Linear response theory and optimal control for a molecular system under non-equilibrium conditions

Wang, Han; Hartmann, Carsten; Schütte, Christof

doi:10.1080/00268976.2013.844370

Cited by 4 publications

(4 citation statements)

References 16 publications

(22 reference statements)

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“…However, from an MSM for a given periodic forcing, optimal control problems may come into play. Using available methods, such as non-equilibrium linear response theory 51 or computational alchemy for MSMs, 52 one can construct the MSM for slightly changed parameters (e.g., period and amplitude) of the external forcing by appropriate reweighting of the MSM for the given forcing. This, in principle, allows for answering questions such as the following: for which parameters of the external forcing does one achieve maximal population of the lefthanded α-helix?…”

Section: Concluding Remarks and Discussionmentioning

confidence: 99%

Building Markov State Models for Periodically Driven Non-Equilibrium Systems

Wang

Schütte

2015

J. Chem. Theory Comput.

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Non-equilibrium molecular dynamics (NEMD) simulations have seen increased interest the past few years, especially for molecular systems with periodic forcing by external fields (e.g., in the context of studying the effects of electromagnetic radiation on human body tissues). Recently, an NEMD method with local thermostatting has been proposed that allows for the study of non-equilibrium processes in a statistically reliable and thermodynamically consistent way. In this article, we demonstrate how to construct Markov state models (MSMs) for such NEMD simulations. MSM building is well-established for systems in equilibrium, where MSMs with only a few (macro-)states allow for accurate reproduction of the essential kinetics of the molecular system under consideration. Non-equilibrium MSMs have not yet been established. This article presents a method for constructing such MSMs and illustrates their validity and usefulness through conformation dynamics of an alanine dipeptide in an external electric field.

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Section: Concluding Remarks and Discussionmentioning

confidence: 99%

Building Markov State Models for Periodically Driven Non-Equilibrium Systems

Wang

Schütte

2015

J. Chem. Theory Comput.

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“…The vectorized solution to ( 34)-( 37) is given by m * as in (26), where α * again denotes the eigenvector corresponding to the largest eigenvalue of the matrix U U . Finally, as for the positive M case, we have that both m * and −m * yield the same Euclidean norm of the response (34). Hence, we choose the sign of the matrix m * so that h M 2 < h M +εm * 2 for small ε > 0.…”

Section: 41mentioning

confidence: 99%

“…This important avenue of research has relatively few precendents in the literature. One exception is [34] who consider optimal control of Langevin dynamics; using a linear response approach, they apply a gradient descent algorithm to minimise a specified linear functional.…”

Section: Introductionmentioning

confidence: 99%

Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

2018

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The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the littlestudied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.

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“…The "inverse problem" of determining perturbations to achieve a particular response in the invariant measure has been studied in [21,31]. Determining optimal perturbations that maximise certain quantities relating to the system has been addressed in [39,2]. In the present paper we apply the idea of linear response to quantify the response of finite-time mixing properties in dynamical systems by developing theory to describe the response of finite-time coherent sets.…”

Section: Introductionmentioning

confidence: 99%

Linear response for the dynamic Laplacian and finite-time coherent sets

Antown¹,

Froyland²,

Junge³

2019

Preprint

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Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.

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Linear response theory and optimal control for a molecular system under non-equilibrium conditions

Cited by 4 publications

References 16 publications

Building Markov State Models for Periodically Driven Non-Equilibrium Systems

Building Markov State Models for Periodically Driven Non-Equilibrium Systems

Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

Linear response for the dynamic Laplacian and finite-time coherent sets

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