t-mixing: from fluctuation relations to response and irreversibility in MD

Abstract

This paper provides an introduction to the theory of steady-state fluctuation relations for molecular dynamics systems, that led to a general theory of response. The main ingredient of this theory is a new dynamical condition now known as t-mixing. We use such a condition to identify necessary and sufficient conditions for the relaxation to a steady state (whether equilibrium or not) of an ensemble of identical systems, as well as of a single system. This allows us to address the problem of the irreversibility of time reversal invariant (conservative as well as dissipative) particle systems.

Keywords:

• Phase space
• probability
• correlations
• ergodicity
• dissipation

Acknowledgements

We would like to thank S. Chibbaro, D.J. Evans, O.G. Jepps, D.J. Searles, A. Vulpiani for countless insightful discussions.

Notes

No potential conflict of interest was reported by the authors.

1 For example, the model (1(1) ) corresponds to .

2 Thanks to TRI, , the average of over segment equals , in agreement with (5(5) ).

3 One notable exception is afforded by gravitational wave detectors and similar experiments.[43,44]

4 The transient FR holds even in this situation, since it expresses the ratio between the initial probability of observing average negative values over a time , compared with the initial probability of observing positive values. For an even initial , both probabilities are positive, although the first rapidly decreases with , due to the dissipative field. The transient FR quantifies precisely this process at all averaging times .

5 Consider all arrangements of black and white pixels. The result is a dull movie: contains pictures, among which the remotely regular ones constitute a very small fraction. The overwhelming majority are grey. If pictures are shown at a rate of 25 frames a second, and the movie goes through all of them before starting again, the period is years! The fraction of time in which one does not see noisy arrangements is ridiculously small.

6 Note: usually this explanation of irreversibility is referred to the growth of the entropy of the universe, intended as a measure of disorder. Because entropy is however a questionable concept in the context of the violently nonequilibrium evolution of our universe, we merely refer to the ‘numerosity’ of states. For rarefied gases, these two notions come together in the definition of the Boltzmann entropy.

7 Actually, even the presence of extra additive constants of the motion would not make a difference: the reasoning can be repeated calling phase space the intersection of the corresponding hypersurfaces. Moreover, what matters is not the exploration of all phase space, which would take super-astronomical times even for systems with moderately large N, but the fact that the phase space regions corresponding to the range of the observables are visited. With this in mind, the reasoning can be continued in phase space, as usually, although imprecisely, done.

8 There are only several exceptions.[45,46]

9 After all, even the Hamiltonian description of atoms and molecules does not do that.[26]

10 Gauss expressed a similar view when he proposed his principle of least constraint: ‘It is always interesting and instructive to regard the laws of nature from a new and advantageous point of view, so as to solve this or that problem more simply or to obtain a more precise presentation’.[47]

12 NEMD models such as SLLOD or colour field are adiabatically incompressible.