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Computation 2018, 6(1), 11; https://doi.org/10.3390/computation6010011

Holonomic Constraints: A Case for Statistical Mechanics of Non-Hamiltonian Systems

1
Institute for Applied Mathematics “Mauro Picone” (IAC), CNR, Via dei Taurini 19, 00185 Rome, Italy
2
University of Rome “La Sapienza”, P.le Aldo Moro 5, 00185 Rome, Italy
3
University College Dublin (UCD), Belfield, Dublin 4, Ireland
4
Dipartimento di Scienze Fisiche, Informatiche e Matematiche, University of Modena and Reggio Emilia, Via Campi 213/A , 41125 Modena, Italy
*
Author to whom correspondence should be addressed.
Received: 8 January 2018 / Revised: 25 January 2018 / Accepted: 25 January 2018 / Published: 1 February 2018
(This article belongs to the Special Issue Computation in Molecular Modeling)
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Abstract

A dynamical system submitted to holonomic constraints is Hamiltonian only if considered in the reduced phase space of its generalized coordinates and momenta, which need to be defined ad hoc in each particular case. However, specially in molecular simulations, where the number of degrees of freedom is exceedingly high, the representation in generalized coordinates is completely unsuitable, although conceptually unavoidable, to provide a rigorous description of its evolution and statistical properties. In this paper, we first review the state of the art of the numerical approach that defines the way to conserve exactly the constraint conditions (by an algorithm universally known as SHAKE) and permits integrating the equations of motion directly in the phase space of the natural Cartesian coordinates and momenta of the system. We then discuss in detail SHAKE numerical implementations in the notable cases of Verlet and velocity-Verlet algorithms. After discussing in the same framework how constraints modify the properties of the equilibrium ensemble, we show how, at the price of moving to a dynamical system no more (directly) Hamiltonian, it is possible to provide a direct interpretation of the dynamical system and so derive its Statistical Mechanics both at equilibrium and in non-equilibrium conditions. To achieve that, we generalize the statistical treatment to systems no longer conserving the phase space volume (equivalently, we introduce a non-Euclidean invariant measure in phase space) and derive a generalized Liouville equation describing the ensemble even out of equilibrium. As a result, we can extend the response theory of Kubo (linear and nonlinear) to systems subjected to constraints. View Full-Text
Keywords: holonomic constraints; non-Hamiltonian dynamics; SHAKE holonomic constraints; non-Hamiltonian dynamics; SHAKE
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ciccotti, G.; Ferrario, M. Holonomic Constraints: A Case for Statistical Mechanics of Non-Hamiltonian Systems. Computation 2018, 6, 11.

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