In molecular dynamics (MD) simulations the need often arises to maintain such parameters as temperature or pressure rather than energy and volume, or to impose gradients for studying transport properties in nonequilibrium MD A method is described to realize coupling to an external bath with constant temperature or pressure with adjustable time constants for the coupling The method is easily extendable to other variables and to gradients, and can be applied also to polyatomic molecules involving internal constraints The influence of coupling time constants on dynamical variables is evaluated A leap‐frog algorithm is presented for the general case involving constraints with coupling to both a constant temperature and a constant pressure bath

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Molecular dynamics with coupling to an external bath.

A new Lagrangian formulation is introduced. It can be used to make molecular dynamics (MD) calculations on systems under the most general, externally applied, conditions of stress. In this formulation the MD cell shape and size can change according to dynamical equations given by this Lagrangian. This new MD technique is well suited to the study of structural transformations in solids under external stress and at finite temperature. As an example of the use of this technique we show how a single crystal of Ni behaves under uniform uniaxial compressive and tensile loads. This work confirms some of the results of static (i.e., zero temperature) calculations reported in the literature. We also show that some results regarding the stress‐strain relation obtained by static calculations are invalid at finite temperature. We find that, under compressive loading, our model of Ni shows a bifurcation in its stress‐strain relation; this bifurcation provides a link in configuration space between cubic and hexagonal close packing. It is suggested that such a transformation could perhaps be observed experimentally under extreme conditions of shock.

Polymorphic transitions in single crystals: A new molecular dynamics method

Three recently proposed constant temperature molecular dynamics methods by: (i) Nose (Mol. Phys., to be published); (ii) Hoover et al. [Phys. Rev. Lett. 48, 1818 (1982)], and Evans and Morriss [Chem. Phys. 77, 63 (1983)]; and (iii) Haile and Gupta [J. Chem. Phys. 79, 3067 (1983)] are examined analytically via calculating the equilibrium distribution functions and comparing them with that of the canonical ensemble. Except for effects due to momentum and angular momentum conservation, method (1) yields the rigorous canonical distribution in both momentum and coordinate space. Method (2) can be made rigorous in coordinate space, and can be derived from method (1) by imposing a specific constraint. Method (3) is not rigorous and gives a deviation of order N−1/2 from the canonical distribution (N the number of particles). The results for the constant temperature–constant pressure ensemble are similar to the canonical ensemble case.

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A unified formulation of the constant temperature molecular dynamics methods

In this article, we present a new LINear Constraint Solver (LINCS) for molecular simulations with bond constraints. The algorithm is inherently stable, as the constraints themselves are reset instead of derivatives of the constraints, thereby eliminating drift. Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules. At the same accuracy, the LINCS algorithm is three to four times faster than the SHAKE algorithm. Parallelization of the algorithm is straightforward. (C) 1997 John Wiley & Sons, Inc.

LINCS : A linear constraint solver for molecular simulations

The authors present a new molecular dynamics algorithm for sampling the canonical distribution. In this approach the velocities of all the particles are rescaled by a properly chosen random factor. The algorithm is formally justified and it is shown that, in spite of its stochastic nature, a quantity can still be defined that remains constant during the evolution. In numerical applications this quantity can be used to measure the accuracy of the sampling. The authors illustrate the properties of this new method on Lennard-Jones and TIP4P water models in the solid and liquid phases. Its performance is excellent and largely independent of the thermostat parameter also with regard to the dynamic properties.

Canonical sampling through velocity rescaling

We derive equilibrium fluctuation expressions for the linear response of many body systems thermostated by the Nose–Hoover thermostat. We show that in the thermodynamic limit this response is the same as that of the corresponding Gaussian isothermal system. Numerical comparisons for shear flow show however that the Gaussian methods provide a significantly more efficient means of calculating the shear viscosity coefficient.

The Nose–Hoover thermostat

1. Introduction 2. Linear irreversible thermodynamics 3. The microscopic connection 4. The Green-Kubo relations 5. Linear response theory 6. Computer simulation algorithms 7. Nonlinear response theory 8. Dynamical stability 9. Nonequilibrium fluctuations 10. Thermodynamics of steady states References Index.

Statistical Mechanics of Nonequilibrium Liquids

We propose a new definition of natural invariant measure for trajectory segments of finite duration for a many-particle system. On this basis we give an expression for the probability of fluctuations in the shear stress of a fluid in a nonequilibrium steady state far from equilibrium. In particular we obtain a formula for the ratio that, for a finite time, the shear stress reverse sign, violating the second law of thermodynamics. Computer simulations support this formula.

Probability of second law violations in shearing steady states.

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamicaluctuations in thermostatted dissipa- tive non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy `thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been inthe last few of years that the generality ofthe Theorem has been appreciated. We now know that the Second Law of Thermo- dynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quanti®es the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observa- tion that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversedanti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been con®rmed experimentally, both using molecular dynamics computer simula- tion and very recently in laboratory experiments.

http://inis.jinr.ru/sl/P_Physics/PT_Thermodynamics,%20statistical%20physics/Evans,%20Searles.%20Fluctuation%20theorem%20(2002)(57s).pdf

The Fluctuation Theorem

We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law-defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales.

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Denis J. Evans

Papers

The Nose–Hoover thermostat

Statistical Mechanics of Nonequilibrium Liquids

Probability of second law violations in shearing steady states.

The Fluctuation Theorem

Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales