constructing landscape in non-equilibrium processes were discussed by Zhou and Li [J. Chem. Phys. 144, 094109 (2016)]. It was speculated that the SDE decomposition would not be generally unique. In this comment, we demonstrate both mathematically and physically that the speculation is incorrect and the uniqueness is guaranteed under appropriate conditions. A few related issues are also clarified, such as the limitation of obtaining potential function from steady state distribution. Current demonstration may lead to a better understanding on the structure and robustness of the decomposition framework.
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