Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Langevin equation with potential of mean force: The case of anchored bath

Alex V. Plyukhin

TL;DR

This work investigates how the potential of mean force $V_*(X)$ shapes Langevin dynamics for a single particle coupled to an anchored, non-passive bath. Using the Mazur-Oppenheim projection, it shows that in general the bath induces a PMF and also makes the dissipation kernel and noise statistics depend on the system position $X$, yielding a non-closed, non-Markovian equation. An explicit exception occurs for a linear bath—the Klein-Gordon chain—where the PMF is quadratic and the standard generalized Langevin equation with PMF is recovered, with a position-independent noise correlation and a well-defined memory kernel. These results clarify the limitations of PMF-inclusive Langevin models and identify the special linear-bath case where a closed, tractable equation of motion is obtained, including explicit expressions for the PMF stiffness and dissipation kernel.

Abstract

The potential of mean force (PMF) is an effective average potential acting on an open system, renormalized due to the interaction with the surrounding thermal bath. The PMF determines the correction to the equilibrium Gibbs distribution, but it is generally unclear how to implement the concept for time-dependent phenomena described by a (generalized) Langevin equation. We study a model where the system is a single particle (so there are no complications related to internal forces) and a non-trivial PMF is due to the presence of on-site (anchor) potentials applied to the bath particles. We found that the PMF does not merely replace the external potential, but also makes the dissipation kernel and statistical properties of noise dependent on the system's position. That dependence is determined by the internal bath and system-bath interactions and is a priori unknown. Therefore, in the general case the Langevin equation with the PMF is not closed and thus inoperable. However, for systems with linear forces the aforementioned dependence on the system's position is canceled. As an example, we consider a model where the bath is formed by the Klein-Gordon chain, i. e. a harmonic chain with on-site harmonic potentials. In that case, the generalized Langevin equation has the standard form with an external potential replaced by a quadratic PMF.

Langevin equation with potential of mean force: The case of anchored bath

TL;DR

This work investigates how the potential of mean force shapes Langevin dynamics for a single particle coupled to an anchored, non-passive bath. Using the Mazur-Oppenheim projection, it shows that in general the bath induces a PMF and also makes the dissipation kernel and noise statistics depend on the system position , yielding a non-closed, non-Markovian equation. An explicit exception occurs for a linear bath—the Klein-Gordon chain—where the PMF is quadratic and the standard generalized Langevin equation with PMF is recovered, with a position-independent noise correlation and a well-defined memory kernel. These results clarify the limitations of PMF-inclusive Langevin models and identify the special linear-bath case where a closed, tractable equation of motion is obtained, including explicit expressions for the PMF stiffness and dissipation kernel.

Abstract

The potential of mean force (PMF) is an effective average potential acting on an open system, renormalized due to the interaction with the surrounding thermal bath. The PMF determines the correction to the equilibrium Gibbs distribution, but it is generally unclear how to implement the concept for time-dependent phenomena described by a (generalized) Langevin equation. We study a model where the system is a single particle (so there are no complications related to internal forces) and a non-trivial PMF is due to the presence of on-site (anchor) potentials applied to the bath particles. We found that the PMF does not merely replace the external potential, but also makes the dissipation kernel and statistical properties of noise dependent on the system's position. That dependence is determined by the internal bath and system-bath interactions and is a priori unknown. Therefore, in the general case the Langevin equation with the PMF is not closed and thus inoperable. However, for systems with linear forces the aforementioned dependence on the system's position is canceled. As an example, we consider a model where the bath is formed by the Klein-Gordon chain, i. e. a harmonic chain with on-site harmonic potentials. In that case, the generalized Langevin equation has the standard form with an external potential replaced by a quadratic PMF.

Paper Structure

This paper contains 8 sections, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of PMF
  3. Anchored bath model
  4. Pre-Langevin equation
  5. Langevin equation
  6. Klein-Gordon chain: the PMF
  7. Klein-Gordon chain: Langevin equation
  8. Conclusion

Figures (3)

  • Figure 1: The setup considered in Sections 6 and 7. The black circle represents the system of interest, which is a classical particle subjected to an (arbitrary) external potential $V_{ex}(X)$. The system is connected by the spring with stiffness $k_0$ to the thermal bath, formed by the Klein-Gordon lattice. The latter consists of $N$ atoms connected by the springs with stiffness $k_0$. Each atom of the lattice is subjected to on-site (anchoring) harmonic potentials with stiffness $k_a$.
  • Figure 2: The Klein-Gordon lattice described by Hamiltonian $H_0'$ in Eq. (\ref{['H000']}).
  • Figure 3: The dimensionless dissipation kernel $\gamma(t)$, see Eq. (\ref{['aux75']}), for three values of the anchor parameter $a=\omega_a/\omega_0$.