The interplay between memory and potentials of mean force: A discussion on the structure of equations of motion for coarse-grained observables

Abstract

The underdamped, non-linear, generalized Langevin equation is widely used to model coarse-grained dynamics of soft and biological materials. By means of a projection operator formalism, we show under which approximations this equation can be obtained from the dynamics of the underlying microscopic system and in which cases it makes sense to introduce a potential of mean force. We discuss shortcomings of previous derivations presented in the literature and demonstrate the implications of our derivation for the structure of memory terms and for generalized fluctuation-dissipation relations. We show, in particular, that the widely used, simple structure which contains a potential of mean force, a memory term which is linear in the observable, and a fluctuating force which is related to the memory term by a fluctuation-dissipation relation, is neither exact nor can it, in general, be derived as a controlled approximation to the exact dynamics.

Position-dependent memory kernel in generalized Langevin equations: theory and numerical estimation

“…For instance, when E is reduced to a basis of linear functions, this projector is also known as the Mori projector [33]. By contrast, when E is chosen as an Hilbert basis of L 2 (R d ), then this projector is known as the Zwanzig projector [12,24]. Lastly for E a finite family of function, P E is a finite-rank projector [8,31].…”

“…In both cases ( 21) and ( 22), as required in [24], the mean force derives from ρ ϕ and the memory kernel is linear in the velocity (although with a non-linear dependency in the position). Indeed, linear reaction coordinates are considered in [24], which means ( 16) holds, eventually up to a linear change of variable (which amounts to the definition of a suitable effective mass matrix).…”

“…The form of the observable in [24] is similar to the one in Section 2.3, but the function set is chosen as…”

“…There is an important literature on linear (harmonic) cases [17][18][19][20][21]. The non-harmonic case, with a non-linear effective force given by the potential of mean force, is discussed in [22,23], but it is observed in [24] that these derivations rely on some implicit and uncontrolled approximations, and it is argued that in general it is in fact not possible to get, within the Mori-Zwanzig formalism, an equation of the form (1) with a non-linear effective force but a friction term which is linear in the velocity. Our first contribution is a nuance to the statement of [24], namely: in general, it is possible to rigorously derive a GLE similar to (1) in a general non-linear coupled case, with a non-linear force and a friction which is linear in the velocity, if the kernel K is allowed to depend on the position, i.e.…”

“…Moreover, in accordance with the standard assumption and in contrast to [24], in this case the memory kernel and the covariance of the noise are related by a form of fluctuation-dissipation relation (see Section 2.5.2 below for the definition), adapted to the position-dependent framework.…”