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.
Highlights
The dynamical and statistical behavior of a mechanical system of many degrees of freedom subjected to holonomic constraints presents specific features that seem worth presenting and discussing in a unified framework
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 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
We address the central question of dynamical non-equilibrium Statistical Mechanics for systems subjected to holonomic constraints: how to get statistical averages when the evolution of the system is no more stationary be it due to time-dependent perturbations or to the study of relaxation processes [8]
Summary
The dynamical and statistical behavior of a mechanical system of many degrees of freedom subjected to holonomic constraints presents specific features that seem worth presenting and discussing in a unified framework. As we have seen before, the constraints are an essential ingredient in the definition of the dynamical system, any acceptable algorithm introduced to solve the dynamics of such a system cannot propagate any error, as otherwise the statistical behavior of the ensemble in the presence of the constraints cannot be properly formulated This last problem in principle is automatically solved for Hamiltonian systems by using generalized coordinates. The paper is concluded by a short outlook in which we try to assess the state of the art in the treatment of the computational classical Statistical Mechanics for systems subjected to holonomic constraints
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