Time reversal and symmetries of time correlation functions

ABSTRACT

The time reversal invariance of classical dynamics is reconsidered in this paper with specific focus on its consequences for time correlation functions and associated properties such as transport coefficients. We show that, under fairly common assumptions on the interparticle potential, an isolated Hamiltonian system obeys more than one time reversal symmetry and that this entails non trivial consequences. Under an isotropic and homogeneous potential, in particular, eight valid time reversal operations exist. The presence of external fields that reduce the symmetry of space decreases this number, but does not necessarily impair all time reversal symmetries. Thus, analytic predictions of symmetry properties of time correlation functions and, in some cases, even of their null value are still possible. The noteworthy case of a constant external magnetic field, usually assumed to destroy time reversal symmetry, is considered in some detail. We show that, in this case too, some of the new time reversal operations hold, and that this makes it possible to derive relevant properties of correlation functions without the uninteresting inversion of the direction of the magnetic field commonly enforced in the literature.

GRAPHICAL ABSTRACT

KEYWORDS:

• Time reversal symmetry
• Hamiltonian system
• correlation functions
• linear response theory
• magnetic field
• electric field

Acknowledgments

We are delighted to dedicate to our dear aging friend Daan this peregrine search of useful new ways to invert time, hoping to find a way to continue to grow young together for many years to come. We acknowledge the warm hospitality provided by CECAM headquarters at EPF-Lausanne, where this work was done. G Ciccotti also acknowledges the Atosim Erasmus Mundus Master Project for financial support.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. This is proved [12] by showing that ⟨Φ, TΨ⟩ = ⟨TΦ, Ψ⟩. Let us in fact consider, for any real Ψ and Φ,

We now perform the change of variables , so (in measure dX = dΓ) to obtain

which completes the proof, noting that Tρ(X) = ρ(X) since the Hamiltonian is invariant under this transformation, and the probability density is a function of the Hamiltonian.

2. It may be worth to note that not all observables have a signature and, more particularly, that the existence of the signature may depend on the specific symmetry considered.

3. If periodic boundary conditions are adopted, the system may (or may not, depending on conditions) reach a stationary state, but we shall not discuss this case in this work.

4. The first two transformations have already been identified as valid time reversal symmetries in [12].

Additional information

Funding

Atosim Erasmus Mundus Master Project.